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Maximum entropy spectral analysis of free oscillations of the earth : the 1964 Alaska event Davies, John Charles 1976-12-31

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MAXIMUM ENTROPY SPECTRAL  ANALYSIS  OF FREE OSCILLATIONS OF THE THE 1964 ALASKA  EARTH:  EVENT  by John C. fi.Sc,  Davies  U n i v e r s i t y o f B r i t i s h Columbia,  1973  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department of Geophysics and Astronomy  We accept t h i s t h e s i s as conforming t o the r e g u i r e d standard  The U n i v e r s i t y Of B r i t i s h March, 1976 (c)  John C. Davies  Columbia  In p r e s e n t i n g - t h i s  thesis  an advanced degree at the L i b r a r y s h a l l I  f u r t h e r agree  for scholarly by h i s of  the U n i v e r s i t y  make  it  written  thesis  of B r i t i s h for  the requirements  Columbia,  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying o f  this  It  is understood that  for financial  gain s h a l l  /^{^X^/ t \\  The U n i v e r s i t y of B r i t i s h  Columbia  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  VflfajL-K.Jlf.  copying or  for  that  study. thesis  purposes may be granted by the Head of my Department  permission.  Department of  Date  fulfilment of  freely available  that permission  representatives.  this  in p a r t i a l  or  publication  not be allowed without my  i  ABSTRACT  The Alaska  DCLA gravimeter  p r o c e s s i n g techniques.  maximum entropy design  method  was  gravimeter  I n p a r t i c u l a r , the  (MEM) of s p e c t r a l e s t i m a t i o n and  utilised.  Prediction f i l t e r s tidal  a f t e r the 1964  earthguake, has been s u b j e c t e d t o v a r i o u s , r e c e n t l y  developed data  filter  r e c o r d i n g obtained  were used t o extend the o r i g i n a l  r e c o r d i n g , so as to avoid i n f o r m a t i o n l o s s  caused by t a p e r i n g the r e c o r d p r i o r t o f i l t e r i n g . Time adaptive  p r e d i c t i o n e r r o r f i l t e r s were then used to l o c a t e  n o i s e b u r s t s , or ' g l i t c h e s ' , which are present  i n the  f i l t e r e d r e c o r d . The r e c o r d was then d e g l i t c h e d using two methods, one a p r e d i c t i v e approach, and the other i n v o l v i n g a d i v i s i o n of the r e c o r d by a weighted envelope f u n c t i o n . Power s p e c t r a using the c l a s s i c a l periodogram approach, as w e l l as MEM, both before  were c a l c u l a t e d f o r the f i l t e r e d  records,  and a f t e r d e g l i t c h i n g .  T h i s a n a l y s i s r e s u l t e d i n new values f o r the s p l i t t i n g parameter ' ^ ' , and the c e n t r e frequency, o s c i l l a t i o n modes. A dramatic  f o r various free  i n c r e a s e i n the s i g n a l t o noise  r a t i o was a l s o observed a f t e r the f i l t e r e d r e c o r d s were d e g l i t c h e d . The presence of core mode o s c i l l a t i o n s was a l s o  investigated,  but no evidence f o r these undertones  I n s t e a d , numerous peaks a t t r i b u t e d t o e i t h e r  was fcund.  instrument  n o n - l i n e a r i t i e s , o r barometric pressure e f f e c t s , were found i n the freguency range 0.12  t o 1.20  c y c l e s per hour.  iii  TABLE OF CONTENTS Page  ABSTRACT  i  TABLE OF CONTENTS  i i i  L I S T OF TABLES  V  L I S T OF FIGURES  vi  ACKNOWLEDGEMENTS  viii  CHAPTER I . 1-1. , 1-2. 1-3. 1-4. I - 5. CHAPTER I I . I I - 1. A. , B. II-2. A. B. C. D. E. .1.1-3. II-4. I I - 5. CHAPTER I I I . I I I - 1. .111-2. III-3.  INTRODUCTION Purpose Data D e s c r i p t i o n Free O s c i l l a t i o n s Spectral Splitting Core Undertones  1 2 7 10 12  THEORETICAL BACKGROUND C l a s s i c a l Spectral Estimation Periodogram Autocorrelation Function Maximum E n t r o p y Method I n f o r m a t i o n and E n t r o p y Maximum E n t r o p y S o l u t i o n Analytical Integration Z - t r a n s f o r m Approach Heuristic Solution Levinson Recursion Autoregressive Models and MEM Time A d a p t i v e MEM  14 15 16 18 19 22 24 26 28 31  DATA ANALYSIS PROCEDURES Necessity of Tapering Prediction Filtering  35 36 36  iv  III-4. A. B. C. I I I - 5. A. B.  Deglitching Identification P r e d i c t i o n i n t o Gaps D e g l i t c h i n g by E n v e l o p e Power S p e c t r a l A n a l y s i s P e r i o d g r a m and MEM S p e c t r a Complex D e m o d u l a t i o n  CHAPTER I V . I V - 1. A. B. C. D. E. IV-2.  61 64  RESULTS Free O s c i l l a t i o n o S ^ mode S mode S^. mode S« mode S mode Core Undertones  0  3  0  0  0  CHAPTER V  46 47 52  fe  SUMMARY  Mode F r e q u e n c i e s 68 78 86 93 99 100 110  REFERENCES  112  APPENDIX  115  1  V  LIST OF TABLES PAGE  Table 1.  0  S^  Frequencies  69  Table 2.  0  S  3  Frequencies  79  Table 3.  0  S  ¥  Frequencies  87  Frequencies  9U  Table 4. Table 5.  S  0  S  „S  6  Frequencies  99  vi  LIST OF FIGURES Page  Figure  1.  Unprocessed gravimeter  data  4  Figure  2.  Periodogram power of unprocessed data  6  Figure  3.  P r e d i c t e d gravimeter  data  38  Figure  4.  High pass f i l t e r e d  p r e d i c t e d data  41  Figure  5.  Band pass f i l t e r e d  p r e d i c t e d data  43  Figure  6.  High pass f i l t e r e d  unpredicted  45  Figure  7.  49  Figure  8.  P r e d i c t i o n e r r o r t r a c e o f high passed r e c o r d P r e d i c t i o n e r r o r t r a c e of band passed record  51  Gap p r e d i c t i o n t e s t case  54  data  Figure  9.  Figure  10.  D e g l i t c h e d high passed r e c o r d  56  F i g u r e 11.  D e g l i t c h e d band passed r e c o r d  57  Figure  12.  Figure  13.  P r e d i c t i o n e r r o r t r a c e of d e g l i t c h e d high passed r e c o r d Prediction error trace of deglitched band passed r e c o r d  F i g u r e 14.  D e g l i t c h e d high passed r e c o r d envelope sguared  59 60  using 63  Figure  15.  Power s p e c t r a of  0  S  mode  73  Figure  16.  Power s p e c t r a of  0  Sj mode  82  Figure  17.  Power s p e c t r a o f  a  S+ mode  89  Figure  18.  Power s p e c t r a of  0  S  96  a  S  mode  S  Figure  19.  Power s p e c t r a o f  Figure  20.  Periodogram of u n d e g l i t c h e d passed r e c o r d  Figure  21.  Periodogram of d e g l i t c h e d passed r e c o r d  0  &  mode band  band  viii  ACKNOWLEDGEMENTS  I would l i k e to thank Dr. T. J . U l r y c h f o r many v a l u a b l e d i s c u s s i o n s t h a t took p l a c e during t h e course of t h i s research  p r o j e c t . Dr. Ulrych a l s o devoted time and e f f o r t i n  the c r i t i c a l a p p r a i s a l o f t h i s t h e s i s , the necessary f i n a n c i a l support  as w e l l as provided  to see t h i s p r o j e c t t o an  end. My thanks a l s o go t o Dr. R. M. Clowes who gave a second o p i n i o n of t h i s t h e s i s and to Rob Clayton  and George Spence  who dropped s e v e r a l u s e f u l programming h i n t s my way. Much use was made o f the U n i v e r s i t y of B r i t i s h Columbia computing c e n t r e and the computing c o s t s were born by a N a t i o n a l Research C o u n c i l of Canada grant made t o Dr. T. J . Dlrych  (#67-1804).  1  I.  INTBODUCTION  1. PUBPOSE This research investigate new  Dr.  earth free  technigues  focused  p r o j e c t was u n d e r t a k e n i n o r d e r t o  i n time  {Slichter  after  utilising The  the Alaska  relatively  investigation  data  obtained  resolution study  e a r t h q u a k e o f March 27, technigues  1964  were used f o r  study:  To d e s i g n  and t o u s e t h e i m p r o v e d  f e a t u r e s o f t h e maximum e n t r o p y  filter  operators  method t o  splitting.  necessary  to identify  and  s e c t i o n s o f t h e r e c o r d which were made  untenable  deglitching  modes,  the a s s o c i a t e d s p e c t r a l  predict  by n o i s e  c f the data  glitches.  s e t was  an i m p o r t a n t  step i n  r e d u c i n g t h e n o i s e power o f t h i s r e c o r d , so t h a t t h e and  by  To compute power s p e c t r a o f f u n d a m e n t a l s p h e r o i d a l oscillation  The  analysis.  1 9 6 7 b ) . Maximum e n t r o p y  two p u r p o s e s i n t h i s  2.  series  data  on t h e w e l l known UCI& g r a v i m e t e r  L. S l i c h t e r  1.  oscillation  detectabiltiy  of core  presence  u n d e r t o n e s c o u l d be i n v e s t i g a t e d .  2  2. DATA. DESCRIPTION The data used i n t h i s study o r i g i n a t e d from UCLA gravimeter  #4,  one  of two Lacoste-Romberg t i d a l  gravimeters  that were i n o p e r a t i o n i n Los Angeles at the time of the Alaska earthquake. T h i s s e i s m i c event i s the l a r g e s t to e x t e n s i v e l y recorded primary  with l o n g - p e r i o d instruments,  1964 be  and i s the  source f o r much of the present i n f o r m a t i o n on normal  modes. The  data has a sampling  extends f o r 18.5  r a t e of 1/12  days. Free o s c i l l a t i o n  superimposed on high amplitude  of an hour  and  information i s  e a r t h t i d e data,*and, i n the  unprocessed form, the normal mode data i s not observable when the r e c o r d i s p l o t t e d (Figure The data s e t was gravimeter,  recorded  which i n simple  spring-supported  #1). on a Lacoste-Romberg  tidal  terms, i s a mass on the end of a  beam. A p h o t o - e l e c t r i c c e l l d e t e c t s motions  i n the mass,• r e l a y s the i n f o r m a t i o n to a servomechanism, which i n t u r n r a i s e s or lowers the upper end of the s p r i n g by means of a screw. The recorded  digitally  gravimeters  p o s i t i o n of the screw i s  i n u n i t s of 0.1  m i c r o g a l s . Lacoste-Romberg  g e n e r a l l y have d r i f t r a t e s of between 1 and  m i c r o g a l s per day. i n c r e a s e s 28 db. hour  angular  (Figure The f i r s t  The  20  noise power f o r t h i s data s e t  from 0.4  c y c l e s per hour to 0.0  c y c l e s per  #2). step i n the data p r o c e s s i n g was  to e l i m i n a t e  the t i d a l f r e q u e n c i e s from the r e c o r d using a f i l t e r i n g scheme. However, s i n c e the f r e e o s c i l l a t i o n s t a r t s at the beginning  information  of the t i d a l r e c o r d , any  frequency  Original recorded  at  unprocessed Los  Alaska earthguake.  Angeles The  gravimeter during  record  the  consists  data 1964 of  5320 p o i n t s with a sampling r a t e of 1/12th of an  hour.  FIGURE 2. Unsmcothed periodogram .power spectrum of the  unprocessed gravimeter data. Ihe Nyguist  freguency i s at  6.0  cycles  location  of  spheroidal  oscillation  filso  shewn  seme  are  semi-diurnal t i d e s .  of  per the  modes  the  basic  hour.  The  fundamental  is  indicated,  diurnal  and  CM  I  0.0  I  '  0.857  I 1.714  1 2.571  FREQUENCY FIGURE 2  I  3.429  CY/HR.  I  4.2B6  1 5.143  I  6.0  cr>  7  domain  filtering  and  tail  not  to taper the record  normal lost.  end  method w i l l  in tapering  of the  of the data s e t . I t i s p a r t i c u l a r l y in this  manner, as h i g h  predictive  filtering  Another evident  this was  problem  amplitude  record  tidal  by  filtering  Miller  immediately  i s accomplished.  errors  1972).  to  fill  i n reasonable data values  3.  FEEJ  OSCILLATIONS  of  and a f t e r s h o c k  To e l i m i n a t e  these was  used  ( O l r y c h and  C l a y t o n 1975) .  Free o s c i l l a t i o n s  have t h e i r  theoretical  beginning  with  ( 1 8 8 2 ) , who  c o n s i d e r e d the problem  of normal  modes  first  a u n i f o r m s p h e r e . Love  development  to include  (1911) t h e n e x t e n d e d  a self-gravitating  B e n i o f f e t a l . (1954) f i r s t after 1961  The  glitches  unwanted n o i s e f e a t u r e s , a method o f d a t a p r e d i c t i o n  Lamb  be  on  by n o i s e b u r s t s o r  digitization  ( W i g g i n s and  sill  ( O l r y c h e t a l . 1973).  i s contaminated  are caused  features  used  a scheme b a s e d  w i t h t h e d a t a becomes  when l o w - p a s s  filtered  problem,  front  important  mode i n f o r m a t i o n a t t h e o n s e t o f t h e r e c o r d , To c i r c u m v e n t  which  result  o b s e r v i n g a normal v a r i o u s groups  modes e x c i t e d al.  1961;  al.  1961).  point,  and  model was  uniform  reported earth  free  mode w i t h a p e r i o d  earthquake  e t a l . 1961;  Theoretical  Bogert  interest  the mathematical  of  o f 57  1960  1961;  increased  and  sphere.  oscillations  r e p o r t e d o b s e r v i n g a number o f  by t h e C h i l e a n  Benioff  the  minutes. normal  (Alsop e t Ness e t  greatly  at  this  theory f o r a heterogeneous  developed, allowing  a comparison  In  between  earth  8  e x p e r i m e n t a l and  theoretical  developed,  when u t i l i s e d ,  for  which  the earth,  parameters Bullen  as a f u n c t i o n  1967;  Gilbert  B a c k u s and  I n v e r s i o n schemes were  yielded  enabled  of depth Gilbert  velocity-depth  s e p a r a t i o n o f Lame's  {Press  1967;  models  and  1968;  Haddon  Dziewonski  and and  1973) .  There a  which i n t u r n  results.  a r e two  fundamental  t y p e s of f r e e o s c i l l a t i o n s  sphere:  1.  Spheroidal: radial  2.  Toroidal;  d i s p l a c e m e n t s always  displacements are e n t i r e l y  S p h e r i c a l geometry can  be  broken  Slichter  1.  modes a s :  1^=0  O X  3.  three radius(r).  (1967a) g i v e s t h e d i s p l a c e m e n t s f o r t h e s e t h r e e  components f o r t o r o i d a l  2.  tangential.  down i n t o  c o m p o n e n t s , l a t i t u d e { 0 ) , l o n g i t u d e ( f) , and  exist.  0  =  (r,£,m) c o s ( e ) - »  P™(sin © )  "dexp f - i ojUt+mcar-*/! \  oy =•-v„ ( r , £ , a )  3 vTisxn  X exp (-i^ft+may-Y} )  e)  of  9  For s p h e r o i d a l modes the displacment  1.  components a r e ;  0^ = 0^ (r,^,m) P ™ ( s i n © ) X exp (-i ct> {t+mittj- /}) 1  s  2.  0  = W^Cr,H m)  e  f  9Posing )  X exp (-idOgCt+mt^- ^}) 1  3.  0? = W ^ r ^ m )  (cos©)- -  pJVin© )  1  X Ji^pl-l^ft-Sra^-i/})  P^: s p h e r i c a l s u r f a c e harmonic of degree *J0* and order 1  . «m« (m=0,1,2...l) .  For the r a d i a l f u n c t i o n s V ^ O ^ a n d  , • n• denotes the  overtone number, where the fundamental overtone i s ' n=0*. S p h e r o i d a l f r e e o s c i l l a t i o n s a r e represented symbols ^S^ and t o r o i d a l .  The  0  S  0  by the  o s c i l l a t i o n s are represented  mode i s e n t i r e l y  by  a r a d i a l motion of the  e a r t h , expanding and c o n t r a c t i n g much as i f a b a l l o o n were being  inflated  and d e f l a t e d p e r i o d i c a l l y .  Another s p h e r o i d a l  mode, S ^ , i s nicknamed the f o o t b a l l mode and r e p r e s e n t s a e  change of shape of the e a r t h from p r o l a t e t o o b l a t e and back again.  Nodal p o i n t s do e x i s t  and i t i s p o s s i b l e some modes  w i l l not be observed when the r e c o r d i n g s t a t i o n i s l o c a t e d a t one  of these  oscillation  p o i n t s of minimum amplitude. Some f r e e modes are not p o s s i b l e f o r the e a r t h . . S 0  (  i sa  10  rigid  body o s c i l l a t i o n  rigid  body t r a n s l a t i o n a b o u t  been are  4.  observed,  and  and  i s not  f o r the  observable.  is a  t h e p o l a r a x i s and  mode T ^  a l l the  n  yet  displacements  egual to zero.  SPECTRAL After  SPLITTING• t h e r e c o r d s from  the  1960  C h i l e a n earthquake  examined, s p l i t t i n g  o f some o f t h e f u n d a m e n t a l  modes was  An a n a l o g o u s  physics  observed.  i s t h e Zeeman e f f e c t  splitting  i n the presence  ellipticity  for this  it  found  was  central  of a magnetic  rotation  peak i n t o  splitting  into  2^+1  ^+1  peaks,  peaks.  t o show s p l i t t i n g ,  guickly,  resulting  The  following  and  (1  \=  +  +  0(^)2  LU^ = s p l i t  t o be  spheroidal  «4  +  2  E£  peak  freguency  developed of  and the  would c a u s e  a  modes have been  of the Alterman  effects to f i r s t  m ^)  +  and  modes a t t e n u a t e t o o  from  order i n r o t a t i o n .  show  possible  the s p l i t t i n g  as t o r s i o n a l  splitting  lines  Rotation  two  ellipticity  formulation taken  second  ^  field.  i n d i s s i p a t i v e broadening  (1974) p r e d i c t s  ellipticity  and  Only  spheroidal  spectral  t h e o r y was  would c a u s e  observed  lines.  where a t o m i c  behaviour. Suitable  that  were  phenomenon i n n u c l e a r  o f t h e e a r t h were b e l i e v e d  causes  al.  has n o t  spectral et  order i n  11  m  = number of the s p l i t  •A- = angular freguency 1X3  e, = eigenfreguency  £  of  = e f f e c t of e a r t h ' s  peak  of the e a r t h ' s r o t a t i o n S  or  T  ellipticity  A,B,C,D,E = constants The c e n t r e frequency d i f f e r by  of a group of s p l i t  from the frequency  because of the e f f e c t of 'm', 2  around the c e n t r a l peak  peaks w i l l  of an u n s p l i t peak. A l s o ,  splitting  (n=0). For lower  w i l l be asymmetric order modes, the  r o t a t i o n e f f e c t i s much q r e a t e r than the e l l i p t i c i t y so t h a t the f i r s t  effect,  order s p l i t t i n g e f f e c t i n e l l i p t i c i t y  i s of  the same magnitude as the second order s p l i t t i n g e f f e c t i n r o t a t i o n . For the higher order modes, t h i s equivalence does not h o l d . L a t e r a l h e t e r o g e n e i t i e s can a l s o give r i s e s p l i t t i n g or widening  of s p e c t r a l l i n e s , but the  cannot be w e l l p r e d i c t e d or  to  effects  quantized.  Up to the present, only c o n v e n t i o n a l power s p e c t r a l techniques have been used i n the study of f r e e s p l i t t i n g . MEM  oscillation  seems to be a l o q i c a l c h o i c e i n s t u d y i n g  e f f e c t s because of the higher r e s o l u t i o n and  these  o p t i m a l l y smooth  nature of the HEM  power spectrum. B o l t and C u r r i e (1975) and  Olrych and  (1975) have shown the improvement i n  frequency  Bishop  resolution  when using MEM,  B o l t and C u r r i e (1975)  s t u d i e d the t o r s i o n a l e i g e n f r e g u e n c i e s recorded at T r i e s t e a f t e r the 1960  C h i l e a n earthquake,  and r e p o r t e d an i n c r e a s e  i n p r e c i s i o n and number of peaks d e t e c t e d .  12  5.  CORE  UNDERTONES  One for  of t h e aims  core  cycles .698  mode  oscillations  per hour.  cycles  He a l s o  per hour  predicted  The  1964  Alaska  now  generally  1976)  I t has been that  periods  Spheroidal  as any r a d i a l  medium  range  and s p h e r o i d a l  periods,  theoretically The  most  which  driving  stable.  gravimeter corresponds  a t the South  of core  t o a core  i s very  Rotational  72  model observation  sub-adiabatic only  when t h e  coupling  between  the theoretical 0 a n d 12  hours,  Rossby  undertones  motion  could  microgals  motion  communication  i s possible  i n the f l u i d  Pole.  was n o t  hours.  At  waves c a n  1975).  i s a translational  a s i g n a l o f 0.08  i n 1961  in a  t o between  a t about  (1974) b e l i e v e t h e y  show  and i t i s  Slichter*s  exist  motion  (Crossley  motion  event.  low.  periods  interesting  a return  Slichter produced  exist  mode  modes r e s t r i c t s  starting  1.0  per hour.  peaks,  reported  between  can only  particle  of the undertone  longer  mode,  i s very  i s gravitationally  torsional  no s u c h  of the Slichter  to  Chile  cycles  (Crcssley, personal  undertones  0.0  again  a s p e c t r a l peak a t  .659  t h e peak  and t h e c o r r e l a t i o n  predicted  core,  yielded  that  shown  the period  dependent, and  believed  range  o f t h e 1960  a t .741-and  earthquake  once  p a r t i c u l a r mode s h o u l d  this  peaks  was t o l o o k  i n the freguency  i n the record  that  splitting  project  (1961) r e p o r t e d  Slichter  spectral  valid.  of this  This  i s the  of the solid outer  detect on a signal  core. such  Slichter  inner  core  Jackson and  a motion  i f  Lacoste-Romberg amplitude  of 6 centimeters  with  an  inner-  i t  13  outer  core  density  detectable  contrast  s i g n a l would be  s t a t i o n s at  other  u n d e r t o n e s u c h as  useful  i n accurately  inner At  exist, of  and  this and  outer  2.  whether t h e y a r e to three  level  the  quite  low  (Smith  density  contrast  overtones,  packed  limit  Crossley  'm*.  '2n+1* g r e a t e r  It i s possible that  core  (1975) who  The  this  record.  to  number of  of  to  the  undertones  corresponding i n d i v i d u a l modes  1975).  many o f  the  undertones w i l l of  12  upon c a l c u l a t i o n s done used a  corresponds to t h a t of  spectral  respect  than the  formulated  scheme t o a r r i v e at t h i s r e s u l t .  power f r o m  modes i s b e l i e v e d  a r o u n d an u p p e r l i m i t  i s based  lack  1974).  number  (Crossley  do  factors:  making i d e n t i f i c a t i o n  difficult  This  of  between  undertones  phenomena. T h i s  non-degenerate with  order  therefore  periods  observation  mode w o u l d - b e - e x t r e m e l y  observable  of e x c i t a t i o n of  azimuthal  3.  larger for  e a r t h . The  Slichter  determining  Undertones are  is  the  the  minimum  3  t i m e i t i s unknown w h e t h e r c o r e  The be  g/cm- . T h i s  core.  knowledge i s due  1.  0.3  correspondingly  l o c a t i o n s on  a core  the  of  tidal  the  This  tend  hours. by  computing 12  semi-diurnal  peak w i l l  have  to  hour tide,  period and  obscure  so the  14  I I . TJJOBJTICJL BACKGROUND  1.  CLASSICAL SPECTBAI ESTIMATION Orthodox methods of s p e c t r a l e s t i m a t i o n  u n r e a l i s t i c assumptions about the extension  o f t e n make o f the sampled  process beyond the windowed r e g i o n . The maximum entropy method of s p e c t r a l e s t i m a t i o n obviate  was developed s p e c i f i c a l l y t o  t h i s r e s t r i c t i v e s u p p o s i t i o n . The two ' c l a s s i c a l *  methods of o b t a i n i n g a power spectrum a r e o u t l i n e d below.  A.  Periodggram A f u n c t i o n f ( t ) can be expanded i n a F o u r i e r s e r i e s i f  f(t) and  i s piecewise continuous i n the i n t e r v a l -1/2 < t < T/2 i s p e r i o d i c with p e r i o d T , Then f ( t ) can be expressed as  a sum of s i n e and cosine  functions.  f (t) = 1/2a + ^ G  (a cos«; t + b^sinuj^t) n  n  -n=i  n=0,1,2...  n=1,2,3...  The  amplitude spectrum f o r the process f (t) i s given as  15  ! f | =Va2 • b* n  Due  t o - i t s ' C o n d i t i o n s of e x i s t e n c e ,  assumes a p e r i o d i c extension  the periodogram method  of the data beyond the sampled  region.  B  »  M i s s ° £ £ § 1 § l i s s fjyssliSJQ Seiner (1930) f i r s t  shoved the r e l a t i o n s h i p between  a u t o c o r r e l a t i o n f u n c t i o n i n the time domain and spectrum i n the freguency domain. The  first  the  the power  step i n  obtaining  a power spectrum using t h i s method i s to o b t a i n a s u i t a b l e estimate of the a u t o c o r r e l a t i o n f u n t i o n 8 ( t ) , where fi (t) — for  | t | > 8. T h i s i s e g u i v a l e n t  to saying t h a t an  0  infinite  non-zero a u t o c o r r e l a t i o n f u n t i o n H (t) i s m u l t i p l i e d by some weighting f u n c t i o n H (t)  where I (t) •- 0 f o r | t j > N,  The  product E(t)W(t) i s then transformed to the freguency domain. The  estimated spectrum i s t h e r e f o r e the c o n v o l u t i o n  F o u r i e r transform  of the weighting f u n c t i o n with the  spectrum. C o n s i d e r a b l e design  the best  transform. the f i r s t and  work has  variance  been put  the  true  i n t o t r y i n g to  p o s s i b l e weighting f u n c t i o n and i t s  There are two being  of  the  d i f f i c u l t i e s i n v o l v e d i n the  design,  t r a d e o f f t h a t e x i s t s between r e s o l u t i o n  i n the freguency domain, and  the second  being  the requirement t h a t the spectrum be non-negative. As Burg(1975) p o i n t s out, although window theory neatness to i t , i t i s an a r t i f i c e t h a t B(t) - 0 - f o r - I t | >  N.  has  imposed by the  a certain assumption  16  "If one elegance unfounded data, and be t o t a l l y hopefully  were not b l i n d e d by the mathematical of the c o n v e n t i o n a l approach, making assumptions as to t h e value of unmeasured changing data values that one knows would unacceptable from a common sense, and a s c i e n t i f i c point of view" . 1  2. MAXIMUM ENTBOPY METHOD  A. I n f o r m a t i o n and Entropy Burg's r a t i o n a l i n developing the theory of the maximum entropy method set  (MEM), was, t h a t s i n c e t h e r e i s an i n f i n i t e  o f non-negative  power s p e c t r a that correspond t o a given  a u t o c o r r e l a t i o n f u n c t i o n , the best spectrum  would be the one  t h a t corresponds t o the most random time s e r i e s . The concept of  maximum entropy s p e c t r a l a n a l y s i s i s ' b a s e d upon  t h e o r e t i c a l developments made i n s t a t i s t i c a l  mechanics and  i n f o r m a t i o n t h e o r y . One of the great advantages  of the MEM  technigue, i s i t s a b i l i t y t o r e s o l v e c l o s e l y spaced  spectral  l i n e s i n a much b e t t e r manner than c o n v e n t i o n a l methods. From a s t a t i s t i c a l  viewpoint, MEM can be shown t o  correspond to a s i t u a t i o n with the most random behaviour. F o l l o w i n g U l r y c h and Bishop t h e r e a r e M* 1  (1975) i t can be seen t h a t i f  d i f f e r e n t t h i n g s •m,-' which could p o s s i b l y  o c c u r , a l l with p r o b a b i l i t i e s ' p ' L  f  then i f a l l the *p^*s'  J . P. Burg, Maximum Entropy S p e c t r a l A n a l y s i s , Ph.D. T h e s i s , S t a n f o r d U n i v e r s i t y 1975. 1  17  a r e t h e same, t h e n no i n f o r m a t i o n a b o u t g a i n e d . I f one  expressed  I  =  b e t w e e n i n f o r m a t i o n and  p r o b a b i l i t y can  be  as  log<VPc)  k  2.  t h e o c c u r r e n c e s a r e summed o v e r a l o n g p e r i o d T t h e n  Iror/ic  The  been  t h e p r o c e s s has been a g u i r e d . A  k = 1 when l o g i s b a s e  If  has  o f t h e ' p ^ ' s * i s d i f f e r e n t , t h e n some  i n f o r m a t i o n about relationship  the system  =  average  = kf  MP,T  logd/P,)  + PT a  l o g ( 1 / P ^ ) + ...)  e n t r o p y g i v e n by Shannon (1948) i s  P.log(P )  II-1  L  l-1  I f a l l t h e • P,;*  b u t one  e x c e p t i o n i s e g u a l t o one, about in  a r e e g u a l t o z e r o , and  t h e n H = 0,  the process i s a v a i l a b l e ,  the system.  A g a i n , a l l knowledge  and t h e r e i s no  uncertainty  I f H i s g r e a t e r t h a n z e r o , a measure o f  uncertainty i s available. u n i t sample i s d e f i n e d as  lim  this  H/ (N + 1)  The  entropy rate or entropy  per  18  and  can be shown to be p r o p o r t i o n a l t o  In P (f) df  II-2  P (f) = power spectrum f.,  =  Nyguist  8. Maximum Entropy  freguency  Solution  To s o l v e the maximum entropy v a r i a t i o n a l problem, the f o l l o w i n g argument i s taken from Burg(1975). The  f u n c t i o n P (f) that maximizes eguation I I - 2 must be  o b t a i n e d so t h a t the c o n s t r a i n t  eguations  (-N < n < N)  are  s a t i s f i e d . B (n) i s the a u t o c o r r e l a t i o n f u n c t i o n .  II-3  Another  c o n d i t i o n i s the F o u r i e r transform r e l a t i o n s h i p between the a u t o c o r r e l a t i o n and the power spectrum.  P(f) = 1/2f„  The  2  B(n) exp {-i2 7T f  t)  II-4  c o n s t r a i n t eguation I I - 3 w i l l be s a t i s f i e d i f the B (n) i n  equation I I - 3 are e q u i v a l e n t solve  t o the B(n) i n eguation 11-4. To  t h i s problem, l a g r a n g i a n  what f o l l o w s i s a simpler  m u l t i p l i e r s can be used, but  derivation.  Substituting II-4 into II-2 gives  19  h << \ l n [ 1/24  Maximizing  N  y  P-» (f) exp<-i2 n fsn t)  for  - 1  II-5  I I - 5 with r e s p e c t to E(n) r e s u l t s i n  h,(.1/2f  where P  2. B («) -«*P ( - i 2 n tn a t) ] df  df = 0  |s| > N  II-6  (f) i s the i n v e r s e of the power spectrum P ( f ) .  Expanding P"~ (f) i n terms of a F o u r i e r s e r i e s g i v e s 1  ^  \ e x p (-i2 n fn a t)  .'. P(f) = 1/j£  A e x p (-i2 ft ts a t)  P-* (f) =  II-7  5  II-7 Is the same eguation that would r e s u l t from v a r i a t i o n a l s o l u t i o n , where the  ? v * s are the  m u l t i p l i e r s . From II-6 i t can be seen t h a t The next task i s to express the methods are presented, one and the other a z-transform  the  Lagrangian A =0 f o r In j > N. 1N  /\ 's i n a s u i t a b l e form.  Two  an a n a l y t i c a l i n t e g r a t i o n scheme, argument.  C. M a l y i i c a l I n t e g r a t i o n S u b s t i t u t i n g eguation II-7 i n t o eguation II-3 g i v e s  J  ft  exp(i27r fnA t) ^ x p ( - i 2 7T f s a t )  L e t t i n g z = exp(-i2ff f / l t )  df =  fi<n)  (-N  < n < N)  II-8  which i s the normal z-transform  e q u i v a l e n c e i t can be shown t h a t  20  dz = - i 2 T i a t e x p ( - i 2 H f A t ) df =  -±2Tf£it  z  df  and df = -dz/i2n£t z.  Equation II-8 can now  1/27Ti4t^  be w r i t t e n  z-^-VJ: A z  dz = R(n)  s  s  as  II-9  T h i s i n t e g r a t i o n curve i s c o u n t e r c l o c k w i s e around the u n i t circle  i n the complex z plane. From II-7 and the p r i n c i p l e of  polynomial f a c t o r i z a t i o n i t i s p o s s i b l e t o say t h a t  2  \z  s  = [P ZJt]-» [1 N  a^z^ [1  • a, z +...+  + a%-»- +...•  - [P At]-» Z a z £ a *  Substituting  w  11-10  S  N  a*z~ 3  s  11-10 i n t o I I - 9 r e s u l t s i n an e x p r e s s i o n f o r  B (n).  fi(n) = P»/ L z~^ 2-iTi ™ i a, z £ s "  dz  1  s  Since £ (n) =  a  z  \ P(f)z-^  -N  and i f 11-11  i s summed over a  that  dz  i\--o  2rri J 2  a  5  2 5  < n < N 11-11  5  Z s " a  z  so  21  it  i s possible to  ,Z.a*  write  R(n-r) = P/o C  ^ =  PP C 2/ri J 1"  Knowing t h a t  a, z 2  z^-*  2~~T3  i  ds  r>0  a  5. a t ^ d z ^ar -5  z  s  Z  .11-12  s  a  z  i s a n a l y t i c f o r z < 1, and  for r >  1,  s«e>  z^  - 1  i s a n a l y t i c f o r z < 1, i t i s evident  lz^-i/-s  a z] 2  dz = 0  that  for r > 1  S-O  and  also that  j"f a^R (n-r)  T\-0  Now  =  t h a t the  0  for  r > 1  case f o r r > 1 has  11-13  been s o l v e d  a l l that  need be done i s o b t a i n a s o l u t i o n f o r the r = 0 c a s e .  [1/27ri] ^ [ f ( z ) / z ]  i t can  dz = f(0)  be shown t h a t  1/2 7Ti ^  where f (z) M",  Since  (Za z )-*/z  dz = f (0)  (2  a h a s been already  s  5  a^z  so t h a t f (0) =  c  1. Therefore  11-12  becomes  defined  as  22  v-o  Tl  B(n-r) = E  for r = 0  w  11-14  Taking the complex conjugates of 11-13 and 11-14 we get  ^B(n-r)  Z  a„= P  w  B(n-r) a~ = 0  r = 0  II-15  r > 1  11-16  An e x p r e s s i o n f o r the power P(f) can a l s o be w r i t t e n by s u b s t i t u t i n g 11-10 i n t o I I - 7 t o give  P(f) = P ^ A t / j r a z £ a * z 5  11-17  5  s  S-.o  which can be r e w r i t t e n i n the f a m i l i a r  form  N  P(f) = P / ( 2 ^ 11 • 2 V  e x  N  P (-i2 77 f s o. t) | )  11-18  2  s-o  D. ^ t r a n s f o r m  approach  An a l t e r n a t i v e method to o b t a i n equations 11-15 and 11-16  i s through  a z-transform approach.  S u b s t i t u t i n g 11-14  and 11-10 i n t o I I - 6 g i v e s  s-o  S*o  = (V2f )2  E  N  <r) ~ zA  A.s-»0  KJ  ^  M u l t i p l y i n g by ^ a ^ z - ^ g i v e s  11-19  23  S=<5  TKO  A.---0O  1  = £  a*z-*£ S (n-r) z*-^  =2  [|<S(n-r)  -n-.o  A* ^  11-20  ] z-^  V\_ .  When s = 0 then  11-21 11*0  and  when s > 0, powers of z do not match so t h a t  0  =2  11-22  a^B (n-r)  Taking the complex conjugates of 11-21 and 11-22, equations 11-15 in  and 11-16 are again o b t a i n e d . W r i t i n g 11-15 and 11-16  matrix form  gives  I B (0)  B(1)  . B(H)  i 1 J  I B(1)  H(0)  .  I a, |  I 0 |  I . I  I • I  I • I  ! •I  B(N-1)  I I  .  I B(N)  .  I  B(0)  R(N-1)  11-23  0 I  which i s the eguation f o r o b t a i n i n g the N-t'h p o i n t prediction f i l t e r .  From 11-18 i t can be seen t h a t to o b t a i n a  s p e c t r a l estimate P (£), error f i l t e r  a l l that i s required i s a p r e d i c t i o n  and the e r r o r power c f p r e d i c t i o n . These twc  values are qiven by equation  11-23.  2H  E. H e u r i s t i c  Solution  From Ulrych e t a l . (1973) a h e u r i s t i c s o l u t i o n t o the maximum entropy problem can be r e p l i c a t e d . I f there e x i s t s a time s e r i e s  x (t)  X(tc),  which has a F o u r i e r transform  there i s some f i l t e r  i (t)  that whitens  x (t).  The maximally  white time s e r i e s t h a t corresponds t o a p a r t i c u l a r  power  spectrum has a l r e a d y been shown t o be the one t h a t  exhibits  maximum entropy. I f the t r a n s f e r f u n c t i o n o f  is  then | X ( L U ) I(u>)| e q u a l s a constant 'k». Also  I (to),  |X(«;)  is  i (t)  |2 = k V I K w )  I  2  a power e s t i m a t e . Any s t o c h a s t i c , n o n - d e t e r m i n i s t i c ,  s t a t i o n a r y time s e r i e s can be represented as a moving-average process.  x(t)  b  Q  = Z*>  T  e^_  s  > 0  b z  + b,2 + b^2 •+...< oo  0  e^. = a white noise s e r i e s .  Therefore wavelet  x ( t ) i s the c o n v o l u t i o n of some f i l t e r or b (t)  with a white noise s e r i e s  the  filter  t h a t whitens  if  a(t) * b(t) =  e(t). b(t)  - 1  is  x ( t ) . In other words,  £ (t) , then  x (t)  a d e c o n v o l u t i o n or s p i k i n g f i l t e r  * a(t) - e ( t ) .  a(t)  is  and can be determined from  25  .8 a •= 1  11-24  B i s an N by a = a, >a^ 1 =  The  N autocorrelation  ,...a  i s the p r e d i c t i o n e r r o r  n  filter  (1,0,0,.. .0)  prediction f i l t e r  written  matrix  (t) =  8(t)  - a (t)  so 11-24  can  be  as  =  ft"  —  P=  1 # -a | , — a ^, •. • — a  (P^,0,0,...0)  N  f»-l  P  w  -  r +2 Q  R  =  L  e  r  r  o  power of p r e d i c t i o n .  r  L-I  From the  power estimate  IX (w) 12 =  i t can  kVIKwil  2  be seen t h a t  u P(f) = P  /H  w  +2  which i s e q u i v a l e n t From the between HEM  r exp (-12 77fk) |2 K  to equation 11-28.  above d i s c u s s i o n , the fundamental d i f f e r e n c e  and  conventional  power s p e c t r a l a n a l y s i s can  be  seen. Maximum entropy s p e c t r a l a n a l y s i s gives the spectrum of a model t h a t f i t s  or approximates a f i t to the p r o c e s s , a  sample of which i s represented by the data. Conventional  26  f o u r i e r t r a n s f o r m methods, on the other hand, o b t a i n the spectrum  3.  of the sample.  LEVINSON  JJCOBSICN  Burg developed  a s o l u t i o n to eguation 11-23  based  upon  the L e v i n s o n r e c u r s i o n a l g o r i t h m . The s o l u t i o n i s a l s o c o n s i s t e n t with the concept of maximum entropy. The of  accuracy  Burg's method i s g r e a t e r t h a t t h a t of the normal Levinson about 1/3  method, and i t i s a l s o executed  faster.  The  development i s shown below. A prediction f i l t e r forward  i s run over the data s e t i n both  and r e v e r s e d i r e c t i o n s , and the power output  t h i s o p e r a t i o n i s minimized. ends of the d a t a , and so no  The f i l t e r  the  from  i s not run o f f the  assumptions about the  c o n t i n u a t i o n of the data o u t s i d e the sampling i n t e r v a l made. T h i s i s the major f e a t u r e t h a t i s r e l a t e d to  are  maximizing  the entropy of the process. For the two  point f i l t e r  running the f i l t e r  I/2(N-D  =  ( 1,Y),  the power output  from  over the data i s  i2  (x  d + (+  yx-)2  •  t-l  The  value of P^  can be shown to be a minimum when  = 2|'(x x )/|'(x.)2 i  U (  11-25  27  For t h e two p o i n t case, eguation 11-23 i s w r i t t e n as  | B(0)  I I Rd)  |  | 1| I Pol 1 1 1 = 1 1 E{0) I | y I I I B{1)  0  ^ i s obtained from  eguation 11-25. The f i r s t  autocorrelation  A;  value B(1)  B(0)  B(0) = 1/H ~Z. (x / ) , and sc  i s estimated by  2  L-i  can be c a l c u l a t e d from  be obtained from  B(1)  P^= B(0) (1 - ^  running the f i l t e r  and  P^ can  ) .  ( 1 , V,, ^ )  For the t h r e e p o i n t f i l t e r after  2  = - ^(B(0)),  the power output  over the data i s  *-a P~ =  (1/2 (N-2)) 2 { ( X  l + a  +X  l+l  V +X : j ^ ) 2 a  (  When ^ i s s e t egual to ]f(1 +^ ), where Y i s t h e f i l t e r 3  c o e f f i c i e n t o b t a i n e d f o r the two-point of  X j t h a t minimizes  { 1#  )fn  *Xi)'^3^  P  3  case, then the value  can be o b t a i n e d . The f i l t e r  w i l l a l s o be minimum phase. Eguation 11-23  f o r the t h r e e point case i s w r i t t e n as  28  1 B(0)  B<1)  B (2)  J  J 1 I  1 B(1)  B(0)  B(1)  |  | y | + Yjl Y  | B(2)  B(1)  E (0)  |  \ 0  I P^l  o | +yi  1  I  1 4,1 3  0 |  =  I  I  =  1 1 |  P I 3  1 0 | 1 0 |  I F I  ! 4*1  | 0 |  A  Zl = B{2) + B ( 1 ) ) f a  & -  -  v  p  *  The advantage of t h i s r e c u r s i v e method i s t h a t only one v a r i a b l e , i n t h i s case  }fs *  n  a  s  t  o  b  e  a d j u s t e d so as t o  o b t a i n the minimum power output. The r e c u r s i o n i s continued until  the d e s i r e d f i l t e r  the p r e d i c t i o n f i l t e r of  l e n g t h i s o b t a i n e d . The values f o r  { 1, x* , o" #» a  p r e d i c t i o n P , are then plugged N  # o ^ j ) and the e r r o r  poser  i n t o equation 11-18 t c  o b t a i n an estimate of the power P ( f ) . As Lacoss(1971) and Burg(1972) p o i n t e d out; the power estimate P(f) i s a c t u a l l y a s p e c t r a l d e n s i t y e s t i m a t e . I f a more r e l i a b l e estimate of the spectrum  (in terms of r e l a t i v e peak amplitudes)  the maximum entropy  i s required,  estimate should be i n t e g r a t e d .  4. ASJ'CREGRESSIVE MODELS AND MEM One  of the c r i t i c a l  entropy spectrum  f a c t o r s i n determining the maximum  i s d e c i d i n g upon which l e n g t h f i l t e r  29  operator  should  Too  an o r d e r l e a d s t o i n s t a b i l i t y  high  be used  i n the spectral  whereas t o o low an o r d e r some c a s e s  gives  prior'knowledge  of the spectrum  principle  of a u t o r e g r e s s i v e time  o f MEM,  autoregressive estimation  problem.  autoregressive  i s being a  who showed t h e modelling  and t h e on  c a n be a p p l i e d t o t h e f i l t e r  In p a r t i c u l a r  filter  length  t h e works o f  and G e r s c h ( 1 9 7 0 ) a r e worth n o t i n g . The  model f o r an o r d e r M, i s w r i t t e n  x^., ,x^. - , . . x^. . the  series  i t c a n be s e e n t h a t t h e p r e s e n t  series  (throuqh  the large guantity of l i t e r a t u r e  modelling  A k a i k e (1969* 1970)  quickly,  useful in picking a  Due t o t h e work o f VandenBos (1971),  equivalence  and  fairly  calculation.  a poorly resolved estimate. In  c o n v e n t i o n a l methods) c a n be v e r y order.  estimate  w  produced  x(t)  from t h e known p a s t  o f t h e time  values  a^ i s an a d d i t i v e w h i t e n o i s e  i n n o v a t i o n and h a s z e r o  the z - t r a n s f o r m  value  series  called  mean and a v a r i a n c e o f t Y ^ .  o f 11-26 i s taken,  2  the r e s u l t  If  c a n be w r i t t e n  as  X(z)  and  - X(z) [ ^ z  +\z* +...+  = A(z)  therefore  |X(z)|  2  = |A(z)|VI  1 -(k,z - ^ z  2  <^zV  H-27  30  Equation 11-27 can be r e w r i t t e n as  P(f).  = 2^/1  I  1 -£^.exp(-i2^fj)  2  which i s e q u i v a l e n t to 11-18, the e x p r e s s i o n f o r the maximum entropy spectrum,  with the e q u i v a l e n c e of a u t o r e g r e s s i v e  modellinq and MEM  shown, the f i n a l  of  prediction error  Akaike can be used to h e l p determine the f i l t e r  FPE i s d e f i n e d as the mean square p r e d i c t i o n  FPE = E [ (x  t  A  - x  t  The FPE can be shown to be  FPE^ » p  + (M~1)/N -  Order. The  error.  11-28  ) 32  = estimated p r e d i c t i o n of  x  criterion  x  f  (Ulrych and Bishop  1975)  (M + 1)} S 2  M i s the order of f i l t e r S  2  i s the minimum r e s i d u a l sum of squares M  In  many cases the FPE c r i t e r i o n  the f i l t e r  q i v e s an e x c e l l e n t i d e a cf  l e n g t h , but f o r the data being s t u d i e d i n t h i s  p r o j e c t , i t was found not to be as u s e f u l . In numerous examples a guess of the optimum order was made, i n some cases a f t e r s e v e r a l t e s t runs were performed. be g r e a t e r than set.  N/2  where  N  M was never taken t o  i s the l e n g t h of the data  31  5.  TIME ADAPTIVE  MEM  As s e l l as handling s t a t i o n a r y time s e r i e s , maximum entropy can be expanded i n t o the realm of n o n - s t a t i o n a r i t y i n space  and time. The  use of time-adaptive MEM  was  considered  here as a p o s s i b l e means of i d e n t i f i c a t i o n of the noise b u r s t s i n the data. These g l i t c h e s can be thought o f as r e v e r b e r a n t peaks superimposed on a n o n - s t a t i o n a r y time s e r i e s . Time a d a p t i v e MEM  i s a d e c o n v o l u t i o n scheme t h a t  designs a p r e d i c t i o n e r r o r f i l t e r i n order to l o c a t e areas of u n p r e d i c t a b i l i t y . Seismic a r r i v a l s i n r e f l e c t i o n r e c o r d s , and the n o i s e g l i t c h e s i n t h i s data s e t , can both be thought as r e g i o n s of u n p r e d i c t a b i l i t y . The  deconvolved  of  r e c o r d should  show s p i k e s c o r r e s p o n d i n g t o the l o c a t i o n of the  glitches,  and a f t e r t h i s i d e n t i f i c a t i o n has been made, the n o i s e r e g i o n s can be removed by some p r e d i c t i v e method. Normal time adaptive schemes are handled by time and two  1.  p o s s i b l e approaches can be  used.  The a u t o c o r r e l a t i o n l a g s f o r each gate can  be  c a l c u l a t e d , the p r e d i c t i o n e r r o r f i l t e r s can determined  gates,  be  from these l a g s using the Levinson  r e c u r s i o n a l g o r i t h m , and the f i l t e r s can then  be  a p p l i e d to the data.  2.  The  Burg approach can be used to c a l c u l a t e the  p r e d i c t i o n e r r o r o p e r a t o r s i n the time gates, and the f i l t e r can be a p p l i e d t o the data.  32  Burg (1971) has developed an a l t e r n a t i v e to the gating  approach, where the p r e d i c t i o n e r r o r c o e f f i c i e n t s  continuously  updated as the f i l t e r  This eliminates  the  only  data segments.  stationary  parameters d i s c u s s e d  of the  program l i s t e d  c o n t r o l s the l e n g t h  i s run  a c r o s s the  are  data.  problem o f p i c k i n g time gates t o cover  A b r i e f d e s c r i p t i o n o f the The  time  processing  are d e s c r i b e d  scheme f o l l o w s .  i n the comment s e c t i o n  i n APPENDIX 1. A v a r i a b l e NWABM  o f a warmup segment o f data. A p r e d i c t i o n  e r r o r operator i s c a l c u l a t e d on t h i s ' s t a t i o n a r y '  data  segment i n the same manner as f o r the normal Burg technigue where the f i l t e r This f i l t e r  length  i s s e t egual t o 'LCN + 1 ' p o i n t s .  i s then run over the f i r s t  the p r e d i c t i o n e r r o r o f the The  filter  * NHARM • 1* p o i n t .  i s new recomputed using  of the segment from  'NWARM' p o i n t s t o f i n d  only  the  process i s continued  'NWARM + 2* p o i n t i s then found. The  u n t i l the  end o f the  data segment i s  reached. I f the parameter 'IFIGAP' i s g r e a t e r only every *IFLGAP* point  the  pcints  'N8ARM - LCN +1* t o * NWARM + 1'. The  p r e d i c t i o n e r r o r f o r the  filter  'LCN*  than 1, then  i s used i n the c a l c u l a t i o n of the  and the d e t e r m i n a t i o n o f the  prediction errors.  p r e d i c t i o n e r r o r t r a c e f o r every •IFLGAP* p o i n t  c a l c u l a t e d , intermediate p r e d i c t i o n error values  After  has been  are  computed;,There i s a l s o a parameter *NZC* which c o n t r o l s number o f r e f l e c t i o n c o e f f i c i e n t s s e t t o z e r o . e f f e c t of creating a f i l t e r distance,  while s t i l l  that  This  the  has the  p r e d i c t s f u r t h e r than u n i t  maintaining minimum phase requirements.  33  As d i s c u s s e d e a r l i e r , the optimum p r e d i c t i o n filter  error  i s the-one with the minimum output power as i t i s run  over the data. For the time a d a p t i v e case, the minimum average squared power must be s e t by the c o n s t a n t updating of the  filter  the  s t a t i o n a r y MEM  filter  c o e f f i c i e n t s . A r e c u r s i v e scheme as o u t l i n e d f o r  from the  N  i s u t i l i s e d to o b t a i n the  N+1  point  point f i l t e r . Only one parameter  C^ is  a d j u s t e d t o minimize the average sguared error.  corresponds t o a r e f l e c t i o n c o e f f i c i e n t i n an  i d e a l l y l a y e r e d medium. Burg (1972) has shown that the average power i s  CV(2N)} f [ F . (k) + C B ^ ( k ) ] 2  P^ =  d  + [Cj F (k) + Bj (k) rf  which i s the sum  ]2  of the sguared forward and backward  p r e d i c t i o n e r r o r s . The value c f power  j> ±  which minimizes the  i s given as  which i s the n e g a t i v e average cross-power of the forward and backward p r e d i c t i o n e r r o r , d i v i d e d by the averaged auto-powers  of the forward and backward p r e d i c t i o n e r r o r s .  handle n o n s t a t i o n a r i t y the cross-powers and auto-powers weighted so t h a t the s t a t i s t i c s c l o s e t o the p o i n t  To  are  predicted  34  will  be more The  important  than  statistics  further  w e i g h t i n g i s e x p o n e n t i a l and i s both  away. time  and  space  variable  H e i g h t (k,m)  The  of it  C:  -  r r  relaxation  time  % -  relaxation  distance  time  between  i s 1/e  speedy  , t h e u s e o f MEM  allows calculation  A"Fortran  #1.  exp(-kdT/  %=  relaxation  tradeoff  =  IV  listing  m/^  )  (seconds) (traces)  t h e decay  time.  Since  a d a p t i o n and a c c u r a t e i s particulary  of coefficients o f t h e program  there i s a  computation  advantageous on s h o r t  i s included  data in  i n that segments. Appendix  35  III.  DATA ANALYSIS PROCEDURES  As d i s c u s s e d i n Chapter  I I , two major problems e x i s t  with the time s e r i e s under i n v e s t i g a t i o n . One i s the f a c t i  that the onset of the f r e e o s c i l l a t i o n i n f o r m a t i o n i n t h i s data corresponds difficulty  t o the.beginning  i s the presence  of the r e c o r d . The second  of high amplitude  which occur i n the f i l t e r e d  noise g l i t c h e s  r e c o r d s . Both of these  can be overcome by use of p r e d i c t i o n f i l t e r s  problems  designed  with  the Burg a l g o r i t h m .  1. NECESSITY OF TAPERING In order t o accomplish  freguency  domain f i l t e r i n g , i t i s  e s s e n t i a l t h a t t h e data s e t be zero mean and tapered i n some manner before t r a n s f o r m a t i o n to the freguency  domain.  Otherwise unnecessary  discontinuities  n o i s e power due t o s t e p  i n the data w i l l be added t o the spectrum. I f a data s e t i s not zero mean and tapered, i t can be broken down i n t o two p a r t s , one being the s i g n a l i n f o r m a t i o n , and t h e other being a D.C.  component. T h i s D.C. s i g n a l i s i n  e f f e c t a step f u n c t i o n which r e p r e s e n t s unwanted i n f o r m a t i o n . The F o u r i e r t r a n s f o r m of a step f u n c t i o n  u (t) = 1/2 + 1/2 sgn (t) is  <p [ u ( t ) ] = ifSlcu) + 1/itu  36  Therefore  a step f u n c t i o n w i l l add a D.C.  as a freguency  component, as  dependent component that i s higher at  f r e g u e n c i e s . Therefore  well  low  the n e c e s s i t y of removing the mean and  t a p e r i n g the data s e t before transforming  to the  freguency  domain, i s c l e a r .  2. J?lJDICTION Since the f r e e o s c i l l a t i o n onset of the r e c o r d , any  i n f o r m a t i o n s t a r t s at the  t a p e r i n g w i l l remove d e s i r e d  i n f o r m a t i o n . To e l i m i n a t e t h i s problem the data s e t p r e d i c t e d forward i n f o r m a t i o n was was  designed  original  backwards i n time so t h a t  preserved.  The  500  necessary  point p r e d i c t i o n f i l t e r  using the Burg maximum entropy  algorithm.  The  5320 p o i n t s were p r e d i c t e d up t o 8092 p o i n t s , which  represents and  and  was  an extension  of the data  backwards. T h i s r e c o r d was  b e l l taper  (Figure  1386  p o i n t s both  then tapered  with a 5%  forward cosine  #3).  3. FILTERING Two original  filtered  data s e t s were obtained  data. One  c u t o f f freguency used to study  was  a high pass f i l t e r e d r e c o r d with a  of 0.66  c y c l e s per hour. T h i s r e c o r d  the fundamental f r e e o s c i l l a t i o n  associated spectral s p l i t t i n g . pass f i l t e r e d  from the p r e d i c t e d  The  hour. T h i s band passed data was  modes and  second data s e t was  r e c o r d with c u t o f f s of 0.-12  and  was  1.20  the  a band  c y c l e s per  used f o r the i n v e s t i g a t i o n of  Predicted forward  and  gravimeter backward  f i l t e r l e n g t h was  500  in  data time.  p o i n t s and  were added i n each d i r e c t i o n . The the o r i g i n a l record i s i n d i c a t e d .  extended Prediction  1386  points  p o s i t i o n of  38  <  Prediction  FIGURE 3  4 0 0 0 yqals  39  the  core undertones. Both f i l t e r i n g o p e r a t i o n s were performed i n the  freguency domain by a smoothed boxcar o p e r a t o r . The seguence of f i l t e r  1.  multiplication  design i s as f o l l o w s :  The boxcar f u n c t i o n with the d e s i r e d c u t o f f s i s c o n s t r u c t e d i n the freguency domain.  2.  T h i s boxcar i s then transformed to the time domain and the time f u n c t i o n i s t r u n c a t e d and tapered to some d e s i r e d  3.  length.  T h i s t r u n c a t e d time f u n c t i o n i s retransformed back to the freguency domain, r e s u l t i n g i n a smoothed boxcar. The amount of smoothing  i n the freguency  domain depends upon the amount the time f u n c t i o n has been t r u n c a t e d i n the time domain. The l o n g e r the time domain f u n c t i o n , the sharper the c u t o f f i n the freguency domain.  F i g u r e #4 i s the high pass f i l t e r e d  p r e d i c t e d r e c o r d and  F i g u r e #5 i s the band passed p r e d i c t e d r e c o r d . The noise g l i t c h e s are immediately e v i d e n t i n both data s e t s , and can be seen to have very high amplitudes i n r e l a t i o n to the s i g n a l . A comparison  should be made between F i g u r e #4 and  F i g u r e #6, which i s the high pass f i l t e r e d  not p r e d i c t e d  uo  High pass f i l t e r e d  predicted record.  The  c y c l e s per hour.  The  are immediately e v i d e n t ,  and  filter  c u t o f f i s at 0.66  noise  glitches  have a s e l l d e f i n e d  character.  12 h r s  FIGURE 4  1  0  0  /QaiS  Band pass f i l t e r e d p r e d i c t e d f i l t e r c u t o f f s are at 0.12  and  record. 1.20  The  cycles  per hour. G l i t c h e s are o b s e r v a b l e , but not well d e f i n e d  as f o r the  high  passed  record.  as  43  High  pass f i l t e r e d  r e c o r d . The  t a p e r i n g , which i s  accomplish evident there  the  filtering,  at the onset is  amplitude.  unpredicted  a  large  of  the  reduction  original  necessary is  to  immediately  record,  where  i n the  signal  45  | 12 h r s  FIGURE 6  100 )/gals  46  Alaska  r e c o r d . T h i s i s the record t h a t would r e s u l t from  t a p e r i n g and f i l t e r i n g the t i d a l r e c o r d before p r e d i c t i o n forward  and  backward i n time. The  d i f f e r e n c e i n the  i n f o r m a t i o n a v a i l a b l e at the beginning obvious,  and  of the r e c o r d i s  much b e t t e r s i g n a l t o noise r a t i o s would be  expected i n the power spectrum.  4. DEGLITCHING As seen on both F i g u r e s #4 and g l i t c h e s are present i n the f i l t e r e d  #5,  high amplitude  r e c o r d s . These g l i t c h e s  c o n t r i b u t e a s i g n i f i c a n t amount c f noise i n t h e range 0.0  A.  t o 2.0  c y c l e s per hour  (Wiggins and  frequency  Miller  1972).  Identification The  first  step i n e l i m i n a t i o n of these g l i t c h e s i s  i d e n t i f i c a t i o n . For t h i s r e c o r d v i s u a l i d e n t i f i c a t i o n i s o l a t e most of the noise areas at l e a s t f o r the record  (Figure #4).  can  high-passed  For the band passed r e c o r d , the g l i t c h e s  do not have the same w e l l d e f i n e d c h a r a c t e r and  are harder  to  spot, p a r t i c u l a r l y i n the e a r l i e r part of the r e c o r d . To a i d i n t h i s i d e n t i f i c a t i o n , i t was u s e f u l to run the time-adaptive records  thought t h a t i t would MEM  r o u t i n e over the  be filtered  to see i f r e g i o n s of u n p r e d i c t a b i l i t y were l o c a t e d .  As mentioned i n Chapter I I , the g l i t c h e s should  show up  areas of r e l a t i v e l y l a r g e p r e d i c t i o n e r r o r . A f t e r experimenting  with the l e n g t h of the  filter  as  47  operator,  and  the  time r e l a x a t i o n f a c t o r  values  100  and  5.0  rule  of  f o r determining  (ZTAU), r e s p e c t i v e  were c h o s e n . T h e r e i s no these  values,  although  make some r e c o m m e n d a t i o n s f o r ZTAU. F i g u r e s the  p r e d i c t i o n e r r o r t r a c e s f o r the  passed  r e c o r d s . The  first  a warm-up p e r i o d where t h e some i n i t i a l up  value.  into  up  very  B.  P r e d i e t i o n Into In  prediction  prediction  segment, t h i s  gap  segments of d a t a is  thus  achieved.  on  one  coefficients  However i n s t e a d o f  and  hoped  #7  show  band represent  are  set  to  are  a l s o warm-  record reguired i t s f o r , the  glitches  glitches  from  records, a  demonstrated  used. I t i n v o l v e s the  filter  #8  show  Gaps  t o remove t h e s e  was  and  examples.  scheme b a s e d  C l a y t o n (1975)  #7  passed  Figure  fast  B u r g (1972) does  coefficients  C on  segments. As  w e l l i n both  order  filter  long l e n g t h of t h i s  three  and  p o i n t s o f each r e c o r d  A r e a s B and  s e c t i o n s , as the  division  100  high  hard  using the  designing the  filter  the by  Ulrych  calculation Burg on  of  algorithm.  data  only i n  p r e d i c t i o n method c r e a t e s a f i l t e r that are  not  connected.  An  and  averaging  one  frcm effect  Time adaptive p r e d i c t i o n e r r o r t r a c e high  pass  show up as time  f i l t e r e d predicted areas  of  data.  of  Glitches  unpredictability.  The  r e l a x a t i o n f a c t o r , ZTAO, equals 5.0 and  the f i l t e r  length  equals 100 p o i n t s . Areas A,  B, and C are f i l t e r prediction  occurs.  warmup s e c t i o n s ,  where no  49  A  k  ^  T B  4-  12 hrs  FIGURE 7  100 ygal  50  Time adaptive p r e d i c t i o n e r r o r t r a c e band  pass  filtered  p r e d i c t e d data. The  r e l a x a t i o n f a c t o r , ZTAU, equals 5,0 f i l t e r l e n g t h equals  100  points.  and  of time the  51  J L .  1  FIGURE 8  7 5  KS ' 3  5  52  Once the f i l t e r data such t h a t the s i d e s . The and  p r e d i c t i o n i s done i n t o the  p r e d i c t i o n s are then tapered  solid  (0.05  l i n e . The  middle 120 first  and  and  0.06  Hz.)  gap  frcm both  #9,  a harmonic  with 20% noise i s shown b j the  dashed l i n e shows the p r e d i c t i o n of  p o i n t s of data using only i n f o r m a t i o n l a s t 40  p o i n t s of data. The  s t u d i e s by U l r y c h and  the  with a ramp f u n c t i o n  combined i n a f i n a l r e s u l t . In F i g u r e  signal  and  has been designed, i t i s run over  Clayton  the  from  the  r e s u l t i s g u i t e good  (1975) have shown that  the  improvement i n the s p e c t r a l r e s o l u t i o n i s e x c e l l e n t . The  r e s u l t s f o r d e g l i t c h i n g both the high  passed  and  band passed data s e t s can be seen i n F i g u r e s  #10  and  #11.  p r e d i c t i o n e r r o r t r a c e s f o r these d e g l i t c h e d  records  are  shown i n F i g u r e s  #12  improvement i n the  «  D e g l i t c h i n g Ey  #13.  There i s a remarkable  r e c o r d s , and  both the d e g l i t c h e d  c  and  and  t h i s can  be seen v i s u a l l y i n  prediction error traces.  Envelope  An a l t e r n a t i v e method f o r d e g l i t c h i n g was Wiggins  (personal communication), and  a p p l i e d to the f i r s t  suggested  t h i s technigue  3000 p o i n t s of the  high  passed  D i v i d i n g the data by the sguare of i t s envelope thought of as-a p o s s i b l e means of removing the amplitudes of the noise b u r s t s , even though the content would not was  The  by  was record. was  high freguency  be a l t e r e d . A m o d i f i c a t i o n of t h i s scheme  applied, in that a function E  1  was  created  where  Test case of the.gap p r e d i c t i o n r o u t i n e . A  mixed  0.06  Hz.  harmonic  l i n e represents and  0.05  with 20S added n o i s e . The  r e p r e s e n t s the o r i g i n a l  40  of  last  data, and  solid the  and line  dotted  the p r e d i c t e d data. The  first  40 p o i n t s were used to p r e d i c t  the middle 120 p o i n t s . The chosen to be  Hz.  35.  filter  length  was  54  a oi  FIGURE 9  FIGURES JO AND  1J. Deglitched  filtered  high  predicted  pass  record.  and  band  Glitches  pass were  i d e n t i f i e d , then p r e d i c t e d out using a f i l t e r operator  determined from the Burg  algorithm.  56  I 12 h r s  FIGURE 10  1 0 0 ygals  57  1IG0HES 12 A N D _13_. Prediction deglitched predicted  high  error passed  traces and  records.  band  The  u n p r e d i c t a b i l i t y have been l a r g e l y by the d e g l i t c h i n g  prccess.  of  areas  both passed of  eliminated  59  -wW  *<—r>  vr>^*~  y  B  c -,/-v>v-V-  1 2 h r s  1  FIGURE  12  100  ygals  FIGURE 13  7 5 ygals  61  E' =  E/W.I. E = envelope  of the process  H.L. = some p r e d e t e r m i n e d  In  t h i s case,  weighted to  t h e water l e v e l  1  the d a t a . function  1.0.  The  E  1  was s e t e g u a l  b u t was  of t h e r e c o r d so as  amplitude  i n this  section of  above t h e w a t e r l e v e l , t h e to  E/W.L. I f any d a t a  the f u n c t i o n  E*  = 1.0 i f E» < 1.0  E«  = E/I.L.  E*  value  fell  was s e t e g u a l t o  i f E» > 1.0  was t h e n  amplitude  divided  was v e r y  a n o t i c e a b l e drop  POHEJ S P E C I E J L  »  Peridogram In  by  (E*)  2  and t h e r e d u c t i o n i n  s u c c e s s f u l ( F i g u r e #14). T h e r e i s  i n the noise content  i n the frequency  (see C h a p t e r I V ) .  5.  the  parameter.  i.e.  domain  both  fell  t h e water l e v e l ,  glitch  A  the high s i g n a l  I f any d a t a  record  also  was n o t c o n s t a n t  exponentially a t the beginning  correspond to  below  water l e v e l  MillSIS  and MEM  Spectra  the c a l c u l a t i o n  o f power s p e c t r a f o r t h i s  c o n v e n t i o n a l and MEM  s p e c t r a were used.  project,  In a l l cases,  p e r i o d o g r a m s were were n o t smoothed. The p e r i o d o g r a m  was  D e g l i t c h e d high passed p r e d i c t e d  record.  D e g l i t c h i n g was accomplished by d i v i d i n g record  by  the  sguare  of i t s envelope. The  amplitudes of the g l i t c h e s have been reduced,  but  the  the  freguency  g l i t c h e s i s s t i l l evident.  greatly  content of the  63  1  FIGURE  14  1  0  0  /9  als  64  extremely u s e f u l i n h e l p i n g to determine the p r e d i c t i o n e r r o r f i l t e r MEM  the optimum order of  needed, i n the c a l c u l a t i o n of the  s p e c t r a , as i t gave p r i o r knowledge of the r e s u l t . In  almost a l l c a s e s , the HIM  s p e c t r a f o r each f r e e  oscillation  mode were c a l c u l a t e d s e v e r a l times u s i n g d i f f e r e n t  filter  o r d e r s and data l e n g t h s . In most cases the FPE gave no realistic  i n d i c a t i o n of the f i l t e r  B. Complex In  order t o be  used.  Demodulation  the a n a l y s i s of the s p e c t r a of i n d i v i d u a l f r e e  o s c i l l a t i o n modes, a technigue known as complex was  demodulation  used. T h i s method i n v o l v e s a s h i f t i n g of the freguency of  i n t e r e s t down to and f i n a l l y  1  u> = 0",  then low pass f i l t e r i n g the d a t a ,  resampling the data at a c o a r s e r i n t e r v a l . T h i s  method i s p a r t i c u l a r l y  u s e f u l when c a l c u l a t i n g MEM  spectra,  as fewer p o i n t s are used i n the c a l c u l a t i o n of the c o e f f i c i e n t s . Two  1.  Roundoff  sources of e r r o r are thus e l i m i n a t e d :  errors leading to inaccuracies  with harmonic  2.  filter  particularly  terms.  Order of magnitude d i f f e r e n c e s between the power of d i f f e r e n t freguency components which r e s u l t s i n numerical i n s t a b i l i t i e s  (Olrych and Bishop  The technigue of complex demodulation  i s expressed  1975).  65  schematically  below.  Suppose a time s e r i e s x(t) i s comprised of two freguency bands, and i t i s necessary t h a t  peak  0  A  The complex time s e r i e s  o f peak  -LO  8  C(t) •= ( x ( t ) , i O ) i s m u l t i p l i e d by A  *A*  •A»  w  exp ( - i ^ n a t ) , where cu i s the angular freguency and  multiplication peak  ou-.o  - u><i-ui  the f a c t o r  be i s o l a t e d .  8  ft -co  'A*  a t i s the sampling i n t e r v a l . This  s h i f t s peak  »A'  t o <w=0, and the n e g a t i v e  to ou - -2«v».  CO  " A  The data can then be low pass f i l t e r e d  so as to only r e t a i n  the freguency band around ^ = 0.  -co  Resampling  ui-.o  i n the time domain can now  a l i a s i n g , and a s t a b l e MEM  be performed  with no  spectrum can be c a l c u l a t e d f o r the  66  freguency component of i n t e r e s t . T h i s above o p e r a t i o n C h i l d e r s and Pao  can be expressed mathematically.  (1972) show t h a t i f a time s e r i e s c o n s i s t s  of a s e t of t r a n s i e n t harmonics  f ( t ) = [e-^-TVV  sinco (t^T)] 0  e-<^~ Vc = t r a n s i e n t decay f a c t o r T  then m u l t i p l y i n g  f (t) by the demodulation f a c t o r  Q-cto -k 0  gives  f (t) = { e-<*" >/r/2i [e °<*- > 7  =  T h i s new can  e-H-T)/t/2i  series  du,  [e-  t a ,  f ( t ) has a  T  e T  (e-t^Vr  /2i)  e  e-^to -t 0  Ca3 e  ° ], r  2 oo freguency component which G  be f i l t e r e d out to leave  f(t) =  - e - ^ o < £-r> ] - i o v i j  |e-^ ) r  67  Iv. RESQITS  1. FREE 0SCI1L&TICN MODE The  d e g l i t c h e d and  FREQUENCIES  u n d e g l i t c h e d high pass f i l t e r e d  s e t s were used i n the examination  of f r e e  m u l t i p l e t s and the a s s o c i a t e d s p l i t t i n g Periodogram and MEM data l e n g t h s and The  c a l c u l a t e d using the f i r s t  0^=  cu+ 0  "V  s  «3o=  parameters. various  orders.  values f o r the s p l i t t i n g  given by Backus and  oscillation  s p e c t r a were c a l c u l a t e d using  filter  data  parameter * ft * were  order s p l i t t i n g  Gilbert  approximation  (1961), where  mfiJX  split  peak  freguency  c e n t r e peak  freguency  m = splitting  order number  /S = s p l i t t i n g  parameter  r o t a t i o n r a t e of the earth  The f o l l o w i n g s e c t i o n s d e t a i l and c a l c u l a t e d s p l i t t i n g s p h e r o i d a l modes S ^ , 0  Q  values  parameters f o r the fundamental  S ', 3  the measured freguency  QS^,  E  S  S  , and  E  S . Comparisons are 6  a l s o made between the s p e c t r a f o r u n d e g l i t c h e d , d e g l i t c h e d , undemodulated, and unenhanced data s e t s . Remarks are a l s c made about the e f f e c t of varying the f i l t e r  order f o r  68  c a l c u l a t i o n of HEM  s p e c t r a , and the higher r e s o l u t i o n  c a p a b i l i t y of MEM  s p e c t r a as compared to periodogram s p e c t r a .  In a l l cases,•the  power s p e c t r a shown were c a l c u l a t e d from  the complex demodulated data s e t s .  A.  mode The  periodogram s p e c t r a f o r the u n d e g l i t c h e d  and  d e g l i t c h e d r e c o r d s are shown i n F i g u r e 15a and F i g u r e The  n o i s e l e v e l s are i n d i c a t e d , and the decrease  power was c a l c u l a t e d to be 5 db. For the spectrum  (Figure 15a)  15b.  i n noise  undeglitched  i t i s o n l y p o s s i b l e to i d e n t i f y the  peaks, whereas the d e g l i t c h e d spectrum ±1 peaks as w e l l as the +2  (Figure 15b)  ±1  shows the  peak.  F i g u r e 15c i s the periodogram power of the unenhanced (not predicted) high pass f i l t e r e d r e c o r d  (Figure 1 ) . T h i s  spectrum shows lower s i g n a l t c n o i s e l e v e l s than r e s u l t of i n f o r m a t i o n l o s s at the beginning The  advantage of p r e d i c t i v e f i l t e r i n g  r e c o r d can  15a, as a  of the r e c o r d .  i n the a n a l y s i s of the  be seen.  Figure 15d i s shown j u s t as a check of the complex demodulation t e c h n i g u e .  For a periodogram t h e r e should be  no  d i f f e r e n c e between the spectrum c a l c u l a t e d f o r the demodulated data and  the one  c a l c u l a t e d f o r the undemodulated  data. I f F i g u r e s 15a  and  are compared, t h i s  15d  can be seen. F i g u r e s 15e and  15f show MEM  d e g l i t c h e d r e c o r d with orders of 63 and  equivalence  s p e c t r a f o r the 150 r e s p e c t i v e l y . The  69  g r e a t e r d e t a i l i n 15f i s the r e s u l t of choosing filter  order.  A p o s s i b l e -2 peak has a l s o been i d e n t i f i e d  t h i s p l o t . F i g u r e s 15e  and  15f show c o n s i d e r a b l y l e s s  and b e t t e r r e s o l v e d peaks than F i g u r e 15g, spectrum  a higher  (order=110) f o r the undeglitched  I n t e r e s t i n g l y enough, the MEM (order=45) , and  F i g u r e 15i  spectra, Figure  r e s o l u t i o n c a p a b i l i t y of the MEM c  S  ±2 peaks,  increased  s p e c t r a l e s t i m a t i o n method.  mode, the • ar=0 peak cannot be observed, as i t ,  a  1200  (Figure 15j) shows only  the ±1 peaks. T h i s i s a good example of the  For the  15h  (order=55) , f or the f i r s t  periodogram  HEM  set.  p o i n t s of the r e c o r d , seem to show both the ±1 and whereas the corresponding  noise,  which i s the data  on  1  corresponds to a nodal p o i n t l o c a t e d at l o s Angeles. Therefore, in  the ±1, and ±2 peaks are a l l t h a t w i l l be  t h i s r e c o r d . The  freguency  of the  '0' peak i s c a l c u l a t e d  as the average of the ±1 peaks. Table f r e q u e n c i e s observed f o r the S ^  TABLE  Periodogram 5320 P o i n t s freg,, Jcy^hrJ. +1 1.130 ± 0 1. 11*4 ± -1 1.097 ±  .001 .001 .001  1 details  mode.  C  observed  JM  Ondeglitched & .360  ±  .007  .432  ±  .007  15a  the  70  Periodogram  5320 P o i n t s  f r e g . l£XZhLk ± ± ± ±  Deglitched  15b  &  +2 +1 0 -1  1.159 1.129 1.114 1.099  .002 .001 .001 .001  .528 ± .008 .336 ± .007  MEM  order=63 5320 P o i n t s freg. Jcy/hrl  .384 ± .007 Deglitched &  +2 1.158 ± .002 +1 1.130 ± .001 0 1.115 ± .001 -1 1.099 ± .001  .516 ± .008 .360 ± .007  MEM  order=J[50 5320 P o i n t s freg_. J c y / h r ) .  Deglitched g  +2 +1 0 -1 -2  1.157 1.130 1.115 1.099 1.074  .504 ± .360 ±  .008 .007  .384 ± .492 ±  .007 .008  MEM  ± ± ± ± ±  .384 ±  .002 .001 .001 .001 .002  order=110  freg  A  5320 P o i n t s  Jcy2kEl  ~  +2 1.159 ± .001 +1 1.130 ± .001 0 1. 114 ± .001 -1 1.098 ± .001 MEM  order=45  +2 1.164 ± .002 1.129 0 1.115 -1 1.100 -2 1.073  ± ± ± ±  .002 .001 .001 .002  .007  Undeglitched  J5f  15g  ~& .528 ± .005 .360 ± .007 .408 ± .007  1200 P o i n t s  f reg.. SciZhLi. +1  15e  Undeglitched  & .588 ± .008 .336 ± .011 .360 ± .007 .504 ± .008  JJ5h  71  order=55 J200 P o i n t s IregA J c y ^ h r l " +2 + 1 0 -1 -2  1.162 1. 134 1.117 1.100 1.078  ± ± ± ± ±  Undeglitched  .002 .002 ^001 .001 .002  Periodogram 1200 "~ l £ S g i c y / h r l  Points  ± .001 ± .001 ± .002  4Z§£§gg V a l u e s from fregj. i S l i h r l +2 +1 0 -1 -2  ± ± ± ± ±  1.160 1.131 1. 115 1.099 1.075  Average  @  MEM  The e r r o r s g i v e n deviations.  Gilbert  .439  project. for  1,115  .007 .008  .384  ±  .007  .384  ±  .011  ±  .540 .374  ± .035 ± .046  .379 .480  .020 ± .032  splitting  c y c l e s per  However, t h e  of the  are  « ^ ' on  the  m o d e ,  0  parameter. Derr  standard  1.116  and  cycles  a value  hour and  0.439 o b t a i n e d 1  As  in  of  average this  t h a t were c a l c u l a t e d a i g g i n s and  a dependence o f t h e  splitting  of  (1969) gave v a l u e s  0.397. T h e s e compare t o t h e  values f o r * 0  t h e r e a p p e a r s t o be  parameter  ±  .081  e a c h peak were n e t c o n s t a n t .  noted,  15j  Backus(1965) gave a v a l u e o f  c y c l e s p e r h o u r and  values of  ± ±  -0-  per hour f o r the freguency  1.114  .360 .444  f o r the average values  and  0.397 f o r t h e  .008 .011  spectra  .003 .002 .001 .001 .003 overall =  ± ±  Undeglitched _£_  A  +1 1.131 0 1.115 -1 1.099  .564 .455  J5i  o r d e r number  Miller  (1972)  splitting  'm*.  The  latter  a.  Periodogram  5320 p o i n t s  Undeglitched  b.  Periodogram  5320 points  Deglitched  c.  Periodogram  5320 p o i n t s  Unenhanced  d.  Periodogram  5320 p o i n t s  Undemodulated  €.  HEM order=63  5320 p o i n t s  Deglitched  f.  MEM  order=150  5320 p o i n t s  Deglitched  g.  MEM  crder=110  5320 p o i n t s  Undeglitched  h.  MEM crder=15  1200 p o i n t s  Undeglitched  i.  MEM  1200 p o i n t s  Undeglitched  j.  Periodogram  1200 p o i n t s  Undeglitched  order=55  73  •x.  ZD CC  f—  CJ". lUc.Q_  LD  a  LLJ LO •—ITJ-  5 ^ >z ce o  r I.oi  T 1.05  r  1.09  FIGURE 15a 1.13  1.17  FREQUENCY IN CY/HR  ZD OC  ^3<o  LJo" Cu  to  O UJ in I—tit  <£° o  r  1.01  -i 1.05  1— 1.09  1 1.13  FREQUENCY IN CY/HR  r  1.17  FIGURE 15b 1.21  I—  1.01  T 1.05  1 1.09  r 1.13  FREQUENCY IN CY/HR  i 1.21  FIGURE 15d  75  03  3= ZD CC CJ" UJo" D_ CO 3  Q UJ CO >—  d° z: cc  o z:  o o"  1.0)  r——  -i 1.D5  1.D9  1  —i—  ].)3  FIGURE 15e 1.21  I.17  FREQUENCY IN CY/HR  UJo" Q_  to a UJ  co l—IT z: cc: o CM  a a"  -2 - 1  1.05  —  -1 FIGURE 15f  ! —  1 .09  FREQUENCY IN CY/HR  1.21  76  77  78  authors gave values f o r  » of 0.408, 0.410, and 0.522 f o r  the -1, +1, and +2 peaks, which compare reasonably the values of 0.379, 0.374, and 0.540 obtained  B.  w e l l to  here.  „ -S„3 mode o The  periodogram power spectra, f o r the u n d e g l i t c h e d and  d e g l i t c h e d high passed records are shown i n f i g u r e s 16a and 16b.  As observed  f o r the  0  mode,  there i s a very  n o t i c e a b l e r e d u c t i o n i n the noise content f o r the d e g l i t c h e d r e c o r d . F i g u r e 16c i s the MEM spectrum  (order=130) f o r the  u n d e g l i t c h e d data s e t , and should be compared to F i g u r e s 16d, 16e,  and 16f, which are the MEM s p e c t r a f o r the corresponding  d e g l i t c h e d data  (orders 70, 130, and 150). The e f f e c t of  v a r y i n g the l e n g t h of the c a l c u l a t e d p r e d i c t i o n operator can be seen i n these l a s t three f i g u r e s , where the s p e c t r a l r e s o l u t i o n i n c r e a s e s as the order of the f i l t e r  increases.  However, going t o too high an order w i l l r e s u l t i n an u n s t a b l e spectrum. T h i s i s s t a r t i n g t o happen with the 150 p o i n t MEM  spectrum, as the n o i s e i s beginning  to take on a  very s p i k y appearance. D e g l i t c h i n g the r e c o r d by d i v i d i n g by the sguare of the envelope i s demonstrated i n the next two f i g u r e s , 16g and 16h.  Figure 16g shows the periodogram power of 3000 p o i n t s of  undeglitched  data, and 16h shows the periodogram power f c r  the d e g l i t c h e d 3000 p o i n t segment. There i s a n o t i c e a b l e r e d u c t i o n i n the noise content  using t h i s method of removing  79  the  glitches,  method  of  Ey in  the  16f, which  spectrum of t h e shows  ± 1 , ± 2 , and a p o s s i b l e observed  freguency  ± ± ± ± ±  .001 .001 .001 .001 .001  order=V30  -3  peak.  5320 P o i n t s  1.704 ± .001 1.694 ± .001 1.685 ± .001 1.680 ± .001 1.667 ± .001  1.705 1.691 1.687 1.682 1.669  predictive  ± ± ± ± ±  .001 .002 .001 .001 .001  Table  #2  #2  Deglitched &  J6b  J6c  Deglitched  ~& .216 ± .192 ±  .005 .007  .144 .228  .007 .005  ± ±  Deglitched &  i s  displayed  o f the major  .144 ± .007 .192 ± .005  MEM order=70 5320 Points Jreg.. J c y / h r l +2 •1 0 -1 -2  mode  3  .204 ± .005 .216 ± .007  f r e g ^ SZlZhEL +2 +1 0 -1 -2  S  multiplets.  Periodogram 5320 P o i n t s freg. Jcy/hrl 1.703 1.695 1.686 1.680 1.670  C  the s p l i t t i n g  TflELJ  +2 •1 0 -1 -2  as the  deglitching.  f a r , the best  figure  into  but i t i s not as e f f e c t i v e  J6d  .228 ± .005 .120 ± .011 .096 ± .007 .204 ± .005  details  peak some o f  80  MEM  order=J30 ± ± ± ± ± ±  5320 P o i n t s  Deglitched .216 ±' .005 . 168 ±- .007  +2 +1 0 -1 -2 -3  1.704 1.693 1.686 1.680 1.670 1.665  MEM  order^l50 5320 P o i n t s Ireg. Jcjr/hrl  Deglitched  +2 +1 0 -1 -2 -3  1.704 1.694 1.686 1.681 1.671 1.665  .216 ± .005 .192 ± .007  ± ± ± ± ± ±  .001 .001 .001 .001 .001 .001  . 144 ± .007 .192 ± .005 . 168 ± .004  .001 .001 .001 .001 .001 .001  1.705 1.694 1.687 1.682 1.673 1.667  ± ± ± ± ± ±  +2 •1 0 -1 -2 -3  1.704 1.693 1.686 1.681 1.669 1.665  Average 0  ± i ± ± ± ±  per  0 .005 .007  .096 ± .007 . 156 ±- .005 . 152 ± .006 spectra  .001 .001 .001 .001 .002 .001 o v e r a l l = . 177 ±  Gilbert  Undeglitched .228 ± .168 ±  .001 .001 .001 .001 .001 .002  A v e r a g e V a l u e s f r o ni MEM f£§3i JSlZferl  .007 .005 .004  . 120 ± .180 ± .168 ±  Periodogram 5320 P o i n t s freg^. J c ^ h r l +2 +1 0 -1 -2 -3  J6e  .220 ±- .007 .168 ± .033 .126 ± .024 .201 ± .020 . 168 ± - .009 .040  and Backus (1965) gave  hour f o r t h e freguency  of  D  S  3  a value  o f 1.688 c y c l e s  and a v a l u e  c f .1839 f o r  81  IIGUEE 16 a.  Periodogram  5320 p o i n t s  Undeglitched  b.  Periodogram  5320 p o i n t s  Deglitched  c  MEM  order=130  5320 p o i n t s  Undeglitched  d.  MEM  order=70  5320 p o i n t s  Deglitched  e.  MEM  order=130  5320 p o i n t s  Deglitched  f.  MEM  order=150  5320 p o i n t s  Deglitched  9*  Periodogram  3000 p o i n t s  Undeglitched  h.  Periodogram  3000 p o i n t s  Deglitched  83  03  ZZi  CC  UJa" CL. CO O UJ CO >r. cc o  o"  FIGURE 16c 1.62  1.58  1 . 6 6  1.7  N  FREQUENCY IN CY/HR  1 . 7 4  1.78  TI  zz> cc I— < o O  UJo'  Q_  CO O UJ CO  »—irt  5?° CC  o V -2 -1 -1 +2 I  1.5B  •  1 1 . 6 2  r-J—  1.66  i  1.7  FREQUENCY IN CY/HR  FIGURE 16d 1.74  1.78  84  or L3<°. UJo"  cn  cu  a  in  UJ ex  a: o  1.58  r1  62  1  1.66 1 -7 FREQUENCY IN CY/HR  11  lis 1  1 62  111,1 1.7 FREQUENCY IN CY/HR 1.66  1.74  , 1.74  1.78  , 1.78  FIGURE 16e  FIGURE  16f  86  the s p l i t t i n g  parameter. Derr  (1969) gave values of 1.687  c y c l e s per hour and .187. These compare with the average values of 1.686 and .177 obtained i n t h i s p r o j e c t , k dependence of * ^ * on »m' i s again  C.  observed.  S„ mode O-Cf.  ;  The periodogram power s p e c t r a f o r the u n d e g l i t c h e d and d e g l i t c h e d h i g h passed 17b.  record are shown i n F i g u r e s 17a and  S p l i t t i n g can be observed  i n both, and although  i d e n t i f i c a t i o n of the number of the s p l i t peaks i s d i f f i c u l t , a fairly  good e x p l a n a t i o n of the peaks has been made. F i g u r e  17c i s the periodogram spectrum of the unenhanced d a t a , and if  t h i s i s compared with F i g u r e 17a, t h e advantage of  extending  the data s e t before t a p e r i n g can be seen.  F i g u r e s 17d and 17e are MEM s p e c t r a of the u n d e g l i t c h e d (order=130) and the d e g l i t c h e d (order=150) r e c o r d s . D e g l i t c h i n g by d i v i d i n g by the sguare o f the envelope record i s i l l u s t r a t e d  of the  i n F i g u r e s 17f and 17g, where 17f i s  the periodogram power c f 3000 p o i n t s of u n d e g l i t c h e d data, and  17g i s the power of the corresponding  There i s however, very l i t t l e  d e g l i t c h e d data.  d i f f e r e n c e between these two  s p e c t r a , i n d i c a t i n g that t h i s method o f removing the noise g l i t c h e s i s freguency some o f the observed  dependent. Table #3 g i v e s values f o r freguency  splits.  5320 g o i n t s f reg. J c y / h r l +4 +2 +1 0 -1 -3 MEM  2.359 2.343 2.335 2.328 2.321 2.309  Deglitched ~&  186 ± .005 180 ± .008 168 ± .007  .002 .002 .001 .001 .001 .002  order=130  168 ± .007 152 ± .006  5320  Points  Ondeglitched  ~  Ireg. J c y ^ h r l  +4 •2 +1 0 -1  2.359 2.341 2.335 2.328 2.321  MEM  order5l50 5320 P o i n t s f reg.. j c y / h r j  Deglitched  +4 +2 +1 0 -1 -3  2.359 2.341 2.333 2.328 2.321 2.310  .186 .156  ± ± ± ± ±  ± ± ± ± ± ±  f reg.. +4  +2 +1 0 -1 -3  2.359 2.341 2.334 2.328 2.321 2.310  Average  The  . 186 ± .156 ± .168 ±  .001 .001 .001 .001 .001  17d  .003 .005 .007  . 168 ± .007  .001 .001 .001 .001 .001 .001  Average Values  17b  . 120  ± ±  17e  .003 .005  .007  .168 ±- .007 .144 ± .004 from MEM  spectra  SQlZhLl ± ± ± ± ± ± 0  .186 ± .003 .156 ± .005 .144 ± .034  .001 .001 .001 .001 .001 .001  .168 ± . 144 ±  o v e r a l l = .161  value  f o r the  Q  .007 .004  ± 0.021 •  S„ centre  freguency  given by  88  a.  Periodogram  5320 p o i n t s  Undeglitched  b.  Periodogram  5320 p o i n t s  Deglitched  *  Periodogram  5320 p o i n t s  Unenhanced  d.  HEM order»130  5320 p o i n t s  Undeglitched  e.  MEM  5320 p o i n t s  Deglitched  £.  Periodogram  3000 p o i n t s  Undeglitched  g.  Periodogram  3000 p o i n t s  Deglitched  c  order=150  89  90  91  93  G i l b e r t and Backus (1965) i s 2.335 c y c l e s per hour and Derr (1969) gave a value o f 2.327 c y c l e s per hour. Derr a l s o gave a value of .103 f o r the s p l i t t i n g parameter which i s s m a l l e r than  the value of .161 obtained  D.  o-$  here.  S„ mode  The  periodogram poser  s p e c t r a f o r the u n d e g l i t c h e d and  d e g l i t c h e d high passed data are shown i n F i g u r e s 18a and 18b. There i s a n o t i c e a b l e improvement i n the s i g n a l to noise r a t i o , but i d e n t i f i c a t i o n o f i n d i v i d u a l peaks remains a difficult  matter. In a l l the s p e c t r a observed,  symmetrical these  s p l i t s sere observed,  are blended  two major  and i t I s b e l i e v e d t h a t  +1, +2, +3 and - 1 , -2,,and -3 s p e c t r a l  l i n e s , which are not p r o p e r l y r e s o l v e d . T h e r e f o r e , f o r the 0  §  was  G  mode, only t h e *m=0' freguency taken  as t h e midpoint  was determined, and i t  of the two major peaks on each  spectrum examined. Figure 18c i s the MEM u n d e g l i t c h e d data MEM  (order=130) spectrum of the  s e t and F i g u r e 18d shows the corresponding  (crder=130) spectrum f o r the d e g l i t c h e d data. The n o i s e  r e d u c t i o n due t o d e g l i t c h i n g i s evident i n the MEM s p e c t r a as w e l l as the periodograms. F i g u r e 18e i s the periodogram power f o r a 760 p o i n t u n d e g l i t c h e d data segment. The observed frequencies f o r the S . 0  g  mode are l i s t e d  i n Table #4.  94  HSU iii £sriodogram  5320 P o i n t s  Undeglitched  18a  lUllhLl  f reg..  0 3.028 ± .002 £S£io^£3ilS 5320 P o i n t s jE£Sai l£lZhLl  Deglitched  18b  0 3.027 ± .002 I I S order=J30 5320 P o i n t s f£§g. i c y ^ h r l  Undeglitched  18c  0 3.029 ± .002 MEM order=130 5320 P o i n t s freg.. Jcvyhr),  Deglitched  1.8d  0 3.028 ± .002 Periodogram ££§3.2.  760 P o i n t s  Undeglitched  18e  Jcy/hrl  0 3.028 ± .002 Average Values from MEM s p e c t r a f r eg.. J c y ^ h r l 0 3.029 ± .002  T h i s average value f o r the c e n t r e freguency of the 0  m o d e  compares with t h a t of 3.034 c y c l e s per hour given  by G i l b e r t and Backus by Derr  (1969).  (1965) and 3.027 c y c l e s per hour given  95  IJGUEE 18 a.  Periodogram  5320 p o i n t s  Undeglitched  b.  Periodogram  5320 p o i n t s  Deglitched  c.  HEH  order=130  5320 p o i n t s  Undeglitched  a.  HEH. o r d e r = 1 3 0  5320 p o i n t s  Deglitched  e.  Perioaogram  760 p o i n t s  Undeglitched  96  97  98  99  E.  S  Q  mode  6  f i g u r e s 19a and 19b show the periodogram power s p e c t r a of the undeglitched  and d e g l i t c h e d high passed data. The  noise r e d u c t i o n due to removing the g l i t c h e s i s not as n o t i c e a b l e as f o r other f r e e o s c i l l a t i o n evidence  f o r s p l i t t i n g i n the d e g l i t c h e d spectrum, but the  splitting are MEM  number cannot be determined. F i g u r e s 19c and 19d  power s p e c t r a f o r the u n d e g l i t c h e d  deglitched o f MEM  modes. There i s some  (order=110) and  (order=55) r e c o r d s . The high r e s o l u t i o n c a p a b i l i t y  i s illustrated  i n Figures 19e and 19f. The f i r s t i s a  p l o t of the periodogram power of a 760 point data segment, and the second i s an MEM  undeglitched  (order=10) spectrum  f o r the same data segment. The i n c r e a s e d r e s o l u t i o n i s g u i t e obvious. Table the  Q  S  fe  #5 gives the observed  center  values f o r  freguency.  TABLE #5  Periodogram 5320 P o i n t s freg^ Jcy/hri  Ondeglitched  J9a  0 3.739 ± .004 £e£i°dojjr.am. 532.0 P o i n t s ~ treg."jc3i/hrl ~ 0 3.739 ± .004  Deglitched  19b  100  MEM orderfjHO 5320 P o i n t s fregT J c y / h r l  OndeoJ-itched  19c  0 3.74 0 ± .001 MEM ord«r=55  5320 P o i n t s  Deglitched  19d  Undeglitched  J9e  0 3.740 ± .001 Periodogram 760 P o i n t s fregj, J c y / h r l 0 3.738 ± .003 MEM ordexfJHD 760 P o i n t s fl§g. J c y ^ h r l  Undeglitched  19f  0 3.743 ± .001  MSiaaS Values from JEM s p e c t r a freg.. J c y ^ h r l  0 3.741 ± .002  T h i s average value compares with the value of 3.749 c y c l e s per hour g i v e n by G i l b e r t and Backus c y c l e s per hour given by Derr  (1965) and 3.735  (1969).  2 i CORE UNDERTONES A f u r t h e r g o a l o f t h i s t h e s i s was t o see i f the d e g l i t c h i n g o f the band passed Alaska r e c o r d could r e s u l t i n the i d e n t i f i c a t i o n of any f r e e o s c i l l a t i o n ' s o f the e a r t h ' s i n n e r c o r e . The p e r i o d range s t u d i e d was r e s t r i c t e d t o between 0.833 and 8.333 hours.  8.333 hours was chosen as the  lower c u t o f f p e r i o d , as i t was f e l t t h a t the overpowering  101  I2GURJ 19 a.  Periodograo  5320 p o i n t s  Undeglitched  b.  Perioaogram  5320 p o i n t s  Deglitched  c.  HEM oraer=110  5320 p o i n t s  Unaeglitchea  a.  MEM order=55  5320 p o i n t s  Deglitched  f.  Perioaogram  760 p o i n t s  Undeglitched  MEM order=10  760 points  Undeglitched  102  FIGURE 19b  FREQUENCY IN CY/HR  103  105  presence of the  d i u r n a l and  contaminate  record  The  the  both before  Figures  and  20  deglitched  21  records  Removing t h e  spectrum  unexpected and  a t the bursts  i n the  at the  peaks i n both  after  f o r the  lower  of the  exact  solar tide,  and  frequency  There are  an  domain  noticeable  become much more o b v i o u s i n The  periods  of  i n t e g e r m u l t i p l e s of the extend  scale.  t i m e domain r e s u l t e d i n  deglitched record.  peaks are  and  the  freguency  power i n t h e  they  were  undeglitched  of the  freguencies.  s p e c t r a , but  and  glitches.  l o c a t i o n of  upper end  noise  bandpassed  removing the  spectra  i n the  was  power s p e c t r a  r e s p e c t i v e l y . The  noise  amazing r e d u c t i o n especially,  and  t i d e s would  periods.  periodogram  show t h e  mode i s marked  lunar  at longer  unsmoothed  calculated  the  semidiurnal  predicted interpolated record  conventional,  S  the  up  t o a b o u t 0.65  these  basic  cycles  per  hour. An  initial  r e a c t i o n was  overtone e f f e c t , h a r b o u r s and high based  order  such  basins.  that these  as s h a l l o w  water t i d e s f o u n d  However, i t i s v e r y  ocean t i d e ? e f f e c t s w i l l  gravimeter.  Because of the  high  to  represent  undertone o s c i l l a t i o n s .  core  explanation linearities, reported The  by  S%  be  peaks r e l a t i v e  0  is that  , i t i s even  they  are  or barometric  due  power f o r t h e  doubtful that  detected amplitudes  by  The  most  a  of  these  land these they  reasonable  to e i t h e r instrument effects,  tidal  in  more u n l i k e l y t h a t  pressure  Warburton et a l .  noise  p e a k s were some  such as  nonthose  (1975). undeglitched  spectrum  goes  from  106  FIGURES 20 AND  21 Periodogram  undeglitched  power  (Figure  (Figure  21)  band  record.  The  filter  1.20 0  S^  cycles free  ter-diurnal  20)  pass  oscillation (M3)  and  of  both  deglitched  filtered  predicted  c u t o f f s a r e a t 0.12  p e r h o u r . The  tide  spectra  locations mode  a r e marked.  of and  and the the  o o lO O  M  o  Q.  a  cc z: cc o ID  o oo  0.125  -1 0.225  1 — 0.325  —1 0.425  0.525  0.625  FREQUENCY  0.725  (CY/HR)  0.825  0.925  -1 1.025  1.125  1.225  109  40.0 x 10*-at .125 Hz. t o 5 x 10 1  6  a t .725 Hz. The n o i s e power  f o r t h e d e g l i t c h e d spectrum goes from 12 x 10* at .125 Hz. to 2 x 10» at .725 Hz. T h i s drop i n noise power i s 5.2 db. at .125 Hz.  and 4.0 db. at .725 Hz.  110  V. SUMHAJY  A t i d a l gravimeter  r e c o r d i n g from t h e 1964 Alaska  earthquake was s u b j e c t e d to v a r i o u s data p r o c e s s i n g t e c h n i q u e s . i n : a n attempt t o improve the q u a l i t y of the r e c o r d . An e f f o r t  was a l s o made t o t r y and l o c a t e core  undertone o s c i l l a t i o n s . The f i n d i n g s of t h i s p r o j e c t can be summarized as f o l l o w s :  1.  A time a d a p t i v e p r e d i c t i o n e r r o r t r a c e provides a u s e f u l t o o l f o r i d e n t i f y i n g and l o c a t i n g n o i s e b u r s t s i n time series  2.  The method o f removing the unwanted noise by p r e d i c t i v e d e g l i t c h i n g , r e s u l t s i n an e x c e l l e n t improvement of the s i g n a l t o noise r a t i o of t h i s data s e t .  3.  The maximum entropy  method o f s p e c t a l e s t i m a t i o n g i v e s  some e x c e l l e n t r e s u l t s f o r the power s p e c t r a of v a r i o u s f r e e o s c i l l a t i o n modes. U n f o r t u n a t e l y , e s t i m a t i o n of the l e n g t h o f the f i l t e r  operator r e g u i r e d i n the s p e c t r a l  c a l c u l a t i o n remains a d i f f i c u l t t a s k . P r i o r knowledge o f the spectrum through is  a conventional s p e c t r a l  technique,  very u s e f u l i n a i d i n q i n the determination of the  optimum f i l t e r  order.  111  In s e v e r a l i n s t a n c e s , the maximum entropy  method of  s p e c t r a l e s t i m a t i o n gave good r e s u l t s f o r the of v a r i o u s f r e e o s c i l l a t i o n  modes. C e r t a i n l y more s p l i t  peaks were observed using MEM periodogram approach. The 0  S , a  0  S ;  oS^ , Sg-, and  3  0  2.328, 3.029, and The  splitting  are .439,  as opposed to the  c e n t r a l f r e g u e n c i e s f o r the  the S Q  fe  modes are  0  and  1. 115,  a  C  S^,  and S Q  modes  f  .161.  f o r core undertones was  discovered  the a n a l y s i s of the band pass f i l t e r e d r e c o r d . freguency  1.686,  c y c l e s per hour r e s p e c t i v e l y .  parameters f o r the S ,  .177,  No evidence  3.741  splitting  band below 0,12  during The  c y c l e s per hour i s s t r o n g l y  dominated by t i d a l f r e g u e n c i e s , and the band from to  0.65  c y c l e s per hour i s dominated by e i t h e r  instrument effects.  n o n - l i n e a r i t i e s or barometric  pressure  0,12  112  REFERENCES Akaike,  H. 1 9 6 9 . Eower spectrum e s t i m a t i o n through a u t o r e g r e s s i v e model f i t t i n g . Ann. Instjs, S t a t i s t . . Vol. 21, p. 4 0 7 .  Akaike,  H.  1970. listi  JLSSJL  Alsop,  S t a t i s t i c a l predictor i d e n t i f i c a t i o n . Statist,. J a t h V o l . 2 2 , p. 2 0 3 . A f  A. L i , G . H. Sutton, and M. Ewing. Measurement of Q f o r very long p e r i o d f r e e oscillations. JAGJ_RJ_, V o l . 6 6 , p. 6 3 1 .  1961.  Alterman, Z. S., Y, Eyal, and A. fl. Merzer. 1 9 7 4 . Cn f r e e o s c i l l a t i o n s of the e a r t h . G e o p h y s i c a l Surveys, V o l . 1, p. 4 0 9 . Backus,  G. and F . G i l b e r t . 1 9 6 1 . The r o t a t i o n a l s p l i t t i n g of the f r e e o s c i l l a t i o n s of the e a r t h . Froc Nat Acad.. Sci.. OJ.SJ.A_. , V o l . 4 7 , p. A  Backus,  A  362.  G. and F. G i l b e r t . 1 9 6 7 . Numerical a p p l i c a t i o n s of a formalism f o r g e o p h y s i c a l i n v e r s e problems. Geophysic.. Jour. Roy. a s t r . Soc.., V o l . 13,  p.  2777  ~~  ~  ~  Benioff, H. F., B. Gutenberg, and C. F. R i c h t e r , 1954. Progress r e p o r t , S e i s m o l o g i c a l L a b o r a t o r y , C a l i f o r n i a I n s t i t u t e of Technology. Transj. Anu Geophys,. On., V o l . 3 5 , p. 979., Benioff, H. F., F. P r e s s , and S. Smith. 1 9 6 1 . E x c i t a t i o n of the f r e e o s c i l l a t i o n s o f the e a r t h . Jjj.Gj.Ri, V o l . 6 6 , p. 6 0 5 . Bogert,  B. P. 1 9 6 1 . An o b s e r v a t i o n of f r e e o s c i l l a t i o n s of the e a r t h . J.G.R., V o l . 6 6 , p. 6 4 3 .  Bolt,  B. A., and R. G. C u r r i e . 1 9 7 5 . Maximum entropy e s t i m a t e s o f earth t o r s i o n a l e i g e n p e r i o d s from 1 9 6 0 T r i e s t e data. Geophysic. Jour. Roy a s t r . Soc.,, Vol. 40, p. 1077. " A  Burg,  J . P. and D. C. R i l e y . 1 9 7 1 . Time and space adaptive d e c o n v o l u t i o n f i l t e r . S.E.G., P r e p r i n t .  Burg,  J . P. 1 9 7 2 . The r e l a t i o n s h i p between maximum entropy and maximum l i k e l i h o o d s p e c t r a . Ge oj_ hj_si cs , Vol. 37, p. 3 7 5 .  Burg,  J . P. 1 9 7 5 . Maximum entropy s p e c t r a l Unpublished Ph.D. T h e s i s , S t a n f o r d .  analysis.  113  Childers, B. G. and Min-Tai Pao. 1972. Complex demodulation fox t r a n s i e n t wavelet d e t e c t i o n and extraction. IAEXEAIA on Audio and Ii§£troacoustics, V o l . AU-20, p. 295 C r o s s l e y , D. J. GeoDhysic. 200." Derr,  1975. Jour..  J . S. 1969. 1968. Bull A  Core undertones with r o t a t i o n . Boy. astr. Soc,, V o l . 42, p.  f r e e o s c i l l a t i o n o b s e r v a t i o n s through Seism Soc. Am., V o l . 59, p. 2079. A  Dziewonski, A. M. and F. G i l b e r t . 1973. Observations of normal modes from 84 r e c o r d i n g s of the Alaskan earthguake of 1964 March 28. Geo£hysie Jour^ Eoy astr. Soc,, V o l . 35, p. 401. A  x  Gersch,  H. 1970. E s t i m a t i o n of the a u t o r e g r e s s i v e parameters of a mixed a u t o r e g r e s s i v e moving average time s e r i e s . l i J i l i J . T r a n s on Automatic C o n t r o l , Vol. AC-18, ~p~ ~267. A  G i l b e r t , F. and G. Backus. 1965. The r o t a t i o n a l s p l i t t i n g of the f r e e o s c i l l a t i o n s of the e a r t h . Beyiews of Geg£hysics, V o l . 3, p. 1. Haddon,  B. A. and K. E. B u l l e n . 1967., D e r i v a t i o n of an e a r t h model from f r e e o s c i l l a t i o n d a t a . Proc Hat. Acad. S c i . JJ S.A , V o l . 58, p. 846." x  A  A  Jackson, B. V. and L. B. S l i c h t e r . 1974. The r e s i d u a l d a i l y earth t i d e s . JLs.G.B., V o l . 79, 1711.  p.  Lacoss,  B. T. methods.  Lamb,  H.  Love,  A. E. H. 1911. Some problgms o f ggodynamics. Cambridge U n i v e r s i t y P r e s s , Cambridge.  Ness,  N. F;, J . C. H a r r i s o n , and L. B. Slichter. 1961. Observations of the f r e e o s c i l l a t i o n s of the earth. J.iG B., V o l . 66, p. 62.  1882.  1971. Data adaptive s p e c t r a l a n a l y s i s GeofhjjsJ.cs, V o l . 36, p. 661. Proc.  Math.  Soc..,  Vol.  13,  p.  189.  A  Press,  F. 1968. inversion.  Earth models obtained by Monte C a r l o j3.G B., V o l . 73, p. 5223. A  Shannon, C. E. 1948. A mathematical theory of communication. B e l l . Syst Tech* Jour., p. 379. A  Vol.  27,  114  Slichter, I . B. 1961. The fundamental f r e e mode of the earth's inner core. P r c c Nat.. Acad.. Sci.. U^S^A.., Vol. 47, p. 186. ~~ A  S l i c h t e r , 1. B. 1967a. Free o s c i l l a t i o n s of the e a r t h . D i c t i o n a r y of Geophysics, ed. S. K. Runcorn, Pergamon P r e s s , Oxford. Slichter, L. B. 1967b. S p h e r i c a l o s c i l l a t i o n s of the earth. Geophysic... Jour. Roy., a s t r Soc., V o l . 14, p. ~ 1 7 l T x  Smith,  M. L. 1974. The normal modes o f a r o t a t i n g e l l i p t i c a l e a r t h . Ph.D. T h e s i s , P r i n c e t o n . U l r y c h , T. J . , D. E. Smylie, 0. Jensen> and G. K. C. C l a r k e . 1973. P r e d i c t i v e f i l t e r i n g and smoothing of s h o r t r e c o r d s by using maximum entropy. JJLGJUIA# Vol. 78, p. 4959. Ulrych,  T. J . , and 1. N. , Bishop. 1975. Maximum entropy s p e c t r a l a n a l y s i s and a u t o r e g r e s s i v e decomposition. Reviews of Geophysics and Space Physics, V o l . 13, p. 183.  Ulrych,  T. J . , and B. 8. C l a y t o n . 1975. Time s e r i e s modelling and maximum entropy. Preprint.  VandenBos, A. 1971. A l t e r n a t i v e i n t e r p r e t a t i o n of maximum entropy s p e c t r a l a n a l y s i s . IJJSAJSAIIA Trans Infcrm.. Theory J C o r r e s p l j . , V o l . I T - l 7, p. 493. A  Harburton, B. J . , C. Beaumont, and J . M. Goodkind. 1975. The e f f e c t of ocean t i d e l e a d i n g on t i d e s of the s o l i d e a r t h observed with t h e superconducting gravimeter. Geophysics. Jourj. Bgy astr.. Soc.__, Vol. 43, pT~ 707." x  Weiner,  N. 1930. G e n e r a l i z e d harmonic a n a l y s i s . Math.. V o l . 55, p. 117.  Acta  A  Wiggins, R. A., and S. P. M i l l e r . 1972. New noise r e d u c t i o n technigue a p p l i e d t o long p e r i o d o s c i l l a t i o n s from the Alaskan earthquake. Bull. Seism. Soc. Am.., V o l . 62, p. 471.  115  APPENDIX 1 F o r t r a n source l i s t i n g of the time adaptive maximum entropy i  d e c o n v o l u t i o n scheme.  116 SUBROUTINE  BAFL(LOUT,CX,IFLGAP,LCN•NWARM,ISTRT,ZTAU,XTAU,NZERO)  C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  80  SUBROUTINE BAFL THE BURG A D A P T I V E F I L T E R L A D D E R : A N A D A P T I V E OR T I M E V A R Y I N G F I X E D L E A D P R E D I C T I O N ERROR PROCESSOR A D J U S T M E N T OF EACH R E F L E C T I O N C O E F F I C I E N T I S MADE E V E R Y JUMP S T A T E A T T E M P T I N G TO M I N I M I Z E THE S T A G E OUTPUT POWER INPUTS X ( L X ) = I N I T I A L DATA LCN=LAST NON-ZERO R E F L E C T I O N C O E F F . I F L G A P = N U M B E R OF G A P S BETWEEN F I L T E R C O E F F I C I E N T S S E T T I N G I F L G A P = 1 DOES NOT GAP THE SPECTRUM AND T R I E S TO O P E R A T E ON THE E N T I R E S P E C T R U M . NWARM= CURAT ION OF S T A T I O N A R Y C Y C L E I S T R T = S T A R T CF S T A T I O N A R Y GATS Z T A U = T E M P O R A L R E L A X A T I O N FACTOR COMMON / B L K 3 / X U 7 7 4 ) X T A U = S P A T I A L R E L A X A T I O N FACTOR OUTPUTS X t L O l T ) = F O R W A R D . ERROR P R E D I C T I O N T R A C E C ( L C N ) = F O R W A R D S T A T E VECTOR B ( L C N ) = B A C K W A R D S T A T E VECTOR C ( L C N ) = R E F L E C T I O N C O E F F . AT EACH S T A G E CX=R E F L E C T ION C O E F F . I N T E G R A T E D IN S P A C E AND D E N ( L C N ) = S T A G E AUTOPOWER N U M ( L C N ) = STAGE CROSSPOWER  TIME  C A L L I N G B A F L F I R S T S E T S UP THE L O O P I N G AND P A S S I N G A R R A Y S FOR THE P A R T I C U L A R PROBLEM AS S P E C I F I E D BY LCNGIFLGAP. THEN IT COMPUTES A SHORT!LENGTH=NWARM) ESTIMATE OF THE R E F L E C T I O N C O E F F . S E R I E S IN ORDER TO START THE A D A P T I O N OUT WITH SOME R E A S O N A B L E NUMBERS THEN IT LOADS UP THE CX A R R A Y WITH THE I N I T I A L V A L U E S AND P A S S E S INTO ENTRY ' B A F L G O * THE U S U A L ENTRY IS B A F L GO WHICH F I R S T I N I T I A L I Z E S THE BACKWARD ARRAY THEN P A S S E S TO T H E MAIN R O U T I N E THE CX S E R I E S I S UPDATED E V E R Y I F L G A P DATA P O I N T S AND IN THE I N T E R M E D I A T E S T E P S THE OUTPUT ARE I N T E R P O L A T E D OF P R O C E S S E D AS THOUGH THE R . C . WERE S T A T I O N A R Y W R I T T E N BY J . P . BURG REAL N U M ( 2 0 O ) , D E N { 2 0 0 ) , B { 2 0 0 ) , F ( 2 0 0 ) t C ( 2 0 0 ) , E M ( 2 0 0 0 ) , *EP(20 0C),CX(LCN,LOUT),A(200) OAT A A , B , C / 6 0 0 * 0 . / FTEST=1. LCNP1=LCN+1 LCNP2=LCN+2 IFGM1=IFLGAP-1 LBSP=LOUT-IFLGAP NZERP1=NZER0+1 NEND=LCN-NZERP1 DO 80 K = L B S P , L O U T EP(K) = 0.  117  10  c  12  11  33  e c  c cc c  1000  c c  1010  B E G I N S T A T I O N A R Y WARMUP DO 10 I=1,NWARM EMU) = X(I*IFLGAP+ISTRT) EP(I) = X(I*IFLGAP+ISTRT) DO 1 1 J = 2 , L C N P 1 DEN(J) = 0. N U M ( J ) = Oc DO 12 I = J , N W A R M DEN(J) = DEN(J)+EP(I)*5P{I)+EM(I-J+i)*EM(I-J+l) N U M U ) = N U M U ) + EP ( I ) * E M ( I - J + l ) DIV=NWARM-J+1 NUM{J) = N L M ( J ) / D I V DEN(J) = DEN(J)/DIV C U ) = -2.*NUMlJ)/DEN( J) DO 11 I=J,NWARM EPI=£P(I> EP(I) = EPI+CUJ*EM{ I-J+l) KS=IFLGAP*LCN+1 EM(I-J+1) = EM(I-J+l)+C(J)*EPI DO 8 J = 1 , L C N DO 8 K = 1 , K S C X ( J , K ) = C(J+1) DO 3 3 J = 1 , L C N DEN { J ) = D E N U + l ) NUMU) = NUMU + l ) END WARM UP C Y C L E SET R E L A X A T I O N T I M E S DLX=EXP(-1./XTAU) DL=EXP(-1./ZTAU) DRX=1.-DLX DR=lo-DL -USUAL  ENTRY-  ENTRY B A F L G O I N I T I A L I Z E BACKWARD VECTOR B(l) = X(KS-IFLGAP) A(l) =l . DO 1 0 0 0 J = 2 , L C N B(J) = X ( K S - J * I F I G A P > A(J) = 0.0 DO 1 0 0 0 I=2,J A(I) = A(I )+CX(J,KS)*A{J-I + l ) 3(J) = B U ) +A C I ) * X ( K S + { I - J + 1 ) * I F L G A P ) B E G I N MAIN LOOP DO 5 0 0 0 K = K S t l O U T , I F L G A P Z=X(K) DO 1 0 1 0 J = l t N Z E R P l FIJ) = Z DO 2 0 0 0 J = N Z E R P 1 , L C N DEN { J ) = ( F ( J ) * * 2 + B ( J ) * * 2 ) * D R + D £ N ( J ) * D L  200C  3000  4000 5000 C C C C  6000 70C0  8000  8050 9000 9050  NUM(J) = F{J)*B(J)*DR+NUM{J)*DL I F ( F T E S T . L E . l . l ) CX(J,K)=-2.*NUM{J)/DEN(J) C X ( J t K ) = -2o*DRX*NUM{JJ/DEN(J}+CX{J,K)*DLX F(J+1) = F { J ) + C X ( J t K ) * B { J ) X(K) = F(LCNPl) DO 3000 JR = 11NEND J=LCN-JR BlJ+1J=B(J)+CX(J K)*F{ J) IF(NZERO.EG.O) GO TQ 5000 DO 4000 JR=1,NZER0 J=NZERP1~JR BCJ+l) = B(J) B(1)=Z t  END  OF MAIN LOOP  NOW GO BACK TO F I L L IN THE I F ( I F L G A P . E Q . l ) GO TO 9050 DO 9000 L = l t I F G M l KSTART=KS+L DO 9000 K=KSTART»LOUT,IFLGAP KC=K-L Z=X(K) DO 6000 J = l , N Z E R P l F(J) = Z DO 7000 J=NZERP1,LCN F ( J + 1 ) = F< J ) + C X ( J,KC)*B('J) XlK) = F ( L C N P l ) DO 8000 JR=1»NEND J=LCN-JR B(J+1) = B(J )+CX(J,KC)*F(J) IF(NZERO.EC.O) GC. TO 9000 DO 8050 JR=1,NZER0 J=NZERP1-JR B(J+l) = B ( J ) B ( l )= Z FTEST=FTEST+1. RETURN END  GAPS  

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