UBC Theses and Dissertations

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UBC Theses and Dissertations

A pkp study of the earth's core using the warramunga seismic array Bertrand, Aimee Elizabeth Surrendra 1972

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A PKP STUDY OF THE EARTH'S CORE USING THE WARRAMUNGA SEISMIC ARRAY  by AIMEE ELIZABETH SURRENDRA BERTRAND B. Sc. U n i v e r s i t y o f  the West  Indies,  1970  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department of GEOPHYSICS  We a c c e p t t h i s  t h e s i s as c o n f o r m i n g to the  r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA August,  197 2  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.  It is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department The University of B r i t i s h Columbia Vancouver 8, Canada  i ABSTRACT PKP core phases r e c o r d e d a t array, WRA,  the U.K.A.E.A. - type  i n Northern T e r r i t o r y , A u s t r a l i a over  seismic  the d i s t a n c e  113° to 17&° are used to determine a v e l o c i t y - d e p t h model f o r earth's c o r e .  Paper r e c o r d i n g s  from analog magnetic filters  bandpass  played out at h i g h speed  mm/sec)  (kO  p r o v i d e d the d a t a a v a i l a b l e f o r a n a l y s i s .  Four d i s t i n c t  Such d a t a  times between s e i s m i c  t r a v e l time branches were o b s e r v e d .  these a r e the well-known  additional  the  tape w i t h amplitude gain c o n t r o l and narrow  enabled p r e c i s i o n measurement o f r e l a t i v e  m i t t e d through  range  Two of  PKP„^ and PKPp.,- branches which are  the o u t e r and inner c o r e s , r e s p e c t i v e l y .  branches a r e f o r e r u n n e r s to the DF branch a t  traces.  trans-  The two distances  a less than 143  .  In r e c e n t y e a r s ,  branches d e s i g n a t e d  PKP  served p r e c u r s o r s .  However,  branch, d e s i g n a t e d  PKP_ , has  precursor  , has  branches p r o v i d e  the e x i s t e n c e of one of  been accepted to account f o r  in t h i s  study of  ob-  the e x i s t e n c e of a d i s t i n c t  second  been a s u b j e c t f o r d e b a t e .  Since  i n f o r m a t i o n about  between the f l u i d o u t e r core and the s o l i d ification  these  two d i s t i n c t  the " t r a n s i t i o n  inner c o r e , the  the  region"  ident-  sets of forerunners  is  of some importance. T r a v e l t i m e and t r a v e l t i m e g r a d i e n t  ( dT/dA  ) measure-  ments o f each phase observed on the seismograms were made. dT/dA  v a l u e s were measured by a l e a s t - s q u a r e s  technique.  The These  measurements were s t r o n g l y  p e r t u r b e d by s t r u c t u r e beneath the  array and i t was n e c e s s a r y  to c o r r e c t f o r  irical  approach.  The c o r r e c t e d  the method o f summary v a l u e s -regression  technique.  the e f f e c t by an emp-  d T / d A v a l u e s were smoothed by  and, f o r comparison,  The smoothed  dT/dA  values  by a polynomial for a l l  four  ii branches were i n v e r t e d by the H e r g l o t z - W e i c h e r t method i n o r d e r t o o b t a i n a v e l o c i t y - d e p t h model f o r  the e a r t h ' s  core.  The f i n a l  v e l o c i t y model, UBC1, determined i n t h i s manner was the one which gave the- best f i t  to a l l  observations  in t h i s  study.  Near a depth o f 4000 km., UBC1 r e q u i r e d a - s l i g h t u c t i o n i n the J e f f r e y s - B u l l e n t a i n t r a v e l times core. °f  velocity  4810 km ,  increases  and 0.92 km/sec,  in order to ob-  t h a t agreed w i t h the o b s e r v a t i o n s  The model e x h i b i t s  4393 km ,  (1940) v e l o c i t i e s  for  the o u t e r  three v e l o c i t y d i s c o n t i n u i t i e s and  5120  km.  at the d i s c o n t i n u i t i e s  respectively.  shells  surrounding  thick,  the v e l o c i t y g r a d i e n t  second s h e l l , 3Q0 km t h i c k ,  is  at  The magnitude of 0.10,  are  These d i s c o n t i n u i t i e s  the inner c o r e .  red-  In the outermost  is  the  0.24  define  two  s h e l l , 417 km  near z e r o i n magnitude.  the v e l o c i t y g r a d i e n t  depths  In the  slightly  negative.  iii ACKNOWLEDGEMENTS I g r e a t l y appreciate the advice and encouragement given to me by my supervisor Dr. R. M. Clowes. S p e c i a l thanks should be given t o Dr. J.R. Cleary, H r - H.G.Doyle and the Department of Geophysics and Geochemistry, A u s t r a l i a n N a t i o n a l U n i v e r s i t y f o r t h e i r h o s p i t a l i t y t o Dr. Cloves w h i l e he obtained the data used i n t h i s study.  The  help of Dr. C. Wright i n p r o v i d i n g many of the computer programs used i n t h i s study i s deeply appreciated. F i n a n c i a l assistance f o r t h i s p r o j e c t was provided by a U n i v e r s i t y of the West Indies Postgraduate Scholarship, a N a t i o n a l Research Council Bursary, and N a t i o n a l Research C o u n c i l operating and computing grants numbered A-7707 and C-1257, r e s p e c t i v e l y o f Dr. R. M. Clowes.  iv TABLE OF CONTENTS Page 1.  INTRODUCTION 1.1  The P r o j e c t  1  1.2  P r e v i o u s Work on Core S t r u c t u r e  2  1.3  2.  1  1.2.1  D i s c o v e r y of C e n t r a l and  1.2.2  Velocity Solutions  Inner Core  for Earth's  S e i s m i c V e l o c i t i e s and P h y s i c a l  4  Core  P r o p e r t i e s of  the Core 1 4  the Core  14  Inner Core  15  1.3.1  Central  D e s i t y of  —  1.3.2.  S o l i d i t y of  1.4  The Warramunga  1.5  Use of an A r r a y i n D e t e r m i n i n g S t r u c t u r e  the  2  Seismic Array  17 19  MEASUREMENT AND ANALYSIS 2.1  2.2  2.3  Determination of Array  dT/dA and Azimuth From a S e i s m i c 21  2.1.1  Some P o s s i b l e Methods  21  2.1.2  D e t a i l s of  23  the Least Squares Method  Measurement Techniques  26  2.2.1  Data A v a i l a b l e f o r  2.2.2  Measurement of d T / d A  and Azimuth  2.2.3  Sources o f S y s t e m a t i c Azimuth Measurements  Error  the Study  Smoothing of C o r r e c t e d dT/dA 2.3.1  2.3.2 2.4  21  Smoothing of Values  dT/dA  Smoothing of  dT/dA  26  in  dT/dA  29 and 34  Values  by the Method o f  52 Summary 52  I n v e r s i o n of Smoothed Wiechert I n t e g r a l 2 . 4 . 1 . D e s c r i p t i o n of  dT/dA  by UBC TRIP  63  Data U s i n g the H e r g l o t z 66  the H e r g l o t z - W i e c h e r t Technique  66  V TABLE OF CONTENTS (cont'd) Pagj. 2.4.2  3.  70  DISCUSSION  73  3.1  General O b s e r v a t i o n s  73  3.2  S p e c i f i c Observations  76  3.3  3.4  4.  A p p i i c a t i o n of the H e r g l o t z - W i e c h e r t Method to (1T/4A Values f o r the E a r t h ' s Core  3.2.1  I d e n t i f i c a t i o n of  Phases  3.2.2.  T u r n i n g P o i n t s and End P o i n t s f o r time Branches  % the T r a v e l 77  3.2.3  Amplitude O b s e r v a t i o n s  80  3.2.4  A Record S e c t i o n f o r  82  Discussion of  the Core Phases  the Analyzed Data  3.3.1  C o r r e c t e d dT/dA  3.3.2  T r a v e l time Measurements  V e l o c i t y Model  Measurements  UBC1  83 83 87  93  CONCLUSION  100  BIBLIOGRAPHY  103  APPENDICES APPENDIX 1. APPENDIX 11.  Seismic R a y Theory L i s t of Earthquakes Used i n Study  107 116  APPENDIX 111. E l l i p t i c i t y and F o c a l Depth C o r r e c t i o n s to Measured T r a v e l times  120  APPENDIX IV.  Method of Summary Values  123  APPENDIX Va Vb  UBC TRIP - T r i a n g u l a r R e g r e s s i o n Package 126 L i s t of dT/dA Smoothed by UBC TRIP 129  vi LIST OF FIGURES Page FIGURES 1.1  V e l o c i t y models o f Gutenberg  and R i c h t e r  3  1.2  B o l t ' s 1964 T 2 v e l o c i t y model  7  1.3  Adams and R a n d a l l 1964 v e l o c i t y model  9  1.4  Buchbinder's  12  1.5  Warramunga A r r a y , A u s t r a l i a  16  2.1  Diagram t o i l l u s t r a t e a l e a s t squares t e c h n i q u e o f d e t e r m i n i n g dT/dA and azimuth  22  2.2  Example o f d a t a used  27  2.3  M a t c h i n g waveform t e c h n i q u e used t o d e t e r m i n e a r r i v a l t i m e s a t each seismometer  1971 v e l o c i t y model  i n the s t u d y relative  30  2.4  Measured t r a v e l times  33  2.5  Reduced t r a v e l times  34  2.6  R e f r a c t i o n o f a p l a n a r s e i s m i c beam i n c i d e n t on a d i p p i n g i n t e r f a c e between a ha I f - s p a c e and the o v e r l y i n g l a y e r .  41  2.7  Measured  49  2.8  dT/dA  c o r r e c t e d f o r l o c a l s t r u c t u r e beneath  2.9  dT/dA  smoothed by t h e method o f summary v a l u e s  dT/dA the a r r a y  50 62  2.10 Comparison o f dT/dA smoothed by p o l y n o m i a l r e g r e s s i o n and method o f summary v a l u e s  65  3.1  T r a v e l t i m e - d i s t a n c e p l o t o f seismograms  81  3.2  F i t o f smoothed dT/dA  84  3.3  F i t o f smoothed t r a v e l t i m e c u r v e s t o d a t a  3.4  F i t o f smoothed reduced  3.5  Traveltime residuals  3.6  Traveltime differences  curves to c o r r e c t e d data  t r a v e l t i m e curves to data  (0 - C)  . 85 86 89 90  vii LIST OF FIGURES (cont'd)  3.7  The v e l o c i t y model  A-1  Seismic  A-2  S e i s m i c r a y theory f o r d i s c o n t i n u o u s a t a boundary.  A-3  UBC1 f o r the e a r t h ' s  core  32  r a y theory  S e i s m i c r a y theory f o r d i s c o n t i n o u s a t a boundary.  108 increase  in v e l o c i t y 1  decrease  in v e l o c i t y  1  2  114  v i i i LIST OF TABLES Page TABLE 1.1  C o o r d i n a t e s o f the i n d i v i d u a l Warramunga a r r a y  seismometers o f the 18  2.1(a)  Summary p o i n t s f o r the DF branch  56  (b)  Summary p o i n t s f o r the GH branch  57  (c)  Summary p o i n t s f o r the IJ branch  58  (d)  Summary p o i n t s f o r the AB branch  59  2.2  L i s t of  2.3  Equations  3.1  E q u a t i o n s f o r t r a v e l times smoothed by polynomial ression  3.2(a) (b) 3.3  dl/dH  smoothed by method o f summary v a l u e s  f o r dT/dA  smoothed by polynomial  P o s i t i o n o f t r a v e l t i m e cusps  regression  60  Sk  reg88  i n v e l o c i t y model UBC1  96  V e l o c i t y - d e p t h r e l a t i o n s h i p f o r v e l o c i t y model UBC1  96  T r a v e l t i m e s f o r model UBC1  97  CHAPTER I INTRODUCTION The P r o j e c t  1.1  The c o r e o f neighbors  and  its  the e a r t h  the e a r t h ' s  made near the s u r f a c e . in this  regard.  the e a r t h ' s has  core  less accessible  p r o p e r t i e s a r e almost  c h a r a c t e r i s t i c s of  ful  is  as d i f f i c u l t  Seismologtcal  provided d e t a i l e d  PKP,  to d e t e r m i n e .  planetary Physical  c o r e can o n l y be i n f e r r e d from measurements investigations  The a n a l y s i s of s e i s m i c  (e.g.  than most o f our  have proved most  success-  body waves which have p e n e t r a t e d  SKS and o t h e r phases)  i s one such method which  i n f o r m a t i o n c o n c e r n i n g the p h y s i c a l  p r o p e r t i e s of  the  core. The p r e s e n t study array  uses PKP phases r e c o r d e d a t  in Northern T e r r i t o r y A u s t r a l i a  model f o r  the e a r t h ' s  tion region  1  -  core.  that r e g i o n ,  'fluid'  o u t e r c o r e from the  through  the e a r t h ' s  or f o r e r u n n e r s  p l a c e d on the c o n t r o v e r s i a l  inner c o r e .  to the PKIKP phases p r o v i d e These p r e c u r s o r s  i d e n t i f i e d on i n d i v i d u a l  The s i g n i f i c a n c e to the major  is  a few hundred k i l o m e t r e s 'solid'  of  the a r r a y  PKIKP phases,  is  PKIKP.  travel  The p r e c u r s o r s  are n o r m a l l y too small  that  large  these p r e c u r s o r s ,  can be i d e n t i f i e d , not j u s t  to be earthquakes.  in  by t h e i r  v e l o c i t i e s or  addition t r a v e l times,  traveltime  (dT/dh). The remainder of  of  The PKP phases which  i n f o r m a t i o n on the n a t u r e of  but even more d e f i n i t i v e l y by t h e i r apparent gradients  'transi-  t h i c k , which s e p a r a t e s  seismograms, e x c e p t f o r  study  seismic  as a means of d e r i v i n g a v e l o c i t y - d e p t h  inner c o r e are o f t e n d e s i g n a t e d  the t r a n s i t i o n r e g i o n . readily  Emphasis  the Warramunga  the work done  this  chapter  in the p a s t r e g a r d i n g  The r e l a t i o n s h i p of  seismic  the s t r u c t u r e of  v e l o c i t i e s w i t h the p h y s i c a l  the c o r e a r e b r i e f l y d e s c r i b e d . i n som? d e t a i l , and mention  is devoted to a b r i e f  The Warramunga  i s made of  discussion  the e a r t h ' s properties  seismic array  the use o f an a r r a y  is  core. of  described  in determining  the  structure.  Chapter 2 i s devoted to d e t a i l s  t e c h n i q u e s used, and the r e s u l t s o b t a i n e d . of  the r e s u l t s .  in t h i s 1.2  study  A brief digression  is  presented  the measurement and  Chapter 3 concerns a  on the b a s i c  in Appendix  seismic  Discovery of  the C e n t r a l  At the timrn o f t h a t up to d i s t a n c e s less  this  discussion  ray t h e o r y  involved  I.  105°.  Short  distances  Inner Core  century, observational  of about  uniform amplitudes;  and  beyond 1 0 0 ° , the a m p l i t u d e s  than about  1 4 3 ° , but at  less  not r e a p p e a r .  than 1 0 5 ° .  rapidly  were d e l a y e d  the t r a v e l time curve f o r  The c o r r e s p o n d i n g s h o r t p e r i o d S waves  The e x p l a n a t i o n  suggested  P waves  fell  by  (Gutenberg,  a p p r o x i m a t e l y 40%.  marked the boundary between s o l i d m a t e r i a l (since  S waves were not  1914) was  a l i q u i d c o r e of It that  the e a r t h was was  not  This  (mantle)  t r a n s m i t t e d throught  than  a g a i n at  did  that  a depth of about 2900 km, t h e r e was a sharp d i s c o n t i n u i t y a t which v e l o c i t y of  noted  greater  and s t r o n g l y  these d i s t a n c e s ,  some two minutes compared w i t h the e x t e n s i o n of distances  decayed  p e r i o d waves a t d i s t a n c e s  p e r i o d P waves appeared c o n s i s t e n t l y  greater  seismologists  1 0 0 ° , both P and S waves were r e c o r d e d w i t h mo  with complete e x t i n c t i o n of s h o r t  it).  at the  discontinuity  and l i q u i d m a t e r i a l Thus,  the e x i s t e n c e of  postulated.  long b e f o r e more s e n s i t i v e  seismographs  showed  the P wave t r a v e l time c u r v e , which extended from 1 4 3 ° to 1 8 0 ° , was  associatednear i n g as was  analysis  P r e v i o u s work on c o r e s t r u c t u r e  1.2.1  or  of  far  that  its  back as  b e g i n n i n g with small 110°.  P-wave o n s e t s ,  The immediate e x p l a n a t i o n of  these  the waves were motion which was d i f f r a c t e d a t a cusp  t r a v e l t i m e curve near of an  amplitude  143°.  A competing h y p o t h e s i s ,  inner c o r e i n which the P v e l o c i t y was g r e a t e r  was advanced  in 1936 by the Danish  seismologist  I.  extend-  observations i n the  s u g g e s t i n g the e x i s t e n c than  in the o u t e r  Lehmann.  This  core,  hypothesis  -3«4i  Q OJ  In > t 0  3 u 110  o  3 z o  6-W.TEH oefto— RICHTER  I'O  6-5  01 .  1 * 1 p.. V e l o c i t y models of J e f f r e y s and  T E F F R E V S  C»U  Gutenberg-Richter  W a t K&- —  RICHTKR  DISTANCE  OlSTONCE  I,lb. Traveltime Curves  I.Ic. Ray paths through core corresponding:'to travelti, e curves i;  Figure I.I V e l o c i t y Models of J e f f r e y s and Gutenberg-Richter,  -improved c o n s i s t e n t w i t h o b s e r v a t i o n s of It  was a c c e p t e d  inS+ead,'  of  the d i f f r a c t i o n  showed by an e x t e n s i o n o f A i r y ' s for  waves w i t h p e r i o d s of  the  hypothesis, after  theory of d i f f r a c t i o n  approximately  more than 3° from the cusp at  t r a v e l t i m e curves at  the  inner  been l o c a t e d ,  o b s e r v a t i o n s c o u l d be e x p l a i n e d  by the  and o u t e r  core, all and the  1.2.2  for  Velocity  the c o r e of  the  Solutions for  of  core.  Earth's  P waves  8 k m / s e c . The v e l o c i t y  a depth of 4850 km, thereupon  at  to e x i s t  a depth of 5150  and R i c h t e r 5120  core.  the  km.  i n that  km, f o l l o w e d  inner  They t h e r e f o r e  between  it  inner  Jeffreys  Thenext  The v e l o c i t y  1.1a,  Figure  1.1b.  which  Note  c o r r e s p o n d s to the p o i n t the  the  velocity-  and  in  distribution mantle-  13-7 k m / s e c . 10.2km/sec  c o r e , with the  11.3  inner  11.4  km/sec,  core  from t h a t of  km/  at  boundary Gutenberg  decrease from a depth of 4980 km to 1 :8 k m / s e c . at  the  boundary of  l e n g t h s of  shallowest angle  inner  The two models are  the c o r r e s p o n d i n g t r a v e l t i m e curve i n the  to  r e g i o n 300 km  i n the zone s u r r o u n d i n g the  A on the c u r v e ,  t r a v e l t i m e curve ABCDEF.  to  a transition  (1939) model d i f f e r e d  i n c r e a s e of  the d i f f e r e n c e  leaves the f o c u s a t  a t r i p a r t i t e earth,  decreasing slowly  and o u t e r  decrease  and the form of  con-  step was to o b t a i n  then i n c r e a s e d s l o w l y t o  needed to e x p l a i n h i s observed t r a v e l t i m e s . Figure  P and S wave  dropped s h a r p l y from  postulated  showed a v e l o c i t y  by a sharp  bulk of  boundaries  increasing r a p i d l y andcontinuously  s e c . o v e r a depth of 300 km, and f i n a l l y  thick  not occur  i n c r e a s e d w i t h depth down to the  to a p p r o x i m a t e l y  center.  the main  (1937) f a v o r e d a v e l o c i t y  about 2900 km, where i t  the e a r t h ' s  that  Core  c o r e boundary at  at  (1939)  earth.  Gutenberg and R i c h t e r i n which the v e l o c i t y  a caustic,  geometry o f a t r i p a r t i t e e a r t h  depth s o l u t i o n s c o n s i s t e n t w i t h o b s e r v a t i o n s f o r particular  Jeffreys  1 s e c , diffraction.could  w i t h i n the e a r t h had a p p a r e n t l y  inner  distances,  143°.  With the d i s c o v e r y of  s i s t i n g of a m a n t l e ,  near  all  the  in  i s shewn  branch BC,  and s u c c e s s i v e l y s t e e p e r the  c o r e was  shown  n e c e s s a r y to enter  Rays c o r r e s p o n d i n g to  the  The the rays  in ray  core trace  branches AB and BC  -5p e n e t r a t e o n l y the outer c o r e , the cusp a t discontinuous  decrease  in v e l o c i t y at  quent, P v e l o c i t y v a r i a t i o n (i.e.  large amplitudes).  rays which undergo core.  total  B being a d i r e c t consequence o f  the m a n t l e - c o r e boundary and the  paths of  internal r e f l e c t i o n at  the boundary of  to a ray t r a v e l l i n g d i a m e t r i c a l l y through  these rays are  emergence of the same  rays  illustrated  caustic  the t r a v e l t i m e curve r e p r e s e n t s the  inner  Rays that r e p r e s e n t the branch DF p e n e t r a t e the inner c o r e ; the  F thus c o r r e s p o n d s  subse-  i n the o u t e r c o r e , and is a s s o c i a t e d w i t h a The branch CD of  in F i g u r e  1.1c,  that correspond to the p o i n t s  point  the e a r t h .  in which the p o i n t s  in F i g u r e  The of  1.1b are shown by  letters. The h y p o t h e s i s o f a s i n g l e  v e l o c i t y jump in the t r a n s i t i o n  region  between the inner and outer c o r e s proved inadequate, however, with the covery  the  (Gutenberg,  1 9 5 7 . 1958) of  i n the approximate d i s t a n c e a t the s t a t i o n s  short period precursors  range  1^3°.  \25°<c&<-  up to 15 seconds e a r l i e r  dis-  to the DF branch  These p r e c u r s o r s  than the DF a r r i v a l s .  arrived  Gutenberg  1  on  (1957)  suggested,  (1953).that  gradual  in a t r a n s i t i o n zone between a l i q u i d outer core and a more  inner c o r e . increase  However, h i s  s u g g e s t i o n became u n a c c e p t a b l e when no  in wave p e r i o d s c o u l d be observed in the time  from the b e g i n n i n g of  the s h o r t p e r i o d p r e c u r s o r s  DF phases. Another p o s s i b l e mechanism f o r by Knopoff and MacDonald magnetic f i e l d  (1958),  They p r e d i c t e d that a t o r o i d a l  differing  at A  time by 12 sees,  s e p a r a t i o n would be independent of f r e q u e n c y . that s t r e n g t h was u n l i k e l y ,  the e q u a l i t y of  the t r a v e l t i m e s of  since  the a x i a l  the discussed  the e f f e c t s of a p o s t u l a t e d  PKIKP pulses  in t r a n s m i s s i o n  of  the e a r l y PKP onsets was  in a study of  in the inner c o r e .  interval  to the a r r i v a l  about 5 x 10^ o e r s t e d was r e q u i r e d to s p l i t  a f i e l d of  by Kuhn and V i e l h a u e r  the PKP wave p u l s e may become d i s p e r s e d i n t o a f r e q u e n c y - depend-  ent wave t r a i n viscuous  on the b a s i s of e x p e r i m e n t s ^ v i s c o s i t y  f i e l d of  i n t o two components  = 1 4 0 ° , and that  the  They observed however, the s e i s m o l o g i c a l and e q u a t o r i a l  pulse that  evidence for  PKIKP waves,  with  -6a l l o w a n c e f o r e l l i p t i c i t y , p l a c e d an upper on the magnitude o f the magnetic f i e l d , Bolt waves  (1959,  (DF branch)  1962,  p r o v i d e d the f i e l d was t o r o i d a l .  suggested t h a t  the p r e c u r s o r s  c o u l d be caused by an a d d i t i o n a l  v e l o c i t y at a shallower inner c o r e .  1964)  l i m i t of o r d e r 5 x 10"' o e r s t e d  level  t o the PKIKP  discontinuous  increase  than the p r e v i o u s l y a c c e p t e d boundary o f the  The replacement o f a s i n g l e  velocity discontinuity  by two a d -  j a c e n t jumps has the f o l l o w i n g consequences f o r the t r a v e l time c u r v e : the jumps o c c u r a t r a d i i where R <C Rj  Rj and R2 i n s i d e  As shown i n F i g u r e  R .  2  c  the core o f r a d i u s  1.2a,  R  c  p = dT/d£ , first  the f a m i l y o f r a y s  steeper angles of  incidence  t o the s p h e r i c a l (smaller  reflected  to generate  magnitude  of the v e l o c i t y jump a t R  total  internal  rays  determines  generate  of ray  the c r i t i c a l  R, and f o r  internally  angle f o r  the p o i n t a t which the p o s i t i o n o f the  o f the c o r e between  branch GH.  surface of radius  the r e c e d i n g branch HD.  described.  The v e l o c i t y c o n t r a s t or  i n the s h e l l  the a d v a n c i n g  t o the s p h e r i c a l  are t o t a l l y  i . e . i t determines  r e f r a c t e d normally  (R|> R2) generate  becomes t a n g e n t i a l flected  f  surface of radius  the r a y s  r e f l e c t i o n , and hence determines  The rays  R, and R2  p)  the r e c e d i n g branch QG.  r e f r a c t i o n beneath R] commences p o i n t G.  through  generate the l o c u s ABC as p r e v i o u s l y  At C , a r a y becomes t a n g e n t i a 1  let  = 3,4-73 km  the mantle and outer c o r e which a r e d e f i n e d by d e c r e a s i n g v a l v e s parameter  in  radii  At H, a r a y once a g a i n  R , and the t o t a l l y r e 2  As b e f o r e , the magnitude  of  the v e l o c i t y jump at R d e t e r m i n e s  the p o s i t i o n o f the cusp a t D, where r e f r a c -  t i o n beneath R  r e f r a c t e d beneath R  the  2  inner c o r e )  begins.  The rays  generate  the a d v a n c i n g  decreasing gradient  which (cf.  the observed p r e c u r s o r s  arguments  1.1  is,  through  p.  i t produces an a d d i t i o n a l  Figure  (that  branch DF which has m o n o t o n i c a l l y  The main p o i n t o f the double-jump h y p o t h e s i s apparent -  2  is  immediately  s e c t i o n o f the PKP c u r v e - CGH - to  to PKIKP (DF branch)  f o r one v e l o c i t y jump).  may be r e f e r r e d  By e x t e n s i o n of the p r e c e d i n g  to a t r i p l e v e l o c i t y jump h y p o t h e s i s ,  i t can be seen t h a t the  m»TEL TIME VELOCITY (KM/SEC)  "  -L-  f  -8r e s u l t s would be the a d d i t i o n o f For any h y p o t h e s i s  two branches of p r e c u r s o r s  to be u s e f u l ,  to the DF b r a n c h .  i t must be shown to lead to com-  p l e t e t r a v e l t i m e curves which do not v i o l a t e accepted c o n s t r a i n t s those p r o v i d e d by ray t h e o r y .  For example, ray theory demands  r e c e d i n g branches  CG and HD generated by t o t a l l y  concave^ upwards,  while  the a d v a n c i n g  must be concave downwards. straight Since that  The branches  from C and G r e s p e c t i v e l y , with  the v e l o c i t y  increases  B.  These c o n s t r a i n t s  Empirical  constraints  with  that  as  the  r e f l e c t e d raysi. must be  generated by r e f r a c t e d  CG and GH must extend  rays  practically  the main c u r v a t u r e near G and H.  discontinuously  the o n l y cusp to be a s s o c i a t e d  one a t  branches  such  at R] and R ,  it  2  large amplitudes  are d i s c u s s e d  more f u l l y  a r e p r o v i d e d by observed  is  expected  would be the  in Appendix  t r a v e l times f o r  I.  the  individual  branches. Using  the v e l o c i t y t a b l e s  of  Jeffreys  (1939) as h i s  B o l t found t h a t he c o u l d r e t a i n the c a u s t i c  at B, and at  eliminate  Jeffreys'  the n e g a t i v e  r e d u c t i o n of His f i n a l this  Jeffreys'  v e l o c i t y gradient velocity just  procedure was  region  to the f i r s t  of  for  Rj and R  solutions, is  2  discontinuity  (i.e.  from 0.52  illustrated  Figure  inner c o r e to be c o n s t a n t ,  2  1.2b.  that " t h e r e  is  c o r e h a v i n g mean r a d i u s material  V  s  near  1,220  10.31  with thickness  km/sec.  This shell  kms and mean P v e l o c i t y  i n the i n t e r m e d i a t e s h e l l  is  or  c  from  depths  solutions shells.  Three  preferred solution,  curve i n 1.2d.  A  caustic.  bounded by Rj  and to c a l c u l a t e  the three superimposed  in the c o r e a d i s c r e t e s h e l l  and w i t h a mean P v e l o c i t y  to 0.48 R  c  The c o r r e s p o n d i n g v e l o c i t y model  1.2c and the r e s u l t i n g dT/^A  by a  10*03 km/sec.  in the s h e l l  and T3 were d e t e r m i n e d , of which h i s  in F i g u r e  R  at  point,  time  s o l u t i o n only  to take the v e l o c i t y as constant  by b u i l d i n g up times f o r  T|, T  the same  beneath the r e g i o n p r o d u c i n g the  of 4,564 km to 4 , 6 0 2 km), assume the v e l o c i t i e s and R£ and in the  starting  Bolt  is  shown  T , 9  in  concluded  of o r d e r 420 kms, surrounds  11.22 km/sec.  not l i k e l y to have marked  the  inner  The  rigidity.  rt  R ,V , -1978 km(0.57Ru) 9.87 -10.00 km/sec 1  20m  30i  W  "~  20  i»  11*  l«m.  A  $ <  s  30s  CDs  30s  F  1  R ,V , -1579 km(0.4555*) 9.98 -10.41 km/sec &  -1249 k a f O . ^ Z B c ) 10.15 -IT.16 km/sec  R ,v 0  Cusp  T I8m I8n; 20m ian 20m ISm  B I J G H D I.3a.  10  o  s  ;  K,  i  t  i i i  /  i ——.  ,  i "  I  s  y  Solution ~i  J  i  •  o — —  J  D  s  —  / '  I.5c.  1.956  i i  i  i  2.161  r ~  i  i  2.764 2.650  ;  i  .a  156.4 125.0 160.5 IIO.O  Velocity  I.'Jb. A-R Traveltime Ourvss,  i  144.1 3.501 130.0 3.455  37.2s 48.1S 13. 2s 46.9s 17.4s 33.2s  Si  «o  1  t  Ro  1  i  1 , 1  *4*  IM  fi»  I.3d.  A~R dt/dA  t i 0  A-R V e l o c i t y Model.  F i g u r e 1.5. Mans a i d Randall I964 V e l o c i t y Kodel  Model.  \ F \  ISO  -10w The  1 ike1y to be sol i d .  inner core is  J u l i a n • , -fit oi  reported that they may have o b s e r v e d the phase at  the  LASA a r r a y  i n Montana.  The PKJKP phase  would prevent the c o n v e r s i o n from P to Adams and R a n d a l l ' s  recorded  i s one which t r a v e l s  through  A liquid  inner c o r e  S.  (1963) a n a l y s i s  of seismograms w r i t t e n from  c e r t a i n New Zealand e a r t h q u a k e s  i n the range  existence o f  the GH branch i n t r o d u c e d by B o l t ,  the l a t t e r p a r t o f  148°.<- A £~ 1 5 6 ° c o n f i r m e d the and o b -  servations p u b l i s h e d by Nguyen-Hai .(1961) a r e now known to b e l o n g to branch.  In  1964, Adams and Randall  together with o b s e r v a t i o n s  to d e t e r m i n e y e t an a d d i t i o n a l  curve d e r i v e d by B o l t .  This additional  branch t o the  branch,  i n t r o d u c t i o n o f an a d d i t i o n a l  r = 0.57  R  c  IJ i n t h e i r n o t a t i o n ,  This d i s c o n t i n u i t y  is above  is  c  the l e v e l o f  the r a d i u s o f  r = 0.54  R, c  was  necessitated  v e l o c i t y d i s c o n t i n u i t y at a  (depth o f 4,390 km) where R  p u b l i s h e d by  traveltime  introduced to e x p l a i n the e a r l i e s t p r e c u r s o r s o b s e r v e d , and the  this  used t r a v e l t i m e data from 24 earthquakes  and one s e r i e s o f n u c l e a r e x p l o s i o n s , other w o r k e r s ,  since  PKJKP on seisinograms  inner core as an S wave and e l s e w h e r e as a P wave.  the  (1972) have  radius  the e a r t h ' s  which i s  core.  the depth  to which r a y s would have to p e n e t r a t e the c o r e b e f o r e a c a u s t i c would be formed at  B, as  feature.  i n B o l t ' s model and c o n s e q u e n t l y t h e i r model does not e x h i b i t The second d i s c o n t i n u i t y was p l a c e d a t a depth of 5,112  r e s u l t s were o b t a i n e d by d i f f e r e n t i a t i n g the t r a v e l t i m e d a t a The  described  then used to o b t a i n a v e l o c i t y - d e p t h model f o r  The d e t a i l s o f  their final  1.3d  respectively.  two s h e l l s ,  i n v e r s i o n t e c h n i q u e , to be  model a r e shown i n F i g u r e  traveltime c u r v e , v e l o c i t y model, and P-A  curves  Thus Adams and Randall  v e l o c i t y gradient  observed t r a v e l t i m e s . several e v e n t s  in the range  130°to  1.3a w i t h the c o r r e s p o n d i n g  in F i g u r e s  surrounding  in each s h e l l  Hannon and Kovach  the c o r e .  1.3b,  1.3c, and  (1964) suggested the e x i s t e n c e of  each between 300 and 400 km t h i c k ,  a small n e g a t i v e  Their  to o b t a i n a  P - ^'relationship. l a t e r , was  Herglotz-Wiechert  km.  this  the  inner c o r e ,  r e q u i r e d to e x p l a i n  the  (1966) v e l o c i t y f i l t e r e d d a t a  160°. The d a t a were r e c o r d e d a t  with  from  the extenaed  Tonto F o r e s t A r r a y of  The v e l o c i t y f i l t e r i n g t e c h n i q u e ,  time d e l a y i n g and summation a c r o s s  conventional  the l a r g e a r r a y ,  IJ and GH, to the DF a r r i v a l  l e a s t one o f these p r e c u r s o r s  at distances  greater  became  than 1 4 5 ° . T h e i r  i n the form  i n c o n j u n c t i o n with  amplitude and t r a v e l time a n a l y s i s , y i e l d e d  period precursors, At  in A r i z o n a .  e v i d e n c e of two s h o r t  at d i s t a n c e s .1 ess than 1 4 0 ° .  an i n t e r m e d i a t e a r r i v a l results  supported  to DF and AB  the model o f Adams  and Randal 1 (1964). From an a n a l y s i s World Wide S t a n d a r d i z e d for  of t r a v e l t i m e and amplitude d i s t a n c e d a t a o f the  Seismograph  Network  (WWSSN), Ergin(1967)  found e v i d e n c  the e x i s t e n c e of the GH and IJ branches o f Bolt and Adams and R a n d a l l ,  and a t t r i b u t e d the branches  to two d i s c o n t i n u o u s  depths of 4,500 km and 4,685 km r e s p e c t i v e l y .  increases  Delays  in v e l o c i t y at  i n PKIKP t r a v e l t i m e  observed at 153° and 162° were i n t e r p r e t e d as the r e s u l t of two l o w - v e l o c i t y layers within as  the inner c o r e .  i n p r e v i o u s models,  Ergin obtained  the second branch being 3-6 seconds  and i n t e r p r e t e d as the r e s u l t of a d e c r e a s e To o b t a i n AB t r a v e l t i m e near  i n the p r o p a g a t i o n o f d i r e c t  this  was put a t around  A  =125°> o  DF branch a t d i s t a n c e s  shorter  than  later  (that a  lower v e l o c i t y was  n  d  the c r i t i c a l  is,  the p o i n t  D i n the  waves observed a l o n g the o  125 , down to 106  were  i n t e r p r e t e d as extinguished  angle.  B o l t (1968) found no e v i d e n c e f o r the d e l a y s r e p o r t e d by Egin (1967). A group o f small DF a r r i v a l s o o than expected i n the range curvature  resulted  Based on a m p l i t u d e  o r d i n a r y r e f l e c t i o n s from the c o r e boundary, which were not q u i t e at  than the f i r s  reduction in v e l o c i t y  P up to 130° and beyond.  the b e g i n n i n g o f the PKIKP branch  p r e v i o u s models)  i n s t e a d o f one,  i n v e l o c i t y at a depth of 4,015 km  180° and beyond, he found t h a t  needed a t the base of the mantle, and t h a t  variations,  two AB branches  a t 153° and 162° about 1 second e a r l i e r  1 5 6 ^ A ^ 160 seemed to i n d i c a t e e i t h e r  that the  i n that range had been o v e r e s t i m a t e d , or the presence o f a s l i g h t  d i s c o n t i n u i t y i n the inner c o r e near a depth of 5,520 km. Gogna (1968) a n a l y s e d t r a v e l times of about 3000 o b s e r v a t i o n s  from 35 e a r t h -  R G - V , - 1 8 2 1 km(0.524Rc) }  &  9.99 -10.00 km/sec RpV,, -1521 km(0.437Rc) 10.15-10.17 km/sec R ,V , D  Cusp  9.0  4000  I.4b.  & J H D  4500 ' 5000 DEPTH (km)  Buchbinder's V e l o c i t y f.'odel.  0  -1226 km(0.353Rj 10.26-10.83 km/see A DT/D/1 (Peg) (Sec/Deg) 140.0 3.18 I42.O 2.61 I5<5.0 2.08 120.0 I.21  I.4a. Buchbinder's V e l o c i t y  F i g u r e 1.4. Buchbinder's 1971 V e l o c i t y  Iylodel.  Solution  -13quakes and c o n f i r m e d , once more, the e x i s t e n c e of  the GH branch of o  p r e c u r s o r s as  postulated  but the p o s s i b i l i t y o f trast,  by B o l t .  On h i s model, GH extended from 130 to 153 >  i t e x t e n d i n g beyond 153° was not r u l e d o u t .  he o b t a i n e d no e v i d e n c e in support of  suggested by Adams and Buchbinder  the a d d i t i o n a l  In h i s  of  is  the mantle  placed at  13.44 km/sec.  =143°.  A  resulting  than  at  D and G near  110° and 125° as of  Since he c o u l d not r e s o l v e  and a m p l i t u d e s . o f 5,145  to  PKP,and a second  are  listed  Buchbinder's of  1.4a.  range o f  to h i s  the model of Adams and Randall 1  traveltimes  Figures  1.4b,  0.576  km/sec.  c and d show  the reduced t r a v e l t i m e s , and the  model c o r r e s p o n d to the  on B u c h b i n d e r s  not  the inner c o r e boundary was p l a c e d at a depth  in Figure  c u r v e s c o r r e s p o nding  forerunners.  i n t o b r a n c h e s , he was  the p o s s i b l e  Buchbinders v e l o c i t y model, P - A  km  discontinuity  the l a t e r  km + 10 km, and r e p r e s e n t e d a d i s c o n t i n u i t y o f  These r e s u l t s  these  p r e c i s e l y , but i n s t e a d produced the two  illustrate  The top of  Together  km/sec. at a depth of 4,550  the f o r e r u n n e r data  to d e f i n e the d i s c o n t i n u i t y  PKP  120° and 140°  the higher m u l t i p l e K phases  the e a r l i e s t f o r e r u n n e r s of  branches o f f o r e r u n n e r s  The c a u s t i c  in p r e v i o u s models.  o f 0.02 km/sec at a depth o f 4,850 km to account f o r  able  gradient,{he  The observed t r a v e l times and a m p l i t u d e s o f  i n a v e l o c i t y d i s c o n t i n u i t y of 0.01  t o account f o r  period  km to 13.44 km/sec.  p l a c e d at a depth of 2,892 km.  w i t h the t r a v e l times and a m p l i t u d e s resulted  2,700  PKP and  the bottom  in a negative v e l o c i t y  were i n t e r p r e t e d to g i v e the t u r n i n g p o i n t s respectively rather  short  r e s u l t i n g v e l o c i t y model, the v e l o c i t y a t  the c o r e - m a n t l e boundary,  B is  con-  IJ branch as  examined t r a v e l times and a m p l i t u d e s of  (1970  v e l o c i t y decreasing from 13.6 km/sec. a t a depth of at  In  Randall.  h i g h e r m u l t i p l e K phases from a worldwide d i s t r i b u t i o n of seismograms.  o  model.  IJ and GH branches,  (1964).  model, the s e c t i o n s  The branches CGG  GG  1  A significant  and J J  1  of  1  and  UJ  1  of  respectively, d i f f e r e n c e is  the r e s p e c t i v e  that  branches  -14represent whilst  phases p a r t i a l l y  they r e p r e s e n t  reflected  off  the t r a n s m i t t e d  the c o r r e s p o n d i n g b o u n d a r i e s ,  phases on the Adams and  Randall  mode I. Seismic V e l o c i t i e s  1.3  Central  1.3.1  and P h y s i c a l  D e n s i t y of  the  The v e l o c i t i e s o f are  g i v e n by the e q u a t i o n s 2  where k is density, obtain  (Bullen,  Core.  o t and ^ r e s p e c t i v e l y ,  19(53).  f^  ~<a  2  i n c o m p r e s s i b i 1 i t y of  and /X i s the c o r r e s p o n d i n g r i g i d i t y .  the medium p. i s  the  From these e q u a t i o n s ,  we can  quantity  = k/p = oC - V3/>0  0 The v a r i a t i o n g i v e n by  P and S body waves,  the b u l k modulus or  the  the  Core  = k + 4/3 A .  foe  P r o p e r t i e s of  2  of d e n s i t y p w i t h depth  the  for  a non-homogenous r e g i o n  is  1963).  (Bullen,  where g i s  2  gravitational  force  per  unit  mass and  & = dk/cfp - c f dtf/riz. Here  the q u a n t i t y  p denotes h y d r o s t a t i c  homogenous medium, 0 a measure o f  is unity.  the a d d i t i o n  cf  0) pressure.  For a c h e m i c a l l y  For v a l u e s g r e a t e r  heavy m a t e r i a l  than u n i t y ,  in excess of  purely  9  gives  hydrostatic  c o m p r e s s i o n of a homogenous medium. The J e f f r e y s ' gradient  i n the  model  transistion  for  the  c o r e , with  its  large  r e g i o n y i e l d s a v a l u e of  negative  dfifdz  velocity !0 e r r / s e c  - ~ '5 x  2 Assuming g r a v i t y equation  (1)  at  a depth of  Reasonable v a l u e s f o r  dk/dp  from 3 to 5 u n i t s .  transistion 30 u n i t s .  600 c . p / s e c  ,  becomes 8  range  5000 km to be a p p r o x i m a t e l y  region,  as  This value  in for  = d k / d p + 25  in  the c o r e ,  based on the  theory  Consequently, with a v e l o c i t y the 8  Jeffrey's  model, Q  i s not o n l y  greater  decrease  becomes than a t  of  solid in  stair,  the  approximately any o t h e r  region  -15in the e a r t h , but ial  which  density  i s denser  increase at  3 20g/cm . in  implies  the admixture of a c o n s i d e r a b l e p r o p o r t i o n o f  than i n t h e o u t e r c o r e .  This e n t a i l s ,  the inner core boundary,  On the o t h e r hand,  the t r a n s i t i o n r e g i o n ,  B o l t ' s - (1964)  and 0  is  T  then of  in t u r n , a  large  and a c e n t r a l d e n i s t y of  about  model has z e r o v e l o c i t y  2  the o r d e r of 5 u n i t s ,  mater-  gradient  implying  3 a c e n t r a l d e n s i t y of about  15 gm/cm .  more c o m p a t i b l e w i t h r e s u l t s shock wave t e c h n i q u e s to These r e s u l t s of  of r e c e n t l a b o r a t o r y experiments u s i n g  infer  the e l a s t i c  p r o p e r t i e s of  is  transient  i r o n at 4 m e q a b a r s the order  13g/cm3.  such as  those  positive v e l o c i t y gradients  in Bmchbinder s  Bolt case.  is  slightly  higher  1.3.2  reasonably  (1964)  small,  than t h a t  is a p p a r e n t ,  result  of  the  to observe  The v e l o c i t y o f  Inner  is  w h i c h , f o r a l i q u i d , has  the v e l o c i t y of  is  in a  S waves  the v a l u e  oc  for  negative the c o r e  •  If  the r i g i d i t y  of  by the  range.  the medium,  zero. region  in the B o l t  (1964)  model,  by 3% and 9%, c o r r e s p o n d i n g to 6%  the t r a n s i t i o n  P waves through  of 6% at  versa.  been e s t a b l i s h e d  in the a p p r o p r i a t e d i s t a n c e  the t r a n s i t i o n  r e g i o n were in a l i q u i d  i t would be given by  k - pc* The jump in  the  gradients,  case.  i n a f l u i d s t a t e has  P waves, oc , i n c r e a s e s 2  18% i n c r e a s e s  velocity  p r o f i l e s w i t h the c o r e , and v i c e  a d i r e c t f u n c t i o n of  At the boundaries of the v e l o c i t y of  6  Core  transmitted  S waves  index  t h e n , t h a t c o r e v e l o c i t y models do p r o v i d e e m p i r i c a l  on a c c e p t a b l e d e n s i t y  Solidity  in a c e n t r a l d e n s i t y  region,  than i n the  model, would, p r o v i d i n g  p r e d i c t e d in the B o l t  That the o u t e r c o r e failure  even lower  The e f f e c t of combined p o s i t i v e and n e g a t i v e  gradient  restraints  i n the t r a n s i t i o n  model would be to reduce the  the c e n t r a l d e n s i t y  in the Adams and Randall  It  (1971)  1  even f u r t h e r , and to b r i n g  and  lower v a l u e of c e n t r a l d e n s i t y  i n d i c a t e a maximum t e r r e s t r i a l c e n t r a l d e n s i t y of  The e f f e c t of  as  This  state,  ( s i n c e /X. = 0)  2  the f i r s t  boundary r e q u i r e s a c o r r e s p o n d i n g  increase  Figure 1.5. The Warramunga seismic array* The inset shows the location of the array i n the stable shield area.  -17in k o f about 6%, s i n c e the d e n s i t y p is not l a r g e amount. magnitude  Bullen  (1963) has  l i k e l y to i n c r e a s e by any  i n d i c a t e d that an i n c r e a s e  is u n l i k e l y , and c o n s e q u e n t l y the e q u a t i o n has  to i n c l u d e the r i g i d i t y  term  in k of  this  to be m o d i f i e d  JJ^ i . e .  k + V3 IL=  fo£  2 With t h i s m o d i f i c a t i o n , some of  the  can be taken up by the r i g i d i t y  term, r e s u l t i n g  in k at the boundary. has s i g n i f i c a n t  This  rigidity,  increase  implies  and i s  that  in  at  ot  the boundary  in a s m a l l e r  the m a t e r i a l  increase  i n the t r a n s i t i o n zone  t h e r e f o r e not c o m p l e t e l y l i q u i d as  in the  2  outer c o r e .  Similarly,  the l a r g e r  jump in ot  the inner c o r e r e q u i r e s an even l a r g e r core,  indicating  the  of  increase  18% a t the boundary of in r i g i d i t y  inner c o r e to be c o m p l e t e l y s o l i d .  p.243) advances f u r t h e r arguments  in f a v o r of a s o l i d  in the  Bullen  inner  (1963i  inner c o r e .  That the inner c o r e is s o l i d  has been c o n f i r m e d r e c e n t l y by o b s e r v a t i o n  of  «tal(l972)  1.4  the PKJKP phase by J u l i a n The Warramunga Seismic  Array  The Warramunga s e i s m i c a r r a y in Northern T e r r i t o r y , A u s t r a l i a  (WRA)  s i t u a t e d near Tennant Creek  i s one of f o u r s e i s m i c a r r a y s  U n i t e d Kingdom Atomic Energy A u t h o r i t y  (U.K.A.E.A.)  three a r r a y s  and i n B r a z i l . of  are  located  in Canada  the o t h e r s ,  is  located  in  (GBA)  than any  It  is  s i t u a t e d on g r a n i t e  outcrops  Precambrian s h i e l d a r e a , a p p r o x i m a t e l y 5 0 0 km from the  n e a r e s t po'mt of  the A u s t r a l i a n c o a s t  medium a p e r t u r e a r r a y c o n s i s t i n g of Willmore Mk II  (EKA), s m a l l e r  India  Scotland.  WRA began o p e r a t i o n in 1965. in the s t a b l e  explosiions.  (YKA), Southern  Another B r i t i s h designed a r r a y  the  type which were deployed  as a means of d e t e c t i n g and i d e n t i f y i n g underground n u c l e a r The o t h e r  of  (see  inset,  1.5).  It  twenty s h o r t - p e r i o d v e r t i c a l  seismometers with a n a t u r a l  damping f a c t o r of 0.6.  Figure  The seismometers  is a component  p e r i o d of one second and a  a r e arranged  in two l i n e s  approx-  Table 1.1 Seismometer  .  '  El B2 133 134 B5 336 B7 B3 B9 BIO Rl R2 R3 R4 R5* R6 R7 R8 R9 RIO  Latitude E 134.34776 134.35250 134.356S1 134.3G050 134.36954 134.36352 134,3S0S0 134.3S213 134.38530 134.39418 134.340S5 134.36555 134.33S31 134.40302 • 134.42764 134.45578 134.47742 134.50191 134.51631 134.54195  Coordinates of the Individual Seismometers of the Warramunga Seismic A r r a y Longitude S  Cartesian Coordinates X , km Y , km.  19,96097 19.94417 19.92453 19.90497 19.87971 19.S5645 19.84227 19.81541 19.79286 19.76862 19,94411 19.94S90 19.95000 19.95181 19.95949 19.95525 19.95667 19.95726 19.95912 19.96109  -0.310 0.183 0. 638 1.025 1.762 1.863 3.150 3.291 3. 623 4.554 -1.033 1.552 3.934. 5.998' 8.052 10.998 13.263 15.826 17.334 20.016  -1.476 0.373 2. 558 4.724 7. 519 10.095 11.662 • 14/635 • 17.132 19.816 0.391 -0.140 -0.264 -0.465 . -1.314 -0.846 -1.004 -1.074 -1.2S1 -1.502  Elevation, • feet • 1294.85 1284.27 1265.89 1252.65 1218.72 1200.07 1203.09 1176.96 1184.94 3188.50 1281.07 1265.19 1236.49 .1228.55 1217.83 1155.42 1167.15 1212.53 1190.GS 11G1.54  * moved to new site 18 September, 1968. R2 and R6 have subsequently been moved to new sites. The geographic coordinates of the origin of the cartesian coordinate system are 19.94777 S, 134.350S1 E .  -19imately at r i g h t  angles  long and c o n s i s t i n g ordinates lative  of  to each o t h e r  of  ten seismometers  the i n d i v i d u a l  to the p o i n t of  in Table  (see F i g u r e  250,000.  and the C a r t e s i a n c o o r d i n a t e s  the two arms of  The s i g n a l s from each seismometer  code on 2 4 - t r a c k FM magnetic system a r e d e s c r i b e d  tape.  in d e t a i l  d a t a a r e forwarded  re-  listed  As a means of each of  seismometers,  improving  installed  T h i s small  a r r a y or  summation,  and t r i g g e r s  in  re-  time  (1965).  England.  array c o n s i s t i n g  and c o n t a i n i n g f i v e v e r t i c a l  in 1 9 6 7 , p a r a l l e l  'cluster'  and Keen e t a l .  e f f i c i e n c y , a small  discriminates  to the main a r m s  a g a i n s t random n o i s e  Universiof  two  component  of  the  array.  by d i r e c t  the tape r e c o r d i n g system o n l y when the c o n e l a t o r  o u t p u t exceeds a s p e c i f i e d t r i g g e r to t h a t of  (1964),  i s o p e r a t e d by the A u s t r a l i a n N a t i o n a l  length 2 . 5 kilometres was  with a d i g i t a l  about  The p r e c i s i o n t i m i n g and FM t e l e m e t r y  by T r u s c o t t  its  of  a r e t e l e m e t e r e d to a c e n t r a l  to the U . K . A . E . A . c e n t e r  A duplicate recorder  lines,  the a r r a y are  co-  are o p e r a t e d a t a d i s p l a c e m e n t m a g n i f i c a t i o n  c o r d i n g s t a t i o n where they are r e c o r d e d s i m u l t a n e o u s l y  ty.  The geographic  1.1. The seismometers  All  each l i n e b e i n g 2 2 . 5 km  e q u a l l y spaced.  seismometers  i n t e r s e c t i o n of  1.5),  the 2k element c l u s t e r  level.  The mode o f o p e r a t i o n  is  similar  i n use at YKA and i s d e s c r i b e d  in  Whiteway  (1965). 1.5  Use of an A r r a y  in D e t e r m i n i n g  Before the development of  Structure seismic  S waves p r o v i d e d the most r e l i a b l e means f o r the e a r t h ' s  interior.  D i f f e r e n t i a t i o n of  c u r v e s y i e l d e d the f i r s t T h i s d e r i v a t i v e was later)  in order  consideration. profiles  arrays,  the t r a v e l times of  investigating  the smoothed  traveltime-distance  i n the H e r g l o t z - W i e c h e r t  integral  to determine a v e l o c i t y - d e p t h p r o f i l e f o r E x t e n s i v e use o f  that are  the s t r u c t u r e of  d e r i v a t i v e , dT/dA , as a f u n c t i o n o f d i s t a n c e ,  then used  P and  A  (to be d e s c r i b e d  the r e g i o n under  t h i s method has y i e l d e d v e l o c i t y - d e p t h  i n r e a s o n a b l e agreement except  in the upper  (T-A)  1000 kms of  -20of  the mantle and the anomalous  t r a n s i t i o n r e g i o n of  a r r a y can be used t o measure dT/dA more f i n e - s t r u c t u r e d e t a i l the s i g n a l  the c o r e .  d i r e c t l y , and is c a p a b l e of  than the c o n v e n t i o n a l  too small  Past use of s e i s m i c a r r a y s the e a r t h ' s  i n t e r i o r has  Aiso,  the d e t e c t i o n and  to be d e t e c t e d on a s i n g l e  These p r o p e r t i e s o f an a r r a y are u t i l i z e d f u l l y  supplying  t r a v e l t i m e method.  enhancing a b i l i t y of an a r r a y makes p o s s i b l e  i d e n t i f i c a t i o n of phases  However, an  in t h i s  seismcgram.  study.  i n the i n v e s t i g a t i o n of  the s t r u c t u r e of  been c o n f i n e d almost e x c l u s i v e l y to s t u d i e s of  (1965)  e a r t h ' s mantle.  Such s t u d i e s were p i o n e e r e d by N i a z i  who i n v e s t i g a t e d  the upper mantle s t r u c t u r e beneath western North America  by measurement o f  P-wave t r a v e l t i m e g r a d i e n t s  in c e n t r a l A r i z o n a .  The f i r s t s t u d i e s  (1968) who  Cleary et al  Wright  of t h i s  c o n c l u d e d , among other  measurements were s t r o n g l y the a r r a y .  at  (1968)  l a y e r a t a depth c l o s e to 800  on the lower mantle u s i n g WRA d a t a (Wright  things,  that  the d T / d A  i n s t r u c t u r e underneath  30  earthquakes  found e v i d e n c e f o r a low  km in the mantle.  His complete work  is c o n t a i n e d in h i s d o c t o r a l  thesis  1970). As the p r e s e n t study  many of  (TFSO)  nature at WRA were made by  measurements a t WRA from  i n the Marianas Islands and s u r r o u n d i n g r e g i o n s velocity  the Tonto F o r e s t A r r a y  p e r t u r b e d by v a r i a t i o n s using dT/dA  and Anderson  the  is of s i m i l a r  type to t h a t made by  the t e c h n i q u e s of measurement and a n a l y s i s  him have been adopted f o r use i n t h i s i n Chapter 2.  study.  Wright,  d e v i s e d and used by  These w i l l  be f u l l y d e s c r i b e d  -21CHAPTER 2 MEASUREMENT AND ANALYSIS  2.1  D e t e r m i n a t i o n o f dT/dA  2.1.1  and a z i m u t h  from a s e i s m i c a r r a y .  Some P o s s i b l e Methods  '  In o r d e r . t o o b t a i n a v e l o c i t y d i s t r i b u t i o n f o r the e a r t h ' s core, knowledge i s r e q u i r e d o f t h e v a r i a t i o n o f the t r a v e l t i m e g r a d i e n t , dT/dA for  , w i t h depth  i n the c o r e .  T h i s i s o b t a i n e d by d e t e r m i n i n g dT/dA  t h o s e s e i s m i c waves w h i c h a r e known t o pass through  the earth's  core, a r r i v i n g a t the s u r f a c e a t d i s t a n c e s i n t h e range 110° t o 180° for  PKP.  Both dT/dA  and t h e a z i m u t h o f a r r i v a l  can be e s t i m a t e d f r o m t h e a r r i v a l  a t a seismic array  times a t the i n d i v i d u a l  seismometers  In at l e a s t t h r e e ways: (1) A C o r r e l a t i o n Method: The a p p r o p r i a t e time d e l a y s c o r r e s p o n d i n g p a r t i c u l a r azimuth  and dT/dA  to tuning to a  f o r e a c h o f the two l i n e s a r e i n s e r t e d  for each seismometer o f the a r r a y , and a summed o u t p u t d e t e r m i n e d f o r each l i n e .  The n o r m a l i s e d o u t p u t a m p l i t u d e s  c r o s s - c o r r e l a t e d assuming a s i n u s o i d a l determined at  as a f u n c t i o n o f azimuth  the s i g n a l a z i m u t h and v e l o c i t y ,  o f t h e summed l i n e s a r e  s i g n a l and t h e c o r r e l a t o r  and dT/dA (Birtill  T h i s method was e x p l o r e d i n some d e t a i l  output,  , givesmaximum v a l u e  and Wh?teway, 1965). (1970) who c o n c l u d e d  by W r i g h t  that i t gave i n s u f f i c i e n t l y h i g h p r e c i s i o n t o be o f any use i n h i s work on m a n t l e P-wave v e l o c i t i e s .  Even  h i g h e r p r e c i s i o n i s r e q u i r e d when  d e a l i n g w i t h the e a r t h ' s c o r e , s i n c e the dT/dA are at most h a l f a s l a r g e as t h e dT/dA c o r r e l a t i o n technique  v a l u e s f o r the c o r e  v a l u e s f o r the m a n t l e .  w h i c h may g i v e more r e l i a b l e a z i m u t h  measurements has been d e s c r i b e d by Muirhead(1968).  A new  and dT/dA -  -22-  Figure 2.1 Diagram to illustrate a least-squares method of estimating d T / d A and azimuth.  -23(2)  A Fourier Transform Technique: d A / d T and azimuth values are determined at a number of  different  frequencies by measurement of differences  of Fourier spectral d e n s i t i e s across a fixed array  in phase angle (SSna et a l , 1964).  Again, however, this method does not appear to be precise enough for use in deep earth -investigations. (3)  A Least-Squares Technique: The best straight  l i n e is f i t t e d to the set of a r r i v a l  times  at the individual seismometers by the method of least squares. individual a r r i v a l arrival  times are functions of the azimuth of a r r i v a l ,  time at the o r i g i n of the array and the traveltime  or apparent v e l o c i t y .  Consequently their  in the best possible estimate of a l l event being analysed. dT/dA  The the  gradient  least-squares f i t  results  three parameters, for the  particular  Since this method was used in obtaining the  measurements in t h i s study, i t w i l l now be described in more  detail. 2.1.2  Details of the Least Squares Method The least squares method has been described by K e l l y (1964)  and Otsuka (1966).  Let the o r i g i n 0 in Figure 2.1 be the point of  i n t e r s e c t i o n of the two arms, of WRA, and take a c a r t e s i a n coordinate system with the y and x axes pointing north and east r e s p e c t i v e l y , in the d i r e c t i o n of the two arms  of the array.  For a medium aperture  array such as WRA, and at the distances in question, the wave front can be assumed to be plane provided the structure of the crust and upper mantle  in the v i c i n i t y of the array is reasonably simple.  a seismometer at Sj , distance R; Suppose a P wave a r r i v a l  from Oj and let  £S;Oy  by  Consider  flj.  crosses the array from azimuth 0 with apparent  -24surface v e l o c i t y  V, and l e t the a r r i v a l  to an e r r o r e ; .  Let the a r r i v a l  be T o .  If  let  y- = R t  £  Cos G ; .  Sin^/v  P=  time a t  can be seen from F i g u r e 2 . 1  *t  where  time at Sj be T-  and  the o r i g i n o f the  To-T; - V-'^RiCos  =  To-Ji  a  - V '  Qiji  Xi=Ri£in&c = Cos tf/v  the number o f seismometers  J*N To o b t a i n best e s t i m a t e s hence best e s t i m a t e s T o at  of  o r , if be  W  e  let;  L  Sin  }  in the  = Cxup  e  2  P, Q, and To, and  the l e a s t squares  }  traveltime  c o n d i t i o n that  be a mimimum i s used.  ?  w  xiP)  o f azimuth 0, apparent v e l o c i t y V, and  x,'tj;  if  array,  the three unknowns  to the t h r e e normal  (fi)  +ben  ?  Q  the o r i g i n of the a r r a y ,  gives r i s e  R 'Sin<f>Sm®i)  £ CTO-T; - C * . P + ^ ) )  o f the e r r o r s , or f (P^ Q., T o ) ,  This  -h Xi  Further m o r e  .  t  N is  Gs&L +  <f>  Cos $  e- = To - T i - (*j,-a+ and i f  array  that  =  and  subject  the sum>  Thus  equations:  ^  ^ i  2  -  CxxU  e  +c  ?  t h f i oju/rhons  written*,  Qxv]p + cvynd  •+  C Y T J -  Tvlx = ©  carv  -25The v a l u e s of x j and y j , the c a r t e s i a n c o o r d i n a t e s o f the i n d i v i d u a l seismometers,are known and the v a l u e s o f T; a r e d e t e r m i n e d by measurement of the a r r i v a l  times a t each o f the s e i s m o m e t e r s .  Consequently  the three normal e q u a t i o n s can be s o l v e d t o g i v e the b e s t e s t i m a t e s of P, Q, and To.  From t h e s e , we o b t a i n :  w=(p  dT/dA  n )"2 * ' = r /V  Z. a p p a r e n t v e l o c i t y o f P wave a t the d e e p e s t p o i n t o f t h e Y ( o ' radius of the e a r t h )  2  2  x  +  d  ra  and 0 = Tan"'  s t  n  e  m  e  a  n  (P/Q) = a z i m u t h o f a r r i v a l a t the a r r a y .  A knowledge o f To, the a r r i v a l  N.O.A.A.  r  (National  time a t the o r i g i n o f the a r r a y , and t h e  Oceanic and A t m o s p h e r i c A d m i n i s t r a t i o n o f the U.S.)  o r i g i n time o f the e v e n t e n a b l e s the t r a v e l t i m e t o be d e t e r m i n e d . Knowledge o f the e r r o r s i n the c a l c u l a t e d q u a n t i t i e s , i n the dT/dA  values, are e s s e n t i a l  i f statistical  particularly  significance  tests  are to be a p p l i e d i n the smoothing o f the d a t a a t a l a t e r s t a g e .  On  the a s s u m p t i o n t h a t the e r r o r s e; i n the measured o n s e t times a r e i n dependent G a u s s i o n v a r i a b l e s , each w i t h mean z e r o and v a r i a n c e K e l l y (1964) showed t h a t the root-mean-square  e r r o r s i n Vj,dT/dA and 0  are g i v e n by:-  (&vM. .. = (S(dT/dA)/ r  ffl  dT/d/s)  s  _ - «- v / ( D ) % a r x Cos ? 2  m  r  N  s  2G>vCr,ij) Sin <?5 Cos <f> + Var tj S m ^ ) 2  tU) '  r. n>'S-  (in rod .cms) u>Ur*  V/CNO)!  =  '  (Varx i  -f Var y Cos* <f>) X D = Var ac Vary  Vary =  VN I  -  z  + 2 G>v(*, y) S)n$G>s $  a.  (.Coy C * » y ) ) '  C y -y) t  Sm $  z  2  -26The  tr  e s t i m a t i o n o f the q u a n t i t y  , poses a s i i g h t problem.  (1970) approached the p r o b l e m by e s t i m a t i n g a random  Corbishley  r e a d i n g e r r o r f o r a number o f e v e n t s o c c u r r i n g W r i g h t (1970) p r e f e r r e d  the e a r t h .  i n a small region o f  to c a l c u l a t e a value of ~ v  f o r each s e t o f r e l a t i v e o n s e t times on the grounds t h a t  the  i n each o n s e t depends on f o u r main f a c t o r s : (1) The waveform  error  o f t h e e v e n t , s i n c e low f r e q u e n c y e v e n t s g i v e l e s s c l e a r l y peaks and z e r o s .  (2) The s i g n a l - t o - n o i s e r a t i o s i n c e random b u r s t s  o f n o i s e cause spurious  changes i n waveform f r o m one  (3) The i n s t r u m e n t a l  the next.  defined  constants  seismometer t o  o f the seismometers w h i c h  v a r y s l i g h t l y from seismometer t o seismometer and (4) r a p i d v a r i a t i o n in local  s t r u c t u r e , an azimuth-and-distance-dependent f e a t u r e .  v a l u e o f r e s i d u a l €.i i s c a l c u l a t e d  a f t e r P, Q. and  To have been d e t e r m i n e d .  Assuming t h a t the r e s i d u a l s have z e r o mean and c h a r a c t e r i s t i c o f that p a r t i c u l a r set of a r r i v a l «• The  value o f  or  instrumental  constants  variations i n local 2.2  times,  variance  o"^  -  then  = . £ «i / N - 3  thus d e t e r m i n e d c o n t a i n s  e r r o r s , but a l s o s y s t e m a t i c  Each  not o n l y random  reading  e r r o r s due t o p o s s i b l e d i f f e r e n c e s i n  o f the  i n d i v i d u a l seismometers and r a p i d  structure.  Measurement T e c h n i q u e s  2.2.1.  Data A v a i l a b l e f o r the  study  As mentioned p r e v i o u s l y , d a t a a c q u i r e d National  by the A u s t r a l i a n  U n i v e r s i t y f o r WRA a r e r e c o r d e d on 24 Channel FM a n a l o g  magnetic tape.  A t the time the d a t a used i n t h i s s t u d y were a c q u i r e d ,  -28a system of d i g i t a l Consequently, chart  c o n v e r s i o n and a n a l y s i s  the usual  procedure was  to p l a y back the data  onto  paper. Firstly,  the s e i s m i c  narrow bandpass f i l t e r s A 16-channel  procedure.  s i g n a l s on tape were f i l t e r e d  (0.4 to 2.0Hz) h a v i n g  the s i g n a l s from the end seismometers o f were reproduced in one p a s s .  and the procedure r e p e a t e d .  It  using  characteristics. at  t r a c e s were t h e r e f o r e played out  The s i g n a l s from one l i n e of  the s e i s m i c  identical  c h a r t r e c o r d e r was the o n l y one a v a i l a b l e  and the twenty s e i s m i c  signal  had not been i n s t i t u t e d .  the time,  in a  two-step  the a r r a y ,  together  the other  l i n e and a  The l i n e p o s i t i o n s  was o f t e n noted that  t r a c e s v a r i e d c o n s i d e r a b l y over  with timing  were r e v e r s e d  the amplitude of  the time range of  interest.  T h i s n e c e s s i t a t e d d u p l i c a t i o n of c e r t a i n p a r t s of the r e c o r d s at or  lower g a i n s e t t i n g s  analysis.  i n o r d e r to o b t a i n amplitudes  The p l a y b a c k speed  (40 mm/sec)  used  best s u i t e d f o r  in o b t a i n i n g  r e c o r d s enabled high r e s o l u t i o n t i m i n g to be a c h i e v e d . the paper seismograms  is  shown in F i g u r e 2.2.  which enabled p r e c i s i o n measurement of peaks or  troughs  range of  113° to 176° a l t h o u g h  this  The m a j o r i t y of events o c c u r r e d over things,  this  range  The s p a r s e n e s s  o f data  between  the a r r a y . used i n t h i s  study were  is  not u n i f o r m l y covered  zones w i t h r e s p e c t t o WRA. the d i s t a n c e s o f  uneven spread o f  i n some d i f f i c u l t y in smoothing  waveforms  They r e p r e s e n t e d data over the d i s t a n c e  due to the l o c a t i o n of a c t i v e s e i s m i c  Among other  An example o f  Note the c l e a r  Seismograms from the 115 earthquakes reproduced in the above manner.  the paper  the r e l a t i v e d i s t a n c e s  of a given c y c l e across  higher  the data  the measured dT/dA  in the ranges  130° to 150°.  later  resulted  values.  120° to 130° and 150° to  led to d i f f i c u l t y in d e t e r m i n i n g p r e c i s e l y the end p o i n t s  of  160° the  -29precursor branches of the traveltime curve, a point which w i l l be discussed l a t e r .  A l i s t of the earthquakes used in t h i s study is  given in Appendix 2.2.2  II.  Measurement of dT/dA  If  and Azimuth  p r e c i s i o n in measurement of dT/dA  comparable to p r e c i s i o n  obtained from an array such as LASA is to be obtained on a medium aperture array such as WRA, onset times v i a t i o n of between 0.01  must be measured with a standard de-  and 0.03 seconds.  The measurement of P onset  times with such accuracy is extremely d i f f i c u l t ,  i f not impossible.  Measurement, instead, of r e l a t i v e P times obtained by matching P waves recorded at each seismometer can y i e l d accuracy of the required degree.  Since the method was f i r s t  described by Evernden (1953),  the technique of measuring r e l a t i v e onset times by matching waveforms has been commonly used in the c a l c u l a t i o n of phase v e l o c i t i e s of surface waves across t r i p a r t i t e  arrays.  Matching surface waves with  periods of 20 seconds or larger to an accuracy of 0.2 seconds is comparable to matching P waves with periods, of about 1 second to an accuracy of 0.01  sec.  With the chart speed of 40mm/sec used for  the  paper records, waveforms could be matched to an accuracy of ± 0 . 5 mm, which corresponds to an average timing error of 0.012  sec.  Thus the  p r e c i s i o n in timing measurement was s a t i s f a c t o r y . The technique used in matching the waveforms is as f o l l o w s : (1) The paper speeds on the two lines of the a r r a y ,  the Red and Blue  l i n e s , were determined by averaging at least f i v e sets of measurements of the distance between 5 - s e c . intervals of d T / d A  , for each l i n e .  in the region of measurement  The two values thus obtained were further  averaged to give an o v e r a l l value for the p a r t i c u l a r event.  It was  -30-  c  »  lure 2.3..  Matching technique for determining; relative arrival times at each seismometer. From the record at pit R4(a), a family of curves with different amplitudes is constructed (b), and matched with tho record at tho other pits (c), (d).  -31found of  i n p r a c t i c e that  this overall  the speed of e i t h e r  c o r r e c t i o n term.  line,  thus e l i m i n a t i n g  Parallax  (2)  speed was w i t h i n 0.10  corrections  determined f o r each t r a c e by o b s e r v i n g in the onset of square wave p u l s e s traces at  the s t a r t  of  the r e c o r d .  mm/sec  the n e c e s s i t y o f a  i n m i l l i m e t r e s were  the d i s t a n c e  deviations  applied simultaneously  to a l l  The c o r r e c t i o n s were never more  than 1 mm w i t h the v a s t m a j o r i t y c e n t e r i n g around 0.5 mm. A t r a c e was s e l e c t e d from both l i n e s of  (3)  the a r r a y ,  appeared to be the l e a s t contaminated by n o i s e or o t h e r ferences was  2.3a).  (Figure  to be measured, was  line corresponding transparent  One c y c l e of  the h o r i z o n t a l  the t r a c e .  c u r v e was used to c o n s t r u c t ,  the f i r s t  were measured. for  onl-yone or  as shown in F i g u r e 2.3b.  phase of  (k)  Using  of  output was determined r e l a t i v e  |n a d d i t i o n  trough of  series  the curves on i t were matched  the r e l a t i v e times of  the  the a p p r o p r i a t e c y c l e f o r each t r a c e  Thus t h r e e s e t s of r e l a t i v e onset  the p a r t i c u l a r  the s e l e c t e d  t h a t e v e n t , and the p o s i t i o n  z e r o of each seismometer  peak and f i r s t  variations  on the same t r a n s p a r e n t paper, a  output for  to the r e f e r e n c e time mark. first  The v e r t i c a l  i n d i c a t i n g the z e r o -  a p a r t i c u l a r event,  paper as an o v e r l a y ,  w i t h each seismometer  line  In o r d e r to compensate f o r  of c u r v e s of d i f f e r e n t a m p l i t u d e s , transparent  paper.  to a chosen r e f e r e n c e time mark was drawn on the  i n a m p l i t u d e between t r a c e s f o r  this  inter-  the phase whose dT/dA v a l u e  t r a c e d onto t r a n s p a r e n t  paper, as was  si gnoJ l e v e l of  which  the p a r t i c u l a r e v e n t .  times were determined In some  cases,  two s e t s of r e l a t i v e times c o u l d be d e t e r m i n e d , depending  on the c l a r i t y of  the waveform.  -32The reason f o r d e t e r m i n i n g more than one s e t o f o n s e t and hence dT/dA  and a z i m u t h , f o r a p a r t i c u l a r  to some e x t e n t f o r the changes another.  These changes  phase  i s t o compensate  i n waveform fromone seismometer  may be due to r a p i d v a r i a t i o n s  a l s o p l a y an important p a r t precursors  PKIKP i n the e a r l y p a r t s set of a r r i v a l  which were o f t e n f a r s m a l l e r  o f the a p p r o p r i a t e d i s t a n c e  phase, was regarded as  By means o f a computer program times were used to c a l c u l a t e a l e a s t 0 and a r r i v a l  If a seismometer  residual  c o r r e s p o n d i n g onset  independent.  (Wright,  1970) the a r r i v a l  Residuals  o f the  time was d i s c a r d e d and the e n t i r e procedure  r e j e c t i o n procedure ensured t h a t  In poorer f i t s  from  the s t a n d a r d d e v i a t i o n s  than 0.03 seconds,  lower  errors  This  o f the o n s e t  and c o n s e q u e n t l y ensured the  high p r e c i s i o n of t h i s method of d e t e r m i n i n g dT/dA The root-mean-square  , azimuth  squares f i t , were d e t e r m i n e d .  a r e j e c t i o n v a l u e of a 0.04 seconds was u s e d .  times were kept l e s s  Each  was found t o exceed 0.02 seconds, the  r e p e a t e d (insing the r e m a i n i n g t i m e s . quality data,  ranges.  square v a l u e o f d T / d A  least  than  and azimuth d e t e r m i n -  time To a t the o r i g i n o f the a r r a y .  measured times r e l a t i v e to t h e i r  could  of waveform f o r the  t i m e s , and c o n s e q u e n t l y each d T / d A  ation for a particular  caused by  Random n o i s e e f f e c t s  in c a u s i n g changes  to the PKIKP phases,  to  in l o c a l  s t r u c t u r e beneath the a r r a y or even to m u l t i p l e a r r i v a l s s t r u c t u r e a t the deepest p o i n t o f the r a y .  times,  i n the l e a s t  squares  and a z i m u t h .  dT/dA  and azimuth  d e t e r m i n a t i o n were a l s o c a l c u l a t e d by the program to f a c i l i t a t e f u t u r e weighting o f the d a t a .  Figure 2.4.  Measured  Traveltimes  REDUCED TRAVEL TIME VS DELTA  a-DF a - GH  g-l  z -  U  x - AB  o 5  So I  B  1  110.0  115.0  1  120.3  I 125.3  :  I 130.3  B  I 135.0  J  1  I*  1 140.0  1 145.0  DELTA f DEG)  Figure 2.5. Reduced  1 150.0  1 155.0  Traveltimes.  T 160.0  •  1 165.0  1 170.0  1 175.0  1  183.3  -35This matching waveform technique and subsequent c a l culations were carried out for any coherent signal observed on the seisinograms within the time range of i n t e r e s t . essary, in the early parts jf  It was often nec-  the appropriate distances ranges,  to resort to the high amplitude data in order to measure the precursors  to PKIKP. If a' p a r t i c u l a r  of the time travel  phase was i d e n t i f i e d as belonging to one  branches on the basis of i t s d T / d A value  a rough estimate of i t s trowel .-hune i , the actual  travel-time  and was  determined by measuring the onset time of a clear wave and subt r a c t i n g from i t difficult ately, all  the o r i g i n time as given by N.O.A.A.  It was often  to pick the onset of the phase on the sei sinogram accur-  because of the background noise and in some cases where  phases arrived c l o s e l y together,  because of interference  preceding phases, for example in the range 140°- 145*.  from  This  resulted in quite a large scatter of travel-times and, as w i l l be seen l a t e r , times.  in the residuals with respect to the smoothed t r a v e l -  To ensure uniformity,  ellipticity  rections as described in Apperidix III  and, focal depth cor-  were applied to the  travel-  times. The 115 events analysed in the above manner yielded a total of 574 phases.  d T / d A and 188 traveltime  measurements for a l l PKP  Figures 2.4 and 2.5 show the traveltime  traveltime Chapter 3.  measurements.  and reduced  These w i l l be discussed further  in  The dT/dA measurements are shown in Figure 2.7. The  large scatter exhibited by the d T / d A  values is mainly a con-  sequence of systematic errors in their measurement. now w i l l be discussed in more d e t a i l .  These errors  -362.2.3.  Sources of S y s t e m a t i c '  Systematic e r r o r s  Error  i n d T / d A and Azimuth Measurements  in dT/dA  be a t t r i b u t e d to f o u r main s o u r c e s : i n seismometer individual  constants  seismometers  and azimuth measurements can  (1) The e f f e c t of d i f f e r e n c e s  (2) D i f f e r e n c e s  i n e l e v a t i o n o f the  (3) El 1 i p t i c i t y of the e a r t h and  (4) The  e f f e c t o f s t r u c t u r e beneath the array. (1)  The e f f e c t o f d i f f e r e n c e s i n the seismometer c o n s t a n t s .  head  (1968)  differences  investigated  the problem o f phase s h i f t s  i s seismometer c o n s t a n t s  sinusoidal  of c r o s s - o v e r  f r e q u e n c y o f 0 . 9 Hz  motion of a p e r i o d o f 1 s e c , the f i r s t  zero  p o i n t of the r e c o r d e d wave would be 0.04 s e c . l a t e r  than i f the instrument had a n a t u r a l  f r e q u e n c y of 1.1Hz.  showed that the e f f e c t o f d i f f e r e n c e s i n the of  due to small  i n a medium a p e r t u r e "array, and  showed that f o r an instrument with a n a t u r a l recording a  . Muir-  He a l s o  damping c o n s t a n t s  the seismometers was of the second order and c o u l d be s a f e l y  neglected.  The seismometer c a l i b r a t i o n scheme i n o p e r a t i o n f o r  U . K . A . E . A . arrays  is d e s c r i b e d by Keen e t a l  (1965).  f r e q u e n c i e s of the i n s t r u m e n t s a t WRA show small one seismometer 0.18  The n a t u r a l  d i f f e r e n c e s from  to a n o t h e r , w i t h a maximum d i f f e r e n c e o f about  Hz between R6 and B9.  In a d d i t i o n , the f r e q u e n c i e s v a r y  slightly  from day to day, a g a i n w i t h a maximum v a r i a t i o n of about 0.20 H? from 0 . 9  to 1.1  Hz.  errors.  Wright  (1970)  t a i n e d c o r r e c t i o n s to deg  -  .  In p r i n c i p l e , i t investigated dT/dA  is possible  this  errors  problem f u l l y , and o b -  with a maximum v a l v e o f about 0.05 sec/  He concluded t h a t " t h e e f f e c t o f d i f f e r e n c e s i n the n a t u r a l  p e r i o d s o f the seismometers on the onset instances,  t o remove such  too small  times  is,  in almost a l l  t o produce a p p r e c i a b l e s c a t t e r o f  i n the v a l u e s o f d T / d A and a z i m u t h " .  systematic  He i n v e s t i g a t e d  -37parametergof  the o r d e r o f 8 -  10 szc/deq  i n magnitude,  l y a c o r r e c t i o n of 0*05 Sec \def^ to these v a l u e s In the case o f the e a r t h ' s has a v a l u e o f d T / d A at  ranging  later)  which amounted  i s more s i g n i f i c a n t .  (2)  Differences  difference the array  and was c o n s e q u e n t l y  i n E l e v a t i o n of  the s y s t e m a t i c in e l e v a t i o n  than 0.05 km.  by c a l c u l a t i n g  tables dT/dA  Seismometers.  study. The  seismometers  time due to t h i s  of  involved, difference  be i g n o r e d .  The e l l i p t i c i t y c o r r e c t i o n  i s g i v e n by the f o l l o w i n g e q u a t i o n  is also possible  intervals  in t h i s  s u r f a c e , and can c o n s e q u e n t l y  6T It  ignored  constants  is l e s s than 0.01 sec. assuming a P wave v e l o c i t y of  E l l i p t i c i t y o f the E a r t h .  traveltimes  (to be  i n seismometer  and lowest  Never-  to as much as  At the l a r g e d i s t a n c e s  d i f f e r e n c e in a r r i v a l  6km/sec at the e a r t h ' s (3)  Individual  i n e l e v a t i o n o f the h i g h e s t  is less  structure  i n some i n s t a n c e s  0.9 Sec/detj, the c o r r e c t i o n f o r d i f f e r e n c e s seemed i n s i g n i f i c a n t ,  negligible.  from about 2.0 Sec/deg -to 0.0 sec/cteg  i n the f a c e of c o r r e c t i o n s f o r l o c a l  described  i s almost  c o r e , however, where the PKIKP branch  1 8 0 ° , a c o r r e c t i o n of t h a t magnitude  theless,  and c o n s e q u e n t -  to  (see Appendix  III).  - -f ( A X h o + h.)  to c o r r e c t the d T / d A  ST f o r a number o f imaginary  values  for e l l i p t i c i t y  epicentres  at- r e g u l a r  o f d i s t a n c e and a z i m u t h , and d i f f e r e n t i a t i n g the r e s u l t i n g  t o o b t a i n S( d T / d A ). - the e l l i p t i c i t y c o r r e c t i o n f o r the measurements.  Wright  (1970) c a r r i e d out t h i s  procedure  and o b t a i n e d maximum c o r r e c t i o n s of about 0.024 sec/olecj . b a s i s of l a r g e r  corrections  to be determined such as those  On the involved  in c o r r e c t i n g f o r s t r u c t u r e under the array and i n r e s t r a i n i n g the smoothed d T / d A prefferred  values  to ignore  to an a c c e p t a b l e t r a v e l  time .curve , he  the e l l i p t i c i t y c o r r e c t i o n s  to d T / d A . The  -38same procedure i s  (4)  E f f e c t of  serious  adopted  in t h i s  study f o r  S t r u c t u r e Underneath the Array.  source of s y s t e m a t i c e r r o r s  d e t e r m i n e d ' a t WRA, upper mantle  similar  is  dT/dA  is  that  the array.  Initially,  i t was expected t h a t  dT/dA  towards  the end of  (1966) and Carder e t a l  dT/dA first  v a l ue was  i n the middle of  1966,  peri  Studies  i n e r r o r by l e s s  by C l e a r y and  and hence the expected  than 1%.  T h i s p r o v i d e d the  From the r e s u l t s  of  values  Unfortunately,  to enable a r e a l l y d e t a i l e d model o f  azimuth measurements a c c u m u l a t e d ,  it  became e v i d e n t that  beneath the array c o u l d not be approximated by a s i n g l e  simple  in f a c t ,  was  the seismic e x -  beneath the:array to be d e t e r m i n e d . As more d T / d A  and  large  (1967) i n f e r r e d the presence of a  n e a r - s u r f a c e dippi'ng s t r u c t u r e underneath the array.  interface,  Hales  s t r u c t u r e under the array, and the  structure.  ment WRAMP, Underwood  region  of  (1966) s t r o n g l y suggested t h a t a t the  the J-B t a b l e s ,  he used too l i t t l e d a t a  tongshot  they y i e l d e d a v a l u e  d i s c r e p a n c y between the measured and expected dT/dA to t h i s  requirement  11% higher than the v a l u e e x p e c t e d on the  h i n t of an anomalous  attributed  and  reasonably  the n u c l e a r e x p l o s i o n  b a s i s o f the J e f f r e y - B u l l e n t a b l e s .  in q u e s t i o n ,  the c r u s t  s h i e l d f u l f i l l e d these r e q u i r e m e n t s .  which was about  distance  values  discontinuities.  the WRA s i t u a t e d  However, when the array r e c o r d s of were a n a l y s e d  and azimuth  the geology should be  and u n c o m p l i c a t e d by major e l a s t i c  stable  the most  One s t r i n g e n t  homogenous  the A u s t r a l i a n  By f a r  t h a t due to the s t r u c t u r e - o f  i n the v i c i n i t y of  of an i d e a l a r r a y s i t e  in  reasons.  the  and the s t r u c t u r e  plane  appeared to d i v e r g e c o n s i d e r a b l y from  dipping that  approximation. Owing to the f a c t  that d e t a i l s  of  the s t r u c t u r e  underneath  -39the array a r e not y e t a v a i l a b l e ,  and that  the s t r u c t u r e  to be oneof g r e a t c o m p l e x i t y , an e m p i r i c a l method of the  dT/dA  and azimuth measurements  for structure  appears  correcting  must be used.  (1970) d e v i s e d such a method of c o r r e c t i o n based e n t i r e l y  Wright  (1966) t h e o r y on the e f f e c t of a d i p p i n g l a y e r measure-  on N i a z i ' s ments o f  dT/dA  and a z i m u t h .  r e a l i s t i c model o f direction,  The method i n v o l v e s  the c r u s t and a d j u s t i n g  selecting a  the d i p angle  k e e p i n g the P wave v e l o c i t i e s c o n s t a n t ,  until  apparent v e l o c i t y and azimuth c a l c u l a t e d by N i a z i ' s w i t h the measured v a l u e s t h i s method l i e s  o f dT/dA and a z i m u t h .  in the f a c t  and d i p the  theory  agreed  The weakness  that any number of models f i t  and any s t r u c t u r e w i t h a s p e c i f i c v e l o c i t y c o n t r a s t  of  the d a t a ,  c o u l d be  r e p l a c e d by an e q u i v a l e n t s t r u c t u r e w i t h a d i f f e r e n t  velocity  contrast  based on  and d i p a n g l e .  observational angle  of  evidence  If,  however, c e r t a i n  limits  a r e p l a c e d on parameters  such as  the s t r u c t u r e , and expected apparent v e l o c i t y ,  method i s c a p a b l e of y i e l d i n g f a i r l y a c c u r a t e r e s u l t s . has shown how data from earthquakes a t o p p o s i t e azimuths  the  in a given  dT/dA  (1966)  Niazi  at almost equal d i s t a n c e s  from the array can be used to s e p a r a t e  e f f e c t s of a s t r u c t u r e c o n s i s t i n g from e r r o r s  the d i p  of a single  curve.  the  plane d i p p i n g  C l e a r y , Wright and  and  interface  Muirhead  (1968) used t h i s technique to work out a simple model of the c r u s t beneath  the array, and determined the d i p p i n g  a d i p a n g l e around r a t i o o f 0.7.  i n t e r f a c e to have  6 - 7°, d i p d i r e c t i o n of around 230°, and a v e l o c i t y  The c o r r e s p o n d i n g parameters deduced by Underwood  (1967) were a d i p angle of 5°,dip d i r e c t i o n o f 200° and v e l o c i t y r a t i o o f about 0.9.  Wright  (1970), u s i n g  the same method as  in 0  1968,  estimated a dipping  o 7 , and d i p d i r e c t i o n 215 to g i v e  i n t e r f a c e w i t h d i p angle e  - 235  o  of about 6.5  , with a v e l o c i t y contrast  the best c o r r e c t i o n s f o r dj"/dA  and a z i m u t h .  of  to 0.7  Consequently,  -40the predominant e v i d e n c e i n d i c a t e s t h a t t h e d i p a n g l e o f the i n c l i n e d p l a n e assumed t o be r e s p o n s i b l e f o r the l a r g e dT/dA and a z i m u t h a n o m a l i e s i s about 6° o r 7°, w i t h a v e l o c i t y c o n t r a s t o f a p p r o x i m a t e l y  0.7.  The o  d i p d i r e c t i o n i s somewhat more f l e x i b l e , but seems t o be near 220 . It should the dT/dA and  be noted t h a t the above s t r u c t u r e s used t o c o r r e c t  a z i m u t h measurements a r e a l l s i n g l e d i p p i n g p l a n e s t r u c t u r e s ,  whereas, i t appears t h a t the s t r u c t u r e underneath the array i s f a r more complicated  and may c o n s i s t o f a number o f d i f f e r e n t  However, an important  dipping  planes.  f a c t , noted by Wright (1970) i s t h a t the e f f e c t  o f s e v e r a l p l a n e d i p p i n g i n t e r f a c e s on  dT/dA and a z i m u t h measurements  i s i n d i s t i n g u i s h a b l e f r o m t h a t o f a s i n g l e i n t e r f a c e , and  consequently,  the a s s u m p t i o n t h a t the s t r u c t u r e underneath the a r r a y c o u l d be approximated by a s i n g l e p l a n e d i p p i n g i n t e r f a c e i s j u s t i f i e d . iplane d i p p i n g i n t e r f a c e i s t o  introduce  The e f f e c t o f a s i n g l e  approximately  sinusoidal  v a r i a t i o n s i n a z i m u t h and dT/dAas a f u n c t i o n o f azimuth t h a t a r e 90° o u t o f phase.  The dT/dA  anomaly i s l e a s t i n the d i p d i r e c t i o n a n d the  a z i m u t h anomaly changes from n e g a t i v e A c l o s e look a t t h e dT/dA t h i s study  to positive. and a z i m u t h measurements made i n  r e v e a l not o n l y s t a r t l i n g d i s c r e p a n c i e s between e x p e c t e d and  observed values  but a l s o g r e a t f l u c t u a t i o n s i n t h e amount o f the d i s c r e p a n c y  (Figure 2.7). Quite  o f - t e n i t was p o s s i b l e t o c o r r e l a t e the magnitude of  the anomaly w i t h a z i m u t h .  A l t h o u g h the azimuth range was not u n i f o r m l y  c o v e r e d , the r e s i d u a l s o f t h e measured dT/dA  v a l u e s w i t h r e s p e c t t o the  expected values e x h i b i t a q u a s i - s i n u s o i d a l v a r i a t i o n .  The a z i m u t h r e s i d u a l s  a l s o i n d i c a t e some degree o f s i n u s o i d a l v a r i a t i o n but changed more r a p i d l y than do the dT/dA  r e s i d u a l s and a r e not c o i n c i d e n t w i t h them.  On t h i s  Eigure .2.6 Refraction of a planar seismic beam incident on a dipping interface between a half-space and the overlying layer. .  -42basis,  the assumption o f a s i n g l e  the a r r a y as used a justifiable  plane d i p p i n g  i n t e r f a c e underneath  i n the e m p i r i c a l method o f c o r r e c t i o n , seems  to be  one.  For purposes o f a n a l y s i s a computer program w r j t t e n by C. Wright was used to work out the e f f e c t on d T / d A combination of d i p p i n g  t h e o r y , which w i l l  The program i s  be b r i e f l y o u t l i n e d  The geometry of a t h i n p l a n a r on a d i p p i n g  i n t e r f a c e from beneath i s  i n t e r s e c t i o n of plane  (i.e.  OQ, are the  the d i p p i n g  i n t e r s e c t i o n s of  beam with the imaginary  is  taken as  horizontal  beam and v e r t i c a l  axis  OZ, i . e .  is  the t r u e azimuth from which the beam has Q  is  f r a c t i o n of not  the angle of the rays  i b e i n g the angle  of  In  the p r o c e s s of  the v e r t i c a l  to the  be r '  but appear  6*' .  Niazi  interface,  the t r u e  angle angle  1  is  inter-  The r e -  law with  the a n g l e  i  and  1  subtended  Consequently,  , which is d i f f e r e n t from r = i . Q  the rays  to be a r r i v i n g a t  new a z i m u t h , the apparent azimuth 0 ' , d i p being  incident  been i n c i d e n t on the  i n t e r f a c e ON.  r e f r a c t i o n , in g e n e r a ] ,  plane,  the  the f r e e s u r f a c e .  i n c i d e n c e , where i  the angle o f r e f r a c t i o n w i l l  OP and  , and the  p l a c e a c c o r d i n g to S n e l l ' s  between the ray and the normal  The  The angle 0 between OP and the x - a x i s  incidence at  takes  the x a x i s .  is denoted by symbol £  incident  horizontal  plane and the d i p p i n g  i n c i d e n c e , is denoted by i .  i  s e i s m i c waves  plane c o n t a i n i n g  of  face.  below.  shown i n F i g u r e 2 . 6 .  the v e r t i c a l  The d i p angle  between the s e i s m i c  beam of  based e n t i r e l y  i n t e r f a c e w i t h an imaginary  the d i r e c t i o n of s t r i k e )  respectively.  any  i n t e r f a c e s f o r a s e i s m i c event whose azimuth  and d i s t a n c e are a c c u r a t e l y known. on N i a z i ' s  and a z i m u t h , of  do not remain  in  the s u r f a c e from a  the c o r r e s p o n d i n g  apparent  (1966) has shown that the d i r e c t i o n  -43cosines  of  the beam r e a c h i n g the s u r f a c e s  OL , 0  t  are  Y  to the apparent  2 ON, and  and  m=  (Cbsr'-rvO>sS)/Sin  r\ -  Cos r'Gos S ± ocS'tnS Sin r'  the d i r e c t i o n c o s i n e s  plane of  incidence,  2.  ot  = *  velocities  in the upper and  since  Cos'i'  =  i.e.  of  S  the  line  to both the  perpendicular  incident  rays  2. +  4/3  by  +n tf) /<x.  (. = (-m0  where  are g i v e n  Sint  0  =  *  -  V j and  are  lower medium r e s p e c t i v e l y ,  the wave  then  Sin£ + COS*-CDSS  Sm ^  oi -  If  Tan  S Cot t  -  S m ^ / R  Cos 4/^ -Tan  £  Ccstf/R _L  where  K  = C '  The apparent a n g l e s o f v e l o c i t y of I,  m, n as  the rays  T a n ^ Cot "-  +  2  2  + T a n ^ C o S ^ - 2.Sin 2  y4  Tan  6" 6^ i )  i n c i d e n c e , apparent azimuth and new apparent  can then be o b t a i n e d from the d i r e c t i o n  follows:  Cos t $  c  - n =Tan~' ( w / 0 -  cosines  2  "  -44The a c t u a l  procedure was as f o l l o w s .  the c r u s t and mantle was d i p d i r e c t i o n of at  113° to 0 . 3 sec/deg. the DF d a t a ) ,  to about 3 8 0 km/sec. a t the v a l u e s  Taking a f i x e d  1 0 ° and a f i x e d d i p angle o f 0 . 5 , the azimuth 0 was  the e x p e c t e d v a l u e s o f dT/dA  range o f  i n t e r f a c e between  taken as having a v e l o c i t y r a t i o o f 0 . 7 .  i n t e r v a l s o f 2 0 ° from 0 ° to 3 6 0 ° .  deg. at  A dipping  for at  It was assumed from the J - B t a b l e s  the DF branch l a y  176  corresponding  varied  (where  in the range o f 2 . 0 2 2  1 1 3 ° to 1 7 6 ° was  the  that sec/  distance  to expected apparent v e l o c i t i e s o f 5 5 km/sec  C o n s e q u e n t l y , k e e p i n g the d i p d i r e c t i o n and a n g l e f i x e d  c h o s e n , f o u r q u a n t i t i e s were c a l c u l a t e d at each v a l u e  azimuth from 0 ° to 3 6 0 ° at 2 0 ° i n t e r v a l s .  These f o u r v a l u e s  (1)  The apparent azimuth 0 , and c o n s e q u e n t l y  (2)  The measured apparent v e l o c i t y V ' ,  (3)  The measured dT/dA  (k)  The r a t i o of measured v e l o c i t y / e x p e c t e d v e l o c i t y  o b t a i n e d from  where the expected v e l o c i t i e s  the azimuth  of  were:  anomaly.  and c o n s e q u e n t l y V'.  used v a r i e d from 5 5 km/sec -  100 km/sec  i n 2 km/sec s t e p s from 100 km/sec - 2 0 0 km i n 10 km/sec s t e p s and from 2 0 0 - 3 8 0 km/sec  in 20 km/sec s t e p s .  expected apparent v e l o c i t i e s c o r r e s p o n d i n g expected d T / d A  values o  then v a r i e d from 10  for  DF  was c o v e r e d . o  . 0  to 3 6 0  at  10  Thus  the e n t i r e range of  to the e n t i r e .range of The d i p d i r e c t i o n was  intervals,  and k e e p i n g the d i p  a n g l e f i x e d a t 0 . 5 , the e n t i r e procedure was repeated a t each new d i p direction.  The d i p a n g l e was  d i p angle v a l u e , 1 0 ° t o 3 6 0 ° as  the c a l c u l a t i o n s  before.  enable c o r r e c t i o n s for  i n c r e a s e d by 0 . 5 ° from 0 . 5 ° to 1 0 . 0 ° ,  the DF bnanch.  and at each new  were a g a i n performed f o r d i p d i r e c t i o n s  Thus, an e n t i r e s e t o f  t a b l e s was generated  to be a p p l i e d to the dT/dA and azimuth  to  measurements  from  -45The c o r r e c t i o n s were made as f o l l o w s : : 0, and measured d T / d A  and a z i m u t h , and  anomaly o f the event were n o t e d .  The t r u e azimuth  t h e r e f o r e the azimuth  The t a b l e s were checked f o r  a p o i n t where a t the same t r u e azimuth 0 the c o r r e s p o n d i n g dT/dA v a l u e and azimuth anomaly  i n the t a b l e s  p o s s i b l e w i t h the measured dT/dA d i r e c t i o n and d i p angle  and azimuth anomaly.  so s p e c i f i e d were then assumed  parameters o f the plane d i p p i n g particular  agreed as c l o s e l y as  s e t o f anomalies  interface responsible  The d i p to be the for  this  i n dT/dA and azimuth measurement.  In  t h i s manner, a d i p d i r e c t i o n and d i p angle were worked out f o r each dT/dA  and azimuth d e t e r m i n a t i o n f o r the DF b r a n c h . On the assumption  local  that  the same method of c o r r e c t i n g f o r  s t r u c t u r e c o u l d be a p p l i e d  to the o t h e r phases as w e l l , the  e n t i r e p r o c e d u r e was r e p e a t e d f o r the measured v a l u e s IJ and AB t r a v e l t i m e b r a n c h e s .  For the GH and IJ d a t a , expected  apparent v e l o c i t i e s of 30 to 37 km/sec c o r r e s p o n d i n g values  o f about 3 . 6  to 3 . 0  a p p r o x i m a t e l y 2 km/sec  of the GH,  to d T / d A  s e c / d e g were used, r e s p e c t i v e l y , a t  intervals.  The range o f d T / d A  IJ branch was taken onthe b a s i s of the measured dT/dA  values f o r values  which  seemed t o have t h e i r m a j o r i t y o f v a l u e s  in t h i s  ponding  per the Adams and Randall  model.  range o f expected dT/dA v a l u e s The range o f expected dT/dA  estimated  values  of expected d T / d A  values  i n the B o l t  i n order  (1968)  but i t was extended to cover  roughly  values  2 km/sec  o f a bout 4 . 4  intervals.  to 3 . 6  the range  model as w e l l .  AB c o r r e c t i o n s expected apparent v e l o c i t i e s o f 25-31 ing to dT/dA  (1964)  f o r the GH branch was  i n the same manner as i n the IJ c a s e ,  to a maximum v a l u e of about 2 . 9 s e c / d e g  range and the c o r r e s -  For the  km/sec c o r r e s p o n d -  s e c / d e g were u s e d , a g a i n a t  Thus, a d i p d i r e c t i o n and d i p angle was  worked o u t f o r each dT/dA and azimuth d e t e r m i n a t i o n f o r the &H, IJ and  -46AB b r a n c h e s . In e f f e c t , a l l and azimuth v a l u e place  that was done was to take a measured d T / d A  t h a t had been a s s i g n e d to a p a r t i c u l a r  l i m i t s on the expected true v a l u e of dT/dA  for  branch,  that  branch,  and s e a r c h the t a b l e s f o r a c o m b i n a t i o n of d i p angle and d i p direction,until the azimuth anomaly and dT/dA of  the t a b l e s ,  for  of  the measured data agreed w i t h  the p a r t i c u l a r 0 (true azimuth)  A c e r t a i n amount of ambiguity of c o r r e c t i o n . dT/dA  For example,  it  value,  involved  is quite possible  value  IJ v a l u e  method  f o r a measured  a m b i g u i t y were  f i x e d range of expected dT/dA the measured v a l u e s  branch.  The two r e s t r a i n i n g  values,  taken from e x i s t i n g  the d i f f e r e n t  responsible for  the anomaly  angles obtained  in the c o r r e c t i o n s should a l l  is close  An important f a c t  the anomaly  interface  to 7 ° > and c o n s e q u e n t l y the d i p  in the measurement,  be c l o s e  the l a r g e r was  anomaly.  t o , or around is  that  the d i p  Consequently,  the angle  if a  IJ v a l u e was d e s i r e d to be c o r r e c t e d to some expected v a l u e  l y i n g , f o r example on the GH or AB branches,  the d i p angle  t o e f f e c t t h i s c o r r e c t i o n would be much l a r g e r correct  branches.  the d i p p i n g  in c o n n e c t i o n w i t h t h i s  the plane needed to c o r r e c t t h i s  measured  models  range, was a s s i g n e d to c o r r e c t a p a r t i c u l a r  Evidence has shown t h a t the d i p angle of  larger  factors  based on the d i s t r i b u t i o n of  These ranges d i d not o v e r l a p f o r  this value.  IJ  (1) As p r e v i o u s l y mentioned, a  and on expected v a l u e s  the a p p r o p r i a t e d i s t a n c e  the  to be c o r r e c t e d to an expected  l y i n g somewhere on the DF b r a n c h .  used to remove t h i s  of  in t h i s  l y i n g somewhere on the GH or  branches, and f o r a measured  (2)  in q u e s t i o n .  and azimuth v a l u e , a s s i g n e d to the DF branch, to be c o r r e c t e d  to an expected dT/dA  for  is  that  the measured v a l u e  In p a r t i c u l a r ,  needed  than t h a t needed to  to some p o i n t on the expected IJ branch.  i t was observed t h a t the measured 61/dA  and azimuth  -47values  for  the e a r l y p a r t of  the  IJ branch c o u l d be c o r r e c t e d to  be on the GH b r a n c h , thus r e s u l t i n g Bolt's  (1964) model.  i n a t r a v e l t i m e model  On the b a s i s o f  than 1 0 ° , needed f o r  this  c o r r e c t i o n , and s i n c e  C l e a r l y observed on a r r a y seismograms did  not o b s e r v e  the l a r g e r  t h i s phase a t a l l ) ,  similar  dip angles, the phase  in t h a t d i s t a n c e  to  greater  IJ was  range  the c o r r e c t i o n t o B o l t ' s  (Bolt model  was  rejected. The  d i p angles of  the c o r r e c t i n g planes were found to be  c e n t e r e d around the v a l u e of 6 ° - 7°. angle  reached as  high as and 2 0 0 °  o 10 .  low as 0 . 5 ° .  In even fewer  in some cases cases,  it  The d i p d i r e c t i o n s were c o n c e n t r a t e d around  but a few d i p d i r e c t i o n s as  were o b t a i n e d . have, c l o s e l y  low as  Often the t h r e e measurements  similar  the d i p  reached as - j 100  1 0 ° on as h i g h as  t h r e e measurements  but  350°  i t was p o s s i b l e  most  though  in any one event with v a s t l y  unusual differing  d i p a n g l e s and d i p d i r e c t i o n s . T h i s d i s p a r i t y was due to the i n the magnitude  of  the anomalies  these c o u l d be due to s y s t e m a t i c  for  changes  the t h r e e measurements.  changes  In  in waveform from one  structure,  deepest p o i n t o f  irregularities  i n the source r e g i o n or  turn  seismometer  to another as a, r e s u l t of random n o i s e e f f e c t s , e x t r e m e l y r a p i d i n the l o c a l  °  in any one event would  d i p a n g l e s and d i p d i r e c t i o n s , as would  events around the same azimuth 0, to have a l l  although  variations  the  the ray p a t h , o r more than one phase a r r i v i n g  at  n e a r l y the same t i m e . The  a p p r o p r i a t e d i p v e c t o r s were d i v i d e d  each group c o n t a i n e d d i p v e c t o r s of a p p r o x i m a t e l y the  d i s t a n c e and azimuth ranges  i n t o groups  so  the same v a l u e ,  in each group were not too l a r g e .  that and The  -48data f o r 15.  the DF branch was d i v i d e d  the GH i n t o  from 5 °  i n t o 25 groups,  the  1 9 , and the AB i n t o 2 1 . The d i s t a n c e  i n t o groups was  f o r each group under particular  to o b t a i n an average  the assumption  that a l l  group were a f f e c t e d by t h i s  d e t e r m i n e d , and t h a t  the e m p i r i c a l  problem  Q  approach  f  of  method of c o r r e c t i o n , reality  obtainable  the d i p v e c t o r s was approached as a  the s t a t i s t i c s  of a s p h e r i c a l  i s commonly adopted  d e s c r i b e d by F i s h e r and d i p  angle  where m;,  A  °f  i * paleomagnetic  t  n  e  mean.dip v e c t o r are  L  = T a n " ' ( 2u>;nu  studies  A  =  Sin"'  so that  their  sum  group.  A measure of  R is  i s equal  The d i p d i r e c t i o n L givenby:  of  the d i p  their resultant  vectors.  The weights  vectors vector are  to the number of o b s e r v a t i o n s  is K  is  t  the d i s t r i b u t i o n o f  r e s p e c t to the mean v e c t o r  and  (^u n;/R;  the length of  the i n d i v i d u a l  This  Li)  n\ and I } a r e the d i r e c t i o n c o s i n e s  the weight of  The h i g h e r  distribution.  (1953) and Watson ( 1 9 5 6 ) .  i n the d i s t r i b u t i o n , is  group  method. The a v e r a g i n g  wj  in the  i n each  and somewhat d i v e r g e n t from the degree of p h y s i c a l by t h i s  di-p v e c t o r  structure  the f i n e s t r u c t u r e v a r i a t i o n  was merely a consequence of  varied  The purpose  the events  'average'  into  range  to 1 0 ° and the azimuth range from 3 ° to 1 0 ° .  o f the d i v i s i o n  IJ  =  the i n d i v i d u a l  g i v e n by the  normalized  N i n the  vectors  with  quantity  N - I / N - a  the v a l u e o f K the more c l o s e l y are the i n d i v i d u a l  grouped around t h e i r mean.  and  vectors  DT/D-DELTF) VS DELTA X  A - DF  (UNCORRECTED) X  X  Q - GH 2 -  x*  IJ  X - RB  Z  j  z  i  1  z ,z  1 I* x  LL)  21  X  z •  2  <%,  in  A A  a *  A  4»  110.0  1)5.0  120.0  125.0  130.0  -1  135.0  Figure  140.0  145.0  DELTA(DEG)  2.7. M e a s u r e d  150.0  -1 155.0  DT/DA  1E0.0  165.0  170.0  175.0  iao  DT/D-DELTR A -  DF  VS DELTA  (CORRECTED)  • - GH Z X  IJ  - RB  z  z  z  ,  UJ •  O  o A  i 1 ^  Q  A *  4  110.0  115.0  120.0  125.0  130.0  135.0  140.0  145.0  DELTA(PEG)  150.0  I  155.0  Figure 2.8. DT/DA c o r r e c t e d f o r l o c a l  160.0  -1  165.0  structure  I  170.0  I  175.0  ieo.0  -51The measured a z i m u t h , apparent v e l o c i t y and d i p v e c t o r each o b s e r v a t i o n angle of  in a p a r t i c u l a r  incidence i  in- N i a z i ' s  theory.  Q  of  group were used to c a l c u l a t e  the waves at  the e a r t h ' s  The d i r e c t i o n c o s i n e s  L  f o r each v e c t o r from i t s '  angle of  and K were c a l c u l a t e d f o r  the e n t i r e group;  rearranged  if  necessary  weights W| were taken as  The o b s e r v a t i o n s  the square of  greater  this  p o i n t that  the problem o f a v e r a g i n g  itself  somewhat a n i s o t r o p i c .  but the procedure then becomes somewhat  the  statistical  is  is  not  known w i t h  probability  A correct s t a t i s t i c a l  would be one i n t e r m e d i a t e between a s p h e r i c a l  and a c i r c u l a r  model distribution  too complex to be j u s t i f i e d by  the problem.  The d T / d A F i g u r e 2.7.  to  corrected dT/dA  the d i p v e c t o r s  p r e c i s i o n than the d i p d i r e c t i o n , and the  the n a t u r e of  root-mean  tables.  should be mentioned at  is  The  in the manner were used  the c o r r e c t one s i n c e the d i p angle  distribution  were  of K. the  L, A  measurement.  as d e s c r i b e d p r e v i o u s l y and the f i n a l  approach used f o r strictly  estimates  the r e c i p r o c a l of  The averaged d i p v e c t o r s o b t a i n e d  It  t  to o b t a i n s a t i s f a c t o r y  v a l u e s were e s t i m a t e d from these  described  were e s t i m a t e d  n'  L  the  i n c i d e n c e , and the q u a n t i t i e s  square e r r o r on the c o r r e s p o n d i n g d T / d A  prepare t a b l e s  s u r f a c e as  for  values  as o r i g i n a l l y  measured are shown i n  They e x h i b i t c o n s i d e r a b l e s c a t t e r and f i t t i n g a smooth  c u r v e through  the v a l u e s  The same d T / d A  f o r each branch would be e x t r e m e l y d i f f i c u l t .  measurements  j u s t d e s c r i b e d are d i s p l a y e d e v i d e n t and the s c a t t e r t h a t smoothing  corrected for structure in F i g u r e 2.8.  in the o b s e r v a t i o n s  t e c h n i q u e s c o u l d be a p p l i e d  in the manner  The improvement has  is  self-  been reduced s u f f i c i e n t l y  meaningfully.  -522.3  Smoothing of Corrected  dT/dA  Values  In the reduction of random errors by smoothing techniques, i t i s highly desirable that the errors be reduced as far as possible without losing valuable reduction  information,  be estimated.  common use are:  (1)  and that the magnitude of the  Three possible techniques of smoothing in  To subtract a twelfth of the fourth difference  from, or, add a quarter  of the second difference to each observed  value except the two end points. two severe disadvantages.  (Jeffreys, 1937).  This method has  F i r s t l y , the magnitude of the reduction  cannot be properly estimated since the errors in neighbouring points tend to be correlated, and secondly, i t does not remove as much of the error as the uncertainties would allow  (Jeffreys, 1934) nor  does i t give any control over the amount of smoothing to be applied. (2) T o . f i t a simple function such as a polynomial or a series of polynomials to the data by least squares.  In this way, i t i s possible  to get as much smoothing as desired or j u s t i f i e d , but the standard errors in the c o e f f i c i e n t s of the function are not a convenient measure of the reduction of the error unless some physical s i g n i f i c a n c e . for comparison as discussed summary values devised  the c o e f f i c i e n t s themselves have  A v a r i a t i o n of this technique was used in Section 2 . 3 . 2 .  The method of  by Jeffreys (1937, 1961) in which the amount  of smoothing can be controlled, no valuable ascertained  (3)  information  is lost as  by a s i g n i f i c a n c e t e s t , and the magnitude of the reduction  can be estimated.  This method i s superior  for smoothing purposes in t h i s study.  to the other two and was adopted  Appendix IV contains a description  of this method. 2.3.1  Smoothing of dT/dA  by the Method of Summary Values  The method of summary values involves f i t t i n g a linear and quadratic curve to each of several groups of the data.  The curvature  -53i n each group  is estimated  appropriate c a l c u l a t i o n of  beforehand to be r e a s o n a b l y  (Appendix  IV)  the c o o r d i n a t e s  small. of  From an  intersection  the l i n e a r and q u a d r a t i c forms can be determined and a r e known as  the summary p o i n t s . range.  They best r e p r e s e n t the data  The summary p o i n t s  i n t e r p o l a t e d to give  determined f o r  i n the a p p r o p r i a t e  the s e v e r a l  groups are  the complete smoothed c u r v e .  A computer program based on the t h e o r y of Appendix w r i t t e n by C. Wright and was used  in t h i s  v a l u e s f o r any one branch were d i v i d e d ranges were s m a l l ,  study.  The c o r r e c t e d d T / d A  i n t o groups  i n which the  and the c u r v a t u r e a l s o e s t i m a t e d  program was used to c a l c u l a t e the summary p o i n t s u n c e r t a i n t i e s f o r each range o f  the b r a n c h .  IV was  distance  to be s m a l l .  and t h e i r  The  corresponding  The summary p o i n t s  were  o then i n t e r p o l a t e d a t 0 . 0 1 the s i z e of of  the  interval  intervals  by means of d i v i d e d d i f f e r e n c e s ,  b e i n g chosen so as  to f a c i l i t a t e  the c a l c u l a t e d v a l u e s w i t h the measured v a l u e s o  c o o r d i n a t e s were given  to 0 . 0 1  and measured dT/dA v a l u e s whether adequate smoothing  .  The r e s i d u a l s  were determi ned^and  comparison  in which the  distance  between the c a l c u l a t e d  in o r d e r to determine  had taken p l a c e a c h i - s q u a r e d  test  was  v a l u e f o r each r e s i d u a l  was  2 performed on these r e s i d u a l s . taken as  the product of  the weight  measurement and the square of range b e i n g the sum An o v e r a l l  of  for  the c o r r e s p o n d i n g  the r e s i d u a l ,  the i n d i v i d u a l x ?  the sum of  of  dT/dA  the total-x? f o r the p o i n t s  the "x? f o r each of the ranges  branch was c a l c u l a t e d a l s o . F o r to be w h o l l y random,  The "x.  adequate smoothing,  each  in that in the  range. particular  and assuming the e r r o r  must be i n the range V + JlV  where V  is  the  -54number o f degrees o f freedom. f i x e d parameters  less  t h i s c a s e "V i s equal number o f one  •j[2(M-2) .  If  the  branch i s  If x  is  M  0  the number o f  the x-  1  for  i s 2N, and  (i.e.  if  if  must be  f e a t u r e s are being l o s t .  (i.e. is  within  the for  and  2,  (M -  must be in the range  2  2)  +  a r e too l a r g e )  in  (M -2N±J2(M -ZN) 0  0  then the  in the range as a c c u r a t e l y  b e i n g u n d e r e s t i m a t e d , and  To c o r r e c t t h i s  s h o u l d be i n c r e a s e d so t h a t  less  the e n t i r e number o f o b s e r v a t i o n s  the r e s i d u a l s  they c o u l d , the c u r v a t u r e i s  In  i n the e n t i r e b r a n c h , then the number  summary p o i n t s do not r e p r e s e n t the data as  used,  summary p o i n t s  that range  , then the o v e r a l l x  too l a r g e  parameters.  the case b e i n g c o n s i d e r e d  t h e r e a r e N ranges  summary p o i n t s  to the number o f  to the number of o b s e r v a t i o n s  range w i t h M p o i n t s , so t h a t  of  the number o f a d j u s t a b l e  summary p o i n t s f o r  V = M - 2,  Thus *v i s equal  significant  problem, the number o f  the summary p o i n t s f a l l  closer  ranges  to the  2 data is  i n any g i v e n range and the r e s i d u a l s  too s m a l l ,  meaningless problem,  i n s u f f i c i e n t smoothing  detail  is  ranges  In p r a c t i c e , owing  difficulty  p l a c e and  increase  To c o r r e c t  this  the  also.  to the heavy c o n c e n t r a t i o n o f data  and r e l a t i v e l a c k  in the o t h e r r e g i o n s ,  was e n c o u n t e r e d in a t t e m p t i n g t o o b t a i n a  v a l u e o f T& f o r each of  If -x,  possibly  s h o u l d be i n c r e a s e d , so t h a t  i n any g i v e n range would  some d i s t a n c e r a n g e s ,  taking  b e i n g r e t a i n e d i n the c u r v e .  the number of  residuals  is  are thereby reduced.  the r a n g e s ,  and a t  in  some  satisfactory  the same time a  satisfactory  2 o v e r a l l •%  for  the e n t i r e b r a n c h .  to obtain a s a t i s f a c t o r y  A f t e r somewhat e x h a u s t i v e  s o l u t i o n with regard  attempts  2  to  the-X,  for  each r a n g e ,  2 the  o v e r a l l oo  for  d i s t a n c e ranges,  the branch and the r e t e n t i o n o f  reasonably  the f o l l o w i n g procedure was a d o p t e d :  that  small  solution  -55for  which the maximum number o f ranges gave s a t i s f a c t o r y v a l u e s of x  overall  z  and ihe  v a l u e o f x, f o r the e n t i r e branch remained s a t i s f a c t o r y was allowed 2  to s t a n d .  The r e s u l t o f t h i s  procedure i s t h a t t h e l a r g e r  p a r t o f the branches  a d e q u a t e l y smoothed but t h e r e were p o s s i b l y one or two r e g i o n s e i t h e r undersmoothed or oversmoothed.  Since a t t h i s  on i t t h a t were  stage the data was b e i n g  smoothed t o determine a c o r r e c t i o n term c o r r e s p o n d i n g to c o n s t r a i n i n g the d a t a t o an a c c e p t a b l e t r a v e l t i m e c u r v e , t h i s the  lack o f adequate smoothing over  e n t i r e branch i s not too important(Wright In order t o c o n s t r a i n  1970).  the smoothed data to an a c c e p t a b l e  traveltime  curve and c o n s e q u e n t l y to reduce any r e m a i n i n g s y s t e m a t i c e r r o r s even f u r t h e r , the  area under each smooth curve  ( i . e . i n each range) was e v a l u a t e d by  Simpson's r u l e o f i n t e g r a t i o n and s u b t r a c t e d from the t r a v e l t i m e d i f f e r e n c e between the end p o i n t s o f the range. end  points  are  the r e s p e c t i v e standard e r r o r s  If T| and T^ a r e the t r a v e l times a t the  o f the range of i n t e g r a t i o n  A  i n T,  under the smooth curve i n that range  and A  t  and T  r e s p e c t i v e l y , and i f c , a n d « r z  a  and i f , a l s o ,  z  the area  i s A w i t h a standard errore^then the  c o r r e c t i o n C i s g i v e n by:  c = ( ( T z ± <%) - On i °0 - O* ± c r ) ) / a - A , ^ 3  = C Ta -  T, -  where the second term on the r i g h t  R  ) / A  2  - A ,  ±  i s the e r r o r  2  6*7  2  +  ^  Z  ^  2  ) %  2  - A .  i n the c o r r e c t i o n term C.  T h i s c o r r e c t i o n C was c a l c u l a t e d f o r each o f the ranges i n the d i f f e r e n t branches, and a p p l i e d to each o f the d T / d A  values  procedure which merely  'involved  the b a s e l i n e of the dT/dA  values  ranges.  in the i n d i v i d u a l  smoothed dT/dA branches Bolt  changing  i n these ranges, a  The t r a v e l t i m e curves t o which the  data were r e s t r a i n e d a r e the J - B t a b l e s f o r the DF and AB  (which branches f o r comparison were a l s o r e s t r a i n e d to the  (1968) t a b l e s )  and the Adams and Randall  (196^)  -56TABLE 2.1(a)  SUMMARY POINTS FOR DP BRANCH 3 14 DATA PiII NTS RESTRAINED TO J . B . TRAVEL-TIME CURVE  INITIAL SUMMARY POINTS: a (Dnfi)  dT/dA (SEC/DEC)  o (SEC/DEG)  115.45  1.956  0.018  118.98  1.946  0.017  122.10  ' 1.858  0.030  123.68  1.857  0.033  125.55  1.890  0.016  131.26  1.862  Q.017  134.06  1.886  0.010  138.08  1.867  0.010  143.06  1.748  0.012  165 .22  0 .896  -0 .016  Total  x  2  = 304.92  X  C  v  2  (SEC/DEC)  49.32  45  +0.000  21.02  18  +0.083  38.06  35  +0.034  48.41  78  -0.039  on 304 degrees o f freedom 148.11 128  +0.023  FINAL SUMMARY POINTS: A (DEG)  dT/dA (SEC/PEG)  o (SEC/DEG)  115.30  1.962  0.016  118.78  1.9S2  0.013  122.56  1.940  0.016  125.38  1.924  0.016  131.58  1.895  0.011  134.93  1.846  0.010  138.04  1.828  0.010  147.81  1.706  0.015  155.34  1.432  0.019  165.32  0.919  0.016  Total  x  2 =  49.48  45  29.64  32  81.53  77  106.27  108  46.09  42  313.01 on 304 decrees o f freedom.  o = ST1). ERROR ON dT/dA VALUE OP SUMMARY POINT v = NO. OP DEGREES OF FREEDOM C = CORRECTION FOR RESTRAINING DATA TO TP.AVF.LT] ME CURVI:  i j  -57TABI.E 2 . 1 ( b )  SUMMARY POINTS FOR Gil BRANCH (a)  3 05 DATA POINTS RESTRAINED TO A-R  TRAVELTJMIi CUR Vii  I N I T I A L SUMMARY POINTS: dT/dA (SEC/DEG)  A  (DEC)  (SEC/DEG)  136.08  2.653  0.020  141.77  2 .624  0 .021  143.97  2 .605  0 .019  LSI.S3  2 .501  0 .037  Total  v  (SEC/DEG)  69.36  84  -0 .008  19.21  17  -0 .008  = 88.57 on 101 d e c r e e s o f f r e e d o m  2  F I N A L SUMMARY POINTS: dT/dA (SEC/DEG)  A  (DEG)  (SEC/DEG)  136.20  2 .645  141.93  2.613  14 3.7 5  2 .601  0.015  151.94  2 .494  0.024  (158.0  2 .300  Total  x  2  =  0.018 0 .017  .  79.42  84  15.21  17  9 4 . 6 3 on 101 d e g r e e s o f f r e e d o m .  * T a k e n f r o m A-R c u r v e i n o r d e r t o e x t e n d b r a n c h b e y o n d 152°  (b)  RESTRAINED TO BOLT TRAVELTIME  C= 0.087  F I N A L SUMMARY POINTS: dT/dA (SEC/DEG)  A  (DEG)  v  2 .740  0.019  141.91  2 .704  0.020  143.75  2 .692  0.020  151 .91  2.588  0.021  (156 .00  2 .100  *  x  Taken 152°  2  =  (SEC/DEG) 2  (SEC/DEG)  136.19  Total  CURVE  77 .59  84  16.1!  17  93.74 on 101 d e g r e e s o f f r e e d o m .  from B o l t ' s  curve  i n order t o extend branch  beyond  -58-  TABLE  2.1(c)  SUMMARY POINTS FOR I J BRANCH 81 DATA POINTS RESTRAINED TO A+R TRAVEL-TIME CURVE  .INITIAL SUMMARY POINTS: dT/dA (SEC/DEG)  A  (DEG)  (SEC/DEG)  135.68  3.455  0.032  140.46  3.317  0 .015  146.62  3.195  0.017  150.82  3.054  0 .030  Total  x  2  ~ 66.96  (SEC/DEG) 47.42  60  +0.024  19 .54  17  +0.003  on 77 degrees o f freedom,  FINAL SUMMARY POINTS: A (DEG)  dT/dA (SEC/DEG)  a (SEC/DEG)  135.68  3.459  0.030  ,140.46  3.320  0.013  146.6S  3.198  0.016  150.87  3.056  0.021  Total  x  2  = 75.84  x  2  57.23  60  18.61  17  on 77 degrees o f freedom.  -59TABLF.  2.1(d)  SUMMARY POINTS FOR AB BRANCH 70 DATA POINTS (a)  RESTRAINED TO J . B .  I N I T I A L SUMMARY POINTS: A (DEG)  dT/dA (SEC/DEG)  a (SEC/DEG)  146.01  3.670  0.017  149.896  4.010  0.024  4.279  0.034  161.599  4.416  0.032  165.716  4.488  0.026  173.441  4.495  0.008  *1S7.642  Total *  x  2  =  66.30  X  C (SEC/DEG)  v  2  33.01  37  3.56  12  29.73  21  +0.145  -0.093  on 70 d e g r e e s o f f r e e d o m .  Not used i n i n t e r p o l a t i o n because of curve i n that region.  of unsatisfactory behaviour  F I N A L SUMMARY POINTS: A (DEG)  dT/dA (SEC/DEG)  o (SEC/DEG)  146.01  3.814  0.015  149.896  4.165  0.020  161 .599  4 .319  0 .030  165.716  4.395  0.028  173.441  4.419  0.033  Total  x  2  =  35.67  37  27.94  26  6 3 . 6 1 on 63 d e g r e e s o f f r e e d o m . (b)  RESTRAINED TO BOLT  F I N A L SUMMARY POINTS: C = -0.055 s e c / d o g A (DEG)  dT/dA (SEC/DEG)  a n d -0.081 s e c / d e g o (SEC/DEG)  146.01  3.625  0.013  149.89  3.968  0.020  161.60  4.335  0.026  165.72  4.406  0.022  173.44  4.409  0.032  Total  x  NOTE:  (].) A l l v a l u e s o f u n s m o o t l i c d  2  x  2  respectively v  36.75  37  24.32  26  = 6 1 . 0 ? on 63 d e g r e e s o f f r e e d o m .  t o 4.5.  /dA above 4.5 r e s t r a i n e d  -60TABLE 2.2.  DISTANCE (DEG)  DT/DA SMOOTHED BY THE METHOD OF SOBS&Rr  i l ! DF i  I 113.0 114.0 115.0 116.0 117.0 118.0 119.0 120.0 121.0 122.0 123.0 124.0 125.0 126.0 127.0 128.0 129.0 130.0 131.0 132.0 133.0 134.0 135.0 136.0 137.0 138.0 139.0 140.0 14 1 . 0 1t2.0 143.0 144.0 145.0 146.0 147.0 148.0 149.0 150.0 151.0 152.0 153.0 154.0 155.0 156. 0 157.0 158.0 159.0 160.0 161.0 162.0 163.0 164. 0 165.0 166. 0 167.0 168.0 169.0 170.0 175.0 180. Q  1.962 1. 9 6 2 1. 962 1. 9 6 2 1. 9 5 9 1. 9 5 6 1. 952 1. 944 1. 941 1 . 94 1 1 . 94 1 1. 9 3 9 1. 931 1. 921 1.915 1.910 1. 906 1.902 1.897 1. 882 1. 878 1. 865 1. 854 1- 8 5 2 1. 849 1. 8 2 9 1. 824 1. 8 1 8 1. 8 0 8 1.794 1. 7 7 3 1.756 1. 7 3 8 1.718 1. 6 9 7 1.673 1.64 8 1.620 1.591 1.559 1.526 1.489 1.451 1.411 1.368 1.323 1. 2 7 5 1.225 1. 172 1. 116 1- 0 5 8 Q. 9 97 Q. 9 3 3 0. 715 0 . 6 23 0 . 6 05 0.598 0.554 0. 300 0. 000  DT/DA GH (A-R)  2.651 2.649 2.649 2. 648 2.646 2. 643 2.639 2.634 2.629 2.622 2. 615 2.606 2.597 2.587 2. 576 2.564 2 . 551 2.537 2. 523 2.507 2 . 491 2.473 2. 455 2.436 2. 415 2 . 394 2. 303  VALUES  (SEC/DEG) IJ  (BOLT)  2.747 2.745 2.744 2.7 43 2.741 2 . 7 38 2.734 2.729 2.724 2 . 7 17 2.710 2.701 2.692 2.682 2.671 2 . 6 59 2 . 6 49 2.632 2.580 2.550 2.500 2.450 2.400 2.305 2.210  AB (J-B)  3.484 3 . 464 3.444 3. 408 3. 378 3 . 351 3.328 3.307 3. 288 3. 270 3. 252 3.232 3 . 211 3 . 187 3 . 159 3 . 128 3. 090 3 . 047 3 . 030  3.600 3. 820 3.920 4.010 4.090 4. 170 4.200 4.200 4.200 4.200 4.300 4.300 4. 300 4. 300 4.300 4. 300 4.320 4. 340 4.360 ' 4 . 3 70 4.390 4. 400 4.410 4.420 4.420 4. 420 4.420 4.420  (BOLT)  3.500 3 . 6 10 3.720 3.810 3.890 3.970 4 . 100 4 . 200 4.300 4 . 300 4.400 4.400 4.400 4.400 4.400 4.400 4.410 4 . 420 4 . 4 30 4. 435 4 . 440 4.446 4.450 4.457 4.458 4.458 4.459 4.459  -61tables  for  the  IJ and GH branches,  r e s t r a i n e d to the B o l t none o f  these t a b l e s  i t was not p o s s i b l e of  the  (1968)  this  tables  latter  for  branch a l s o  the sake of  p r o v i d e d standard e r r o r s  being  interest.  in t h e i r  Since  travel times,  to e s t i m a t e the e r r o r on the c o r r e c t i o n C f o r  any  ranges. With the a p p r o p r i a t e c o r r e c t i o n C a p p l i e d to each group  the p a r t i c u l a r  b r a n c h , the c o r r e c t e d dT/dA  values  of  in  the e n t i r e branch  were once a g a i n smoothed by the method of summary v a l u e s  to o b t a i n  the  o new summary p o i n t s . the f i n a l  These were i n t e r p o l a t e d at  smoothed d T / d A The i n i t i a l  final  smoothed dT/dA  i n T a b l e s 2.1(a)  and f i n a l values  to 2.1(d)  summary p o i n t s  for  summary p o i n t s  to smooth the data  separately,  and to f i l l  respectively.  branch that were g r e a t e r it  is  Note t h a t  the GH branch has  both Adams and  this  In a d d i t i o n , v a l u e s than 4.5  Randall  was not  region,  145 -  possible  ) 5 0 ° and 160 -  smoothed dT/dA v a l u e s  From the f i g u r e ,  it  i n the  are l i s t e d  1968  in Table  r e s t r a i n e d t o the d i f f e r e n t is  seen t h a t  of  175° J-B  the AB  s e c / d e g were r e s t r a i n e d to C  Owing  so the procedure  of dT/dA f o r  P P phases  the  Listed  in the gap so c r e a t e d with the v a l u e s  F i g u r e 2.9 shows these v a l u e s branches.  for  in the ranges  the maximum given f o r  The f i n a l  give  in o b t a i n i n g  150°to 160° f o r AB, i t  adopted was  as  used  the r e s p e c t i v e branches are  inclusive.  i n the range  to obtain s a t i s f a c t o r y  value,  to  The AB branch was r e s t r a i n e d to both J - B and B o l t .  to l a c k of d a t a  and B o l t  intervals  values.  been r e s t r a i n e d to the t r a v e l t i m e curves of and B o l t .  1  the GH branch  this tables. 2.2  traveltime restrained  DT/DA S M O O T H E D B 7 M E T H O D O F SUMMARY  VALUES  in  Figure 2.9  A rrjEG)  DT/DA smoothed by the method of Summary Values  -63t o B o l t ' s d a t a has g r e a t e r  c u r v a t u r e near  t o t h a t o f Adams and R a n d a l l .  The AB branch r e s t r a i n e d t o the J - B  t r a v e l times has g r e a t e r c u r v a t u r e near r e s t r a i n e d to B o l t ' s near the l a t t e r t o the d a t a ,  times.  and R a n d a l l ' s  Also,  i t s beginning  than the one  i t i s s h i f t e d t o a lower*  part, o f the range.  the d T / d A  150° than the one r e s t r a i n e d  level  On the b a s i s o f the best  fit  curve f o r the GH branch r e s t r a i n e d to Adams  t r a v e l times and the  dT/dA  c u r v e f o r the AB branch  r e s t r a i n e d t o the J - B t r a v e l times were s e l e c t e d f o r the purpose o f inversion.  The c u r v e s as shown f o r branches  IJ and DF were used  in  the i n v e r s i o n p r o c e d u r e . 2.3.2  Smoothing o f d T / d A Package - TRIP  d a t a by UBC T r i a n g u l a r  M a i n l y f o r the sake o f c o m p a r i s o n , for  local  the d T / d A  s t r u c t u r e were a l s o smoothed by polynomial  the U.B.C. computer program U.B.C. TRIP described  Regression  (1972).  values  corrected  regression  The method i s  using briefly  i n Appendix V . a . The r e g r e s s i o n e q u a t i o n s determined as g i v i n g the best  t o the d a t a a r e l i s t e d  i n T a b l e 23.  The independent v a r i a b l e s  DF, GH, IJ and AB branches were taken as and  (A  -145°)  ( A  r e s p e c t i v e l y where the numerical  t u r n i n g p o i n t s of the r e s p e c t i v e branches. taken as t h a t d i s t a n c e , t r u n c a t e d individual for  -113°),  phaseswere f i r s t  ( A  quantities  shown i n F i g u r e 2 . 1 0 .  -134°)  a r e the  The t u r n i n g p o i n t s  were  Smoothed c u r v e s a t 1° i n t e r v a l s  are l i s t e d  For comparison,  the method o f summary v a l u e s  ( A  to a whole number, a t which the  observed.  The c a l c u l a t e d v a l v e s  f o r the  -132°)  the d i f f e r e n t branches were generated from the a p p r o p r i a t e  equations.  fit  regression  i n Appendix Vb, and a r e  the smoothed c u r v e o b t a i n e d by  i s shown i n dashed  lines.  The l e s s e r  -64TABLE 2 .3:  FOR SMOOTHED dT/dA  RECRESS 1 ON EQUATIONS  DAT A  314 data  DF BRANCH  points  R = 0.9643 2  ]NDEPENPEN'T VAR1ARLF.  COEFFICIENT  STD . F.RROR  CONSTANT  1.9S87  0.0180  (A-113°)  -0.1384/10  (A-113°) (A-113°)  0.1095/10  2  3  (4-113°)* P =  d T  /dA  0.2528/10  F - PROB  0.0000  3  0 . 1384 /10''  2  -0.3608/10''  0 .1146/10=  0.2432/10  0 . 5876/10'  6  = 1.9587 - 0.01384 (A-113)  + 0.1095  TOLERANCE  1.0000  0.0000 0.0000  1.0000 1.0000  0.0001  1.0000  / A - U 3 \ 2  10 -0.0360 / V 1 1 3 V 10 /  + 0.0024 / A - 1 1 3 V V 10  GH BRANCH  106 data points  R = 0.7573 2  INDEPENDENT VARIABLE CONSTANT (A-132") (A-132")  COEFFICIENT  STD.F.RROR  2 .GS78  0 .01990  -0.1694/10  0.S931/10  0.1326/10  2  (4-132°)3  2  -0 .4839/10"  P - 2.6878 - 0 .01694 ( A - 132 )  F-PROB  TOLERANCE  0.0000  1.0000  0.84 86/10'*  0.0000  1.0000  0.1301/101  0.0004  1.0000  3  + 0 . 1326/A-132V/'A-132V- -0 -0.0484 .04 84 /A-1321 :  v  io  v io y  / 81 data  I J BRANCH  points  R = 0.8156 2  INDEPENDENT.VARIABLE  3.4556  CONSTANT 2  STD.ERROR  0.1376/10  -0.3761/10  3.4556 - 0 .0183 (A-134)  3  0.307G/10  2  TOLERANCE  2  3  0.0000  1.0000  0.0223  1.0000  0.03761  70 data points  AB BRANCH R=  F-PROB.  0.0255  -0.1833/10  (4-134°) (A-134°)  COEFFICIENT  0.9756 COEFFICIENT  STD.ERROR  CONSTANT  3. 5480  0.0194  (A* 145°)  0.1172  INDEPENDENT VARIABLE  (A-145°) (A-145 ) 0  2  3  -O.S9M/10  2  0.1203/10  P » 3.5480 + 0.11 7?. ( A - H i ; ;  3  F-PROB  TOLERANCE  0.8280/10  3  0.0000  1.0000  0.2310/10  3  0.0000  1.0000  0.0050  1.0000  0.4134/10''  - 0.5964 ^£_-JJJ:^  7  * 0.1203 ^ A -1 4 r,j  3  D T / D A SMOOTHED BY POLYNOMIAL REGRESSION ANO METHOD OF SUMMARY VALUES POLVtvlCrtlAL  SUMMARY  E.£(JRl S5 l o M ' r  VALUES.  G CD  i  O LU CO  <  D  no.o  n 115.J  -1  120.0  I  125.0  130.0  -1  135.0  140.0  DELTA (DEG)  145.0  ~1 150.0  155.0  I  160.0  -1 1BS.0  F i g u r e 2.10 C o m p a r i s o n o f DT/DA s m o o t h e d b y P o l y n o m i a l R e g r e s s i o n and method o f Summary V a l u e s .  "I  170.0  -| 175.0  160.0  -66-  c u r v a t u r e near regression  the s t a r t  o f the AB branch f i t t e d by polynomial  can be a t t r i b u t e d t o t h e l a c k o f d a t a  150° t o 1 6 0 ° .  Thus e x c e l l e n t agreement  wee-i o b t a i n e d .  However,  i n t h e range  between the s e t s o f curves  the smoothed d T / d A  values  from the summary p o i n t s were used f o r the i n v e r s i o n I n v e r s i o n o f Smoothed d T / d A  2.4  values  using  interpolated process.  the H e r g l o t z - W i e c h e r t Integral  2.4.1  D e s c r i p t i o n o f the H e r g l o t z - W i e c h e r t As was shown i n Appendix  inversion  I for a spherical  technique and symmetrical  e a r t h model, the r a y parameter p i s g i v e n by  p = In Sm I / v = dT/dA Also,  the i n t e g r a l s  r a y o f parameter where n  f o r the d i s t a n c e  p terminating  A , and t r a v e l t i m e T . f o r a l 2  2  at levels  rj  and r £  > 2> were d e r i v e d : r  (0  i&, = P I r - ' C r f - p ) " * dr 1  2  • ± T «  I  =  where i \ = r / v f o r c o n v e n i e n c e .  tfr-'Cf-p^dr  These  integrals  -  (Z)  may be s o l v e d e i t h e r  by numerical a p p r o x i m a t i o n or by d i r e c t  i n t e g r a t i o n when the v e l o c i t y  function is suitably defined.  the l a t t e r case f o r a m u l t i -  Consider  l a y e r e d e a r t h where the v e l o c i t y power  i n each l a y e r  i s d e s c r i b e d by a  law o f the form v = ar* , w i t h a and b c o n s t a n t s . 5  is  i d e n t i c a l w i t h ^ of Appendix  is  i n general  i n the l a y e r  discontinuous  Thus v ( r ) i s c o n t i n u o u s  a t each t a b u l a t e d p o i n t .  bounded by r a d i i  (1) and (2) y i e l d s  I.  Note t h a t b  r^ and r^,  (Engdahl e t a l , 1968)  substitution  but d v / d r  If v = ^ . ' r  >12  )2  i n equations  "  -67-  A , z - O-^T'CCos'Xf/ri,)  i  - COS^'CP/IZ))  ....  i T, = 0 -b^-'a^-p^ - C<-P^  -00  2  where  r\ and t\ t  a  r  e  t  n  t h e ray which reaches of  values  e  z  P i s equal  to r[  it's  A,2  of n at r] and r2 r e s p e c t i v e l y .  deepest p o i n t w i t h i n the l a y e r ,  , so that  (3) and  (k)  (3)  For  the v a l u e  become  = 2 0-b.i)"'<:os"'Cp/ri,)  T« = 2 0 - b , x ) - ( n f - ) ,  2  --ft)  C«  i  P  The r a d i u s o f p e n e t r a t i o n rp f o r  this  particular  from the v e l o c i t y r e l a t i o n s h i p and is e a s i l y  Using equations  (3) to  (6),  ray  is  determined  seen to be given  the t r a v e l t i m e and d i s t a n c e of any  or ray segment can be determined by b u i l d i n g up the times and over the proper number of new v a l u e s o f a ,  layers,  b, and T^.  and  r  The v a l u e s  o f a and b are that  calculated  l a y e r bounded by r a d i i  r  a  l z  = In  (V,/v-z)  ( i  = v,r,- *  (?)  of v at r] and r2 r e s p e c t i v e l y .  The t h e o r y embodied i n e q u a t i o n s important a p p l i c a t i o n s . complexity  It  can be used  (3) to  low v e l o c i t y  f o r f o c a l depth is  that  several investigate  of high v e l o c i t y  l a y e r s or to c o r r e c t e p i c e n t r a l  (see S e c t i o n 2.2.1).  the f a c t  (9) has  i n ray t r a c i n g to  i n t r a v e l t i m e c u r v e s caused by r e g i o n s or  <8)  /Jn (r./ra)  b  and v] and V2 are the given v a l u e s  however,  distances  ( l ? 2 ) where b  gradients  ray  each l a y e r b e i n g r e p r e s e n t e d by  from a g i v e n v e l o c i t y d i s t r i b u t i o n f o r rj  by  distances  Of importance to t h i s  i t can be used to d e r i v e T - A  curves f o r a s p e c i f i e d v e l o c i t y d i s t r i b u t i o n ,  and  study, ?-A  thus e n a b l i n g models  to  -68-  be generated for the  equations  comparison w i t h the o b s e r v a t i o n a l  can be used to s t r i p  h by c a l c u l a t i n g  the c o n t r i b u t i o n  total  in q u e s t i o n .  distance A  corresponding adjusted  Equations (Wright,  (T-T to  (3)  *Ck of i;L  The a d j u s t e d d i s t a n c e or the is  g i v e n by  (  (9)  formed the b a s i s of  the computer by the  the  (1) can be s o l v e d  program  Herglotzdistribution.  i s a monotonic d e c r e a s i n g f u c t i o n of to y i e l d  depth  (Bullen,  A,  1963).  la ro/r, ~ TT ' / C o s h o e  is  v e l o c i t y of penetrates A|  ) and  ).  ( i  beneath the s u r f a c e , e q u a t i o n  where r  distance  A - A,2.  Wiechert method and so generate a c o r r e s p o n d i n g v e l o c i t y TK= r/v  well,  the r e g i o n down to h to the  1970) w r i t t e n to i n v e r t the dT/dA v a l u e s  Provided that  As  the e a r t h to any r e q u i r e d depth  to the s t r i p p e d e a r t h  t r a v e l t i m e by  data.  the mean r a d i u s o f the wave  is  to a depth  the e a r t h , r,  V, , P, = T\  is  t  r -r, Q  'Cp/p.)ciA .  (  .  that r a d i u s where the  the parameter o f a ray  and reaches  from the s o u r c e , and V, = r, / p  is  (iq)  that  the s u r f a c e at a d i s t a n c e  For convenience  (10) can be  written At  ±  r, = r expC-TT'f tn(p/ D  To e v a l u a t e v, , d T / d A i s required so that  r,  Pl+  can be c a l c u l a t e d from ( 1 1 ) . If  the r e g i o n  depth to the d i s c o n t i n u i t y observational  data over  V =  a discontinuity exists,the  s t r i p p e d of the r e g i o n down to the d i s c o n t i n u i t y can proceed for  2  as a f u n c t i o n of d i s t a n c e from 0 to  can then be e s t i m a t e d .  the d i s t a n c e  values range  A•,  H  /pi  e a r t h must be  b e f o r e the  beneath the d i s c o n t i n u i t y . and d T / d A  00  C(P/P.) -l) )d:A) Z  If  calculations  h is  are known from the  A% ^° ^ 1  >  the  -69some e x i s t i n g model  is  Then the c o n t r i b u t i o n s  used to f i l l  in the d i s t a n c e range from 0 ° to  to the d i s t a n c e s  i n the range  t h a t r e g i o n to depth h, f o r v a l u e s of d T / d A  due to  A 2 . *°  between p ( A )  and  2  a r e c a l c u l a t e d by summation over the a p p r o p r i a t e number of  +0, A1  radius  so t h a t a dT/dA  ro -  h is  equation  (11)  distance  range of  to  obtained.  The computations  integration  The f i r s t  should be equal  o f parameter  technique is  pC^Az)  is  is  Q  t h a t of  .  that  then proceed as  between  now r e p l a c e d by r  before, Q  the s t r i p p e d d i s t a n c e s  stripped distance,  to 0, meaning  i n the s t r i p p i n g  The  c u r v e c h a r a c t e r i s t i c of a s t r i p p e d e a r t h of  w i t h the d i f f e r e n c e that r  A2. to A-l .  p^A,)  layers.  c o n t r i b u t i o n s a r e then s u b t r a c t e d fromthe a p p r o p r i a t e d i s t a n o e s A2  A^.  using  - h, and the corresponding  i . e . corresponding to A 2 .  the depth h to the d i s c o n t i n u i t y  the maximum depth a t t a i n e d  The s t r i p p e d d i s t a n c e s  by the ray  used path  i n c r e a s e from A z t o A l .  . The computer program mentioned p r e v i o u s l y enabled the e a r t h to be stripped  to any a p p r o p r i a t e d i s c o n t i n u i t y  v e l o c i t y - d e p t h model above refers  to the o b s e r v a t i o n a l  the dT/dA  values,  discontinuity for t a b l e o f dT/dA interpolated first radius  the d i s c o n t i n u i t y . dT/dA  versus  to g i v e v a l u e s  of p e n e t r a t i o n r j  range of  at  stripped distance 1° d i s t a n c e This  in equation  is  V] and  is obtained.  intervals,  by means of  the  Thus a It  beginning  is at  to f a c i l i t a t e e v a l u a t i o n o f  (11)  (3)  to be i n v e r t e d , f \ j and TI2 r e p r e s e n t  is e v a l u a t e d f o r every t h i r d p o i n t  integration  0° to 6 ° e t c .  from an assumed  The q u a n t i t y p in e q u a t i o n  the c o r r e s p o n d i n g v e l o c i t y v a l u e s  values  r^  values  (3)  c a l c u l a t e d from the v e l o c i t y model assumed above  s t r i p p e d distance of 0 ° .  integration,  by e v a l u a t i n g  the the  Simpson's r u l e of  in the t a b l e  (i.e.  the  i s from 0 ° to 2 ° , then 0 ° to 4 ° , then  to o b t a i n v a l u e s o f  and both r and p are known,  r ] , at 2 ° , 4 ° , 6 ° , e t c . )  the v e l o c i t y v f o r  Sin§e v = r/p,  the a p p r o p r i a t e depth c o u l d  -70be c a l c u l a t e d .  Consequently,  a v e l o c i t y versus  d e r i v e d from the measured d T / d A v e l o c i t y model above Finally, distances  values  depth t a b l e can be  in c o n j u n c t i o n w i t h an  assumed  the d i s c o n t i n u i t y .  equations  (3)  and  and t r a v e l times a t d T / d A  (4)  are used to c a l c u l a t e the t r u e  i n t e r v a l s of 0.02  sec/deg.  for  the assumed and the c a l c u l a t e d v e l o c i t y d i s t r i b u t i o n .  These  together w i t h  and depth of  the c o r r e s p o n d i n g d T / d A  p e n e t r a t i o n as w e l l 0.02  as  the v e l o c i t y v a l u e s  s e c / d e g . were the f i n a l  examination of  this  values,  output of  radius  at each dT/dA  values  interval  the computer program.  t a b l e enabled assessment of any adjustments  for additional 2.4.2  t o the d a t a .  in  discontinuities.  The boundary between the e a r t h ' s w i t h a sharp d e c r e a s e km/sec.  to  The e n t i r e procedure can be repeated  A p p l i c a t i o n of the H e r g l o t z - W i e c h e r t method to d T / d A f o r the e a r t h ' s c o r e .  to 8.1  in compressional  (J-B v a l u e s ) .  mantle and core  values  is  discontinuous  wave v e l o c i t y from about  Consequently,  if  13.6  the smoothed dT/dA  this  the e a r t h must be s t r i p p e d  purpose,  assumed  values for  velocity-depth d i s t r i b u t i o n for continuous decrease  This  was  distribution  the AB branch were used to determine the  the o u t e r c o r e .  in v e l o c i t y at  For  to the J-B v a l u e s  to the m a n t l e - c o r e boundary a t a depth of 2894 km.  and the smoothed dT/dA  the  the  to the m a n t l e - c o r e boundary.  the v e l o c i t y d i s t r i b u t i o n c o r r e s p o n d i n g  km/sec  for  AB branch are to be i n v e r t e d to o b t a i n a v e l o c i t y d i s t r i b u t i o n f o r outer core,  of  Close  v e l o c i t y and d i s c o n t i n u i t y - d e p t h v a l u e s which might be n e c e s s a r y ensure a b e t t e r f i t  both  Because of  the c o r e - m a n t l e boundary,  was encountered in o b t a i n i n g a s a t i s f a c t o r y  the l a r g e  dis-  some d i f f i c u l t y  velocity-depth distribution,  an a l t e r n a t i v e but by no means p o o r e r , p r o c e d u r e was a d o p t e d .  The o u t e r  and core  -71was a s s i g n e d an i n i t i a l J-B t a b l e s .  From t h i s  v e l o c i t y - d e p t h d i s t r i b u t i o n c o r r e s p o n d i n g t o the initial  v a l u e s were c a l c u l a t e d . of  model  the c o r r e s p o n d i n g d T / d A  The model was a d j u s t e d u n t i l  and  traveltimes  a satisfactory  the c a l c u l a t e d and the measured v a l u e was o b t a i n e d . The smoothed d T / d A  the depth to the f i r s t  data f o r  discontinuity  the  U  branch were used to determine  i n the t r a n s i t i o n r e g i o n .  done by u s i n g the v e l o c i t y - d e p t h d i s t r i b u t i o n determined f o r  This  the e a r t h to the depth where i t was assumed t h a t the r e c e d i n g Bl branch was s i t u a t i o n . distance for  the f i r s t  the advancing this f i r s t  too s h a l l o w ,  IJ branch would s t a r t e a r l i e r would be c o r r e s p o n d i n g l y meaning t h a t short,  too l o n g .  the a d v a n c i n g  that  results  branch and the length of  data.for  They  134°.  that  If  the  advancing  stripped  branch distance  i n a r e c e d i n g branch The depth to the  regarding  the  too  the  turning The  IJ branch were  were used i n c o n j u n c t i o n w i t h the  the d i s c o n t i n u i t y  GH.  T h i s ensured  the r e c e d i n g b r a n c h .  and t r a v e l t i m e s f o r  t o determine a s a t i s f a c t o r y  the smoothed d T / d A  with  w i t h the r e s u l t  to g i v e s a t i s f a c t o r y  c a l c u l a t e d v e l o c i t i e s above  the s t r i p p e d  i t meant t h a t the depth to the  the advancing b r a n c h .  c a l c u l a t e d as p r e v i o u s l y d e s c r i b e d .  strip  producing  the d e s i r e d d i s t a n c e of about  Likewise, a negative  corresponding v e l o c i t y - d e p t h values  adjustments  possible.  the d i s c o n t i n u i t y was too deep would r e s u l t  d i s c o n t i n u i t y was a d j u s t e d  JB model)  near 0 ° as  than d e s i r e d , w h i l e the r e c e d i n g Bl  and a l a t e r t u r n i n g p o i n t f o r  point of  at  s t r i p p e d d i s t a n c e was p o s i t i v e ,  d i s c o n t i n u i t y chosen was  the d i s c o n t i n u i t y  to  T h i s depth was a d j u s t e d u n t i l  IJ data p o i n t was as  IJ branch would s t a r t  was  the AB b r a n c h ,  together w i t h i t h e J - B d i s t r i b u t i o n abouve the m a n t l e - c o r e boundary,  fit  fit  (i.e.  the AB v e l o c i t i e s  turning point f o r  T h i s was repeated f o r  of c a l c u l a t e d v e l o c i t i e s were necessary  the c a l c u l a t e d t r a v e l t i m e s w h i l e a t  and  the GH branch u s i n g the DF b r a n c h ,  to o b t a i n  the best  the same time e n s u r i n g  fine possible  that  the  -72-  turning points for  the v a r i o u s  the measured d T / d A  values  In t h i s manner, t r a v e l t i m e branches earth's  core.  procedures used  r e q u i r e d no s i g n i f i c a n t the measured d T / d A  and  is discussed further  on the r e s u l t s  in t h i s  study.  of  that  adjustment.  values for a l l  four  were i n v e r t e d to o b t a i n a v e l o c i t y model f o r  T h i s model  a general discussion  branches remained s a t i s f a c t o r y ,  i n Chapter 3,  observed the  along  the measurement and a n a l y s i s  with  -73CHAPTER 3 DISCUSSION  General  3.1  Observations  During the measurement and a n a l y s i s procedures d e s c r i b e d in  Chapter 2, a number of p e c u l i a r o b s e r v a t i o n s  were made.  Generally,  there has been no documentation i n the p u b l i s h e d l i t e r a t u r e of observations. workers  Because such o b s e r v a t i o n s  i n the f i e l d ,  this  section  similar  may be r e l e v a n t to o t h e r  is devoted to a d i s c u s s i o n  of  them. While a n a l y s i n g a phase w i t h a s l i g h t l y  the DF branch i n the range  lower  than the phase i d e n t i f i e d as than one second a f t e r  1 1 3 ° to 1 2 0 ° ,  tiT/dA v a l u e and h i g h e r PKP  amplitude  was c o n s i s t e n t l y observed  0F  the DF phase a r r i v e d .  less  T h i s second a r r i v a l  w i t h d i f f e r e n t apparent v e l o c i t y may s i m p l y be a d i f f r a c t e d e f f e c t due  to i r r e g u l a r i t y  the  path of  near  the a r r a y .  Rapid  lateral variation  in e l a s t i c  the a r r a y s i t e may a l s o be a p o s s i b l e e x p l a n a t i o n .  arrival the  in s t r u c t u r e at the source or a t some p o i n t  with higher  dT/dA  v a l u e and e a r l i e r  presence of an a d d i t i o n a l  c o r e boundary, phase.  More d a t a  investigated. dependent i s from azimuths  is  The  first  t r a v e l t i m e may even suggest  in t h i s  inner  case being the true DF  needed b e f o r e such a p o s s i b l i l i t y c o u l d be  That t h i s anemalous o b s e r v a t i o n may be a z i m u t h a l l y suggested by the f a c t near  300°  and  o In the range  120  60°,  that  is  in q u e s t i o n a r r i v e d  o to 130 , s i m i l a r  with r e s p e c t to the s e c o n d .  range, however,  the events  and a t d i f f e r e n t d i s t a n c e s .  what s p o r a d i c a l l y , w i t h great v a r i a t i o n arrival  parameters  d i s c o n t i n u i t y q u i t e near to the  the second a r r i v a l  in  that at d i s t a n c e s  e f f e c t s were noted some-  i n the amplitude of  Of p a r t i c u l a r of  the  i n t e r e s t in  120.6°, 1 2 1 . 7 °  and  first this  121.8°,  -74the second a r r i v a l , a p p r o x i m a t e l y had higher well  dT/dA  of  of  the Df phase agree w i t h  Four events  in p a r t i c u l a r deserve s p e c i a l  are events numbers 2 5 , all  26,  and 30  29,  a r r i v i n g from azimuths  a n a l y s i s of  near  150°.  p r e c u r s o r s were observed j u s t  arrival  of  results  v a r y i n g from 1.0  to 2 . 0  the DF phase.  sec/deg for  1 to 2 seconds  f o u r events  to 1.5  of  their  sec/deg for  the t h i r d .  The d T / d A  - approximately  observation for a l l  1ik  events  T h e i r dT/dA  than the DF v a l u e s ,  the GH t r a v e l t i m e branch.  to t h i s  value for  to 1.5  PKP  yielded  two events  The f o c a l  was  in the azimuth range presursors  as  the above  low f o r  all  1 0 0 ° to  160°.  some 5 to 7  seconds  slightly  that the DF v a l u e s  i n the extreme, taken as  belonging  were to  T h i s was not done, however, because  depths of H  low to be m e a n i n g f u l .  their belong  the events r u l e d out the p o s s -  arrivals.  A t h i r d s e t of  precursors  f o u r events a p p r o x i m a t e l y 8 to 9 seconds  15 seconds)  and  showedaprecursor  values 0F  b e f o r e the  s e c / d e g - and t h i s was a gen-  and on the b a s i s  them b e i n g pPKP^  was noted on a l l one case  the f i r s t  first  a c c o r d i n g to any e x i s t i n g models were too slow to  branch.  i b i l i t y of  values  v a l u e s were, on the a v e r a g e ,  they c o u l d have been,  traveltimes,  dT/dA  but  For the  3 to h seconds  to  first  precursors,  The f o u r t h event  f o u r events showed a second s e t of  b e f o r e PKP^.  low,  Analysis  Those 125°  These were the  b e f o r e DF, and w i t h s i m i l a r  mentioned p r e c u r s o r s .  higher  range  the p r e c u r s o r s y i e l d e d p u z z l i n g r e s u l t s .  three e v e n t s ,  All  Buchbinder's  mention.  in the d i s t a n c e  events over the e n t i r e range to show any s i g n of  eral  amplitudes  predictions.  (1971)  only  These may  the r e c e d i n g branch HD, and t h e i r  w i t h r e s p e c t to the a m p l i t u d e s  1.7  the DF a r r i v a l  v a l u e and amplitude than the DF phase.  be o b s e r v a t i o n s  130°,  seconds a f t e r  1.5  b e f o r e DF, but t h e i r d T / d A These o b s e r v a t i o n s  ( and  v a l u e s were  too  c o u l d be an azimuthal  in  -75e f f e c t as the events a l l occur a t azimuths near distances.  No s a t i s f a c t o r y e x p l a n a t i o n f o r the p r e c u r s o r s can be g i v e n . At l e a s t  ten events  c u r s o r s which were u n u s u a l . seconds  before P K P  others a r r i v e d only sec/deg.  2.6  with  D F  i n the range  1 t o 2 seconds  values  b e f o r e DF w i t h d T / d A  d T / d A v a l u e s o f about 3 . 3  dT/dA  b e i n g pPKP,-j. and p P K P  &H  structure.  v a l u e , would r e s u l t  i n t h e i r magnitude).  It may be p o s s i b l e  that  deep i n the c r u s t  near  (Wright,  the base of the c r u s t it  is  1970)).  i n about a  they a r e due t o some  (as f o r example  faulting  On the b a s i s o f t r a v e l -  the r e c e d i n g branches 61 andJG-. 131.6°  to 1 3 8 . 9 ° ,  showed phases a r r i v i n g 2 to 4 seconds a f t e r m a j o r i t y of c a s e s , and a m p l i t u d e s  these 2nd a r r i v a l s  than the DF phase.  the DF a r r i v a l s .  This  some 20 events  the DF phase.  had s l i g h t l y  higher dT/dA  (1971)  communication,  that  The h i g h - a m p l i t u d e d a t a o f 9 o f  d T / d A v a l u e s a p p r o x i m a t e l y equal  lower than the dT/dA  than  lower a m p l i t u d e behaviour was observed  these events a l s o showed p r e c u r s o r s some 2 to 4 seconds with  values  is c o r r e c t , their  i n q u e s t i o n should be lower  i n o n l y f o u r o f the 20 e v e n t s .  arrival,  In the  They c o u l d p r o b a b l y be o b s e r v a t i o n s  the r e c e d i n g brand HD, but i f Buchbinder  a m p l i t u d e s a t the d i s t a n c e s of  Such a c o r r e c t i o n ,  l i k e l y t h a t some o f them are o b s e r v a t i o n s  In the d i s t a n c e range  of  sec/deg  The p o s s i b i l i t y o f these phases  source o f m u l t i p l e a r r i v a l s  of  to 3 . 7  were r u l e d out on the b a s i s of f o c a l depth of  in question.  times and d T / d A v a l u e s  values of  (The above mentioned v a l u e s  uncorrected f o r local  on the b a s i s o f the PKPQF 5% i n c r e a s e  pre-  d T / d A v a l u e s o f about 3 t o 4 s e c / d e g w h i l e  Yet o t h e r s had  a r e the dT/dA  134° t o 1 3 9 ° showed  Some o f these a r r i v e d a p p r o x i m a t e l y 4  but a r r i v e d o n l y one second b e f o r e DF.  the events  o  150 , but a t d i f f e r e n t  val ues f o r the DF phase.  1972) has made s i m i l a r  b e f o r e the DF  to or o n l y Wright  slightly  (personal  o b s e r v a t i o n s o f the p r e c u r s o r s  -76over  the same d i s t a n c e range from seismograms  owknife s e i s m i c a r r a y .  S p e c i f i c Observations  3.2.1  the  Yell-  However, no s a t i s f a c t o r y e x p l a n a t i o n f o r  these phases can be made at 3.2  recorded at  I d e n t i f i c a t i o n of  the p r e s e n t t i m e . Concerning PKP  Phases  Phases  S i n c e u l t i m a t e l y the model generated upon the i d e n t i f i c a t i o n o f  the i n d i v i d u a l  in t h i s study depends  phases,  a few comments on  t h i s s u b j e c t would seem to be a p p r o p r i a t e .  The DF phase was  i d e n t i f i e d up t o d i s t a n c e s  b e i n g the most prominent  arrival  on the r e c o r d .  b a s i s of  dT/dA  \k0° to about  is  i n a d d i t i o n to i d e n t i f i c a t i o n on the  a r r i v a l s of a l l  the phases.  i d e n t i f i e d on the b a s i s o f the AB phase  1 5 0 ° to 176° because o f  i n the e a r l y p a r t of  its  about  to have s i m i l a r  its  arrived after  the  ways s l i g h t l y  higher  d T / d A values  flS  and longer  than t h a t o f  145 ° to 1 5 0 ° , no problem  145°.  The AB phase always  IJ  However,  i n the range  a r a t i o n of  the two phases  their  dT/dA  the  IJ and GH  and t r a v e l t i m e meas-  138° to 142° where the time s e p -  narrowed g r e a t l y and the t r u e dT/dA  were sometimes masked by l o c a l  al-  phase.  1 3 2 ° to 1 3 8 ° , s e p a r a t i o n of  urements.  Even  u n c o r r e c t e d d T / d A v a l u e was the  proved easy on the b a s i s of  travel-  from PKP.j , both o f which tend  near  IJ phase and i t s  In the range  was  i n the d i s t a n c e  i t from the o t h e r phases.  PKP  v a l u e s of d T / d A  it  t r a v e l t i m e and apparent v e l o c i t y .  range from about  was encountered i n s e p a r a t i n g  Beyond 146° ,  l i k e w i s e proved easy  time which e f f e c t i v e l y s e p a r a t e d  phases  At d i s t a n c e s of  1 4 6 ° , i d e n t i f i c a t i o n became somewhat dtifficultdue to the  I d e n t i f i c a t i o n of range  This  140° as  and t r a v e l t i m e measurements.  near-simultaneous easily  of about  easily  structural  effects,  values  identification  became more d i f f i c u l t and was o f t e n made a f t e r c l o s e e x a m i n a t i o n o f the  dT/dA v a l u e s o b t a i n e d u s i n g  r e j e c t i o n l e v e l s of  both 0.02 and 0.04  -77seconds on the seismometer  residuals.  The g r e a t e s t weight was  ached to the 0.02 s e c . r e j e c t i o n measurement s i n c e p r e c i s i o n measurement.  In one or two i n s t a n c e s ,  was made s o l e l y on a t r a v e l t i m e b a s i s d i s c r i m i n a t e from one dT/dA branch was masked by the  if  the  is  the higher  identification  i t was not p o s s i b l e  v a l u e to a n o t h e r .  IJ b r a n c h .  this  Near  att-  to  o 145 , the GH  In the l a t t e r p a r t of  the range  o beyond 145 , the two phases the  c o u l d once more be e a s i l y  separated on  t h e i r dT/dA and t r a v e l t i m e measurements.  basis of  On the b a s i s of l i m i t e d amplitude  information  3.2.3) and an o v e r a l l believed that  dT/dA v a l u e s and t r a v e l t i m e s ,  the measured  (which w i l l  detailed analysis  the PKP phases  be d i s c u s s e d  of a l l  used i n t h i s  in  Section  seismograms,  it  is  study have been c o r r e c t l y  identified. 3.2.2  Turning Points At t h i s  and End Points  point,  for  the T r a v e l t i m e  i t should be noted t h a t  t o determine p r e c i s e l y the t u r n i n g p o i n t of  Branches  i t was not  possible  the branch DF due to  lack  .0  of d a t a the  i n the immediate r e g i o n b e f o r e 113.6  t u r n i n g p o i n t was  taken as  i n the i n v e r s i o n t e c h n i q u e s . Randall near the  (1964)  place i t at  As p r e v i o u s l y mentioned,  the -truncated v a l u e of This  is  110°, Bolt  113.6  for  use  not unreasonable as Adams and (1964)  at  1 1 1 . 3 ° , Shufbet  (1967)  1 1 0 ° , and E r g i n  (1968) near 1 1 4 ° . Buchbinder (1971) p l a c e d o t u r n i n g p o i n t near 120 on the b a s i s of amplitude d a t a . The GH phase was f i r s t  The  .  dT/dA  values  i d e n t i f i e d a t a d i s t a n c e of  (uncorrected) f o r  this  model.  132.2 .  phase were lower than the  expected v a l u e by a p p r o x i m a t e l y 23%, but i t a r r i v e d a t time per the Adams and Randall  the expected  Since the DF dT/dA  values  on the same event were a l s o 23% lower than e x p e c t e d , the phase was accepted as  belonging  to the GH b r a n c h .  second o b s e r v a t i o n of GH at a d i s t a n c e of  o  The same a p p l i e s 134.8  o  .  to the  The f i r s t  clear  -78and s t r o n g GH phase was observed a t a d i s t a n c e o f turning point for  the GH branch was s e l e c t e d as  Possible  134.8  o  . Thus,  the  c  b e i n g 132.0 .  IJ p r e c u r s o r s o c c u r r i n g a p p r o x i m a t e l y  12 to 13 o  seconds  b e f o r e DF c o u l d be b a r e l y r e c o g n i z e d a t d i s t a n c e s of  133.11 , 134.5 , and 134.7 » but were too s m a l l , tude d a t a ,  to be a n a l y s e d .  The f i r s t  tude was s i m i l a r c o u l d be made. ing p o i n t  to t h a t of Thus,  it  134.8°.  PKP  Although  the background n o i s e , dT/dA  is  i n the range o f  even on h i g h a m p l i -  i d e n t i f i c a t i o n of  a l s o made on the r e c o r d a t a d i s t a n c e of  l i k e l y t h a t the  132° to 1 3 5 ° .  132.0 ,  was  {J  its  ampli-  measurements  IJ branch has  its  For the purpose of  turn-  inversion,  o i t was s e l e c t e d as Failure distances  b e i n g 134 . to observe e i t h e r  may be due, a p a r t from the obvious  phases s i m p l y do not e x i s t other reasons. by an a r r i v a l ing,  if  GH or  in t h i s  For example, a f t e r  would t r i g g e r  it.  precursons.  less  but the DF phase  is  increase at greater distances  the 13 seconds  to enable r e c o r d i n g o f bearing  the branches f o r , a c c o r d i n g to  the t u r n i n g p o i n t  record-  than 132° t h e i r  array,  T h i s c o u l d have an important  the t u r n i n g p o i n t s of  triggered  loop f o r  U n f o r t u n a t e l y a t these d i s t a n c e s , be s u f f i c i e n t  the  to s e v e r a l  1967» the a r r a y was  to t r i g g e r . t h e  shorter  explanation that  the range,  IJ were p r e s e n t f o r d i s t a n c e s  would not n e c e s s a r i l y  (1971)»  p a r t of  IJ phases a t  a t the c l u s t e r with a 13 second d e l a y  a m p l i t u d e s were i n s u f f i c i e n t  liest  the GH or  the e a r -  in d e t e r m i n i n g  Buchbinder  t h a t d i s t a n c e at which the  amplitudes  but d e c r e a s e r a p i d l y a t s m a l l e r  distances,  Any phases observed b e f o r e the t u r n i n g p o i n t are a t t r i b u t e d to phases p a r t i a l l y r e f l e c t e d from the d i s c o n t i n u i t y and transmitted  through  i n the e a r l y p a r t of  it.  Consequently,  the range,  if  the GH and  t h e i r amplitudes  by B u c h b i n d e r ' s c r i t e r i o n they do not  those  partially IJ phases  are so s m a l l ,  i n any case belong to the  existed that totally  -79t r a n s m i t t e d a d v a n c i n g branches which were determined in t h i s  study.  Another p o s s i b l e  but which  essentially  reason f o r f a i l u r e to observe these phases,  leads  to the same c o n c l u s i o n as above,  h i g h - a m p l i t u d e d a t a was a t times f a r signals  to be p i c k e d o u t .  high-amplitude data a l l  In S e c t i o n 3 . 1 ,  None of of  the  i n the range  four  events  o  125  to 130  were d i s c u s s e d .  these p r e c u r s o r s appeared even remotely to belong to e i t h e r IJ or GH b r a n c h e s .  not e x i s t  T h i s f a c t suggests  i t was not p o s s i b l e and GH b r a n c h e s . d i s t a n c e of  i t seems 151°.  l a c k of d a t a  y i e l d e d a phase at  served at  region.  150.89°.  The l a s t  The event a t  belonging  clear  157.2°  dT/dA  Since the DF  low by a p p r o x i m a t e l y the same amount,  the end p o i n t f o r  157°showed  l i k e l y t h a t the end p o i n t  the expected time of GH, but w i t h  157.2° was a c c e p t e d as  IJ  IJ o c c u r r e d at a  1 5 2 . 7 ° and  lower than the expected v a l u e by about 25%. v a l u e was a l s o  the  T h i s cannot be s a i d with any  in t h i s  GH a l s o o c c u r r e d at  150° to 160°  the end p o i n t s of  Events at d i s t a n c e s  somewhere near  because of  o b s e r v a t i o n of  i n the range  The l a s t c l e a r o b s e r v a t i o n of  the phase so t h a t  the branch i s  of d a t a  to d e f i n e a b s o l u t e l y  150.89° .  no e v i d e n c e of  certainty  t h a t these phases do  i n the r a n g e . Owing to the sparseness  of  the  there was a l a c k of  o which showed p r e c u r s o r s  that  too n o i s y f o r any c o h e r e n t  In o t h e r c a s e s ,  together.  is  values d /dA T  the phase o b -  to the GH b r a n c h .  the branch had t o be g r e a t e r  than t h i s  Thus,  distance  and was chosen to be 1 5 8 ° . The t u r n i n g p o i n t f o r where i n the range looked but  the AB branch was assumed to be some-  143° to 1 4 5 ° .  Strong a r r i v a l s  l i k e AB, were r e c o r d e d a t d i s t a n c e s  i n c l u s i o n of  data r e s u l t e d  these v a l u e s  i n the  of  of a phase  1 4 2 . 3 1 ° and  smoothing of  in an u n d e s i r a b l e f l a t t e n i n g out of  that  142.59°  the AB the AB b r a n c h .  -80These a r r i v a l s cause o f  possibly  t h i s ambiguity,  for  the AB b r a n c h .  for  this  belong to the r e c e d i n g Bl b r a n c h . they were not  For purposes o f  Be-  i n c l u d e d i n the d T / d A  i n v e r s i o n , the t u r n i n g  branch was s e l e c t e d as b e i n g ' a t  145°.  data  point  The phase was observed  o o u t t o 175 , the g r e a t e s t d i s t a n c e f o r any e v e n t a n a l y s e d 3.2.3  Amplitude  in t h i s  study.  Observations  A l t h o u g h a b s o l u t e a m p l i t u d e s from one event t o another c o u l d not  be compared because of  on each seismogram,  lack of  i t was p o s s i b l e  i n f o r m a t i o n as  to the g a i n used  to compare •  the  rel-  a t i v e a m p l i t u d e s o f the d i f f e r e n t phases f o r any one p a r t i c u l a r event.  T h i s showed t h a t a t d i s t a n c e s  on the seismogram then the  is  IJ phases.  the DF phase, f o l l o w e d  a m p l i t u d e v a r i e d from about  of  the  range  that of  than 1 4 2 ° , the l a r g e s t i n magnitude  DF near  1/10  140°.  t h a t of DF near  to 1/4  that o f  Thus the v a r i a n c e  GH phase at  than 142°, although these d i s t a n c e s .  became more prominent than than the DF phase.  PKP , &H  Then, a t about  i n the  greater  distance  t h a n ' the  than-142°  but both were s t i l l  145 ,  tude w h i l e GH was a p p r o x i m a t e l y equal DF i n a m p l i t u d e .  135° to  i t always rema i ned sjnal l e r At d i s t a n c e s  IJ  in amplitude  IJ phase was more pronounced than the GH phase less  1/3  The  \kQ°.  DF near  phase  by the GH,  The GH a m p l i t u d e v a r i e d from about £ to  t h a t o f DF near 1 3 5 ° , to about 4/5  5- t o 3/4  less  PKP  I7  smaller  IJ exceeded DF i n a m p l i -  to or o n l y s l i g h t l y  larger  than  IJ c o n t i n u e d to be the most prominent phase f o r  the  r e s t of  its  d i s t a n c e range a t t a i n i n g a maximum a m p l i t u d e 10 times  that of  DF.  The GH a m p l i t u d e u s u a l l y  tude. greater  The AB phase, e n t e r i n g at about  l a y j u s t beneath the  145° d i d so w i t h a m p l i t u d e  than the DF a m p l i t u d e but by 1 6 0 ° , i t s '  to a p p r o x i m a t e l y the same or s l i g h t l y Of p a r t i c u l a r  interest is  the f a c t  IJ a m p l i -  less  a m p l i t u d e had f a l l e n  than the DF a m p l i t u d e .  t h a t w h i l e Adams and Randall  (1964)  Figure 3.1 A Record Section for the Phases DF,GH,IJ,AB.The IJ Phase i s shown darkened.  -82i n d i c a t e that  the GH phase i  distances  iour o f  the most prominent on the r e c o r d at o  o  between 145  most prominent a t  is  and 153 , i n t h i s  these d i s t a n c e s .  study  the  C l o s e e x a m i n a t i o n of  is  A Record S e c t i o n f o r F i g u r e 3.1  Core  Phases.  is a traveltime - distance section for  In c o m p i l i n g t h i s  record s e c t i o n , only 5 of  20 t r a c e s have been reproduced f o r The phases are record.  and i t s  the sake of  i d e n t i f i e d by t h e i r  In p a r t i c u l a r , the  the r e c o r d .  (1970  discussed.  mograms i n the range where the p r e c u r s o r branches GH and present.  T h i s phase has  clear observation  the  the behav-  the t h e o r e t i c a l a m p l i t u d e s determined by Buchbinder  i n d i c a t e good agreement w i t h the o b s e r v a t i o n s j u s t 3.2.4  IJ phase  IJ phase  is  not e a s i l y  is one of  seis-  IJ were  the  possible  clarity.  dT/dA  values  shown darkened  on the  throughout  been observed in the  the s i g n i f i c a n t  past,  results  of  this  study. Many of sections  the o b s e r v a t i o n s  discussed  are e x e m p l i f i e d i n t h i s f i g u r e .  in the p r e v i o u s  The f i r s t  three  seismogram a t  • o is  134.80  the one on which the IJ phase was f i r s t o b s e r v e d , and the  GH phase was c l e a r and prominent.  The a m p l i t u d e of  the  IJ branch  w i t h r e s p e c t to the o t h e r branches can be seen t o i n c r e a s e u n t i l at a d i s t a n c e o f about  1 4 6 ° i t exceeds the DF phase  in a m p l i t u d e .  It  becomes the most prominent phase, f o l l o w e d by GH, f o r  of  its  d i s t a n c e range.  phases near possibly the is  The d e c r e a s e i n time s e p a r a t i o n between the  1 4 3 ° can be c l e a r l y o b s e r v e d .  GH) phase becomes a l a t e r a r r i v a l  IJ phase  is  seen a r r i v i n g a f t e r  f i r s t observed at  time branches from IJ  the r e s t  145.24°.  By 146.17°, the  (and  to the DF phase. At  the GH phase.  The d e c r e a s e  IJ  147.6  The AB phase  in s l o p e of  to DF can be c l e a r l y o b s e r v e d .  the t r a v e l -  -83Discussion of  3.3  3.3.1  the Analysed  dT/d.A  Corrected  The c o r r e c t e d of  Data  Measurements.  dT/dA  measurements  Summary v a l u e s and by a polynomial  shown i n F i g u r e s 2.9 and 2.10. c u r v e to both the B o l t  smoothed by the method  regression  technique were  The e f f e c t o f r e s t r a i n i n g the AB  t r a v e l t i m e t a b l e s and the JB t r a v e l t i m e s  i n d i c a t e d that  the r e s t r a i n t to the J - B times gave a b e t t e r f i t  the c o r r e c t e d  dT/dA data  particularly  range where the c u r v a t u r e i s data  in this  Steep.  in the e a r l y p a r t of  Even so,  the best f i t  the AB to the  r e g i o n would have been one w i t h g r e a t e r c u r v a t u r e than  t h a t g i v e n by the J - B r e s t r a i n t but t h i s p o i n t s a t almost  necessitated having  summary  the same d i s t a n c e but with w i d e l y d i f f e r i n g  dT/dA  values.  T h i s c o u l d not be o b t a i n e d with the d i s t r i b u t i o n of  data f o r  the r e g i o n .  lies  to  The AB c u r v e smoothed by polynomial  between the two c u r v e s smoothed by summary v a l u e s ,  e a r l y p a r t of  the range  J-B r e s t r a i n e d c u r v e .  it  regression  but  in the  tends to lean h e a v i l y  i n f a v o u r of  the  Owing to the l a c k of d a t a  in the range  150° to  160° the curve here has s l i g h t l y For the same r e a s o n ,  available  l e s s c u r v a t u r e than would be e x p e c t e d .  the c u r v e s smoothed by summary v a l u e s  in  this  r e g i o n behaved somewhat e r r a t i c a l l y and were s i m p l y r e p l a c e d by the expected J - B v a l u e s . The IJ branch smoothed by both methods agree well,  as does  and Randall  the DF b r a n c h .  The GH branch r e s t r a i n e d to the Adams  t r a v e l t i m e c u r v e and smoothed by the summary  method agrees  values  almost e x a c t l y w i t h that o b t a i n e d by the polynomial  r e g r e s s i o n method. this  reasonably  Owing to the f a c t that, the B o l t r e s t r a i n t  branch was t r u n c a t e d a t  GH end p o i n t at  156° i n s t e a d of  156°) and that h i s  Adams and Randall  traveltimes for  158° ( B o l t p l a c e s  traveltimes a l l  his  l i e beneath the  the same d i s t a n c e s ,  r e s t r a i n e d to h i s model was s h i f t e d to higher  for  dT/dA  the c u r v e values  and at  DT/D-DELTR VS DELTR (CORRECTED)  A - DF  A  D - GH Z -  IJ  X - RB  ID  UJ O <_> _ Ujru  i  oo  m  B  I m  a a  110.0  115.0  120.0  -1 125.0  -I  130.0  13S.D  -|  140.0  1  145.0  DELTA(DEG)  150.0  Figure 3.2 F i t of smoothed DT/DAcurves  -1 155.0  -1 160.0  165.0  -1 170.0  on t o c o r r e c t e d d a t a .  ~T~— 175.0  180.0  TRAVEL A -  TIME VS DELTA  DF  Q - GH z -  IJ  x - RB  CJUJ IT.  1JD.D  115. D  ^  "I  125.0  I 130.0  I  135.0  140.0  145.0  DELTA(DEG)  "T  150.0  155.0  160.0  165.0  120.0  Figure 3.3 F i t o f smoothed t r a v e l t i m e curves on t o d a t a .  170.0  —I 175.0  I BO  REDUCED TRAVEL  TIME  VS DELTA  C - DF D - GH z - IJ x - HB  UO.O  -1  115.0  1  120.0  "~l  125.0  -|  130.3  -1  I  135.0  140.J  145.0  150.0  155.0  -I  160.0  I 165.0  T  170.0  DELTAfDEG)  Figure 3.4 F i t o f smoothed reduced  t r a v e l t i m e curves on t o d a t a .  T  175.0  iao  the same time f e l l  o f f more s h a r p l y  The smoothed d T / d A procedure were the AB v a l u e s  near  values  156 .  used  i n the a c t u a l  r e s t r a i n e d to J - B , the IJ and DF d a t a ,  and the GH data r e s t r a i n e d to the Adams and Randall fit is  o f these c u r v e s F i g u r e 3.2.  final  t o the c o r r e c t e d  good f o r a l l  used  in d e r i v i n g  the branches observed  after corrections for  These e q u a t i o n s  were used t o c a l c u l a t e  The c o r r e s p o n d i n g  The independent  are l i s t e d  in Table  the t r a v e l times a t  t r a v e l times and reduced t r a v e l t i m e  c u r v e s a r e superimposed on the d a t a  in Figures  3.3 and 3.4  That the f i t i s a good one i s e v i d e n t from the parameter R  Basically,  the c l o s e r  R  (which  i s to the v a l u e u n i t y , the  i s the f i t of the r e g r e s s i o n e q u a t i o n to the d a t a .  the lowest v a l u e o f It  0.986  R " has 2  f o r the GH branch.  should be.noted that no c o r r e c t i o n s f o r s t r u c t u r e be-  neath the a r r a y were a p p l i e d assumption  respectively.  i n Appendix Va) f o r the d i f f e r e n t branches as shown i n  T a b l e 3.1. better  ellip-  in the smoothing of the  d a t a , and the r e s u l t i n g r e g r e s s i o n e q u a t i o n s  discussed  is  study.  i n Appendix Va.  v a r i a b l e s were the same as those used  1° i n t e r v a l s .  the f i t  depth had been a p p l i e d , were smoothed u s i n g the  U.B.C. TRIP program as d e s c r i b e d  3.1.  i s shown  Measurements  The t r a v e l t i m e measurements, t i c i t y and f o c a l  The  ( S e c t i o n 2.3.t)for the  these c u r v e s ,  in this  model.  measurements  2  Travel-time  3.3.2  dT/dA  As determined by the y . t e s t s  summary p o i n t s  inversion  to the measured t r a v e l times on the  t h a t a t the l a r g e d i s t a n c e s  i n v o l v e d , such c o r r e c t i o n s  would be n e g l i g i b l e . The r e s i d u a l s  (0 - C ) , between the smoothed and measured  times a r e shown i n F i g u r e 3.5. arrival  with r e s p e c t  A positive residual  implies a late  to the smoothed times and v i c e v e r s a .  The  is  -88TARi.'.-: J . I  REGRESSION I:QIJATIO.\'S IOR SMOOTHED TRAVELTIME DATA :  105  DP BRANCH r.= 2  data points  0.99S5  INDEPENDENT VARIARIE  coi;n icii:MT  STD.ERROR  F-•I'ROi!  TOLIiRANCI:  CONSTANT  1119.4653  0.3729  0..0000  1.0000  ;  (A-113)  1.7656  (A-113)  0.1222/10  1  T». 1119.4653+1.76S6(A -113)  1.0000 1.0000  0..0000  1.0000  3  -0.2681/10  (4-113)'  0..0000 . 0. 3536/10 . 0 ,0000 0.2779/10"  3  +1.222 /'A-113.0\' - 0,.Z6S1 <  f A-U3)3  V  J  10  CU DRANCII  34 d a t a  1 0 /  points  R = 0.9S63 2  INDEPENDENT VARIABLE  COEFFICIENT  STD.ERROR  CONSTANT  1141.2908  1.1931  (A-132") (A-132°)  3.2137 -0.2175/10  2  T -= 1141 .2908 + 3.2137 (A-132)  F-PROB  TOLERANCE  0.0566  0.0000  1.0000  0.7953/10*  0.0000  1.0000  - 2 .175/^132^ \ 10  I J BRANCH  /  26 d a t a p o i n t s  R = 0.9915 2  INDEPENDENT VAR I AH J.E •CONSTANT (A-134°) (A-134")  2  COEFFICIENT  STD.ERROR  1140.2857  1.1746  3.7487 -0.19410  T «= 1140 .2857 + 3 .7487 (A-134)  F-PROB  TOLERANCE  0.0655  0.0000  1.0000  0.0142  0.01808  1.0000  -1.9410 ^A-134 J  AE JiP.ANCII  23 d a t a  points  R •= 0.9979 2  INDEPENDENT VARJAI'.LE  COEFFICIENT  STD.ERROR  CONSTANT  1180.8818  0.7517  (A-J4S") (A-145")  3.8276  T = 1130.8818 *  0.0432  0.13923/10  3  3.8276  fA-145 )  P-PROR  +  0.6171/10 1 .3923  / A O  V.  10  2  TOLERANCE  0.0000  1.0000  0.0338  1.0000  -1J.Y  /  -89-  TRAVEL-TIME  RESIDUALS  AB  00  -2C —  IJ •20J—  • .  0 0< •  0 • •  •  tf*ui  GH £ 20J— Ct  OC • 9 -20 —  • 200C  *—rr  _a_  1  •• • ; .  • •  »  -20  110  120  130  140  150  160  DELTA COEGJ  F i g u r e 3.5 T r a v e l t i m e R e s i d u a l s  (0-C).  I70  -90-  TRAVEL-TiriE.  AB  DIFFERENCES  BRANCH J- B  -CfllC.  BOLT—CflLC  IJ  BRRNCH fl.R -  GH  CPLC.  BRUNCH  B O L T — CRLC.  OF  BRANCH J . 8 — CflLC.  eon - cfluc.  120*  DISTANCE  Figure  160'  140*  3.6  rDE&)  Traveltime  Differences  -91residuals  show the l e a s t  consequence of  the f a c t  scatter for that  the DF b r a n c h .  the unsmoothed times f o r  c l o s e l y d e f i n e d the r e g r e s s i o n c u r v e . unsmoothed  IJ and GH times  corresponding r e s i d u a l s .  The l a r g e r  scatter  The s c a t t e r o f  the l a r g e d i s t a n c e range of  phase  the r e s i d u a l s  the sparseness  i n F i g u r e 3.6.  lesser  the onset  of  with  curvature  is  but the smoothed AB times may i n d i c a t e  r e q u i r e d in the J-B c u r v e near  1 second l a t e r  should be remembered, however, that  owing to the l a c k o f data over  c u r v a t u r e near  its  155°, so  the smoothed AB t r a v e l t i m e  the range.  end p o i n t .  the t r u e  situation  The IJ branch r e q u i r e s model near  its  The GH branch is  beginning shifted  t r a v e l t i m e than those d e f i n e d by the Adams and Randall  and t o higher  times with r e s p e c t to the B o l t model.  greater o v e r a l l the behaviour of  the smoothed t r a v e l times o f  i n which too l a r g e emphasis the b r a n c h e s .  tend t o b i a s  It  the p r e c u r s o r  be a consequence of i s p l a c e d on v a l u e s  Obviously,  any e r r o r  model,  requires  c u r v a t u r e than t h a t demanded by both models.  near t h e i r end p o i n t s may s i m p l y  that  than the present J - B t i m e s .  g r e a t e r c u r v a t u r e than the Adams and Randall  points of  regression  i n e x i s t e n c e are shown  c u r v e may not be an adequate r e p r e s e n t a t i o n o f  fit  Also  points  The smoothed AB and DF times agree more c l o s e l y w i t h  the new times are about  to lower  the d a t a  the smoothed times  t r a v e l t i m e models a l r e a d y  the c o r r e s p o n d i n g J - B v a l u e s ,  and l e s s e r  the AB  is o f t e n d i f f i c u l t to p i c k .  respect to various  It  of  for  the branch, so t h a t the  The t r a v e l t i m e d i f f e r e n c e s of  that  in the  r  c u r v e c o u l d not be p r e c i s e l y d e f i n e d by the d a t a . this  the branch  ( F i g u r e 3.3) are d u p l i c a t e d by t h e i r  branch may be more a consequence of over  Thisaa natural  Again,  branches  the l e a s t  squares  near the end  in these end p o i n t s  the r e s u l t i n g smoothed times near these r e g i o n s  in a  would  Figure 3.7  V e l o c i t y Model UBC1.  f o r E a r t h ' s Core.  -93-. c o r r e s p o n d i n g manner. justing  the f i n a l  This f a c t  is  v e l o c i t y model  important when i t comes to a d -  i n order to o b t a i n agreement  the c a l c u l a t e d and the smoothed  times.  of  times near the end p o i n t s  the p r e c u r s o r b r a n c h e s ,  of  the model  times and smoothed  e x c l u s i v e l y demanded. agreement  It  themselves,  V e l o c i t y Model  times  in these r e g i o n s  for  imposed by the s c a t t e r the r e s p e c t i v e  biasing  exact is  of  that  described  to a l l  branches.  UBC1  data analysed  in S e c t i o n  1.2,  by Adams and Randall  study.  Of a l l that  represents  the models suggested  (1964).  of  the t u r n i n g p o i n t s  of  the change  for  in the two models  lies  i n the  the d i f f e r e n t branches and i n the  dT/dA a l o n g the r e c e d i n g b r a n c h e s .  two d i f f e r e n c e s a r i s e  the o t h e r d i s s i m i l a r i t i e s :  esponding d i s c i n t i n u i t i e s , dicontinuities,  in this  i t most c l o s e l y resembles  The b a s i c d i f f e r e n c e  in  the  the t r a v e l -  F i g u r e 3.7 shows the v e l o c i t y model, UBC1, which the best f i t  agreement  not to be  i s of more importance, however,  be w i t h i n the l i m i t s  time r e s i d u a l s 3.4  of  On account o f t h i s  between  magnitude  of  location  magnitude  From these  depth to the c o r r -  the v e l o c i t y jumps at  and t r a v e l time g r a d i e n t s  i n the two s h e l l s  these  defined  by the d i s c o n t i n u i t i e s . In  the p r o c e s s of f i t t i n g  was found t h a t  the J - B v a l u e s  be reduced s l i g h t l y  the AB branch to the d a t a ,  of v e l o c i t y f o r  it  the o u t e r c o r e had  from a depth of about 4000 km downwards,  to  in order  to get agreement  between the c a l c u l a t e d and measured t r a v e l times and  dTjdA  Too g r e a t a r e d u c t i o n at  values.  in an u n d e s i r e d f e a t u r e , part of  this  depth however,  the f o r m a t i o n o f a c a u s t i c  the AB branch and t r a v e l times  resulted  in the e a r l y  that were too slow.  Thus, a  good r e s t r a i n t was put on the magnitude o f the v e l o c i t y r e d u c t i o n .  -94The f i r s t v e l o c i t y discontinuity  in UBC1 giving r i s e to  the receding branch Bl of the traveltime curve is at a depth of 4393 km (0.569-Rc).  The end point of the branch AB is at 144.4°.  The  turning point of the advancing branch IJ is located at 134.2° and the magnitude of the jump at this discontinuity The value of  is 0.108 km/sec.  d T / d A near B is 3.52 sec/deg and near  I it  i s 3.48  o sec/deg.  The turning point  I in UBC1 is 4.2  later than that in  the A & R model and this results in the s l i g h t l y deeper for the UBC1 model.  The change in  discontinuity  dT/dA along the receding branch  Bl for UBC1 is s l i g h t l y less than that in the A & R model, and t h i s , together with the later turning point v e l o c i t y jump.  The depth to the  I for UBC1, results  in a smaller  f i r s t discontinuity, in model UBC1  is well above the level 0.54 Rc (depth 4497 km) to which rays would have to penetrate in order to form a caustic at a distance of about  o 143 .  It  should be noted that the large amplitude a r r i v a l s  this distance  near  (see Figure 3.1) were associated with a r r i v a l s of  IJ phase, rather  the  than as the result of a caustic at B.  The second d i s c o n t i n u i t y giving r i s e to the receding JG branch.occuned at a depth of 4810 km. in UBC1 .  The magnitude of  v e l o c i t y jump at the d i s c o n t i n u i t i e s 0.237 km/sec. the IJ branch occurs The later  at 131.9° with dT/dA  the  The end point of  value of 2.66 sec/deg.  starting point for G, and the smaller change in dT/dA  long JG for model UBC1 results  in a smaller v e l o c i t y jump for  a-  this  model than in the A & R model. The e f f e c t of using the higher dT/dA values for the GH branch,due to r e s t r a i n i n g the smoothed data to the Bolt traveltime curve  rather  than the A & R curve, was to raise the discontinuity to about 4765 km and to cause a s l i g h t l y higher v e l o c i t y jump of 0.287 km/sec at  -95the d i s c o n t i n u i t y .  The change  in dT/dA  as b e f o r e , but the t u r n i n g p o i n t s later  respectively.  measurements  along  J and G were s l i g h t l y  i n the f i r s t  r e q u i r e d a more n e g a t i v e larger  shell  o f UBC1 i s i n i t i a l l y p o s i t i v e  than z e r o and the t h i c k n e s s o f the s h e l l  their  dT/dA  r e s t r a i n e d to the A & R c u r v e .  two d i s c o n t i n u i t i e s  of  e a r l i e r and  However, UBC1 was generated from the  The v e l o c i t y g r a d i e n t  Randall  JG remained the same  d e f i n e d by the f i r s t then s l i g h t l y  i s 417 kms.  gradient.in  Adams and  the r e g i o n on account  v e l o c i t y jump a t the d i s c o n t i n u i t y .  The t h i r d d i s c o n t i n u i t y a t the inner c o r e boundary, d u c i n g the r e c e d i n g branch HD, o c c u r s a t e x a c t l y t h a t o b t a i n e d by J e f f r e y s core of radius discontinuity  1251 km.  D is  (1939)  - 5120 km.  with  dT/dA  the same depth as  This  i m p l i e s an inner  the end p o i n t H or branch GH i s  value of 2.16 sec/deg.  l o c a t e d a t 1 1 3 . 2 ° with  The t u r n i n g  dT/dA v a l u e o f 1.96  sec/deg.  The second and t h i r d v e l o c i t y d i s c o n t i n u i t i e s second s h e l l  surrounding  and has s l i g h t l y  the inner c o r e .  negative  gradient.  o n l y be e l i m i n a t e d by p l a c i n g  This shell  This negative  the second d i s c o n t i n u i t y would  data.  the g r a d i e n t  dist-  The depth to  i n the second  i n the inner c o r e c l o s e l y  follows  (1971).  A d i r e c t consequence of the s m a l l e r gradients  g r a d i e n t can  Such requirements cannot be met by the  The v e l o c i t y d i s t r i b u t i o n  t h a t o f Buchbinder  i s 310 km. t h i c k  i n c r e a s e and the v e l o c i t y jump a t the  d i s c o n t i n u i t y would d e c r e a s e , e n a b l i n g t o be n o n - n e g a t i v e .  define a  the t u r n i n g p o i n t D a t a l a t e r  ance and the end p o i n t H a t a t an e a r l i e r d i s t a n c e .  shell  pro-  The magnitude o f the v e l o c i t y jump a t t h e  i s 0 . 9 2 km/sec w h i l e  l o c a t e d a t 158.6 point  less  and s m a l l e r  v e l o c i t y jumps  negative  i n the t r a n s i t i o n  velocity r e g i o n of  -96-  TABLE  3.2a  POSITION OF CUSPS FOR MODEL UBC1*  CUSP  DT/DA  T  (DEC) B  BIN  1111.1 131.2 152.6 131.9 158.6 113.2  I J  S H 0  19 19 20 19 20 18  SEC  (SEC/DES)  38. 7 2.2 2. 3 1.1 12. 1 39.4  3. 52 3.U8 2. 71 2.66 2. 16 1.96  RESULTS OBTAINED BY RESTRAINING GH DATA TO BOLT'S TRAVELTIME CURVE: 151.6 132.3 158.9  J  G H  19 19 20  58.9 1.8 11.9  2. 82 2.7U 2. 16  TABLE 3.2b VELOCITY-DEPTH VALUES FOR UBC1.  VELOCITY  DEPTH  VELOCITY  DEPTH  VELOCITY  (KM)  (KM/SEC)  (KB)  (Ktt/S EC)  (KM)  (SH/3EC)  289'! 7 894 2961 3033 3103 3 172 3212 3311 3381 3150 3520 3589  13.6700 8 . 1000 8.1800 8.2600 8.3500 8.110 0 8.5300 8.6300 8.7100 8.8300 8.9300 9.03Q0  3659 3728 3793 386 8 39 37 1076 1 146 1215 1393 1393 4188 1500  4810 4810 4829 4900 5000 5120 5120 5400 5600 5800 6200 6371  10 .00001 10.2370 10 . 2 6 9 3 10 . 2 4 4 0 10 . 2 0 7 0 10 . 1600| 11 . 0 3 0 0 ! 11 . 1 300 11 . 2 4 0 0 11 . 2 6 0 0 11 . 2 8 0 0 11 . 2 8 0 0  DEPTH  9. 9. 9. 9. 9. 9. 9. 9.  1 100 2000 2800 3 700 4800 5 000 5500 6200  9. 8000 9 . 9 080  1 0 . 0 105 10. 0000  RESOLTS OBTMNED BY RESTRAINING GH DATA TO BOLT'S TRAVELTIME CURVE  4500 4765 4765 4829  10.0000 1 0 . ooooi 10. 2870J 10. 2698  TABLE 3.3  TRAVELTIMES FOR MODEL 'J BC 1. (SURFACE FOCUS)  GH  DF  DISTANCE (DEG)  MIN  SEC  130.0 131.0 132.0 133.0 134.0 135.0 136.0 137.0 130.0 139.0 140.0 141.0 142.0 143.0 144.0 145.0 146.0 147.0 148.0 149.0 150.0 151.0 152.0 153.0 T54.0 155.0 156.0 157.0 158.0  19  12.0 13.9 15.8 17.7 19.5 21.4 23.2 25. 0 26.9 28.7 30.5 32.3 34.0 35.7 37.4 39.2 4 0.9 42.6 44.2 45.3  MIN  19  SEC  4.4 7. 1 9.7  20  15. 1 17.7 20.3 22.9 25.5 28. 1 30.7 33.3 35.9 38.5 41.1 43.7 46.3 48.8 51.3 53.8 56.3 58.7 1.2 3.6 6.0 8.4 10.7  AB  BIN  SEC  19  1.6 5. 1 8.5 11.9 15.3 18.7 22.0 25.6 28.9 32. 3 35.6 38.8 42.0 45. 2 48.4 51.5 54.6 57.7 0.6  12.1  17.4  48.9 50.4 51.8 53.2 54.5 55.8 57.2 58.5  IJ  20  MIN  SEC  19  40.4 44.2 48. 1 52.0 56. 1 0.2 4.4 8.6 12.8 17.0 21.2 25.5 29.8 34. 1  20  -98model UBC1 is that they would cause the central density of the earth's core to have a lower  value than\ that necessitated by  the Adams and Randall model (see Section 1.3.1) without  greatly  a f f e c t i n g the r i g i d i t i e s of the t r a n s i t i o n region and the inner core (Section 1 . 3 . 2 . ) . The position of the traveltime  cusps and the v e l o c i t y -  depth r e l a t i o n s h i p for UBC1 are shown in Tables 3.2a and 3.2b respectively.  The uncertainties  in the depths to the d i s c o n t i n u i t i e s  are estimated at being of the order of 10 km on the basis of obtaining s a t i s f a c t o r y turning points for the various branches. The traveltimes of model UBC1 l i s t e d in Table 3.3 agree with the smoothed traveltimes the l i m i t s  determined in this study within  imposed by the scatter of the r e s i d u a l s .  The greatest  discrepancy was obtained for the early part of the GH branch.  It  appeared that the smoothed times were too early by about two seconds in this v i c i n i t y .  This was undoubtedly the r e s u l t of a biasing of  the regression curve towards early times owing to the f i r s t measured arrival  time being e a r l y , c  possibly as a r e s u l t of  epicentre mislocation.  The model times for the GH branch tend to be shifted downwards by an average of about 1 second with respect to the Adams and Randall times, and upwards by about 3 to 5 seconds with respect to B o l t ' s traveltimes  in the latter and e a r l i e r  part of the branch r e s p e c t i v e l y .  The model times for the IJ branch agree almost exactly with the model times of Adams and Randall, the only difference being that o  the UBC1 times are about 0.2 seconds e a r l i e r 0.1 seconds later  beyond t h i s .  up to about 146 , and  The model times for the AB and DF  branches agree with those of the J-B traveltime  tables.  It must be emphasized that model UBC1 is that model which  -99gave the best f i t this  to the  p a r t i c u l a r study.  t h a t used changes  d l / d f l and t r a v e l t i m e measurements made A more e f f i c i e n t d i s t r i b u t i o n of d a t a  in the study c o u l d r e s u l t  in l o c a t i o n of  sulting effects)  i n small  the t u r n i n g p o i n t s of  to t h i s model.  modifications  in  than  (such as  the branches, w i t h r e -  -100CHAPTER k CONCLUSION  Measurements have been made at 115  of  t r a v e l t i m e and t r a v e l t i m e g r a d i e n t s  the Warramunga  earthquakes over  seismic array for  the d i s t a n c e range  been used to o b t a i n a v e l o c i t y - d e p t h model f o r The dT/dA by l o c a l  measurements  s t r u c t u r e beneath  correcting for  this  the e a r t h ' s  a t WRA were s t r o n g l y  e f f e c t was employed.  technique.  have  core.  perturbed  The c o r r e c t e d d T / d A  summary v a l u e s  The t r a v e l t i m e g r a d i e n t  values  and a polynomial  Both methods gave r e s u l t s  a b l y good agreement.  The d a t a  the a r r a y and an e m p i r i c a l method of  were smoothed by the method of gression  re-  which were i n r e a s o n measurements  by the summary v a l u e method were the ones a c t u a l l y  used  smoothed  in  inverting  the d a t a t o o b t a i n a v e l o c i t y model s i n c e they had been r e s t r a i n e d acceptable t r a v e l t i m e curves The smoothed  i n order  dT/dA  ured  dT/dfl  v a l u e s were i n v e r t e d by the  The s i g n i f i c a n c e the e x i s t e n c e of  branches, cursor  containing  of  this  study  is  branches -  the e a r t h ' s  core.  that  in a t r a n s i t i o n  two v e l o c i t y d i s c o n t i n u i t i e s  the inner c o r e .  On the other hand,  b r a n c h e s , GH as w e l l  as  IJ,  t o the meas-  it  Previously,  region for  o n l y one of  This single the e a r t h ' s  and one " s h e l l "  the e x i s t e n c e of  gives r i s e  revealed c l e a r l y  IJ and GH - to the main DF  the GH one, had been w i d e l y a c c e p t e d .  branch r e s u l t e d  core.  data.  two p r e c u r s o r  t r a v e l t i m e branch f o r  Herglotz-  the e a r t h ' s  the one which gave the best f i t  and t r a v e l t i m e  to  to remove any random e r r o r .  Wiechert method to d e r i v e a v e l o c i t y - d e p t h model f o r T h i s model, UBC1 , was  >  PKP phases from  to 176°.  113°  dT/dA  these  precore  surrounding  two p r e c u r s o r  to t h r e e v e l o c i t y  discontinuities  -101and two s h e l l s  surrounding  PKP,j , a c c o u n t i n g f o r l e s s than 1 4 5 ° , has corrsequences o f  the inner c o r e .  This additional  the e a r l i e s t p r e c u r s o r s  to P K P  been c o n s i s t e n t l y observed  in t h i s  i t s o b s e r v a t i o n are m a n i f e s t e d  triple-discontinuity,  double-shell  transition  at  r e g i o n f o r model  at  the e a r t h ' s  core.  This  s l o p e of 3.52  generates  branch has  and second d i s c o n t i n u i t i e s , the e a r l i e s t  1 3 4 . 2 ° w i t h an i n i t i a l  the AB t r a v e l -  its'  turning  a shell  p r e c u r s o r branch  v e l o c i t y jump  The' r e g i o n between the  a p p r o x i m a t e l y 417 km.  IJ.  This  s l o p e of 3.48 s e c / d e g and t e r m i n a t e s at  between the second and t h i r d d i s c o n t i n u i t i e s ,  has  its  at 152.6°  sec/deg. 0.92  the inner c o r e boundary a t a depth o f 5120 km.  300 km t h i c k ,  thick  branch begins  The t h i r d v e l o c i t y d i s c o n t i n u i t y of magnitude occurs at  point  sec/deg.  o f 0.24 km/sec o c c u r s at a depth o f 4810 km.  w i t h a s l o p e of 2.74  UBC1.  The r e g i o n between the c o r e - m a n t l e boundary  The second d i s c o n t i n u i t y w i t h an a s s o c i a t e d  generates  The  at  1 4 4 . 4 ° and i n i t i a l  first  study.  km/sec o c c u r s  a depth o f 2894 km. and t h i s d i s c o n t i n u i t y time branch f o r  distances  i n the form of a  The f i r s t v e l o c i t y d i s c o n t i n u i t y of 0.11 a depth of 4393 km.  at  DF  branch,  generates  the second p r e c u r s o r  t u r n i n g p o i n t at  a shell  The  region  approximately  branch GH.  1 3 1 . 9 ° w i t h an i n i t i a l  km/sec  This  s l o p e of 2.66  branch sec/deg  o at that  point  and extends  The r e g i o n c o r e of r a d i u s  to 158.6  w i t h a s l o p e of 2.16  beneath the t h i r d d i s c o n t i n u i t y ,  1251 km., generates  sec/deg.  i.e.  the  the main t r a v e l t i m e branch,  inner DF,  o for  the e a r t h ' s  and i t s '  initial  core. slope  The t u r n i n g p o i n t of is  1.96  sec/deg.  this  branch i s  113.2  The branch t e r m i n a t e s  at  1 8 0 . 0 ° with z e r o s l o p e . The consequences of a d o u b l e - s h e l l  t r a n s i t i o n region  would  -102be manifested than PKP  in other core phases as well.  Thus, phases other  (such as SKP,PKS,SKS) should be c l o s e l y examined for cor-  roborative evidence of a t r i p l e - d i s c o n t i n u i t y t r a n s i t i o n region. Such a region would have the same e f f e c t , producing  two additional  branches to the traveltime curve, on a l l core phases although the magnitude of the e f f e c t would vary.  Some problems could a r i s e in  such  and  a study.  a r r i v a l s and  The phases SKS,  PKS,  SKP never occur as f i r s t  the f i r s t two require horizontal component seismometers  for optimum recording. are considerably  Also, since the periods of these phases  longer than that for PKP,  i d e n t i f i c a t i o n and sub-  sequent measurement of the precursors could be more d i f f i c u l t . Amplitude measurements in addition to dT/dfl  and  travel-  time studies similar to those made in this project would be most useful  in defining c l o s e l y the turning points of the traveltime  branches.  This could have important consequences on the f i n a l  core  model derived. While i t is not expected that the proposed v e l o c i t y model, UBC1,  for the earth's core w i l l  be accepted  as the ultimate  major changes to this model are not presently envisaged. used in this study have enabled a more d e f i n i t i v e of the traveltime branches and  one,  The methods  identification  i t is upon such i 'identification that any  r e l i a b l e v e l o c i t y model depends.  -103BIBLIOGRAPHY Adams, R.D. and M.J. Randall, 1963. Observed T r i p l i c a t i o n of PKP, v o l . 200, NO. 4908, 744-745 Adams, R.D. and M.J. Randall, 1964. The f i n e structure of the E a r t h ' s c o r e , B . S . S . A . , v o l . 54, NO. 5, 1299-1313 Adams, R. D. and M. J . R a n d a l l , 1969. Distance corrections f o r deep focus earthquakes. Geophys. J . R. a s t r . S o c , v o l . 18 329-330. Arnold, E.P., 1968. Smoothing traveltime t a b l e s . B . S . S . A . , v o l . 58, 1345-1351 B f r t f l l , J.W. and F . E . Whiteway, 1965. The a p p l i c a t i o n of phased arrays to the analysis of seismic body waves. P h i l . Trans. Roy. Soc. Lond. A, v o l . 258, 421-493 Bolt, B . A . , 1959. Traveltimes of PKP up to 145 . Geophys. J . R . a s t r . S o c , v o l . 2, 190-198 Bolt, B . A . , 1962 Gutenberg's e a r l y PKP observations. Nature v o l . 196, 122-124 Bolt, B . A . , 1964. The v e l o c i t y of seismic waves near the E a r t h ' s centre. B . S . S . A . , v o l . 54, 191-208 Bolt, B . A . , 1968. Estimation of PKP traveltimes. 58, NO. 4, 1305-1324  B.S.S.A., vol.  C l e a r y , J . R . , C . Wright and K. J . Mulrhead, 1968. The e f f e c t s of l o c a l structure upon DT/LYimeasurements at the Warramunga seismic a r r a y . Geophys. J . , v o l . 16, 21-29 Buchbinder, G . R . R . , 1971. A v e l o c i t y structure of the E a r t h ' s core. B . S . S . A . , v o l . 6l_, NO.2, 429-456 Bullen, K . E . , 1938. E l l i p t i c i t y corrections to waves through the e a r t h ' s central core. M . N . R . A . S . , Geophys. S u p p l . , v o l . 4, 317-331 Bullen, K . E . , 1963. An introduction to the theory of Seismology, 3rd E d i t i o n , Camb. Univ. Press, Cambridge Carder, O . S . , D.W. Gordon and J . N . Jordan, 1966. Analysis of surface f o c i traveltimes. B . S . S . A . , v o l . 54, 815-840 Cleary, J . R , and A . L . Hales, 1966. An analysis of the traveltime of P waves to North American s t a t i o n s , in the distance range 32 to 100 . B . S . S . A . , v o l . 5 6 , 467-489 Corbishley, D.J., 1970. Multiple array measurements of the P wave traveltime d e r i v a t i v e . Geophys. J . , v o l . 19, 1-14 Engdahl, E . R . , J . Taggart, J . L . L o b d e l l , E . P . Arnold and G. E . Clawson, 1968. Computational methods. B. S . S . A . , v o l . £8 1339-1344.  -104BIBLIOGRAPHY (cont'd) E r g f n , K.,1967. S e i s m i c e v i d e n c e f o r a new l a y e r e d s t r u c t u r e o f the E a r t h ' s c o r e . J . G . R . , v o l . 72, NO.14, 3669-3687 Evernden,  Fisher,  J . F . , 1953. D i r e c t i o n of approach of R a y l e i g h waves and r e l a t e d problems - 1. B.S.S.A., v o l . 4 3 , 335-374  R.A.,  1953.  D i s p e r s i o n on a s p h e r e . P r o c . Roy.  Soc. Lond,  A, v o l . 217, 295-305 Gogna, H . L . ,  1968. T r a v e l t i m e s of PKP from P a c i f i c e a r t h q u a k e s . Geophs. J . , v o l . _[6, 489-514  Gutenberg, B., 19l4 Ueber Erdbebenwellen, VIIA Nach. Ges. Wiss Goettingen Math. Physik, KI., 166-218. Gutenberg, G . , 1957- The Boundary of the E a r t h ' s Amer. Geophys. U n i o n . , v o l . 38., 750  Inner C o r e . T r a n s .  Gutenberg,  B., 1958. C a u s t i c s Produced by Waves Through the Core. Geophys. J . , vol^3_, 238  Gutenberg,  B. and C F . R i c h t e r , 1937. P' and the E a r t h ' s M.N.R.A.S., Geophys. S u p p l . , v o l . 4 . , 363  Earth's  Core.  Hannon, W.J. and R.L. Kovach, 1966. V e l o c i t y f i l t e r i n g of c o r e phases. B.S.S.A., v o l . 5 4 , 441-454  seismic  Jeffreys,  H., 1934. On smoothing and d i f f e r e n t i a t i o n o f P r o c . Camb. P h i l . S o c , v o l . 3 0 , 134-138  tables.  Jeffreys,  H., 1937. On the smoothing o f observed d a t a . P r o c . Camb. P h i l . S o c . , v o l . 3 3 , 444-450  J e f f r e y s . H . , 1939. The times o f c o r e waves (second p a p e r ) . Geophys. S u p p l . , v o l . 4 , 594-615  M.N.R.A.S.,  Jeffreys, H., 196l Theory of probability. Third Edition, O.U. P. 2l4-2l6. r Jeffreys,  Julian,  H.,and K.E. B u l l e n , 1940, 1967Seismological Tables B r i t i s h A s s o c i a t i o n f o r the Advancement o f S c i e n c e , Gray M i l n e T r u s t , London  B.R.,  D.Davies and R.M.  v o l . 235,  Sheppard,  1972.  PKJKP. Nature,  317-318  Keen, C . G . , J . Montgomery, W.M.M. Mowat, J . E . M u l l a r d and D.C. P i a t t , 1965. B r i t i s h seismometer a r r a y r e c o r d i n g systems, The Radio and E l e c t r o n i c E n g i n e e r , V o l . 3 £ , 297-306 Kelly,  E . J . , 1964. L i m i t e d network p r o c e s s i n g of s e i s m i c Massachusetts I n s t . Techno1. Group Rept. 44  Knopoff,  signals.  L. and J . G . F . Macdonald, 1958. Magnetic f i e l d and c e n t r a l c o r e of the e a r t h . Geophys. J . , v o l . 1, 216  -105BlBLIOGRAPHY (cont'd)  .  Kuhn, W. and S. Vielhauer, 1 9 5 3 - Boziehungen zwischen der Ausbreitungvon Longitudinal und Transversalwellen i n relaxierden medien 2. Phyzik. Chem., v o l . 202,214. Lehmann,  I.,'1936. P ' , P u b l . Bur. C e n t r a l Seism. T r a v . S c i . , v o l . _I4, 87-115  Muirhead, K . J . , 1968.  Intern.,  Ser A,  '  E l i m i n a t i n g f a l s e alarms when d e t e c t i n g  s e i s m i c events a u t o m a t i c a l l y .  Nature, v o l .  217, 533-534  Nguyen-Hai, 1961. P r o p a g a t i o n des ondes dans l e noyeau t e r r e s t r e d ' a p r e s l e s seismes profonds de F i d j i . Ann. Geophys.,  v o l . J 2 , 60-66  1966. C o r r e c t i o n s t o apparent azimuths and t r a v e l t i m e gradients f o r a dipping Mohorovicic d i s c o n t i n u i t y , B . S . S . A . , v o l . 56, 491-509  Niazi,  M.,  Niazi,  M., and D.L. Anderson, 1965. Upper mantle s t r u c t u r e o f western North America from apparent v e l o c i t i e s o f P waves. J . G . R . , v o l . 70, 4633-4640  Otsuka, M., 1966. Azimuth and slowness anomalies o f s e i s m i c waves measured on the c e n t r a l C a l i f o r n i a S e i s m i c A r r a y . , B . S . S . A . , v o l . 56, 223-239 and 655-675 Shima, E . , K. McCamy and R.P. Meyer, 1964. A F o u r i e r method o f apparent v e l o c i t y measurement.  v o l . 54, 1843-1854  D.M., 1967. The earthquake P phases which p e n e t r a t e the e a r t h ' s c o r e . B.S.S.A., v o l . 57, 875-890  Shurbet,  Stacey,  F . D . , 1969. York  Truscott,  P h y s i c s o f the e a r t h ' s  J . R . , 1964. The Eskdalemuir  v o l . 9, 59-67  Underwood, R.,  i n t e r i o r . W i l e y , New  sesmic a r r a y . Geophys. J . ,  1967. The s e i s m i c network and i t s  Ph.D. T h e s i s , A u s t r a l i a n N a t i o n a l Watson,  applications.  University,  Canberra  G . S . , 1956. A n a l y s i s of s i s p e r s i o n on a s p h e r e . M.N.R.A.S., Geophys. S u p p l . , v o l . 7, 153-159  Whiteway,  Wright,  transform B.S.S.A.,  F . E . , 1965. The r e c o r d i n g and a n a l y s i s o f s e i s m i c body waves u s i n g l i n e a r c r o s s a r r a y s , The Radio and E l e c t r o n i c E n g i n e e r , v o l . 29, 33-46  C , 1968. E v i d e n c e f o r a low v e l o c i t y l a y e r f o r P waves a t a depth c l o s e t o 800 km. E a r t h P l a n e t . S c f . L e t t e r s ,  v o l . 5, 35-40  -106BlBLIOGRAPHY (cont'd)  Wright, C , 1970. P wave i n v e s t i g a t i o n s o f the E a r t h ' s S t r u c t u r e u s i n g the Warramunga S e i s m i c A r r a y . Ph.D. T h e s i s , A u s t r a l i a n N a t i o n a l U n i v e r s i t y , Canberra  -107APPENDIX I A Short  Review o f If  then seismic  Seismic  Ray Theory  t h e v e l o c i t i e s of e l a s t i c waves were u n i f o r m w i t h i n the e a r t h , rays would be s t r a i g h t  lines following/Chords a s ' i n  and t h e t r a v e l t i m e from a s u r f a c e f o c u s tance  to an o b s e r v a t o r y  Figure  A.la  a t an a n g u l a r  dis-  from the f o c u s would be  A  T = 2R/V However, o b s e r v a t i o n s  of  S t r o n g l y with distance more curved than t h i s thus greater  than at  Sin  traveltimes  than  is  A/2  (1)  indicate that  i n d i c a t e d by  (1),  e q u a t i o n would s u g g e s t . the s u r f a c e and s e i s m i c  they  increase  and the T-A  The  less  curves  v e l o c i t y a t depth  r a y s a r e r e f r a c t e d as  in  are is Figure  A.lb. Bullen  (1963) has d e v e l o p e d the t h e o r y a p p l i c a b l e  w h i c h the v e l o c i t y ray  i n a multi-1ayered earth  f r o m the s u r f a c e . symmetrical are  increases with depth.  about  ignored.  A -1 b.  It its  that  centre in a l l  Applying  Snell's  the E a r t h  its  spherical  and  law to each of  the b o u n d a r i e s  A,  depth  completely effects  B i n Figure  From e q u a t i o n  = r, Smf,  i  more' generally, is  the parameter  the r a y .  family of  rays,  =  r Sin z  i  ,,a  z  2  z  of  the r a y , p differs  and  s.r Sinf fa z  2  = Constant = p  between the ray and the r a d i u s  The v a l u e of and a t  r,S.r.f,/V2 = r Si'n i / v z r Sm i / v  the a c u t e a n g l e  i s called  ( )  {Jjx)  .Srti./v,or  Sm v-i/Vi = Sinfi/v^  z  triangles,  ^  along  is  of  then  But from the two  p  v increases with  p r o p e r t i e s , and d i f f r a c t i o n  Sin t, /v, = Sinf,/v  where  in  F i g u r e A . l b shows the geometry  i n which the v e l o c i t y  is assumed  to an e a r t h  (2) at any  is c o n s t ant f o r a l l  point.  points  from member to m ember of a  the deepest p o i n t of p e n e t r a t i o n of any r a y ,  given where  -108-  A - I f e j j a t h o f a seismic r a y through a h y p o t h e t i c a l uniform e a r t h .  A-I(b).Seisa:ic r a y i n t h r e e - l a y e r earth.  Figure A - l Seismic Ray Theory.  -109Sln i " 1 , is equal to the value of r / v  at that point  Consider two rays of a family of rays with parameters p, and p + 4p traveltimes  T and T + d f subtending angles of A  Figure A . l c , r e s p e c t i v e l y .  Q  « NQo/PoQo  0  as shown in Figure A. Id" and let r  = dT/dA  (  3  0  Then, from  = (dr)  2  relation +  2  (rdo)  2  y i e l d s the r e l a t i o n s h i p do - +  pr  •  -I  , 2 (n  where for convenience, the parameter Integrating (5)  we obtain the expression for  2 *i p ) dr  -  n. = r/v  between P  Q  , V-  (5|  and the deepest point of the ray,  A,  (;  2  and the corresponding expression for T  6  where  ir -- jVr-'tf-p)" ao 1  where H  it  (7;  4  is convenient to introduce the important  ^ : d l n v / d Inr  = r/  v  dv/dr,  and  | - 2 / i - £  functions ^ Ond ^  - 2d\r-  r/d.  la r\ ,  can be shown that the downward curvature of a ray with i t s iowest  at some level r  in the earth  be less than the curvature This condition implies that Introducing  (Bullen,  1963)  is giv«in by  l/v dv/dif\  ' ^ r of the level <J <• I > * § >°)  the functions  and then d i f f e r e n t i a t i n g  ;  (= p at the deepest point cf  i A = P jr"V-p)~* dr  At t h i s point,  (2)  (4)  / v <*%U=P  (ds)  )  is p, with polar coordinates  the arc length P P be s .  and s u b s t i t u t i n g for ds from the polar  the r a y ) .  0  (2),  Let P be any point of a ray whose parameter  equation (4)  Then  = v .i.dT/r .^dA  p = rS.ru/v  (r,e)  d A at 0 as shown in  +  Let P N be the normal to the ray through Qo-  Smio and hence from equation  and A  7  olf  and £  and this curvature must  surface at r, <r dr\/dr  point  if  the ray is to e x / s t  >O . m  into equations (6)  and  the r e s u l t i n g equations lead to the expressions  (7)j  -110-  where the s u b s c r I p t s 0 r e f e r the e a r t h ' s  to the v a l u e s of the parameters  at  surface.  These e q u a t i o n s are f u r t h e r s i m p l i f i e d by the d i m e n s i o n l e s s  variables  u/to -  M = n/n.o  , A = P/>\  - -f 0-x ;-4-  t {  a  o  ...  0  introducing  •> T = t / t t i  to become  - \  -x-f-y  0  ( M  x  z  r ± d ^  •  IP  *  {W-A^j  dT/dA « - f A ^ i - A * ) - > ; 0  (ro) (H)  :  te  Consider  the r e l a t i o n s h i p between A  types of v a r i a t i o n o f v w i t h r. v with r e n t a i l s  and T c o r r e s p o n d i n g to d i f f e r e n t  Normal or  'ordinary'  t h a t v i n c r e a s e s s l o w l y as r  r a t e of change of v i s changing  slowly.  This  behaviour of  d e c r e a s e s , and that implies  that  ^  the is  n e g a t i v e and moderate i n v a l u e , d^/dr and d|/dr a r e s m a l l , a l s o dr^/dr  is  p o s i t i v e and  p decreases 90° ( i . e . when  A  is  Q  negative.  s t e a d i l y between n A  '  ^  s  =T = o)  zero,  to 0 ° .  and z e r o as  Q  Since  i t follows  From the r e l a t i o n  ^  i  £in i  p =  goes from  0  is n e g a t i v e and P = f\  Q  that f o r small  0  v a l u e s of  A  0  , X in  e q u a t i o n (10) i s l a r g e and p o s i t i v e . Owing to the s m a l l n e s s of d|/dr and the range of i n t e g r a t i o n f o r small A , Y i s a l s o s m a l l , so t h a t f o r small 2  A  ,  is  negative  is  l a r g e and n e g a t i v e ,  , then, e q u a t i o n s  t e n d i n g to o f o r small (10)  and  rays  &  can be w r i t t e n  dLT(d\  =-|  0  t  *  |  T  - f o o-A ) «f sinCA/f ;  terminating at  For '  o  AO-A )"^  -  1  0  Cos  0  f o r normal behaviour o f Q  l y i n g w h o l l y on one s i d e of the l e v e l s r , and  03)  05)  4  the s u r f a c e o f the e a r t h r = r .  and T a r e from (14)  Ol)  04) 0  taken as  -  "*A z  lowest p o i n t and t e r m i n a t i n g a t  the v a l u e s o f  .  = | (i-A*)-!  that segment o f a r a y , ray's  (11)  A  dA/dA  The above e q u a t i o n s h o l d f o r small A and f o r  and c o n s e -  2  q u e n t l y d T/dA small  dA/dh  A  (15).  and r  2  v,  For the say,  -111-  and the c o n t r i b u t i o n s  to  A  , T and  whole r a y which l i e between l e v e l s *  j =|  o  distance level  beneath r^.  r j , so t h a t  velocity just constant  Rays f o r which  ,UQ,  (15) a p p l y f o r initially negative  this  negative as  Suppose  in F i g u r e A.2a.  A  /Jib i s  °  region.  increases  less  these r a y s ,  from e q u a t i o n s  than  (12),  The b e h a v i o r  (18)  A =|  Jx b  is  to t oO  6- A  -  For r a y s whose  0  =  some p o i n t  lowest  being  points  -  (Yes-'A  Let ^  .  is  6. U<\  given  have  i n N,  the  the  £~ ^ c t •  J  also  (12)  d&/dA  seen t h a t  less  and  rj  is  less  the c u r v e  lie entirely  the boundary a t  and  in L  for  (Zl)  CCS~'(*/M«))  dt\ J d\  by  the  than vb,  in s e c t i o n A B of  (  ?  are  less  at  some  we have  (20),  thereby d e c r e a s i n g through f i n i t e  becoming z e r o at  by  | c)  r  i n L and e q u a t i o n s  it  shown  T = $.  A  Mo.  is  p r e v i o u s l y d e s c r i b e d , becomes  internally reflected off  at  P C  discontinuously  so t h a t  1,  From e q u a t i o n  and f i n i t e as  of  (23)  V ~ Vo  r  lie entirely  £ |  but c o n s i s t  From e q u a t i o n  t^wet\ respectively  immediately above r j , v a ,  Rays f o r which rays  § ©  the v e l o c i t y  Thus  decreases.  A  =  i n l a y e r N where k ^  0  of a  r j , to be u n d e r l a i n by a l a y e r N e x t e n d i n g f o r  beneath r j . k(j  segments  L i n which the b e h a v i o r o f v i~ normal £  the v e l o c i t y  value  f\ xx.ce.  and  0  the p a i r o f  A  ?  e x t e n d i n g down to a l e v e l  r ,  for  d\  ( C o s " A - Cos"' ( //Ut))  Consider a r e g i o n r e p r e s e n t e d by  d&/  positive  o  \  A b  t  jumps di s c o n t i nuous l y values  ( I — A "^ ~ ^ A A a 2  a  as  }\  decreases,  "~ A ^ )  and the a p p r o p r i a t e  )  A-2fl>).  Figure A-2. Behaviour of dA/dA- A, A -A, and T-A, curves for dircontinous increase i n velocity at a boundary,.  -113equat i ons are •'  ^  =f„ ((  T  * ^  0"«-  2  ) 1  As A d e c r e a s e s to the v a l u e  yU|> IM l a y e r N, d A / d A  di s c o n t i n u o u s l y to -00, and as  \  increases  that  A  curve. a  ri'- T a r e  d A /d\  The  importance  Equations  and T - A  i s the f a c t  (26)  jumps dA/dA  the v a l u e ,U.b  (15),  (21),  tor rays  A  d A /dX  , and t h e r e f o r e  i s g i v e n by p = T\Cc&e.  i n L and N.  l a y e r s L and N r e s p e c t i v e l y .  these p o i n t s are  d.A / d A  not  b e i n g near  and  through therefore  iX fc> and AX<x  =  BC was v i r t u a l l y s t r a i g h t  f o r some  distance  C, and branch CD l i k e w i s e  staight  near  It can be seen t h a t the above d i s c u s s i o n a p p l i e s to t h e . v e l o c i t y  discontinuities the  t r a v e l t i m e curve mentioned  the m a n t l e - c o r e  case,  l/fc)  Va.  in section  boundary, and  the s i t u a t i o n  (12) -  1.2.  (15) s t i l l  i s somewhat d i f f e r e n t .  AX b > M a. jX  the bottom of N i s r e a c h e d .  equations  on  the case of a d i s c o n t i n u o u s " d e c r e a s e i n v e l o c i t y ,  extends downward f a r enough f o r before  being  :n the t r a n s i t i o n r e g i o n , and imposes the c o n s t r a i n t s  For at  from C, any marked c u r v a t u r e  runs  virtually D.  some d i s t a n c e  with  (where  to rays r e f r a c t e d  On the branch BC,  r e s p e c t i v e l y ) and c o n s e q u e n t l y  Of  c M j d(L o  ) at  c>  i s z e r o at both B and C (where A  from B, any marked c u r v a t u r e  corres-  to r a y s r e f l e c t e d o f f the d i s c o n t i n u i t y  ( r e c e d i n g branch) w h i l s t AB and CD c o r r e s p o n d  for  (25) show  o f the cusps B and C are a s s o c i a t e d  Branch BC c o r r e s p o n d s  2  (24),  i n F i g u r e s A . 2 b and A , 2 c r e s p e c t i v e l y  that neither  e., the a n g l e of emergence  d*T" j d A "  (22) and  °f  i s i l l u s t r a t e d i n F i g u r e A . 2 a , and the  curves  large amplitudes since  to z e r o .  (14),  c o n t i n u o u s f u n c t i o n s of  (ordA/dp)curve  ponding p - A  equal  d e c r e a s e s beneath  from  ( 25 )  through f i n i t e n e g a t i v e v a l u e s r e s u l t i n g i n the s e c t i o n C 0  dk jd\  the  ^  k  l  I- A  apply.  to r e t u r n  >  a  n  d  w  e  assume  as  In t h i s  that  layer N  through the v a l u e XiJa,  For r a y s w i t h lowest p o i n t s  i n L,  For rays which p e n e t r a t e N , (24) -  (26)  -114-  -115-  apply, but since  \ 4. Ma.  f o r these r a y s , t h e r e a r e no r a y s w i t h  p o i n t s i n t h e upper p a r t o f N f o r w h i c h A — Ako,  there a r e d i s c o n t i n u i t i e s  Ma.  A  ^  b"A and ST  lowest  • Consequently a t in  and T g i v e n  'A  by s u b t r a c t i n g (14) f r o m (24) and ( 1 5 ) f r o m ( 2 5 ) r e s p e c t i v e l y :  SA  For r a y s p e n e t r a t i n g N, A = Aja  and d e c r e a s e  r a p i d l y as A d e c r e a s e s  as  = k j Cos  dA/dh A  Ou/Ab)  e  from equation  decreases  from/Ua.  (26) has t h e v a l u e +00 a t Thus £\  ( | - A  2  s a t i s f y i n g the equation.  ) " ^ k ( . u b  2  z e r o a t t h i s p o i n t , D corresponds t o a c a u s t i c . i s z e r o , and c o n s e q u e n t l y  t h e branch  A-A  A  Since  T ' dA  /d\  i s v i r t u a l l y s t r a i g h t some d i s t a n c e The c o r r e s p o n d i n g  c u r v e s a r e shown i n F i g u r e s A.3a and A.3b.  The above d i s c u s s i o n s c o v e r  the v a r i a t i o n  invelocity  obtained  i n t h e e a r t h f r o m the base o f t h e m a n t l e t o t h e c e n t e r o f the e a r t h ' s c o r e . It i s i n t e n d e d t h a t they p r o v i d e curves generated straints  i  As b e f o r e , a t p o i n t C  f r o m C^any marked c u r v a t u r e b e i n g near the c a u s t i c D.  nd  ^  - A > ( « a -  T h i s minimum i s t h e cusp D shown i n F i g u r e A.3c and  a  and T d e c r e a s e  f r o m / j a , and p r o v i d e d N i s deep enough r e a c h  minima g i v e n by t h e v a l u e o f A  dA/dA~A  (27)  insight  i n t o the n a t u r e o f t h e t r a v e l t i m e  by v e l o c i t y d i s c o n t i n u i t i e s  i n t h i s r e g i o n and t h e con-  imposed upon these c u r v e s by r a y t h e o r y .  s  APPENDIX 2 LIST OF F.ARJKQ.UAKES USED' IN STUDY Event N o .  1 2 3 4 5 6 7 8 9 . 10 11 12 13 14 15 16 17 18 19  20  .  21 22 23 24  25 26 27 28 29 30 31 32  33 3^  35 36 37  •  Date  Lat i tude . Long!tude (Degrees) (Degrees)  Or i gi n Time Hr. Hi n. Sec  J a n . 1 4 / 6 9 23 8 Oct.24/69 1 0ct.l8G7 June 1 2 / 6 9 15 13 Nov.23/67 16 • Dec.19/68 14 Sept.16/69 Oct.8/69 14 O c t . 2 9 / 6 9 22 Sept.6/68 14 Dec.25/68 12 Nov:21/67 17 Nov.30/67 7 A p r ! 1 18/69 17 S e p t . 2 6 / 6 7 11 Jan.21/70 17 Nov.I/69 11 D e c . 1 3 / 6 9 21 8 Oct.27/69 Sept.20/69 15 6 Apr.26/69 5 Nov.27/67 June 23/69 7 1 Oct.7/67 Nov.15/67 21 Nov.13/69 7 Jan.6/68 23 A p r . 17/&9 17 Sept.2/69 3 Aug.20/67 15 8 Sept.8/67 Sept.21/69 2 16 Dec.29/68 !0 Sept.13/69 May 1 6 / 6 8 22 10 Dec.25/67 22 June 6/69  2 29  11 13 42 30 30 30 1  0 17  2 23 43 1 1 51 8 33 10 26  2 13 8 • 14 31 51 27 37 ' 47 3 59  0 29 52  45 41 25  7.9  12. 1 . i,v8  31 1 1 6 0 0 0 0  0  c  49 12 27 4 51 29 21  C  51 4 0 1 20 £ 25 C 51 5 53 7 23 1 38.5 20 S 21 c-> 58.3 41 c.  22  6, 7 1 5 c 1 <•  L  9 1 36 2 59 I 54 • 31 1 58 C 19 31 i 37 •  13.175N • 33.292N 79.8 N 34.405N 80.2 N 37.232N 37.3I4N 35.257N 37.143N 37.135N 35.-130N 72.70 N 4).50 N 24.283.M 33.60 S 7.017N 23.148N 32.708S 44.923N 1.785N 30.645S 30.8 S 18.373N 29.6 S 28.7 S 27.8 S 27.8 S 28.26 S 27.745S 25.2 S 23.4 S 23.552S 23.975S 22.883S  22.883S  21.50 S 22..511S  29.203W 119.'93W 2.4 E 25.061 E 1.0 W 116.477W 116.460W 116.450W 116.064W 116.047W 24.330E 8.5 E 20.5 E 107.738W • 70.5 w 104.293W 107.934W 69.972W 17-232E 101.031 w 71 .542W 71.00 w • 104.546W 71.1 V/ 71.2 W 71.65 W 71.1 w 67./86V/ 66.491W 69.0 W 70.7 W 68.080W 66.673W 63.367W 68.6 W 70.4 W • 63.418W  Focal  Di s t a n c e (Degrees) 113.47 114.11 116.50 116.60 116.70 1 16.80 116.82 116.83 117.12 117.13 117.27 119.20 120.6 120.92 121.70 121.80 122.36 122.71 122.91 123.0 123.87 124.0 124.59 125.0 125.70 126.29 126.50 127.11 128.47 129.70 130.50 131.55 131.77 132.00 132.00 132.20 132.27  .  Depth  (KIT  •)  33 10 33 25 10  ON  33 33 62 36 42 15 33 33 82 174 109 33 120 205 106 104  ON  0 0 OD  0 0 0 0 0 0 0 0 0 0.0G 40 0 51 0 29 0 • 33 0 ' 84 0 33 ON 33 ON 105 OD • 33 ON ON 0 0 0 0 ON 0 0 00 0 0 OD OG OD 0 53 0  125 OD  Magni t u d e  5 5 5 1 5 7 5 8 5 8 6 3 6 .2 5 5 5 7 5 6 5 .0 5 5 6 .0 5 .0 5 .8 6 .2 5 .6 5 .6 5 .3 5 .5 5 .9 5 .4 5 .3 5 .3 6 .2 5 .8 5 .8 C  .0  5 .5 5 .6 5 .5 5 .5 5 .5 5 .4 5 .0 5 .8 5 .0  A z i muth (Degree  .  301 . 6 61 . 6 351 . 5 259.2 352.3 57.6 57.5 57-6 57.3 57.8 299.9 343.9 307.0 73.1 155.6 94.1 7^.7 . 155.6 311.1 101.2 • 153.0 153.6 8 ! .0 152.8 152.2 151.2 151.7 154.2 156.2 152.2 149.2 152.1 153.9 151.4 151.1 148.2 151.1  APPENDIX 2 :nt  38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74  No,  Date  J u l y 25/69 Oct 20/69 Sept 17/67 Oct.22/69 J u n e 15/68 J u l y 19/69 June 2 2 / 6 9 Aug 23/68 Sept.3/67 Nov 7 / 6 8 O c t 31/68 jan 8 / 6 8 Sept.25/68 Sept.28/68 Sept.15/69 Oc t . 5 / 6 7 Sept.27/6? Aug 23/68 June 1 1 / 6 9 April 21/69 •J-y.i 1 2 / 6 9 July24/69 Oct.20/69 Oct.6/69 Oct 6/69 Nov.16/69 Sept.6/68 Nov.18/67 Dec.27/66 J u l y 18/69 Junel1/68 June6/69 Nov.9/68 June 20/68 June21/68 Junel9/68 Oct.4/67  La t i :ude (Dt-q " )  O r i g ) n T i Pfe Hr . Min Se 6 15 7 10 5 4 14 22 21 . 10  6 20 56 21 11 54 30 36 ~i V  9 18 10  13 15 44 38  13 7 9 6  53 14 41 2  23 21 2  14  14 2 18 0 6 10  15 19 12 59 0 37 36 30  7 12 21  49 IS 22  23 5 1.6  17 52 16 1  17 15 0 8 6  51 26 13 2  42 4 36 5 22 .7 52 . 1 17 . 2 54 1 10 7 51 30 39 46 24 38 35 25 3! 39 52 32 7 53 21 28 59 45 1 42  L i ST OF EARTHQUAKES USED  25 17 17 18 14 17 16 21 10 16 16 18  3 8 8 9 5 4 3 8 4 5 7 6 1 0 0 7 6 2  7 0 55 4 1.4 8 10 6 33 . 5 1 7 41 1 56 5 7 .8 35 .0 16 4  •  15 13 18 14 /  21 8 14 14 11 11 11 11 13 5 13 13 18 13 12 37 5 5 5 10  552 S 300N 2 N 086 S 4 N 254S 929N 986 S 6 S 388s 289S 6 S 571N 160S 552S 5 S 3 S 819S 590S. 098N 142S 85 IS 859S 737S 782 S 351N 767S 37N 20N 241 S SON 4 6 7N 96 I N 6 S 7S 6S 7N  Longi tude (D< g r e o s )  low  63.3 95.188W 94. 1 W 71 . 5 4 2 W 92 9 W 72 519W 93 608W 63.547W 79.8 W 73 .460V/ 73 .298w 69 9 W 92 '638W 76 382W 6 9 . 0 2 1 V/  w  75 4 81.3 V 63.530W 79 688W 91 . 0 1 5 W 72 74 7W 75.130W 75 164W 75.100W 74.999W 89.650N 80.261W 89.07W 88.81W 53 314W 88.81W 88.000W 88.464W 77.3 W 77 3W 77 2W 8 6 . OW  IN STUDY  D i s tanc.e (Degrees) 131 133 134 134 134 134 134 134 134 134 134 134 135 135 135 135 135 134 136 136 136 137 137 137 137 137 137 138 138 138 138 138 138 139 139 139 140  Focal (km.)  Depth  63 11 10 46 60 53 52 70 72 70  579.OD 87.0 45.0 . 23 0 25.0  89 90 15 31 40 50 70 96 10  6 7 OD 116.0 137.8 70.OG 177.OD  33 89 10 07 20 23 43 44 00 20 28 40 75 37 80 80 .90 10  54.OD 151.0 38.0 50.0  1C0.0 37,0 541 OD 81 . 0 82.0 113.0 1 .0 12.0 3.0 4.0 79.0 6 6 . CD 78.0 66.0 19.OD 199.0 48.0 19.0 33.0 22.0 28.0 33.0  Ma oil 1 -tude  A>-' <-'.!t (Degre  c 5 4 2 4  158 5 84 7 85. 1 144 ! 89.2 142.7 85 6 129 1 !40 9 156 3 141 0 146 6 8/ 7 134 9 147 6  5 5 5 5 '5 5 6  LK  9 1 5 5 0 5 8 5 7 5 *T 5 7 6 0 5 2 5 6 5 I 5 2 5 0 5 5 5 5 5 5 4  5 2 9 1 1  3 9 5 3 5 1 5 5 5 6 5 3 5 0 5 _> 5 8 5 6 .6 . 4 5 3  13 7 . 2 124. 1 156 2 12 7 . 0 90 3 139. 7 135.0 134 9 134.9 134.0 91.8 123-3 92 . 0 92 . 4 154.3 91 . 4 93.7 54.8 125.9 126.1 126.0 97.2  APPENDIX E v e n t No. D a t e 75 76 77 78 79  80 81 82 83 84  85  86  Oct.3/67 Sept.20/69 Aug.27/67 Oct.15/67  June24/69  Junel5/68  O c t . . II/67  May 1 6 / 6 8 Jan.1/70 Nov.8/67 M a y 31/69  88 89  Dec.16/68 May28/69 Apr i125/69 M a y 5/69 Sept.24/69  92 93  Feb.28/69  87 90 91  94 95 96 97 98 99 100 101 102 103  104  105  106  107  108 109 110 111  Dec.24/69  Sept.24/69  Sept.6/69 Nov.14/69 Aug.31/68 Sept,22/67 Nov.21/67 Sept.1/68 Sept.1/68 Oct.20/69 Oct.20/69 Oct.16/68 Sept.20/68 Oct.22/69  Sept.22/69 Sept,3/68 May 15/69 Dec.24/67  Dec, 25/69 Dec.25/69  Origin Time Hr . Min. Sec. 16 3.2 18 5. 8 57.6 13 8 55,9 0 50.3 8 0 35 5.5 7 8 48.1 20 10.2 28 8 25 9.2 1 43 46.7 10 3 53.3 11 7 17.1 7 3 24,1 30 13 8.9 34 17.7 3 5  34 58  5  4 40  3  2 4 14  6  21 8 21  4 8  13  13  1  6  12 13  15  20 20 22 21  20 30 52 '47  8  50 48  23.5 56.5  44.5  32.5  52.5  39.5  5.3  33.5  4.3  24.3 52.2 57.2  19 1 11 37.0 !1 33.5  55 0  3.2.7  52  22.2 52.2  47 37 43  3  31 32  3.5  0.2  LIST O FEARTHQUAKES USED I N STUDY  Lat i tude (Degrees.)  Longi tude (Degrees)  10.9N  85 .9W  12,3N  86.2W  Di stance (Degrees) 140 3 0 140,46 140.40  85.717W  140.66  58.297N 11.9N 1 1 . 6 6 IN 5.6N 10.3S  3.7S  8.598N  16.8N  1.753S 7-123N  2,036s  7.450N 36.023N  52 5 5 9 N  35.953N . 36.008N  52.499W 36.341N  4.934N  4.505N 0.7S 43.2N  0.995S 0..889S  10..796N 10.796K  19.152N 10..735N  •10.92 I N 4.977N  33-4  20.631N 16.573N  2.3 27.-3  16.083N  10.9  2  17-4N  15.772N  32.189W 86.  OW  140.50  140.80  82.6w 71 . 2 W 76.6W  141,40  141.60 141.69 141.70  83.525W 85.9W  141 . 8 4 142.31  77,744w  82.243W 76.927W 82.075W 10.390W 31.841W 10.396W 10.573W 31.849W  142.30  142.59  145.21 145.72  145.24  145.36 145.78 146.05  1 1 .887W  76.846W 76.359W 20,1W  27.8W 24.512W 24.519W 72.480W 72.4S0W  69.838W  62.663W 62.552W  32.625W 62.238W 61 ,341W 61.1 w 59.771W 59,650W  .  146.1 7 146.39  147,60  148.30  150,63 150.70  152.6!  152.68  157.19 161 . 2 2  161.41  160.45  164.42 164.77 165.10 166.03  166.04  Foca! Depth (km.) 21 . 0 33 .ON 183 .0 1 6 2 .0 100 . O G 16 ,0 585 . 0 113 . 0 4 8 ,0 28 . 0 1 7 2 ., 0 16 . 0 177 . O D 25 .OG 2 9 .0 33. . 0 3 3 ..ON 2 2 .0 3 3 ..ON 33. . O N 53. ,C 97. c 33. . 0 33. .0 33, . O N 33. . O N 40. OG 55. 0 36. ,0 1 0 7 ..OD 79. 0 • 3 3 ,.0 33. 0 50. , 0 2 4 , ,0 8.C) 7.0  Magni tude Azi muth (Degrees) 5 , r8 5 .0 5 2 6 2 5 .. 3 6 .0 5 ,, 0 5 5. 5 '4 5,1 5 .3 5 .5 5 ,. 4 5. 5 5 . ,2 5 .1 l±  ~J I  5 .0 5, 4 .. 7 4 . .6 5 ,. 3 5 , .0 1 . 1  5 .2  5 .0 5 .7 5. 1 ..2 6. 2 5 . ,4 5 .. 7 5 . .5 5. 7 6 ., 4 6 . .0 6. 4  97.0 348.9  94.7  94.5  95.9  106.6  137.8  124.3 101  .6  87.9 120,7 104.6  122.0 104,2 304.9 345.0 304,3  304.9  345.0 307.1 112.0 113.1 233  .6  237.0 227.3 22 7 . 4 105.4 iC5,4 87.9  116.7 i 16.4 222. I 84.6  99.5 97.3 103.7  105.1  APPENDIX LIST Event  112 113  114 115  N o ,.  Date  Dec.26/69 Dec.26/69 Jan.8/68 Sept.24/69  Or i g i n T i me Hr. Mi n . Sec.  8 20 20 18  46 3 22  3  15.2 28.8 15.6 19.0  2  OF EARTHQUAKES USED Lat itude (Degrees)  15.839N  15.791N 8.2 N 15.237N  Long! tude (Degrees)  59.57W 59.555W 38.2 W  45.776W  IN  STUDY Distance (Degrees)  166.13 166.13 166.30 175.31  Focal Depth (km.)  22.0  33.ON  33.0  33.ON  M a g n ii t u d e  5 ,. 2 5 .. 4 5. . 4 5 ,. 8  Azimuth (Degrees)  105.1  104.9 212.7 178.5  -120APPENDIX I 11 ELLIPTICITY AND FOCAL DEPTH CORRECTIONS TO MEASURED TRAVELTIMES  Bullen (1939, 1938, 1963) has shown that the el 1 i p t i c i ty correction 6T to the travel time •foryeocQntric l a t i tudes is given approximately by the formula -  .....0)  ST - f (A)Cho + Hi) where h  e  and h, are the differences between the actual and mean  r a d i i of the earth at the epicentre and observing stations r e s p e c t i v e l y , and  dlso  are l i s t e d by Bullen, (1937).  given by Bullen (1938).  The value of h ,  ranges from 0.10 to 0-095 for PKP  DF  -fC ) A  i s  for WRA is +5.0 km while  and 0.100 to 0.074 for  fY&)  PKP AB  Time corrections for focal depth were determined for earthquakes with focaj depth less than 2 0 0 km. from the tables given by Bolt (1968), and the corresponding distance corrections r e s u l t i n g in a surface focus for the event were estimated from near-earthquake P tables.  For focal depths greater  by Adams and Randall  than 200 km, the equation developed  (1969) for deep-focus earthquakes, and derived  e s p e c i a l l y for use with PKP phases was used: £ -  (0-05  +002.5H  +..o.-;oo,2 8 h ) p a  ••--•CO  where S is the correction in degrees, h is the focal depth in units of hundredths of ^the earth's radius, and P is the ray parameter d T / d ^ 6 is determined from the above formula and the corresponding time correction found from tables of near-earthquake traveltimes using the appropriate values of 5 and h.  -121APPENDIX  III  ELL IPTI CITY AND FOCAL DEPTH CORRECTIONS TO MEASURED TRAVELTIMES (cont.) Thus a l l  the events were adjusted to have surface f o c i .  A check was made on the distance and time corrections obtained for all  the earthquakes in the above manner by using a computer program  designed to calculate these corrections for ray paths of specified parameter t r a v e l l i n g through the earth. program was to calculate the quantities  The method used in this A12  O-^d T  l 2  .  a  s  described in Appendix I for radii down to the focal depth given, using an assumed v e l o c i t y model for the crust and mantle.  It was  found that good agreement was obtained at most focal depths except for those in the v i c i n i t y of about 500 kilometres.  In these cases  the corrections given by the program were smaller by a maximum of about 0.12 deg. than the value obtained using the Adams and Randall equation.  The latter values were assumed to be the correct ones.  In either case, at the large distances in question, an error of deg. is small, compared to the error  0.12  in epicentre determiniation which  can be as large as 0 . 5 ° All  the epicentre coordinates l i s t e d in Appendix II  are the  preliminary determinations complied by N.O.A.A. on the basis of the J-B traveltime tables.  Recent traveltime studies, for example Cleary  and Hales (1966), have revealed systematically shorter  traveltimes  at teleseismic distances than those of Jeffreys and Bullen.  Con-  -122APPENDIX I I I ELLIPTICITY AND FOCAL DEPTH CORRECTIONS TO MEASURED TRAVELTIMES (con't) sequently, s i g n i f i c a n t systematic errors in epicentre'and focal depth determination may exist but until  the traveltime controversy  is s a t i s f a c t o r i l y resolved, very l i t t l e can be done about t h i s . The e f f e c t of errors in epicentre location on dT/dA measurements is to increase the scatter of the data e s p e c i a l l y in those regions of the curve where d T / d A neighbourhood of a c a u s t i c .  is large as in the  Errors in focal depth determinations,  p a r t i c u l a r l y for deep focus earthquakes, may well produce as much scatter  in the dT/dA  values as epicentre mislocations.  The view  is commonly held that focal depth determinations of greater 60 km are somewhat u n r e l i a b l e .  than  In this study, more than 1/3 of  the events had focal depths greater  than 60 km, and many of these  were restrained in some manner to a p a r t i c u l a r  level.  However,  as discussed in Section 2 . 2 . 3 , systematic errors in the dT/dA measurements are more s i g n i f i c a n t than any others.  -123APPENDIX IV METHOD OF SUMMARY VALUES  Jeffreys  (1937. 1961) devised the method of summary values  as a smoothing technique.  The following d e s c r i p t i o n follows more  c l o s e l y that of Arnold (1968). Consider a number of observed values of the function p = f (A) for a certain range of the argument in which there is very l i t t l e curvature. If a straight  line is f i t t e d to the points in this range by the method of  least squares, any two points (Pi, D]) and (P£, D2) on this  straight  line can be taken to represent the data, but in general, the errors in P| and P2 w i l l not be independent thus making estimation of the magnitude of the reduction of error d i f f i c u l t .  Moreover, if  in this  range the curvature is appreciable, the errors have been overestimated and valuable information may be l o s t .  If curvature is accounted for by  passing a quadratic through the data and choosing the points (Pi,  Di)  and (P2, D2) to be the intersections of the linear and quadratic forms, then curvature may be safely neglected and at the same time the uncertaint i e s in P]  and P2  are rendered independent.  Arnold (loc c i t )  has shown  that the ith equation of condition for a quadratic form is  0, If  - Dz  v,  -  the weight of one observation is w  error  in  PC  Uz  =  where  , the three normal equations for P i , P2  07 is the standard can be formed  by multiplying equation (1) by the respective c o e f f i c i e n t s .  -124APPENDIX IV METHOD OF SUMMARY VALUES (con't)  The conditions that the errors in P] , P  and  2  a be independent, are that  the c o e f f i c i e n t s of the off-diagonal elements of the left-hand side of the equation be zero,  i.e.  £u>; (A;- O.X^'- *) = o  _  0  £ ou; (Ai - O./fAi- Oz) - o £  (2) (3)  .  U K (fAj - D . f O i ~ Dz) = o  -(4)  Under these conditions, a cannot contribute to Pj and P  2  and i t may  assume any value, but there are three equations and only two unknowns. However, i f either  (k) is subtracted from ( 3 ) , i t reduces to (2) so that  (3) or (k) is superfluous, and a unique solution j a r Dj and D  is p o s s i b l e .  2  To f i n d D, and D^, the moments are formed: \=-  = £ u>; A;  ;  j  Equation (2) then becomes, putting dj = D] - /i  x  /U  t  + d,d  2  Si  - &l  and  d^ =  ->*-t  D^-^z (5)  = o  and equation (3) reduces to M-3 Substituting for either dj or d  2  JtXx C ^ i + ctz) = O  - - - - _  from (5) into (6), we have  Solution of equation (7) y i e l d s the values of D] and D2, and Arnold  - (6)  -125APPENDIX  IN/,  METHOD OF SUMMARY VALUES (con't)  has  shown how by s u b s t i t u t i o n  normal  equations,  the v a l u e s P,  -  £ w- ( A t - 0 )  Z  2  COi-Oi)  J(£»>L(&  "D,)Pt ±  JTWt  square  Pj and ?2 .  and r e p r e s e n t p  (L\)  used.  (Jeffreys,  method of 1937)  smoothing  can a l s o be used  so that the e r r o r s at  independent of each other and of a p o s s i b l e  T h i s method appears  three  c u b i c term.  to be somewhat of an improvement on the  method of summary v a l u e s ,  i - 0??)  | 0,-021  independent, and reduce to the  The summary v a l u e s  points are  the  10,-02.1  t  Pj and ?2 a r e c a l l e d the summary v a l u e s  to f i t a q u a d r a t i c  into  (3)  Pj and ?2 ace  of the c o e f f i c i e n t s of  i n the range o f argument  and  l  where the u n c e r t a i n t i e s are the i n v e r s e  of  (2)  CD -O3L^£uJcCAi-02)p V  p _ r o o t s of  of e q u a t i o n s  but  it  d i s t r i b u t e d or are not capable of  breaks down i f  the data are  first unevenly  precisely defining a curvature.  -126APPENDIX Va. U.B.C. TRIP TRIANGULAR REGRESSION PACKAGE  One method of smoothing data is to f i t by the method of least squares. designed with this  a polynomial to the data  The computer programme, U.B.C. T r i p , was  (among other things)  in mind.  A polynomial of degree, say m, is requested, being thought by visual examination to give the best f i t  to the data.  The T r i p  program uses the data to calculate the c o e f f i c i e n t s of m regression equations, the f i r s t equation being a simple linear equation of the form y=bo + b)x» where y is the dependent and x the independent v a r i a b l e . Successive equations are this linear equation augmented by a term in the next higher power of the independent variable^ up to. the power m.  2 Thus, the second regression equation would be y = b the f i n a l After  Q  + bjx + b x  regression equation would be y = b +• b,x + b x 0  and  2  2  2  .-  -t-b x  m  m  the c o e f f i c i e n t s of each of the equations have been calculated by  obtaining the best possible f i t least squares technique ( i . e .  of the equation to the data by the usual  by c a l c u l a t i n g the equation to minimise the sum  of the squares of the distances of the predicted values of y from the observed values  ) the program calculates several s i g n i f i c a n t  that give an indication of the quality of the f i t . One of these quantities  is the parameter  c o r r e l a t i o n c o e f f i c i e n t , where and y  ;  R  =  is the ith predicted value of ij,  quantities  ,  R , thesquare of the multiple w  i  C b£ / " y) A  ^ Oil "  (ij. = b + D,Xj) . 0  y.  is the i th  observed value of y, wj is the weight of the ith observation, and y is the  2 mean of the observed values of y.  R  always takes a value of between zero and  unity, and i t s s i g n i f i c a n c e l i e s in the f a c t that the closer value unity,  it  is  to the  the better does the corresponding regression equation f i t  the data,  -127-  2  s i n c e the p h y s i c a l of  \he  total  bj,  by the r e g r e s s i o n  known as  already  that  than o r equal  is  line.  In a d d i t i o n ,  d i f f e r e n t from 0,  it  the r e g r e s s i o n the degree m)  g i v e n by 2  b;=o.  If  this  is c o n c l u d e d that the c o e f f i c i e n t b^ i s  greater  probability significantly  equation  (provided  independent v a r i a b l e  not y e t  the p r e s e n t e q u a t i o n has not yet This quantity  is  in  attained  the p r o p o r t i o n of  the  the independent v a r i a b l e x which cannot be e x p l a i n e d by the  In a d d i t i o n ,  already  in the e q u a t i o n , and i f  the next higher  power o f x is  the F - p r o b a b i 1 i t y  i n the e q u a t i o n  the maximum degree m) the c o e f f i c i e n t of  is  c a l c u l a t e d , and t h i s  the next higher  however,  power of  is  and  if  power of  power o f  falls  i n the e q u a t i o n .  not y e t  x,  attained  s i m p l y the F - p r o b a b i 1 i t y  x would have  if  that  i t were admitted to the  the c r i t i c a l  value  the c a l c u l a t e d v a l u e of F - p r o b .  is  x would be added to the e q u a t i o n  repeated u s i n g the new e q u a t i o n .  the next higher  tolerance  independent v a r i a b l e  Unless r e q u e s t e d o t h e r w i s e ,  the next h i g h e r  and the e n t i r e process  not allowed  of each p o t e n t i a l  F - p r o b . i s assumed to be 0.05,  l e s s than 0.05,  this  (again p r o v i d e d the e q u a t i o n has  present regression equation.  0.05  f o r each cf the c o e f f i c i e n t s  the p r o b a b i l i t y of o b t a i n i n g a v a l u e o f F?  is a l s o c a l c u l a t e d .  beneath 0.001,  :  is  and t h e r e f o r e meaningful.,,  independent v a r i a b l e s  of  proportion  the program" c a l c u l a t e s  = ( b ; / S t d . E r r o r In b i )  t  The t o l e r a n c e of each p o t e n t i a l  not y e t  is  to the one c a l c u l a t e d , g i v e n that  l e s s than 0.05,  that  the dependent v a r i a b l e y which  i n the e q u a t i o n where the F r a t i o  and the F - p r o b a b i l t y  v a r i a n c e of  i t represents  the F - r a t i o and F - p r o b a b i 1 i t y  F  is  is  observed v a r i a n c e o f  accounted f o r quantities  meaning o f R  x is  If  F-prob.  is  greater  not added to the e q u a t i o n and the  than  -128-  c a l c u l a t i o n s stop. of  F-prob. be 1 . 0 ,  In p r a c t i c e ,  i t was r e q u e s t e d  t h a t the c r i t i c a l  value  so t h a t a l l the powers o f X up t o the degree r e q u e s t e d ,  were used t o c a l c u l a t e a f i n a l  r e g r e s s i o n e q u a t i o n , i r r e g a r d l e s s o f whether  the c o e f f i c i e n t s o f the powers were s i g n i f i c a n t or not as i n d i c a t e d by t h e i r F-prob. v a l u e s .  The r e s i d u a l s y j - <y- were c a l c u l a t e d , and a p r i n t e r  p l o t o f the d a t a t o g e t h e r w i t h p o i n t s p r e d i c t e d by the f i n a l r e g r e s s i o n equation obtained. coefficients  The v a l u e s o f R  i n the f i n a l  2  and F-prob. f o r each o f t h e  r e g r e s s i o n e q u a t i o n were then examined,  w i t h the r e s i d u a l s and the p r i n t e r p l o t . coefficient  I f the F-prob. o f t h e  along last  i n the e q u a t i o n was found t o be g r e a t e r than 0 . 0 5 , a new p o l y -  nomial f i t o f degree m-l was r e q u e s t e d . a p o l y n o m i a l f i t o f degree  I f i t was found t o be l e s s than  m + 1 was r e q u e s t e d . The f i n a l  0.05,  regression equation 2  chosen as best r e p r e s e n t i n g the d a t a , was "that one i n which R  was the  n e a r e s t t o the v a l u e u n i t , the F-prob. o f more o f the c o e f f i c i e n t s were g r e a t e r than 0 . 0 5 ,  the r e s i d u a l s appeared t o be e v e n l y d i s t r i b u t e d about a mean  v a l u e , and f r o m v i s u a l e x a m i n a t i o n o f the p r i n t e r p l o t , the f i t o f the r e g r e s s i o n e q u a t i o n t o the d a t a seemed  satisfactory.  -129APPENDIX V b . DT/DA SMOOTHED BY POLYNOMIAL RE3RESSI3N  DISTANCE (DEG) 113.0 114.0 115.0 116.0 117.0 118.0 119.0 120.0 121.0 122.0 123.0 124.0 125.0 126.0 127.0 128.0 129.0 130.0 131.0 132.0 133.0 134.0 135.0 136.0 137.0 138.0 140.0 141.0 142.0 143.0 144.0 145.0 146.0 147.0 148.0 149.0 150.0 151.0 152.0 153.0 154.0 155.0 156.0 157.0 158.0 159.0 160.0 161.0 162-0 164.0 165.0 166.0 167.0 168.0 169.0 170.0 175.0  1  DT/na  i  j  DF  1.959 1.950 1. 94 5 1.936 1.929 1.922 1. 918 1. 914 1.911 1.908 1.906 1.904 1.903 1.901 1. 900 1. 898 1. 896 1.893 1. 889 1. 885 1. 880 1. 874 1. 867 1.859 1. 849 1. 838 1. 812 1.7 97 1.780 1.762 1. 74 2 1.720 1. 696 1.671 1.644 1.615 1. 584 1.550 1.517 1.481 T.443 1. 403 1.361 1. 3 18 1.273 1.226 1. 178 1-129 1. 077 Q . 971 Q- 916 0.859 0. 802 0.743 0. 684 0. 624 tt. 3 16  GH  2. 688 2.678 2. 668 2.658 2. 648 2.640 2.628 2.622 2. 617 2.613 2. 607 2.602 2.595 2.588 2. 579 2.568 2. 555 2.540 2.523 2.502 2. 479 2.451 2. 421 2.386 2. 347 2.300  (SEC/DEG) IJ  3.456 3.437 3.417 3.397 3.355 3.3 32 3. 309 3.285 3. 260 3-235 3.208 3. 181 3. 154 3.125 3.096 3.066 3.035  3. 548 3. 659 3. 760 3. 849 3. 929 4. 000 4. 063 4. 118 4. 166 4.208 4. 244 4.276 4. 304 4. 329 4. 351 4.371 4. 390 4.409 4.448 4. 470 4.494 4. 501 4.501 4.503 4.506 4.509  

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