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Phase and amplitude variation of Chandler wobble Linton, John Alexander 1973

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PHASE AND A M P L I T U D E VARIATION OF  CHANDLER  WOBBLE  by  John Alexander B . A . Sc.,  Linton  U n i v e r s i t y of B r i t i s h C o l u m b i a ,  1970  A T h e s i s S u b m i t t e d in P a r t i a l F u l f i l m e n t of the R e q u i r e m e n t s f o r the D e g r e e  of  M a s t e r of Science  i n the  Department of  Geophysics  W e a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the required  standard  T h e U n i v e r s i t y of B r i t i s h C o l u m b i a May,  1973  In p r e s e n t i n g an  advanced  the  Library  I further for  this thesis  i n partial  degree a t t h e U n i v e r s i t y shall  f u l f i l m e n t o f the requirements f o r of British  make i t f r e e l y a v a i l a b l e  agree that  permission  Columbia,  f o rreference  f o r extensive  copying  I agree  that  and s t u d y .  of this  thesis  s c h o l a r l y p u r p o s e s may b e g r a n t e d by t h e Head o f my D e p a r t m e n t o r  by  h i s representatives.  of  this thesis  written  I t i s understood  f o rf i n a n c i a l gain  permission.  Department  G>e<?£?/^cj.-r/c  of  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Date  shall  Z7o^7<e  f  ,  5  Columbia  that  copying  or publication  n o t be a l l o w e d w i t h o u t  my  i  ABSTRACT  Normally,  the w o b b l e o f the e a r t h h a s b e e n d e a l t w i t h i n a  m a n n e r that a s s u m e s that the two m a i n p e r i o d i c c o m p o n e n t s h a v e c o n s t a n t p h a s e and a m p l i t u d e . parameters  An  initial assumption  can v a r y with time.  to d e t e r m i n e  o f t h i s t h e s i s i s that b o t h t h e s e  A technique of p r e d i c t i v e filtering is u s e d  the C h a n d l e r c o m p o n e n t o f the w o b b l e f r o m b a s i c l a t i t u d e  m e a s u r e m e n t s at the f i v e I. L . S. o b s e r v a t o r i e s .  A simple analytic proce-  d u r e i s e m p l o y e d to o b t a i n the p h a s e a n d a m p l i t u d e v a r i a t i o n o f the p e r i o d i c Chandler motion.  The  r e s u l t s i n d i c a t e that m a j o r c h a n g e s i n b o t h p h a s e  a m p l i t u d e o c c u r i n the p e r i o d associated  with earthquake  1922. 7 to 1949.  T h e s e changes are  and  possibly  a c t i v i t y , a l t h o u g h t h e r e i s n o t h i n g to i n d i c a t e  that t h e r e i s a c o r r e l a t i o n b e t w e e n i n d i v i d u a l e a r t h q u a k e s a n d e v e n t s i n the Chandler motion.  The  c a l c u l a t e d p e r i o d o f the C h a n d l e r w o b b l e i s 437  a n d the d a m p i n g t i m e i s so u n c e r t a i n unlikely.  days  that a v a l u e a p p r o a c h i n g i n f i n i t y i s n o t  ii  T A B L E OF C O N T E N T S  Chapter I  II  Page INTRODUCTION  1  E A R L Y HISTORY OF W O B B L E R E S E A R C H  2  The Annual Wobble The Chandler Wobble Chandler Damping  3 7 10  Chandler Excitation  12  SUMMARY  16  THEORY OF E A R T H WOBBLE  17  Choice of Axes  III  Perturbation Equations Rotational Deformation Special Solutions to Equation (2.9) T H E L A T I T U D E SERVICES Method of Observation Reduction of Data by the I. L . S. . . . Instantaneous Pole Path The Problem  IV  19 23 24 30  , .  32 33 34 35  ..  37  PREPARATION OF E Q U A L L Y SPACED L A T I T U D E MEANS A T E A C H O B S E R V A T O R Y Time Variable Noise Removal  V  19  F R E Q U E N C Y DOMAIN F I L T E R I N G TO OBTAIN C H A N D L E R L A T I T U D E VARIATION Motivation Predictive Filtering The Effect of Phase and Amplitude Shifts on the Fourier Spectrum Band Reject Filtering of Annual and Noise  45  56 56 58 62 70  iii  T A B L E O F CONTENTS (Continued)  Chapter VI  Page T H E P H A S E AND A M P L I T U D E R E S U L T S FOR T H E 5 I. L. S. OBSERVATORIES Phase Results Amplitude Results Interpretation with Respect to Established Theories  78 87 87 90  Conclusions  95  BIBLIOGRAPHY  96  APPENDIX I  99  iv  LIST O F T A B L E S  Table  I  Page  Annual Components of the Rotation Pole in Units of ". 01  II  III  Estimates of Chandler Period  5  5a  Earthquakes of magnitude greater than 7. 9 (after Richter, 1958)  9Qa  V  LIST OF FIGURES  Figure 1  2  3  4  5  6  Page Power spectrum of Polar Motion and Latitude Data (taken from Munk and McDonald, 1960)  4  Power spectrum of Polar Motion (taken from Rudnick, 1956)  4  Chandler polar motion amplitude characteristics (taken from Yumi, 1970)  9  Chandler period variation (taken from Iijimi, 1971)  9  Power spectrum showing split Chandler peak (taken from Rochester and Pedersen, 1972)  11  Smylie and Mansinha's fitting of smooth arcs to observed polar motion (taken from Scientific American, Dec. 1971)  15  7  Orientation of reference axes  8  Polar motion corresponding to damped free wobble of the earth  9  .  20  26  Response of damped earth as a function of frequency  10  Longitude of I. L. S. observatories  11  Configuration of star pair with respect to  27 31  zenith and celestial pole 12  Filter to Remove Random Noise  13  Output from variable noise filter (solid line) and constant noise filter (solid line with circles). The dotted line is the original series Plots of basic smoothed data from each of the observatories  14-18  32 46  ....  49 51-55  vi  LIST O F  F I G U R E S (Continued)  Figure 19  20  21  22  23  24  25  26  27  28  29  Page F o u r i e r t r a n s f o r m of 10 y e a r s of a p e r f e c t sinusoid (period = 435 days). The dotted v e r t i c a l line r e p r e s e n t s t r a n s f o r m of an infinite sinusoid of the same frequency  57  F o u r i e r t r a n s f o r m of synthetic wobble data of F i g u r e 22  59  Method for P r e d i c t i v e E x t e n s i o n of a Data Series  61  Synthetic latitude v a r i a t i o n composed of 2 m a i n frequency components (annual and Chandler) plus r a n d o m noise  63  P r e d i c t i o n of time s e r i e s i n F i g u r e 22 to five times its o r i g i n a l length  64  The F o u r i e r amplitude s p e c t r u m of the extended s e r i e s ( F i g u r e 23)  65  Chandler v a r i a t i o n u s e d to of F i g u r e 22. C o m p o s e d a l l of the same frequency but each v a r y i n g i n phase  66  create the s e r i e s of sine wavelets (1/435 days"*) and amplitude  F o u r i e r amplitude s p e c t r u m of F i g u r e The dashed line is the s p e c t r u m of a sinusoid with constant phase and amplitude  25.  67  Synthesis of the F o u r i e r t r a n s f o r m of a truncated sinusoid  69  E f f e c t of band-reject f i l t e r i n g of annual on the s p e c t r u m of the extended synthetic latitude v a r i a t i o n ( F i g u r e 23 and F i g u r e 24)  72  Chandler v a r i a t i o n r e s u l t i n g f r o m bandr e j e c t f i l t e r i n g of the synthetic data. The dashed line r e p r e s e n t s the known synthetic Chandler v a r i a t i o n  73  vii  L I S T O F F I G U R E S (Continued)  Figure 30  31  32-36  37  38  39  40  41  42  43  Page P h a s e v a r i a t i o n of the synthetic C h a n d l e r . The s o l i d line i s the phase of the synthetic Chandler motion r e c o v e r e d by b a n d - r e j e c t f i l t e r i n g and the dashed line i s the phase of the known o r i g i n a l input Chandler  76  Amplitude v a r i a t i o n of the synthetic Chandler. Solid and dashed lines r e p r e s e n t the f i l t e r e d and actual versions respectively . .  77  Chandler latitude v a r i a t i o n for each of the o b s e r v a t o r i e s  79-83  Chandler amplitude v a r i a t i o n for a l l observatories  85  Chandler phase v a r i a t i o n for a l l observatories  86  Amplitude v a r i a t i o n a v e r a g e d o v e r number of o b s e r v a t o r i e s  88  P h a s e v a r i a t i o n a v e r a g e d o v e r number of o b s e r v a t o r i e s  89  Chandler amplitude v a r i a t i o n after Guinot (1972)  91  Chandler phase v a r i a t i o n after Guinot (1972)  91  F o u r i e r amplitude s p e c t r u m of s i m p l i f i e d undamped Chandler motion  94  viii  ACKNOWLEDGEMENT  I wish to thank Doug Smylie and Tad Ulrych for their generous support and encouragement.  1  INTRODUCTION  At f i r s t glance, its a x i s w o u l d s e e m a s i m p l e r o t a t i n g s p h e r e , one as d e v e l o p e d b y  the d e s c r i p t i o n o f the e a r t h ' s r o t a t i o n a b o u t task.  T h a t i s . i f the e a r t h i s t r e a t e d a s a r i g i d ,  c o u l d s i m p l y a p p l y the r e l e v a n t c l a s s i c a l l a w s of p h y s i c s  Euler.  However,  the r e a l e a r t h i s n e i t h e r r i g i d n o r  spherical.  These  d e v i a t i o n s f r o m the ' i d e a l ' b o d y r e s u l t i n c o m p l e x p e r t u r b a t i o n s i n the r o t a t i o n o f the e a r t h .  The  b u l g i n g o f the e a r t h at the e q u a t o r p r o d u c e s the v a r i a -  t i o n i n the d i r e c t i o n of the a x i s o f a n g u l a r  momentum in space(precession  n u t a t i o n ) . A n o t h e r e f f e c t i s the v a r i a t i o n i n the l e n g t h o f the day. important  and  The  most  p e r t u r b a t i o n to g e o p h y s i c i s t s , a n d the s u b j e c t o f t h i s t h e s i s ,  i s the  p e r i o d i c v a r i a t i o n i n the o r i e n t a t i o n o f the i n s t a n t a n e o u s  rotation axis with  r e s p e c t to the g e o g r a p h i c f r a m e o f the e a r t h - o r a l t e r n a t e l y , the m o v e m e n t o f the r o t a t i o n a l p o l e ( p o i n t n e a r N o r t h p o l e w h e r e i n s t a n t a n e o u s  rotation  a x i s p i e r c e s s u r f a c e ) o v e r a h o r i z o n t a l p l a n e p a r a l l e l to the e q u a t o r . motion i s c a l l e d wobble.  M o s t m e a s u r e m e n t s o f w o b b l e a r e i n t e r m s o f the  m o t i o n of the r o t a t i o n a l p o l e i n u n i t s o f d i s t a n c e - the c o m p l e t e p l a n a r b e i n g d e s c r i b e d b y two  direction components.  a n d the p o l e p a t h data,  to the a n a l y s i s o f w o b b l e d a t a .  motion  S u c c e e d i n g c h a p t e r s of this  t h e s i s w i l l d e a l w i t h the h i s t o r y o f e a r t h w o b b l e r e s e a r c h , theory,  This  the  classical  a l l w i t h a v i e w o f j u s t i f y i n g a new  approach  2  E A R L Y HISTORY OF  WOBBLE RESEARCH  Since the h i s t o r y of r e s e a r c h involving the earth's wobble is thoroughly documented by Munk and M c D o n a l d (I960) i n their book on the rotation of the earth, only the highlights are d i s c u s s e d here. (1765) who  It was  Euler  f i r s t p r e d i c t e d that a r i g i d sphere would, in addition to its n o r m a l  rotation, wobble f r e e l y about its axis of greatest rotational i n e r t i a . dicted that the p e r i o d of the wobble should be 305 days.  Pe'ters,  He  pre-  Bessel,  K e l v i n and Newcomb a l l s e a r c h e d without s u c c e s s (or with false success) for a motion of the p r e d i c t e d period.  In 1888,  K d s t n e r d i s c o v e r e d a v a r i a t i o n in  the constant of a b e r r a t i o n that he attributed to a . 2 second annual v a r i a t i o n i n latitude.  To show that this was  i n fact a v a r i a t i o n due to wobble, m e a s u r e -  ments were made at stations in B e r l i n and W a i k i k i - roughly 180 degrees apart - and the m e a s u r e m e n t s showed the expected opposite change i n latitude. S. C. Chandler i n 1891  announced that the wobble, i n addition to an annual  component, also contains a motion of p e r i o d 428 days.  Newcomb i n the same  y e a r showed that the 428 day p e r i o d could, i n fact, be the f r e e wobble p e r i o d lengthened f r o m 305 to 428 days by the n o n - r i g i d i t y of the earth. c o v e r i e s and the lack of confidence led to the establishment  These dis-  i n the o b s e r v a t i o n a l evidence to that time  of the International Latitude S e r v i c e (I. L. S. ) which  functioned f r o m 1899. 0 to 1962. 0 when it was  r e p l a c e d by the International  Polar Motion Service. Since the time the I. L. S. was  established, the m a j o r thrust  3  o f r e s e a r c h h a s b e e n c e n t e r e d o n two 1.  a s p e c t s o f the w o b b l e p r o b l e m :  A n a l y s i s o f the d a t a a n d d e t e r m i n a t i o n i n d e t a i l o f the c h a r a c t e r i s t i c s o f the two m a i n f r e q u e n c y c o m p o n e n t s ( C h a n d l e r a n d a n n u a l ) .  2.  E x p l a n a t i o n o f t h e s e c h a r a c t e r i s t i c s b y a p l a u s i b l e m o d e l f o r the real earth.  The  annual wobble The  a n n u a l w o b b l e has,  for convenience's  sake,  been u n i v e r s a l l y  treated as a s p e c t r a l delta function which, i n m o r e f a m i l i a r terms, s i m p l e c i r c u l a r o r e l l i p t i c a l m o t i o n o f the r o t a t i o n a l p o l e .  On  i s just  f i r s t g l a n c e at  the p o w e r s p e c t r u m s d e p i c t e d i n F i g . 1 & 2 (both a r e t a k e n f r o m M u n k McDonald), this a s s u m p t i o n the a n n u a l p e a k .  The  and  w o u l d s e e m to b e j u s t i f i e d b y the n a r r o w n e s s o f  power spectra for perfect circular motion  o f p e r i o d one  y e a r w o u l d h a v e a p e a k w i d t h o f a p p r o x i m a t e l y . 04 c y c l e s / y e a r c o m p a r e d to the p e a k w i d t h i n t h e s e p l o t s o f . 03 to . 06 c y c l e s / y e a r . s i m p l i c i t y this i m p l i e s i s not quite a c h i e v e d .  U n f o r t u n a t e l y , the  W a l k e r a n d Y o u n g (1957) p o i n t  out. i n t h e i r e x h a u s t i v e a n a l y s i s o f I. L . S. d a t a that,  "the c o m p o n e n t s o f the  a n n u a l m o t i o n v a r y a c c o r d i n g to t y p e o f s e r i e s a n d i n t e r v a l ,  there being  n o t i c e a b l e d i f f e r e n c e s i n b o t h the a m p l i t u d e s a n d p h a s e s o f the s e v e r a l nents. "  compo-  T a b l e I g i v e s v a r i o u s e s t i m a t e s o f the a m p l i t u d e f o r e a c h o f the  directional components of annual polar motion towards ninety degrees  east of Greenwich,  -  The  a f a c t o r o f two b e t w e e n m i n i m u m a n d m a x i m u m .  two  rrij t o w a r d s G r e e n w i c h , r r ^ amplitude v a r i e s by  almost  T h i s v a r i a t i o n m e a n s that  i f the a n n u a l w o b b l e i s to b e m o d e l l e d a s s i m p l e c i r c u l a r m o t i o n ,  the m o d e l  I-  -E&gure 1 i Power spectrum of Polar Motion and Latitude Data (taken from Munk and McDonald, I960)  Figure 2 : Power spectrum of Polar Motion (taken from Rudnlck,1956)  5  TABLE I ANNUAL COMPONENTS OF THE ROTATION POLE IN UNITS OF ". 01 # Interval  Source  m  i  m  2  Jeffreys 1952  1892-1938  - 3.6cos® -8. 5 sin© 7. Ocos ® -2. 9s  Pollak 1927  1890-1924  -3. 7  -8.9  7. 0  -3.9  Rudnick 1957  1891-1945  -3. 2  -8. 2  6. 7  -2. 8  1899-1954 1900-1934 1900-1920  -6.4 -5. 5 -4.8  -7. 1 -7. 0 -6. 0  7. 0 7. 5 6.6  -4.6 -4.6 -3. 7  1912-1935) ) 1916-1940)  -3. 2  -7. 8  5. 6  -1.6  Walker & Young 1957 Jeffreys 1940 and Markowitz 1942  * ® is the longitude of the mean sun measured from the beginning of the year. # after Munk & McDonald, I960.  5a  TABLE II ESTIMATES OF CHANDLER PERIOD^  Time Interval 1890- 1915 1890- 1924 1890- 1924 1890- 1922 1922- 1938 1892- 1933 1908- 1921 1900- 1940 1891- 1945 1900- 1920 1890- 1924 1929- 1953 1897- 1957 1897- 1922 1930- 1957 1891 - 1952  Values of the Period, Years 1.13, 1.20, 1.27 1.20 1.19 1.08, 1.14, 1.19, 1.27 1.13 1. 223 1. 202 1.108, 1.170, 1.208, 1.250 1. 196 1. 193 1. 191 1. 172 1. 186 1.190 1.180 1. 193  * after Wells, 1972.  Source Witting (1915) Pollak (1927) Stumpff (1927) Wahl (1938) II  Jeffreys (1940) n Labrouste (1946) Rudnick (1956) Walker & Young (1957) Danjon & Guinot (1954) 11  Pachenko (I960) it  n Arato (1962)  6  must be fitted to as short a data length as possible - short enough that the wobble does not depart significantly from circular motion over the interval. Yashkov (1965) and Iijimi (1971), using harmonic analysis on ten and twelve year data intervals respectively, claim to detect variations in period as high as , 05 years from the mean period of one year. Considering the vagaries of harmonic analysis, these results are somewhat questionable. Wells (1972) discovered that the annual component of the latitude variation at an observatory (two or more latitude observatories are required to produce a pole path see Chapter 2) is very inconsistent among observing stations in both phase and amplitude. He goes on to suggest that "the annual polar motion as determined from the usual least squares solution from the latitude series, does not accurately reflect the true annual wobble of the earth. " If the annual latitude variation is markedly different from station to station, one must assume that a portion of the annual variation at any station is due to local effects. This implies that the annual latitude variation at a station is not entirely due to wobble of the earth. What the exact cause of these local variations might be is a matter of conjecture - possibly variations in local vertical, temperature, and 'seeing' . Whatever the cause, it is extremely desirable that, if one is going to remove the annual component in order to examine the Chandler component, the removal should be done on the latitude series before they are used to calculate a pole path. Also, if one wishes to 'see' a real annual wobble, the local effects must somehow be considered. Despite this, the best practical model for the annual component of wobble remains a spectral delta function adapted to vary in amplitude with time. However, care must  7 be taken in equating the annual wobble that best fits this model to real wobble of the earth. The annual wobble has, ever since its discovery, been attributed principally to a seasonal shift in air mass over the surface of the earth - along with less significant variations in snow, vegetation and ocean mobility. Spitaler (1901), Rosenhead (1929). and Munk and Hassan (1960) all conclude that the seasonal air mass shift has the phase and magnitude to account for the observed wobble. However, on a careful reading of the Munk and Hassan paper, we find little evidence to support their conclusion. The excitation function required to produce the observed wobble and the excitation function calculated from meteorological records differ by more than an order of magnitude - even if looking at just the annual component of each of these functions. Their subsequent conclusion that "the annual is principally due to seasonal air shifts", is, at best, premature. If air shifts are the cause of the annual wobble, then variations in climate from year to year would reflect themselves as variations in the frequency, amplitude and phase of this component. This emphasizes our earlier conclusion that the annual wobble should be treated as time-variable in phase and amplitude. The Chandler wobble The Chandler wobble represents the free motion (Jeffreys, 1957) of the earth in response to some perturbation of the instantaneous rotation axis away from its equilibrium position. Since free motion is sensitive to the physical characteristics of the system involved, detailed knowledge of the  8  C h a n d l e r m o t i o n e n a b l e s one to d r a w c o n c l u s i o n s about the r i g i d i t y ,  shape  a n d n o n - e l a s t i c i t y o f the r e a l e a r t h . The  e s s e n t i a l f e a t u r e o f the C h a n d l e r w o b b l e i s that i t s s p e c t r a l  representation resembles  that o f a d a m p e d o s c i l l a t o r - that i s ,  the s p e c t r a l  p e a k i s s p r e a d o v e r f r e q u e n c i e s n e a r the p e a k v a l u e ( s e e F i g . 1 & 2). II ( t a k e n f r o m W e l l s ,  1972)  s h o w s the C h a n d l e r p e r i o d  v a r i o u s e p o c h s a s c a l c u l a t e d b y the a u t h o r s i n d i c a t e d .  ( p e r i o d at p e a k ) d u r i n g T h i s table i l l u s t r a t e s  that the C h a n d l e r w o b b l e m i g h t h a v e t i m e - v a r i a b l e a m p l i t u d e a n d P l o t s b y Y u m i (1970) a n d I i j i m i (1971) ( F i g , 3 & 4), i n d i c a t e how a n d a m p l i t u d e o f the two  Chandler components v a r y with time.  variation is especially remarkable  Table  frequency. the p e r i o d  The  - b e t w e e n ten and t h i r t y feet;  amplitude  It s h o u l d  b e p o i n t e d out that t h e s e a u t h o r s h a v e u s e d v a r i o u s f o r m s o f h a r m o n i c  analysis -  s o m e t i m e s - o n data lengths as s h o r t as ten y e a r s ( Y u m i ) - and t h e r e f o r e r e s u l t s that i n d i c a t e a s h i f t i n f r e q u e n c y a r e s u b j e c t to l a r g e e r r o r s due to r e s o l u t i o n p r o b l e m s o f the t e c h n i q u e . E s t i m a t e s o f the d a m p i n g t i m e ( t i m e f o r s i g n a l to d e c a y to o f i t s o r i g i n a l v a l u e ) v a r y f r o m two y e a r s ( W a l k e r a n d Y o u n g , y e a r s (Pachenko).  T h e s e e s t i m a t e s o f d a m p i n g t i m e l e a d to a  1/e  1957) to n i n e t y Q  factor  ( d i m e n s i o n l e s s m e a s u r e o f e n e r g y d i s s i p a t i o n ) o f t h i r t y to f i f t y , w h i c h i s s e v e r a l t i m e s s m a l l e r than  Q  values calculated f r o m  F e l l g e t t (I960) c a l c u l a t e d a d a m p i n g t i m e o f 12.4  s e i s m i c data.  y e a r s but c o n c l u d e d that  the v a l u e i s " u n c e r t a i n b y a f a c t o r o f at l e a s t ten. a n d d a m p i n g t i m e s a s s h o r t a s 2. 5 y e a r s o r a s l o n g a s s e v e r a l h u n d r e d  y e a r s a r e not e x c l u d e d . "  The  9  « g  -io\  a. E o l  aoL—i  1993  Figure  3  L  i  ISOS  j  i  _j  L  1929  1317  19S3  19*/  : Chandler polar motion amplitude (taken from Yuml, 1970)  1M\  1  i  I96ST  characteristics  T — i — i — i — i — r  i — i — r  l.20\  aa> T3  o  **Z  1.16  1.14-  <u a  L  rn.  J  190072  Figure  4  : Chandler  JL  i  i  period  i  J_  2 4 3 6  variation  (taken  from  Iijiml,  1971)  10  apparent disparity between Q  factors still remains.  1972) would be a frequency dependent Q.  One  solution (Wells.  That is, the earth would have low  damping for seismic frequencies and high damping for wobble frequencies. There is no evidence to support this hypothesis.  Some (Melchior, Niccolini,  1957) explain the broadened spectral peak of the Chandler component in terms of a variable frequency - of which we have already some evidence.  This  explanation requires a variation of four percent in the Chandler period. is not apparent in the observations.  This  Yashkov (1965) carried out harmonic  analysis on polar motion data and produced a split Chandler peak. F i g . 5 shows a similar split peak obtained by Rochester and Pedersen (1972) from a seventy year record of monthly mean pole positions.  Colombo and Shapiro  (1968) assume the split peak represents two real neighbouring frequencies and on that basis resolve the Q  factor disparity.  Unfortunately for them, Fedorov  and Yatskiv (1965) proved that this apparent splitting could in fact be caused by a simple phase shift in a pure harmonic of the Chandler frequency.  Thus,  the general characteristics of the Chandler wobble can be summarized as: a time-variable wide-band periodic function of central period 435 days and a spectral width typical of a damped system with a damping time estimated to be ten to thirty years.  Chandler damping It is generally accepted that the apparent damping of the Chandler wobble, as indicated by its spectral width, is a real damping due to inelastic response of the earth to the free wobble.  The major problem  11  100 90 80  ct o  7 0  •"60 O x 50 o  cc 40 UJ  * ? 30 20  BAND WIDTH  10 .5  .7 .8 FREQUENCY  L A  .9 (CPY)  • /V.  1.0  VY.  I.I  F i g u r e 5 s Power spectrum showing s p l i t Chandler peak (taken from Rochester and Pedersen, 1972)  TOWARD 90 DEGREES  F i g u r e 6 : Smylle and Manslnha's f i t t i n g o f smooth a r c s to observed p o l a r motion (taken from S c i e n t i f i c American, Dec. 1971)  12  has been to explain the discrepancy between the damping required by this wobble characteristic and that observed in seismic work. The conclusions reached by Munk and McDonald in this regard are: 1.  Solid friction in the mantle could produce a Q 200 - as opposed to a Q  2.  factor of 100 to  factor from wobble data of 30 to 50.  The additional damping could be due to: (a) the oceans. (b) the lower mantle having a viscosity that is especially important on a time scale of the order of the Chandler period. (c) core-mantle interaction.  They conclude by stating that "the situation is appallingly uncertain".  There  the matter rests (or perhaps wobbles).  Chandler excitation Even without observational evidence that the free wobble is damped, it can be assumed that damping does occur (as in all real systems). Thus, left undisturbed, the Chandler wobble would eventually dissipate completely.  Therefore, an excitation mechanism is required.  The form of this  excitation has been the cause of much conjecture in recent years. It was thought that irregularities in the annual atmospheric loading were the cause of the excitation until Munk and Hassan (I960) proved that this mechanism was several times too small.  Today, two possible causes of the excitation  remain: 1.  Core-mantle interaction of some type.  13 2. Redistribution of mass due to earthquakes and subsequent perturbation of the moment of inertia. In regard to the first of these causes, the main interaction occurring at the core-mantle boundary is thought to be electro-magnetic coupling. Munk and Hassan (I960) and Rochester and Smylie (1965) found that this mechanism fails by several orders of magnitude. However, Rochester (1970), Stacey (1970) and Runcorn (1970) argue that effects of secular variation and precession in concert with coupling could possibly account for observed wobble. Until the physical parameters at the core-mantle boundary are better defined, electromagnetic coupling will remain a somewhat confusing issue. The form of the excitation mechanism is likely to be either random catastrophic events that cause sharp discontinuities in the free (circular) motion of the rotational pole or a periodic forcing function of frequency equal to the observed wobble frequency. Earthquake excitation of the wobble has been considered for many years (Cecchini, 1928), but calculations based on simple models yield a negligible effect, (Munk and McDonald, I960). Press (1965) applied dislocation theory to earth models and produced much larger displacement fields than had been anticipated. Smylie and Mansinha (1967) used Press's techniques to calculate theoretical displacement fields of earthquakes and the associated change in the moment of inertia. They concluded that the perturbations were large enough to maintain the Chandler wobble. If the excitation is due to random earthquakes, then the rotational pole path will be smooth, circular arcs (spirals if damping is considered) separated by sharp breaks.  14 and therefore, Smylie and Mansinha  (1968) fitted circular arcs in a least  squares fit to the observed Chandler pole path. As can be seen in Figure 6, they allow random shifts in the origin, radius and phase of the motion - a shift occurring when the data and the fitted curve no longer agree within some preset  circle of validity.  They found excellent correlation between these  'breaks' and major earthquakes.  In fact, the 'breaks' tended to occur just  before the earthquake so that if the effect was real, earthquakes could be predicted.  Haubrich (1970) found that a re-analysis of the same data produced  quite different events and so he concluded that the strong correlation found by Mansinha and Smylie was merely a coincidence.  Later work by Smylie et al  (1970) using a deconvolution method tends to support the original correlation though not so strongly.  Smylie and Mansinha (1971) calculated a pole path  displacement caused by the 1964 Alaskan earthquake on a real earth model. Their result of one foot compares well with similar work by Dahlen (1971). Displacements of this order could quite easily sustain the Chandler wobble. However, so far, observational evidence of such shifts is not conclusive. It should be kept in mind that both the damping and excitation mechanisms of the Chandler wobble are based on the assumption of a damping time of about twenty years. An order of magnitude shift in this value, which is perfectly possible according to Fellgett (I960), would have an enormous effect on all the models that are now used to describe the Chandler wobble. F o r example, if the damping time is actually 200 years, the conflict between seismic and wobble damping factors would no longer exist and the excitation required to produce the wobble would be far smaller - allowing a variety of mechanisms.  now  thought to be i n s i g n i f i c a n t ,  to a f f e c t the C h a n d l e r w o b b l e .  16  SUMMARY  The  m o s t i m p o r t a n t f a c t s that c a n be g l e a n e d f r o m t h i s b r i e f  h i s t o r y a r e that: 1.  b o t h the a n n u a l a n d C h a n d l e r c o m p o n e n t s a r e t i m e v a r i a b l e i n amplitude and perhaps in frequency.  2.  no a s s u m p t i o n Chandler  3.  s h o u l d be m a d e about the d a m p i n g t i m e o f the  motion.  the C h a n d l e r m a y amplitude.  have r a n d o m sudden changes i n phase and  17 C H A P T E R II  THEORY OF E A R T H WOBBLE  The rotation of bodies about their centre of mass is governed by Euler's equation of motion which states that the time rate of change of angular momentum  L  is equal to the applied torque  P  if all measure-  ments are made in an inertial system.  ~F~  _  T~  1  inertial  Since it is convenient to use a coordinate system attached to the rotating earth (angular velocity *w in space), the transform «  L  •  =  ~~~  —  ~~~  L  inertial  :w  4-  x  i i  L  rotating  is used. The equation of motion is now  7^  «  =  w  x  L~  4-  L~  (2.1)  where the time derivative is taken in the rotating frame.  Total angular  momentum is given by  L  =  /{{  earth tYi  [ r x fvv"x r 4- v (r)) p (7)]  dV  where v(r) is velocity of material at point "r with respect to the rotating frame and p(r) is the density of material at that point. The term r x (w x T) can be expanded by the vector identity ~  18 r  x  ( w x r )  =  w ( r . r ) -  r ( w . r )  which becomes r x (w  x r) =  e. [ r &.. L ij 2  1  -  x. x. "1 w. J J J 1  when the standard summation convention (repeated subscripts indicate summation over that subscript) is used. directions and  =  arth earth  L  1  earth  e. I., w. 1  I.. = J X  iJ  J  The angular momentum is now  ij  fff  4-  where  e"- are unit vectors in the coordinate  is the Kronecker delta.  c$ij  L  The  4-  fff \r earth  2  L  1  J J  w. p fr) dV J  r x v ( r ) p ( r ) dV '  J  cf'.. - x. x. 1 p ( r ) dV *J JJ ~  (Z. 2)  1  is the inertia tensor of the earth, and  =  JyT earth  7 x v ( r ) p ( r ) dV '  is the angular momentum relative to the rotating frame. Substituting equation 2.2 into equation 2. 1 gives the Liouville equation:  where  t£  is the normal alternating tensor.  19  Choice of Axes The most convenient choice of axes is one in which the coordinate system is attached to the observatories - relative motion of the observatories due to nonrigidity of the earth being the only complication. The axes are oriented so that the origin is at the centre of mass, the x^ axis is reasonably close to the axis of rotation and the x^ axis passes through the Greenwich meridian. See F i g . 7. If steady, equilibrium rotation about the x^ axis is assumed, the above coordinate system will be aligned with the principal axes of inertia which are defined as the axes that make  I.. = 0 if i 4 j.  Thus, if the  principal moments of inertia for the earth are denoted A, A and C, then  I  11  =  I  22  =  A  A  N  D  l  33 =  C  where two of the principal moments are equal because of cylindrical symmetry.  Perturbation Equations The development of the equations governing the case where the axis of rotation is perturbed from its equilibrium position follows the method of Munk and McDonald  (1960).  The departures from uniform, diurnal rotation  are almost undetectable in practice, so that first order perturbation theory should work extremely well.  Therefore, let  20  Z  instantaneous  (geographic  North  pole)  rotation  axis — fixed in space except for effects of  Iml  precession earth s centre of mass Y (90°  E  of Greenwich)  (to  Figure  Greenwich)  7  : Orientation  of reference  axes  21  w  =  J\  I  l l  X  13  5  =  s  !  A  +  31  C  =  +  l l  B  ij  JX  4-  hi-  l l  +  C  1 3  l  I  b  m1  e^f\.  4-  Z3  =  = I  1 2  A  I  +  32  2 2  C  =  2 3  B  = b  2 1  +  2 2  B  +  C  m3)  4-  X  1  <<  33 *  C  +  C  33  +  B  3 3  3  2  +  1 2  (1  r r i | , m^.  - angular velocity of earth;  b  1 3  f\  c  1 2  is the contribution to I., from rotational deformation above that due to J X  steady, diurnal rotation, and c-j is deformation due to other causes.  Then,  the angular momentum components are L  L  l  2  L  hi  =  A-a  hi  =  3  1  =  W  w  m i  I  AAm  =  I  =  C A  2  (b.j  4- C . J ) J T L  .  2  +  W  2  +  A ( b  +  1 3  I  3  W  1 3  wj + I  (1  W  j\ ( b + c  *22  3 2  + m)  where terms like (b.j 4- c.j) k J\ m  1 2  +  l+  =  3 1  +  3  w  2  2  4-  J  3  23  W  c  2  4-  I  3 3  A  1 3  3  +  3  A  +  ^  +  )  w  X  4-  z  X  4-  3 3  1  2  +  3  (c  x  ) +•  z  b  3 3  ) 4-  /  3  are neglected in comparison to those like  Substituting into the Liouville equation  (2. 3)  yields  22 P  1  ~ £ 1  + A J\ m  4- _TL (b , 4- c ) 4- C yL m_ 13 13 2  1  -jT_ Am -  Jl?(b  2  2  p  2  =  A jT.m  ^  + SL  =  4- c ) - y i _ i 2 3  2  (2.4)  4- JT (b 4- c ) 4- i + A A m 23 23 ^ *  (b  CJlm  2 3  1 3  4-  3  4- c ) 4- /I i 1 3  y i(c  3 3  4-  b ) 3 3  i  - C Jl m  (2.5)  /  (2.6)  2  4-  3  where, in addition to the previous approximation, we now neglect products of l j and mj.  Now, let m  =  rrij 4- i  r = p —  + i 1  i  - ix ±  By multiplying equation 2. 5 by  p  2  az  i = (V-l ), and adding the result to equation  2. 4. we obtain m  71 A  - i j / l - i(c  - im  - (c  1 3  4- i c  l 3  2 3  -A. ( C - A.) 2  4- i c ) _ f L  )/l  23  2  - (b  u  - i( b ^ + ib  =  _Q -  4- i b'  2 3  )_/!  23  )yi_  2  (2.7)  23  T h i s i s the e q u a t i o n o f m o t i o n f o r the w o b b l e - m d i s p l a c e m e n t o f the a x i s o f r o t a t i o n a w a y f r o m m o t i o n of the instantaneous  g i v e s the c o m p l e x a n g u l a r  "e^. o r . i f o n e w i s h e s , the  p o l e o f r o t a t i o n o v e r the s u r f a c e o f the e a r t h ,  . 1 s e c o n d o f a n g u l a r m o t i o n e q u i v a l e n t to t e n f e e t o f p o l a r m o t i o n . maining  with  The r e -  equation  m  77T [  =  3  P  3  "  ^ 3  - <  '33 3 3 >^-]  < '  + b  2  8)  g o v e r n s the c h a n g e i n l e n g t h o f d a y .  Rotational Deformation S i n c e the t e r m s  b.. a r e i n f a c t f u n c t i o n s o f m, !J —  t h i s p o i n t to o b t a i n t h e e x a c t f u n c t i o n a l r e l a t i o n s h i p .  b  k_ d 12 l *  23  B  3 G  where k  d  d  J\  m7  2  "  = 13  c.  I  =  k  b.  It c a n b e s h o w n that  m , _TL  m  e.  i t i s u s e f u l at  3 G  k  2  d  5  m  l ^  2  3~G  e q u a l s the r a d i u s o f the e a r t h , G i s the g r a v i t a t i o n a l c o n s t a n t a n d  i s the L o v e n u m b e r - d e f i n e d s u c h that i f t h e s y m m e t r i c e a r t h i s s u b j e c t e d  to a d i s t u r b i n g f o r c e f i e l d e x p r e s s i b l e a s the g r a d i e n t o f a s e c o n d s p h e r i c a l h a r m o n i c p o t e n t i a l U^. the a d d i t i o n a l p o t e n t i a l V s u r f a c e i s V = k 2 U-,.  order  at the d e f o r m e d  T h e s e r e s u l t s a r e b a s e d o n the a s s u m p t i o n o f l i n e a r  s t r e s s - s t r a i n r e l a t i o n s , a n d a r e d e r i v e d o n p a g e 25 o f M u n k a n d M c D o n a l d  24 (1960). Substituting for b.^ in equation 2.7 gives  m/L(A+ -  = H  -  1  ) - i rn j \ -  — 3G  "  "  If we neglect k^d _ / l  /3G  +  ) 3  '  1  1 (c  13  +  1  c  G  23 ^ )  compared to A, and allow (C-A)/A to be replaced  c 2 k2<i A /3GA  by (C-A)/C and  ^13  (  ( C - A - _2  by  k,,d  5  2 .A. /3GC  (introduces an error of only  one part in 300 in each case), then  2-i<r 2. o  =  [JZ  X A  - i (C  4-  13  -ii^  -1  <TQ  H  The quantity H  1 3  + iC  2 3  )yL  i C ) J1 2 J  (2.9)  2 3  k dA ^  where  -(C  =  (H -  =  C - A (3  ?  5  y,  Z  )  determines the rate of precession and from direct observa-  tion has the value  H = 1 /305. 5.  Special Solution to Equation 2. 9 A.  Free Wobble For this case, the right hand side of (2.9) is zero and so we  have m  - i G~  Q  i <r from which  m  =  A e  m  =  0  t °  (A = constant)  25  This represents a circular motion of the instantaneous pole of rotation with angular velocity sional constant H  (T 0  and radius A.  For a rigid earth with the same preces-  as the real earth, the angular velocity is given by  i  CT  Q  = J\  H  and this corresponds to a period of 305. 5 days.  No periodicity  of this value has been detected in the wobble. For a non-rigid earth, the angular velocity is k  cr  A H -  =  ?  d  5  i l  2  —  o  3GC  The only unknown in this equation is k^. and Takeuchi (1950) has computed k^ = . 281 for one of Bullen's earth models.  Using this value, the period of  the free wobble is 2 TT  435 days  H - ( k d JT. /3GC) 5  2  2  which is close to the observed value for the Chandler wobble.  However, the  good agreement is probably more due to good luck than anything else, since no consideration is made for the effect of oceans or for core-mantle coupling. According to Munk and McDonald (I960), these two effects seem to just balance each other with regard to the period of the wobble.  B.  Damped Free Wobble Once again, the right hand side of equation 2.9 is zero.  allow for damping of the motion,  cr o  =  0  s  w. °  is allowed to be complex:  +  i/t  To  26  w  o  f  i s the a n g u l a r f r e q u e n c y of the f r e e w o b b l e a s c a l c u l a t e d p r e v i o u s l y , i s the d a m p i n g t i m e .  and  E q u a t i o n 2.9 t h e n y i e l d s the s o l u t i o n  . A e  m  t/Z  e  i wQt  a n d the m o t i o n o f the p o l e i s a w e s t to e a s t s p i r a l a s s h o w n i n F i g . 8.  Fig.  8.  P o l a r m o t i o n c o r r e s p o n d i n g to d a m p e d f r e e w o b b l e o f the e a r t h .  C.  H a r m o n i c E x c i t a t i o n with D a m p i n g Allow  <PQ  to b e c o m p l e x a s a b o v e , a n d a s s u m e the r i g h t  h a n d s i d e o f e q u a t i o n 2.9 ( w h i c h r e p r e s e n t s the e x c i t a t i o n o r f o r c i n g  function  of the d i f f e r e n t i a l e q u a t i o n ) i s o f the f o r m  i o<  where  o(  and  <T  <rt  e  a r e the a m p l i t u d e a n d f r e q u e n c y o f the e x c i t a t i o n .  27.  T h e n equation 2.3 b e c o m e s  - i CT  m  rn_  =  CTt  <=< e  The steady-state solution o f this equation i s  (Tt m  i (  CT- CT ) 0  It f o l l o w s that  V ( cr  m  wQ)2  A s s u m i n g that the a m p l i t u d e o f the e x c i t a t i o n , f r e q u e n c i e s , a plot of  /m  {  + (i/r )  2  o £ , i s the s a m e f o r a l l  v e r s u s f r e q u e n c y w o u l d l o o k l i k e F i g . 9'.  By  f i t t i n g a c u r v e o f t h i s t y p e to the a c t u a l d a t a , o n e o b t a i n s a n e s t i m a t e o f the  parameter  C  a n d thus o f the d a m p i n g f a c t o r  Q  w o t = —£—.  t I ml  F i g . 9.  Response of damped earth as a function of f r e q u e n c y .  D. Random Instantaneous Excitation of the Damped Wobble This is the case of primary interest to us, since earthquake excitation will correspond to this type.  The excitation function F(t) (right  hand side of (2. 7) can be modelled in two ways:  (a)  F(t) = JH(t)  where  H(t)  J  (b)  F(t)  cf (t)  =  heaviside  function  =  O  t < O  =  1  t > O  =  a complex constant  =  Jcf(t)  is the dirac delta function.  The general solution of equation 2.9 is given by  m(t)  =  i e  Q"  j  t  '  [m  -  Q  F ( t ) e~  1  d  "tJ  - oo  Upon substituting for F(t) from case (a), a simple integration gives  The  m(t)  =  m(t)  =  e (m —  Q  . -  i  t < 0  J ~  <r '  )  o  n  i cr t n  e  4-  :  i  cr  .t > 0  -  o  spiral motion after t = O has modified equilibrium pole position J  U  x  . new amplitude  m  0  -  i cj-  o  and new phase 0  given  C a s e (b) g i v e s m(t)  m  Here,  (t)  =  m  =  Q  e  (m0  t <  t  - J ) e  the e q u i l i b r i u m p o l e p o s i t i o n i s u n a l t e r e d  0  >  0  f o r t > 0, b u t b o t h the p h a s e  a n d a m p l i t u d e a r e c h a n g e d b y the a d d i t i o n o f the q u a n t i t y J .  From  these  e x a m p l e s , one w o u l d e x p e c t the C h a n d l e r m o t i o n due to e a r t h q u a k e e x c i t a t i o n to b e s m o o t h s p i r a l s i n t e r r u p t e d b y s u d d e n s h i f t s i n p h a s e a n d a m p l i t u d e possibly, also i n equilibrium pole position.  and  30  C H A P T E R III  THE  L A T I T U D E SERVICES  To an observer fixed with respect to the instantaneous rotation axis, the geographic coordinate system attached to the earth will wobble in exactly the same manner as the rotation axis wobbles with respect to an observer on earth.  Since the rotation axis is essentially fixed in space, the  relative motion of the geographic frame manifests itself as a changing orientation of the earth in space - that is, relative to the distant stars. The sidereal time and latitude of any point on earth are both affected by this apparent motion of the stars. The latitude of a point on earth is defined as the conjugate of the angle between the point where the rotation axis pierces the celestial sphere (celestial pole) and the point where the local vertical pierces the same sphere (zenith). Therefore,  any motion of the local vertical (assumed  fixed relative to earth) with respect to the distant stars causes a variation in latitude.  Sidereal time is similarly affected, but until recently time could  not be measured accurately enough to detect polar motion, so we will deal exclusively with the measurement of latitude variation. The International Latitude Service operates five latitude stations in the northern hemisphere.  They are all located very close to  39° 08' North in order that all stations can observe the same stars - thus making errors with respect to the star's position common to all observatories and thus correctable.  These stations are located at the longitudes indicated  in Fig. 10.  Misisowa,  Japan  140° 0 8 '  Kitab,  East  /  /  60°  Russia 2 9 ' East  "  "  \ Gaithersburg, Ukioh, 123°  California 13'  \  77°  12'  m  to  Greenwich  Maryland West  West  Fig. 10. Longitude of I. L. S. observatories  The stations have operated somewhat intermittently since the inception in 1899.  Gaithersburg was closed from 1914 to 1931.  formerly located at Tschardjin - some 3° to the west.  Kitab was  Carloforte was close  for nearly four years during World War II. F o r the epoch of interest to us (1922. 7 to 1949. 0), only Misisawa and Ukiah operated continuously.  Owing  to precession of the equinoxes, the catalogue of observing stars has been altered at various times:  1906.0. 1912.0, 1922.7, and 1935. 0. This factor  along with the changes in method of observation and reduction that have accompanied the changes in program director (1922.7, 1935.0), causes the  data to be somewhat inhomogenous. Partly to minimize these inhomogeneities and partly because of accessibility of data, we have chosen to look at the epoch 1922. 7 to 1949. 0.  The only 'break' in this period occurs at 1935. 0.  Method of Observation The I. L. S. stations use the Talcott method of measuring latitude.  This involves selecting a pair of stars with small, nearly opposite  zenith angles (angle from star to zenith), which also transit within a few minutes of one another.  The configuration is then as shown in F i g . 11. P  = celestial pole  Z  = zenith  E  = celestial equator  S. = l Z. l  =  <fi =  Fig. 11.  The latitude 0  The difference  Z^ - Z j  zenith  angle to  declination of  S.  Configuration of star pair with respect to zenith and celestial pole.  is given by  or upon addition by  star i  0  0  =  =  cfl +  f  ( l  cT2  - Z  l  0 = cf 2  and Z  2  -  Z  +  Z  2  l  (3. 1)  is. by choice of star pairs, a very small angle.  By  fixing the telescope on the first star and then rotating 180° about the vertical and measuring with a micrometer the angle adjustment required to align the telescope with the second star,  -  is obtained directly. The  declination  v a l u e s m u s t be f o u n d i n s t a r c a t a l o g u e s .  T h e i m p o r t a n t p a r t of this p r o c e s s i s  m a i n t a i n i n g the t e l e s c o p e l e v e l (or a constant zenith).  T h i s was originally-  d o n e v i s u a l l y , b u t m o s t m o d e r n m e a s u r e m e n t s a r e m a d e w i t h the P h o t o g r a p h i c Zenith Tube ( P Z T ) .  In t h i s i n s t r u m e n t ,  the l e v e l l i n g d e v i c e i s a f r e e s u r f a c e  o f m e r c u r y w i t h r e f l e c t s a s e r i e s o f f o u r s t a r i m a g e s onto a p h o t o g r a p h i c p l a t e f r o m w h i c h the z e n i t h d i s t a n c e  z  is measured directly.  R e d u c t i o n o f D a t a b y the I. L.'S. U n f o r t u n a t e l y , the d a t a p u b l i s h e d b y the I. L , S. a r e n o t the basic m e a s u r e m e n t s of latitude.  B a s i c latitude values  0  a r e given by  e q u a t i o n 3. 1 w h i c h i n t e r m s o f m e a s u r e d q u a n t i t i e s b e c o m e s  0  where  cf =  =  d"  + R ' M  +  (3.2)  I. S.  a d o p t e d v a l u e o f m e a n d e c l i n a t i o n o f the s t a r p a i r (^  ( <$i + <^2 )  a t  'he t i m e o f m e a s u r e m e n t - t i m e  v a r i a b l e b e c a u s e of s t a r p a i r p r o p e r  motion.  R  =  m i c r o m e t e r m e a s u r e m e n t (turns).  M  =  a n g l e o f t e l e s c o p e s h i f t p e r o n e h a l f r e v o l u t i o n o f the m i c r o m e t e r - including a t e r m for temperature dependence.  °  I. S. = a t e r m to a c c o u n t f o r i n e q u a l i t i e s o f the m i c r o m e t e r screw. The  I. L . S. u n d e r K i m u r a a d d e d to t h i s e q u a t i o n s e v e r a l c o r r e c t i o n s b a s e d o n  the d a t a i t s e l f .  A c o r r e c t i o n f o r d e c l i n a t i o n i s i n c l u d e d i n o r d e r to m a k e the  34  m e a n l a t i t u d e g i v e n b y e a c h s t a r p a i r e q u a l to the m e a n v a l u e f o r the s t a r p a i r group o f w h i c h it is a p a r t .  T h e group m e a n is obtained by weighting  a c c o r d i n g to the t r e n d o f the d a t a i n a m a n n e r t o t a l l y i n c o m p r e h e n s i b l e . c o r r e c t i o n f o r the t i m e v a r i a b l e p a r t o f M (1935).  i s also obtained.  A  See K i m u r a  F o r t u n a t e l y , o n l y the l a t t e r c o r r e c t i o n h a s a l o c a l e f f e c t - the  correction  to d e c l i n a t i o n p r o d u c e s the s a m e e f f e c t at e v e r y s t a t i o n .  C a m e r a o p e r a t e d the I. L . S .  f r o m 1 9 3 5 . 0 to 1 9 4 8 ,  reduction is straight-forward.  Luigi  and happily his m e t h o d o f  H e i n t r o d u c e s two c o r r e c t i o n s  s i m i l a r to the  f i r s t a n d l a s t d e s c r i b e d a b o v e f o r K i m u r a - the d i f f e r e n c e b e i n g the s i m p l i c i t y of h i s a s s u m p t i o n s a n d subsequent c a l c u l a t i o n s .  Instantaneous P o l e  See C a m e r a ( 1 9 5 7 , page 2 5 6 ) .  Path  M o s t o f the a c c e s s i b l e  d a t a p u b l i s h e d b y the I. L . S.  consists  o f m o n t h l y c a l c u l a t i o n s o f the two d i r e c t i o n a l c o m p o n e n t s o f the p o l a r m o t i o n . i , & 0 j , due to p o l a r m o t i o n i s j u s t  L a t i t u d e v a r i a t i o n at s t a t i o n  ^  where  m^  a n d m^  0^  =  rrij cos  axis.  t .  i s the l o n g i t u d e o f  m e a s u r e d p o s i t i v e l y i n the e a s t e r l y d i r e c t i o n .  variation can be caused by e r r o r s and  sinf •  a r e the r e s p e c t i v e d i r e c t i o n a l c o m p o n e n t s o f a n g u l a r  p o l a r d i s p l a c e m e n t a w a y f r o m the x ^ station i.  +  i n star positions,  Since latitude  temperature  fluctuations  o t h e r s i m i l a r v a r i a t i o n s that do n o t d e p e n d o n p o l a r m o t i o n , l a t i t u d e  variation is written  0. X  =  m. A  c o s "C*. + m_ s i n 1 Cd  7j. 1  f-  Z  35  z  is a term added to account for non-polar latitude variation.  is fitted to the observed  ^ 0-  This equation  at each station by choosing m obs  m_ and \  c  z which make  5  a minimum.  The  Problem To this point, we have been describing other people's accomp-  lishments. 1.  F r o m these are drawn the major objectives of this thesis: •  To achieve a clear separation of the two spectral components from relatively short data records.  2.  To obtain this separation without destroying time variable characteristics of either of the components; for example, a separation that will retain sudden phase and amplitude shifts of the Chandler component.  3. To 'search' for these sudden events once such a separation is achieved. 4.  To use for this analysis the original latitude series for the five different I. L. S. stations.  Hopefully, this will remove distortions in the pole path  due to local effects of a single observatory, in addition to eliminating the effect of the reduction required to produce monthly mean pole positions. Also, the occurrence of sudden events at each of the stations would be overwhelming proof that they are real.  Here I might mention that the  I. L. S. data are not readily available and the data used were painstakingly  36  collected by D. Smylie and typed onto computer cards by four very patient young ladies at the U. B. C. computing centre.  37  P R E P A R A T I O N OF MEANS AT  Our  CHAPTER  IV  EQUALLY  SPACED  EACH  LATITUDE  OBSERVATORY  b a s i c data c o n s i s t of latitude o b s e r v a t i o n s f r o m  s t a r p a i r s - i n c l u d i n g the c o r r e c t i o n s to d e c l i n a t i o n a n d m i c r o m e t e r described above.  S i n c e o b s e r v a t i o n s c a n n o t be m a d e at  to w e a t h e r c o n d i t i o n s , m a n y n i g h t s h a v e no r e a d i n g s at a l l a n d  s o m e have v e r y few.  In o r d e r to do t i m e s e r i e s a n a l y s i s , i t i s e s s e n t i a l to  have u n i f o r m l y spaced data.  A l s o , i t i s d e s i r a b l e to a t t a c h g r e a t e r w e i g h t to  nights where m a n y observations were m a d e . l e a s t one  variation  A m a x i m u m of s i x t e e n s u c h o b s e r v a t i o n s a r e m a d e e a c h  n i g h t at e a c h o f the f i v e o b s e r v a t o r i e s . a l l t i m e s due  individual  m e a s u r e m e n t was  e s t i m a t e of latitude on night  Looking  f i r s t at n i g h t s w h e n at  m a d e at a p a r t i c u l a r s t a t i o n , the m e a n v a l u e j  is given  by N  (4-1) i=l  where N = n u m b e r of o b s e r v a t i o n s on night x  i  -  j  .th , th 1 l a t i t u d e o b s e r v a t i o n on j night  j N o t e that the s y m b o l The  s\  i n d i c a t e s that the q u a n t i t y i s a n e s t i m a t o r  u n b i a s e d v a r i a n c e e s t i m a t o r f o r any o b s e r v a t i o n  x1 i s j  only.  38 N i=l . Assuming each x j value s . is J var  (Bendat and Piersol, 1966, • pg 126) is independent, the variance estimator of the mean  N  ft] = Nirhr 2 ['r'H i=l (Bendat and Piersol, 1966, Pg 135)  For nights when N=l, f ^ l  var Is J j  =  2 .1 (second)  This value was chosen to make the standard deviation include the estimated maximum possible error in a single measurement. To achieve a uniformly spaced time series, it is necessary to interpolate for nights without data from the above mean and variance estimates. To this end, let  Zk  =  I * J=N  *  J  ^i,.  (4-3)  k+j  k  where the summation is only over nights with observations and k refers to a night with no observations. Taking expected values of Z^.: N EA Z. J-  =  >. j=-N  . a E S s f *" "' J  J  N =  L j=-N  s. , . represents the actual latitude variation at time k+j. k+j * J  a k  j  s k + J  Let s, . = s, + E k+j k k  3  39  where E''  k  time  is a term to account for the small variation in actual latitude from  k until time k+j.  Now  N  ~  r  =  E ( z j  N  . aJ  Y  s +  . a  £  j = -N  • E  J  J  J=-N N  and, if in addition the a?  are constrained by  £ j=-N  k  f E  i  Z  1 k  l  S  k  2  +  k  j  N  =  s? - 1, then  \  (4  \  "4)  j=-N If N  is not too large (20 days), then E^  is very small (assuming minimum  k period of variation of s^ equals 365 days) and z^ is a good estimator for s^.  j If the variation of latitude with time as represented by E  is considered in  k determining the variance of z  with respect to s , the foregoing assumption  k is unnecessary.  k  Therefore, it is desirable to estimate 2 D = E k I k kJ  Upon expanding, and utilizing equation (4-4),  becomes N  D  •  k  E  {  z  l  }  -  \  -  z  I \ '  <->  E  \  4 5  k  j=-N which is quite unwieldy.  However, if var(z^) is expanded in the same manner,  the solution is simplified somewhat. var(Z  k )  =  =  That is,  E{[z -E(Z )] k  k  „  ?  2  }' N  N  .  E [ Z |- s - 2 s I a E [ - £ a J E ? l I k J k k k k L ^ k k J j =-N j=-N J  40  Combining this result with equation (4-5) yields N D  = k  2  J  var (Z, ) + [ £ j=-N  k  k  N  K  3  k  1  (4-6)  J  2  For convenience, let I V L j=-N  a'' E I k k J  =  A  . Then D. = var (Z. ) +• A, . k k k'  The value of var (z^) can be determined directly from the identity N  _  N  a s j=-N  k  k  2- ( ) a  ) = +  J  j=-N.  (  v a r  k  k  +  s  J  )  (Bendat and Piersol,  1966  Pg 67) which is valid if the Is^  are independent - as is assumed. Thus, we have  finally  2  N  .  N  2  °k •{ K - \] }• i <%>-«v, [ £ < K\ e  2  v  J=-N  The unknowns in the above include the coefficients  1+  j=_N j a and the terms k  j E . k  The coefficients a are chosen so that, in addition to satisN j fying the condition 51 a =1, they minimize var (z ) . Normally, one would j=-N choose to minimize D^. However, this leads to a set of up to 2 N+1 linear k  k  k  equations, which makes solving for the coefficients a tedious and expensive business. In addition, sample calculations indicate that var (Z )  <<  for most cases - which means that D  k  «  n  k respect to the a? yields k  var (Z, ). Minimizing var (z, ) with k k  -  (4 7)  41 a'  = v a r  (  \+j>  w h e r e C i s a c o n s t a n t d e t e r m i n e d b y the c r i t e r i a N  j aJ k  j=-N  =  1 2  T o m i n i m i z e the e f f e c t o f the t e r m  A,  =  Y a j=-N L k  k i a  w.  Once again  =  w  IJI N+1  o  k  of A ^ one.  _  8 )  C  is determined by  £  a .  i n the d e t e r m i n a t i o n o f  where  E  j  a-' k  i s l i k e l y to b e l a r g e .  k  a n d thus o f D^.  w o  •  j=-N  of Z  let  ( 4  N  w. J  \ kJ  i s a triangle function with equation w. J  factor  3  w.C J var (?k+.)  =  k where  E  =  1.  T h e e f f e c t o f i n c l u d i n g the  k  i s to g i v e l e s s e m p h a s i s to t e r m s  T h i s a m o u n t s to a r o u g h m i n i m i z a t i o n  T h e t r i a n g l e s h a p e i s a l m o s t c e r t a i n l y n o t the o p t i m u m  H o w e v e r , i t i s u n q u e s t i o n a b l y b e t t e r than the b o x c a r f u n c t i o n w h i c h i s  assumed if  j \  C varft' ) k+j  '  The  o n l y r e m a i n i n g u n k n o w n s f r o m e q u a t i o n (4-7) a r e the t e r m s  E^ k  i s a f u n c t i o n o f the a c t u a l l a t i t u d e v a r i a t i o n , i t c a n n o t b e e v a l u a t e d w i t h o u t  k n o w i n g the l a t i t u d e s i g n a l e x a c t l y . ted v a l u e o f A  k  = \Y  a? k  L  E  | kJ  E"' .  Since  H o w e v e r , i t i s p o s s i b l e to d e f i n e a n e x p e c i n t e r m s of a u t o - c o r r e l a t i o n s calculated on  j the b a s i s that the a c t u a l l a t i t u d e s e r i e s i s o n l y o n e d e t e r m i n a t i o n o f a s t o c h a s t i c p r o c e s s that c a n b e s i m p l y r e p r e s e n t e d a s s where  =  x c o s (wk +  6)  xC  x  =  constant amplitude of latitude v a r i a t i o n  w  =  a n g u l a r f r e q u e n c y o f the m a i n p e r i o d i c v a r i a t i o n  9  =  a r a n d o m variable with u n i f o r m probability distribution f r o m  o  to 2 n  •  42  Values  x = .2 seconds w = 2 II /365 radians/day-  were chosen to reproduce the greatest rate of variation with time that might occur in the actual data.  The auto-correlation function 0 for the above  process is given by 2 0(X ) =  Since  p(0) =  E js s k  J p(9) cos (wk + 0) cos (wk + w\ 0 + 0) dO  ^J=  k +  , integration yields 0(X)  2 I . cosw\  =  To simplify the calculations, it is desirable to expand the above equation for the case where w\  is small:  e u > * f [ i  4^]  -  2  This is further simplified by assuming a linear  (X) = where  n  [l -  0  such that  n|X|]  (4-9)  is chosen so that 0(X)  over the range - N  <  0 (X)  < X < N. This choice is made so that the approximations  made will bias the result in a way that overestimates the value of A . This k condition is fulfilled by simply putting  0=0  when  X = N. This gives  43  wN 2  n  It remains now to write  A,  =  in terms of the above auto-correlation function:  =  E  {  k  l  N  N  V  V  j=-Nm=-T*  Subsituting E * = s, . . - s yields k +j k ' N  a  a E k k k j  m  j  ) k i m  E  1  6  k  A  E{  =  N  2  £ m=-N  j=-N  a  a  J  (s  m  k  . - s ) (s  k+i  - s )}  k+m  k  k  J  which, after taking expected values inside the summation reduces to N ^  =  N m= N  J*  *k \  Z  ^  ^  +  '  ^  - { k k+m} E  But.  E {s  and similarly for other terms.  A  k  =  k + j  s  s  k + m  s  }  +  ^ V j  (  E  s  k  S  ^  }]  2  0(k-m )  =  Therefore N N  ^ j=-N  I  m = -N  a  a  j  k  [ 0 (k  m  -m ) - 0  (j) . 0 ( )  k  0  (0)]  N Remembering that  Z j=-N  a  J  = 1 , and taking non-indexed terms outside the  k  summation: N  = 0 (0) - 2  £ j=-N  a 0 (j) + k j j  N  N  I  I  =  _  N  m  =  _  a a k k j  N  m  0 (m - j)  m  44 Substituting  i n the l i n e a r a p p r o x i m a t i o n  f o r the a u t o c o r r e l a t i o n ( e q u a t i o n 4 - 9 ) ,  gives N  2  A  F o r the v a l u e s  k  x  =  If-  [z  2  L  and w  N a  I  (j, -  k  I  g i v e n a b o v e , a n d f o r N=9  T h u s i n s u m m a r y : T h e l a t i t u d e at t i m e  k  k  I  k  j=-N  E  {[ k z  - \]  aJ k  k  |j - m |  k  (4-10)  t . +  (4-3)  J  > N  )-  a  days,  is estimated by 2  k K  i s estimated by  N z  a  2 s e c /day  .00003  2  z  I  j=-Nm=-N  j=-N  2  a n d the v a r i a n c e o f  N  £  2 N  [<] ™ r « V , >  +  \  ( 4  -  7 )  It s h o u l d b e r e m e m b e r e d that a l l t h e s u m m a t i o n s a b o v e a r e o n l y o v e r n i g h t s with data. mum  T h e value of N  w a s c h o s e n to b e n i n e d a y s - b a s e d o n the m i n i -  f i l t e r l e n g t h r e q u i r e d to b r i d g e a l l b u t the m o s t e x t r e m e d a t a g a p s .  B e c a u s e o f the s u m m a t i o n p r o c e s s i n v o l v e d i n c a l c u l a t i n g of  z, k  z^, the v a r i a n c e  w i t h r e s p e c t to s, i s m u c h s m a l l e r t h a n the v a r i a n c e o f the l a t i t u d e K  estimate  "s^ f o r nights w i t h d a t a .  T o r e m e d y t h i s s i t u a t i o n , the i n t e r p o l a t e  t i o n f i l t e r w a s a p p l i e d a l s o to o b s e r v a t i o n n i g h t s - i n t r o d u c i n g a n a d d i t i o n a l c o e f f i c i e n t a.^ so that the m e a n v a l u e o f l a t i t u d e , "3 , o n the n i g h t i n q u e s t i o n , k k i s i n c l u d e d i n the c a l c u l a t i o n o f z ^ . C a r r y i n g o u t the c a l c u l a t i o n s  described  in the s u m m a r y above y i e l d latitude a n d latitude v a r i a n c e e s t i m a t e s for e a c h night.  S i n c e o n l y the v a r i a t i o n i n l a t i t u d e (as o p p o s e d to the a b s o l u t e l a t i t u d e  given by  z ^ ) , i s r e q u i r e d , the m e a n l a t i t u d e o f the s t a t i o n a s d e t e r m i n e d b y  C a m e r a ( 1 9 5 7 , p g Z13) w a s  subtracted f r o m each  r e s u l t i n g s e r i e s w i l l not n e c e s s a r i l y  z^, k = l ,  , N.  The  h a v e z e r o m e a n v a l u e b e c a u s e the  w o b b l e m o t i o n c o n t a i n s a s l o w l y v a r y i n g c o m p o n e n t that i n e f f e c t m a k e s the m e a n value v a r y with  time.  Time Variable Noise Removal The  l a t i t u d e v a r i a n c e e s t i m a t e s f r o m the i n t e r p o l a t i o n  process  p r o v i d e a n e s t i m a t e o f the m e a s u r e m e n t n o i s e i n the r e d u c e d o b s e r v a t i o n s . Therefore,  allow z  k  =  sk  +  n  (4-11)  k  >  where  nk  i s a white n o i s e p r o c e s s w i t h c h a r a c t e r i s t i c s s u c h that  =  n  k  if  k= j  0  if  k 4j  A f i l t e r d e s i g n e d to e x t r a c t the n o i s e f r o m the s i g n a l c a l l y i n F i g u r e 12.  T h e output  *  y  The  =  optimum filter coefficients  f i t o f the f i l t e r output equation  y, k  y  k  of such a  h^  i s shown  schemati-  filter i s given by  X * " -i h  z  k  "  (4  12>  a r e d e t e r m i n e d b y the l e a s t s q u a r e s  to the d e s i r e d o u t p u t  s . k  T h i s r e s u l t s i n the  46  Figure 12. Filter to Remove Random Noise N dp sz  £ j=-N  (k. k-m) =  hj  $  (k - j , k - m)  m = -N  where  <I> zz  ( X, t) = E - (  0,  .N  (4-13)  z J  z  X  fc  is the time dependent autocorrelation. The autocorrelations in equation (4-13) can be defined in terms of s^ and n^ by substituting for z^ from equation (4-11): $  *  zz z  3> ( , j) 4ss  (k, j) =  (k. j) =  <J> (k, j) nn  k  %  s  (k. j)  (4-15)  <p (k. j) and sn * •" the independence of s^ and n. and because  The cross autocorrelations  $ (k, j) are zero because of ns E n. =0. Since  x  *nn< ' J) = k  (4-14)  E  - n  {  n  k  J  n  J  s.  l }  ^  c  a  n  be w r i t t e n  k  n  =  *  k  (k. k) zzv ' '  E  { [Zk -  D  The  quantity  Dk  tude v a r i a t i o n  * (k. k) ss  ^k]2}  k  i s known d i r e c t l y f r o m  the i n t e r p o l a t i o n p r o c e s s .  i s a s s u m e d to b e a s t a t i o n a r y p r o c e s s . *  where  -  (k. j)  s  =  The lati-  This means  that  0 (k - j)  0 i s the n o r m a l t i m e i n d e p e n d e n t a u t o c o r r e l a t i o n o f the l a t i t u d e s e r i e s .  E q u a t i o n s ( 4 - 1 4 ) a n d (4-15) c a n n o w be w r i t t e n  zz  (k, j) =  0 (k - j) + D k  (k, j) =  0 (k - j)  sz  j  (4-16)  0  (4-17)  W r i t i n g e q u a t i o n (4-13) o u t i n f u l l f o r v a r i o u s v a l u e s o f m  and incorporating  the a b o v e s u b s t i t u t i o n s f o r the a u t o c o r r e l a t i o n s , p r o d u c e s the f o l l o w i n g s e t of equations: -N  0  0 (0)  h k  +  0(-  + h k  N) +  = 0(-  -  h  N  0 (N) + . . . + h  k  0 k  0 (0)  •  h"k  N  0 (2N) +  •  +N + ... + h . k  *  . . . + hg  N + h, 0 k  _  (-  N  2 N) + h k  D, „ k+N  N)  0 (N) + nkh  D, K  =  0  (0)  •  0 (N) +  . . . + h™  0 (0) +  h* D  k  N  =  0 (N)  48  To  s i m p l i f y , let  0  0(0) .  -N ^k  (-2N)  0  H  0  0(2N)  i  (0)  k  •N  0 B  D  (-N)  n  and  0/0)  0  k+N 0  D  0  0  0  k+N-1  0  0  0  0  0  *0  D  (N)  Then  ( T  and  H  + JJ ) H = B  =  ( T  +  U)  -1  (4-18)  B  -1 where  (T + E )  ( T + H)  E q u a t i o n (4-18) w a s point  k.  = identity m a t r i x .  u s e d to o b t a i n the n o i s e f i l t e r c o e f f i c i e n t s  A f t e r the f i l t e r was  h^  a p p l i e d at that p o i n t , the f i l t e r w a s  for a given m o v e d ten  d a y s a l o n g the d a t a a n d r e - c a l c u l a t e d .  T h u s , a s e r i e s of latitude m e a n s with  ten  Ten  day s a m p l i n g i n t e r v a l i s o b t a i n e d .  days was  c h o s e n i n o r d e r to b e  c o m p a t i b l e w i t h the d a t a o f the B u r e a u I n t e r n a t i o n a l e de l ' H e u r e .  Autocorre-  l a t i o n s w e r e o b t a i n e d f r o m the u n s m o o t h e d d a t a b y u s i n g the B u r g a l g o r i t h m a s d e s c r i b e d i n A p p e n d i x I, a n d the m a t r i x i n v e r s i o n w a s standard IBM  p r o g r a m SL.E.  accomplished  T y p i c a l r e s u l t s a r e s h o w n i n F i g u r e 13.  p l o t t e d i n F i g u r e 13 a r e the r e s u l t s o f a n o t h e r  t e c h n i q u e w h e r e b y the  s e r i e s i s smoothed a s s u m i n g a constant n o i s e factor  1967,  page 43).  by Also  same  ( e q u a l to the a v e r a g e  v a r i a n c e f o r the c o m p l e t e s e r i e s ) a n d u s i n g the L e v i n s o n i n v e r s i o n (Robinson,  k-N  technique  Figure  13  : Output from v a r i a b l e noise f i l t e r (solid line) and c o n s t a n t n o i s e f i l t e r ( s o l i d l i n e w i t h circles). The d o t t e d l i n e i s t h e o r i g i n a l series. sO  50  T h i s , and other non-plotted c o m p a r i s o n s c l e a r l y n i q u e i s s o m e w h a t i n f e r i o r to the m e t h o d u s e d . numerical  i n d i c a t e that the l a t t e r t e c h T h i s i s p e r h a p s due to  i n s t a b i l i t y i n the L e v i n s o n i n v e r s i o n f o r a m a t r i x  In a p p l i c a t i o n , the s m o o t h i n g f i l t e r l e n g t h w a s  so l a r g e (37 x 3 7 ) .  c h o s e n a r b i t r a r i l y to b e 37  d a y s (N=18) - the o n l y c r i t e r i a b e i n g t h a t the f i l t e r m u s t b e s h o r t e n o u g h not to c a u s e h a r m f u l h i g h - f r e q u e n c y  cutoff.  the m e a s u r e m e n t n o i s e h a s b e e n f i l t e r e d .  It s h o u l d b e k e p t i n m i n d that o n l y A s c a n b e s e e n f r o m the r e s u l t i n g  t i m e s e r i e s f o r the v a r i o u s o b s e r v a t o r i e s ( F i g u r e s 14 to 18), t h e r e i s s t i l l a f a i r l y l a r g e n o i s e f a c t o r w h i c h c a n p o s s i b l y be a t t r i b u t e d to v a r i a t i o n s i n a i r p r e s s u r e , t e m p e r a t u r e , v i s i b i l i t y and l o c a l v e r t i c a l .  T h e ten day m e a n  v a l u e s that a r e output f r o m E q u a t i o n (4-12) a n d that a r e p l o t t e d i n F i g u r e s to 18, r e p r e s e n t the b a s i c d a t a f o r s p e c t r a l a n a l y s i s .  14  CO  o  F i g u r e lit- : B a s i c smoothed data f o r Misisawa.  F i g u r e 15  : B a s i c smoothed data f o r U k i a h . Ul  ro  Figure  16  : Basic  smoothed  data  for Carloforte.  Figure  17  : Basic  smoothed d a t a  for  Kitab.  F i g u r e 18 : B a s i c smoothed d a t a f o r  Gaithersburg.  56  CHAPTER  FREQUENCY  V  DOMAIN FILTERING TO  CHANDLER LATITUDE  OBTAIN  VARIATION  Motivation T w o o f the o r i g i n a l o b j e c t i v e s o f t h i s t h e s i s w e r e : 1.  T o s e p a r a t e the two m a i n f r e q u e n c y c o m p o n e n t s f r o m s h o r t r e c o r d s .  2.  T o r e m o v e the a n n u a l c o m p o n e n t w i t h o u t d e s t r o y i n g i n f o r m a t i o n a s s o c i a t e d with sudden phase and amplitude  s h i f t s i n the C h a n d l e r  m o t i o n and without  f o r c i n g the a n n u a l c o m p o n e n t to b e a c o n s t a n t s p e c t r a l d e l t a f u n c t i o n . T r a d i t i o n a l m e t h o d s o f h a r m o n i c a n a l y s i s i n c l u d e p r i o r a s s u m p t i o n s about the frequency components being dealt with. p e r f e c t h a r m o n i c that time.  E a c h c o m p o n e n t i s a s s u m e d to b e a  does not v a r y i n p h a s e , a m p l i t u d e o r m e a n value with  In a d d i t i o n , a l l the t r a d i t i o n a l m e t h o d s a s s u m e that o u t s i d e the k n o w n  i n t e r v a l , the d a t a s e r i e s  i s either zero o r c y c l i c .  B e s i d e s the o b v i o u s  i n a c c u r a c y o f these e x t e n s i o n s , they c a u s e s e r i o u s i n t e r f e r e n c e e f f e c t s i n the f r e q u e n c y d o m a i n that m a k e t h e s e p a r a t i o n o f c l o s e f r e q u e n c i e s v e r y difficult. 435  F i g u r e 19 i s the F o u r i e r t r a n s f o r m o f a p e r f e c t s i n u s o i d o f p e r i o d  days - b a s e d on a p p r o x i m a t e l y ten y e a r s of d a t a .  If t h i s s i n u s o i d w a s  c o n t i n u e d to i n f i n i t y i n b o t h d i r e c t i o n s , the F o u r i e r t r a n s f o r m w o u l d be the v e r t i c a l l i n e i n the m a i n p e a k o f the s a m e p l o t .  T h e d i f f e r e n c e b e t w e e n the  F o u r i e r t r a n s f o r m s o f the t r u n c a t e d s i n u s o i d a n d the c o n t i n u o u s  sinusoid  i l l u s t r a t e s q u i t e d r a m a t i c a l l y the u n d e s i r a b l e e f f e c t o f e x t e n d i n g the k n o w n  Figure  19  : F o u r i e r t r a n s f o r m o f 10 y e a r s o f a p e r f e c t s i n u s o i d The d o t t e d v e r t i c a l l i n e r e p r e s e n t s t r a n s f o r m o f an s i n u s o i d o f t h e same f r e q u e n c y .  (perlod=^35 infinite  days)  data with z e r o e s .  F o r the c a s e o f the C h a n d l e r a n d a n n u a l c o m p o n e n t s o f  l a t i t u d e v a r i a t i o n , a p p r o x i m a t e l y f i f t y y e a r s o f d a t a a r e r e q u i r e d to o b t a i n e n o u g h s e p a r a t i o n o f the f r e q u e n c y c o m p o n e n t s so that the t r u n c a t i o n e f f e c t s o n the r e s p e c t i v e p e a k s do not i n t e r f e r e too s e v e r e l y . o f t e n y e a r s , w h a t s h o u l d be two one  F o r a r e c o r d length  d i s t i n c t f r e q u e n c y p e a k s b e c o m e m i r e d into  v e r y b r o a d p e a k . F i g u r e 20, w h i c h r e p r e s e n t s the F o u r i e r t r a n s f o r m  of  t e n y e a r s o f s y n t h e t i c w o b b l e d a t a (see F i g u r e 22), e x e m p l i f i e s t h i s u n d e s i r a b l e phenomenon. The  d o t t e d l i n e s r e p r e s e n t the a c t u a l p e a k l o c a t i o n s .  o b v i o u s that i f the o b j e c t i v e s d e s c r i b e d a b o v e a r e to be f u l f i l l e d , s p e c t r a l technique  Predictive  It s e e m s  some  new  is required.  Filtering  r  A reduces  m e t h o d o f p r e d i c t i v e f i l t e r i n g d e v e l o p e d b y U l r y c h et a l (1973)  the m u t u a l i n t e r f e r e n c e p r o b l e m s o f h a r m o n i c a n a l y s i s .  Essen-  t i a l l y , the m e t h o d i s j u s t the p r e d i c t i o n o f the d a t a s e r i e s s e v e r a l t i m e s i n b o t h the b a c k w a r d a n d f o r w a r d d i r e c t i o n i n s u c h a m a n n e r t h a t , i f the o r i g i n a l s e r i e s i s c o n s i d e r e d to be one  determination  of a s t a t i s t i c a l p r o c e s s ,  o r i g i n a l and extended s e r i e s a r e s t a t i s t i c a l l y i d e n t i c a l . new  i n f o r m a t i o n i s added by extending  the t e c h n i q u e to b e  the  T h i s m e a n s that no  the o r i g i n a l s e r i e s .  The  a d v a n t a g e of  i s that the i n c r e a s e d d a t a l e n g t h c a u s e s the i n t e r f e r e n c e e f f e c t s  greatly reduced. The  d i e t i n g the s e r i e s d e f i n e d b y the  prediction coefficients  z  at the t i m e  equation  of  ot  g., j = 1, 2,  J  N,  for pre-  i n t e r v a l s a h e a d o f the d a t a p o i n t  k  are  0.0  I 0.DB3  I 0.125  1  0.188  1—  :  0.25  1  1  U„3]3  0.375  FREQ U/DflY) CX30" )  1  0.438  i  0.5  2  Figure  20  : Fourier transform  o f s y n t h e t i c wobble  data  of Figure  22.  <•£  60' N Z  N  k+a  =  Z  A  k+1 - j  i s l i m i t e d to v a l u e s l e s s t h a n the t o t a l n u m b e r of d a t a p o i n t s a v a i l a b l e .  S i m i l a r l y , the f i l t e r c o e f f i c i e n t s f o r b a c k w a r d s p r e d i c t i o n , b . , a r e d e f i n e d J N by  K- a  U l r y c h et a l (1973) h a s  a  ^Z  b. J  Z. , . . k+j - 1  s h o w n that c a l c u l a t i n g t h e s e c o e f f i c i e n t s i s e q u i v a l e n t  to a p p l y i n g the u n i t p r e d i c t i o n f i l t e r ( a the p r e d i c t e d p o i n t a s the new  = 1)  or  times, each time incorporating  last point of d a t a .  A p r o o f i s g i v e n i n Smylie  et a l ( 1 9 7 3 ) .  T h e r e f o r e , o n l y one  c a l c u l a t i o n o f the c o e f f i c i e n t s  is n e c e s s a r y  to o b t a i n a p r e d i c t i o n f o r a l l v a l u e s of  o  .  g?  1  b:  C o m p u t a t i o n of the  c o e f f i c i e n t s i s s i m p l i f i e d s o m e w h a t b y the f a c t that f o r a r e a l t i m e  g  and  series  1 =  b  T h i s e q u a l i t y r i s e s out o f the m e t h o d of c o m p u t a t i o n - w h i c h i s j j the m i n i m i z a t i o n o f N E  lI  ( ,x! z  k+1  N  - f-I J =  l  g1 i J  )2  z  &  k + l - jJ '  w i t h r e s p e c t to the c o e f f i c i e n t s  linear equations.  +  (Z  1 g . and •>  -  k-1  b  1  .  X  j=i  bj J  Z  J  This minimization yields N  This algorithm  O n c e the u n i t p r e d i c t i o n c o e f f i c i e n t s h a v e b e e n  o b t a i n e d , the s e r i e s i s e x t e n d e d one The  iterative  simultaneously produces autocorrela-  tions and unit p r e d i c t i o n coefficients for a given data s e r i e s . i s d e s c r i b e d i n A p p e n d i x I.  + 1  j  F o r t u n a t e l y , J . B u r g (1967) h a s d e v e l o p e d a n  m e t h o d ( c a l l e d the B u r g a l g o r i t h m ) that  wards directions.  ) J 2  k - l +Jj  p o i n t at a t i m e i n b o t h f o r w a r d a n d  back-  extension procedure is illustrated schematically in  .61  F i g u r e 21.  Continuing  i n t h i s m a n n e r , the s e r i e s c a n b e p r e d i c t e d to i n f i n i t y  1  1 g  4  Step 1  4-  x  1 g  3 X  1 g  2  +  X  l x  4-  Z  Z  Z  N-3  Z  N-2  Z  N-1  N+1  N  V o r i g i n a l data m o v e f i l t e r f o r w a r d one interval 1  1 g  Step 2  5  x  •  Z  g  4-  1 g  4 4-  x Z  N-3  N-2  3 x  Z  1 g  N- 1  4-  2 x  Z  g  N  l i  4-  x Z  Z  N4-1  N4-2  o r i g i n a l data F i g u r e 21. Method for P r e d i c t i v e Extension of a Data Series.  i n b o t h d i r e c t i o n s (if o n e i s r i c h e n o u g h ) .  According  to U l r y c h e t a l (1973)  i n the l i m i t , a s t h e s e r i e s i s p r e d i c t e d to i n f i n i t y , the s p e c t r a l d e n s i t y f u n c t i o n o f the s e r i e s a p p r o a c h e s the m a x i m u m (MESE).  T h i s i m p l i e s that the e x t e n s i o n  that i n h e r e n t i n the d a t a .  entropy spectral estimator  a d d s no s t a t i s t i c a l i n f o r m a t i o n to  T o a p p l y t h i s m e t h o d , the d a t a r e c o r d w a s d i v i d e d  into o v e r l a p p i n g e p o c h s o f n e a r l y t e n y e a r s  (360 p o i n t s o r 3600 d a y s ) .  The  l e n g t h o f p r e d i c t i o n f i l t e r w a s c h o s e n to b e 250 p o i n t s (2500 d a y s ) o n the b a s i s that the r e s o l u t i o n o f the a n n u a l a n d C h a n d l e r f r e q u e n c y c o m p o n e n t s r e q u i r e s a t l e a s t 225 c o e f f i c i e n t s - a c c o r d i n g to the t h e o r e t i c a l r e s o l u t i o n o f the M E S E t e c h n i q u e  - a n d that i f m a n y m o r e t h a n o n e h a l f a s m a n y  as d a t a p o i n t s a r e u s e d , t h e n the B u r g a l g o r i t h m i s n o t s t a b l e .  coefficients  T h e s e ten  y e a r d a t a r e c o r d s w e r e e x t e n d e d two s e r i e s of n e a r l y fifty y e a r s .  To  t i m e s i n e a c h d i r e c t i o n to c r e a t e a t i m e  i l l u s t r a t e the v a l u e o f t h i s t e c h n i q u e , the  syn-  t h e t i c s i g n a l o f F i g u r e 22 ( c o m p o s e d to p r o d u c e a s i g n a l r o u g h l y c o m p a r a b l e to the o b s e r v e d one b y u s i n g the t i m e s e r i e s o f F i g u r e 25 to m o d e l the  Chandler  v a r i a t i o n a n d a p e r f e c t s i n u s o i d o f a m p l i t u d e . 15 s e c o n d s to r e p r e s e n t a n n u a l a n d r a n d o m l y g e n e r a t e d n o i s e o f s t a n d a r d d e v i a t i o n .02 r e p r e s e n t m e a s u r e m e n t n o i s e ) was d u c e the s e r i e s o f F i g u r e 2 3 .  the  s e c o n d s to  e x t e n d e d i n the p r e s c r i b e d m a n n e r to p r o -  It i s c l e a r that the c h a r a c t e r i s t i c s o f the p r e d i c -  t e d p o r t i o n o f t h i s p l o t a r e s i m i l a r to the o r i g i n a l - the m a i n d i f f e r e n c e b e i n g that the e x t e n s i o n a p p e a r s to b e c o m e s i n g l e - f r e q u e n c y v a l u e d t o w a r d s the e x t r e mities.  The  t r u e w o r t h o f the t e c h n i q u e i s i m m e d i a t e l y o b v i o u s w h e n the f a s t  F o u r i e r t r a n s f o r m S ( F F T ) o f the o r i g i n a l a n d e x t e n d e d s e r i e s a r e See  F i g u r e s 20 a n d 24.  The  t r a n s f o r m o f the o r i g i n a l s i g n a l h a s one  the o n l y i n d i c a t i o n that t h e r e m i g h t be two o n the h i g h f r e q u e n c y  the k n o w n input f r e q u e n c y  m a i n peak:  r e a l components is a slight bulging  s i d e o f the m a i n p e a k .  the e x t e n d e d s e r i e s h a s two  compared.  By  c o m p a r i s o n , the t r a n s f o r m  very well separated m a i n peaks corresponding  components.  of  to  T h e r e f o r e , having a m e t h o d that a c h i e v e s  the o b j e c t i v e o f c l e a r r e s o l u t i o n , i t r e m a i n s to d e v e l o p a b a s i s f o r f i l t e r i n g out the a n n u a l c o m p o n e n t a n d u n w a n t e d n o i s e s u c h that the s e c o n d o b j e c t i v e a b o v e i s met. The  E f f e c t o f P h a s e a n d A m p l i t u d e S h i f t s o n the F o u r i e r S p e c t r u m Synthetic w a v e l e t s c o n s i s t i n g of t r u n c a t e d s i n u s o i d s of constant  p e r i o d e q u a l to 435  d a y s , but v a r y i n g i n a m p l i t u d e a n d a b s o l u t e p h a s e w e r e  a d d e d e n d to e n d to p r o d u c e the f u n c t i o n the F o u r i e r t r a n s f o r m o f a c o n t i n u o u s t r a n s f o r m of  X(t)  X(t)  s h o w n i n F i g u r e 25.  s i n u s o i d (as i n F i g u r e 19) a n d  Comparing the  ( w h i c h a r e s u p e r i m p o s e d i n F i g u r e 26) d e m o n s t r a t e s the  typical effect of i n t r o d u c i n g sudden phase and/or amplitude shifts.  That i s .  F i g u r e 22 : S y n t h e t i c l a t i t u d e v a r i a t i o n composed o f 2 main f r e q u e n c y components ( a n n u a l and Chandler) p l u s random n o i s e .  Figure  23  : P r e d i c t i o n of time s e r i e s its original length.  i n Figure  22  to  five  times  Figure  2k  : The  Fourier  amplitude  spectrum  of  the  extended  series  (Figure  23).  us o  0  fN  UJo"  5=  H  ct: CE  UJ ZDo. I— •  cr  i  -1.9  46.1  1  ~I  981 . TIME  1481 .  ~~r  1  1981 .  ( J U L I A N DAYS)  1  ,248.1 2981 .  I  3481 .  (XlO ) 1  F i g u r e 25 : Chandler v a r i a t i o n used t o c r e a t e the s e r i e s o f F i g u r e 2 2 . Composed of s i n e wavelets a l l o f the same frequency (1/^35 days but each v a r y i n g i n phase and amplitude.  )  Figure  26  : F o u r i e r amplitude spectrum o f F i g u r e 25. The d a s h e d l i n e i s t h e s p e c t r u m o f a s i n u s o i d w i t h c o n s t a n t p h a s e and a m p l i t u d e .  0  68  the F o u r i e r a m p l i t u d e b e c o m e s m u c h m o r e s p r e a d o v e r f r e q u e n c i e s n e a r the known central frequency.  In a d d i t i o n , the p e a k s h a p e t e n d s to be  asymmetric  a n d the p e a k f r e q u e n c y i s s h i f t e d a w a y f r o m the k n o w n v a l u e (1/435 d a y s T h i s b e h a v i o u r c a n b e e x p l a i n e d i n g e n e r a l t e r m s b y v i s u a l i z i n g the s y n t h e t i c d a t a i n t e r m s o f the w a v e l e t s that c o m p o s e i t .  E a c h o f the w a v e l e t s c a n be  r e p r e s e n t e d a s a c o n t i n u o u s s i n e f u n c t i o n m u l t i p l i e d b y the ' b o x c a r ' f u n c t i o n of a p p r o p r i a t e width.  See left hand side o f F i g u r e 27.  The  Fourier transform  o f the p r o d u c t o f two f u n c t i o n s i s the c o n v o l u t i o n o f the i n d i v i d u a l F o u r i e r t r a n s f o r m s a s s e e n o n the r i g h t h a n d s i d e o f F i g u r e 2 7 .  T h e b r e a d t h o f the  s i n e f u n c t i o n ( F o u r i e r t r a n s f o r m of b o x c a r f u n c t i o n ) i s i n v e r s e l y p r o p o r t i o n a l to the l e n g t h o f the b o x c a r f u n c t i o n ( o r l e n g t h o f w a v e l e t ) .  T h e r e f o r e , the  data s e r i e s c o m p o s e d of s h o r t w a v e l e t s w i l l be a c o m p l e x s u m  of b r o a d e n e d  sine functions.  that i s some-*  T h i s results in a F o u r i e r amplitude spectrum  w h a t s m e a r e d out c o m p a r e d to the s p e c t r u m o f the c o n t i n u o u s s i n u s o i d . F o u r i e r t r a n s f o r m o f s u c h a s e r i e s o f w a v e l e t s c a n be d e t e r m i n e d H o w e v e r , e x c e p t f o r the c a s e o f two w a v e l e t s , the n u m b e r o f m a k e s the s o l u t i o n e n o r m o u s l y c o m p l i c a t e d . been examined by F e d o r o v  The  analytically.  parameters  c a s e o f two w a v e l e t s  has  a n d Y a t s k i v (1965) a n d t h e y s h o w that e q u a l  a m p l i t u d e w a v e l e t s 180 d e g r e e s out o f p h a s e w i l l p r o d u c e peak.  The  a split s p e c t r a l  T h u s , the a p p a r e n t s p l i t t i n g o f the C h a n d l e r c o m p o n e n t m e n t i o n e d  p r e v i o u s l y i s now  a t t r i b u t e d to a l a r g e p h a s e s h i f t ( W e l l s 1972).  It i s a p p a r e n t  t h e n , that i n o r d e r to p r e s e r v e p h a s e o r a m p l i t u d e s h i f t i n f o r m a t i o n ,  fre-  q u e n c i e s n e a r the c e n t r a l C h a n d l e r f r e q u e n c y m u s t be l e f t u n a l t e r e d (if p o s s i b l e ) b y the p r o c e s s o f f i l t e r i n g out the a n n u a l c o m p o n e n t .  A l s o , it i s  TIME  FREQUENCY  DOMAIN  DOMAIN  co  Figure  27  : Synthesis  of  the  Fourier  transform  of a  truncated  sinusoid.  70  o b v i o u s that b y m e r e l y i n t r o d u c i n g s u d d e n p h a s e a n d a m p l i t u d e s h i f t s i n a p e r f e c t h a r m o n i c , that i t i s p o s s i b l e to c a u s e : (1)  the F o u r i e r s p e c t r u m o f s u c h a s i g n a l to be t o t a l l y d e v o i d  of meaning  so f a r a s h a r m o n i c a m p l i t u d e a n d f r e q u e n c y i n f o r m a t i o n i s (2)  concerned.  s p u r i o u s a r t i f i c i a l e f f e c t s that c o u l d b e i n t e r p r e t e d a s b e i n g r e a l - f o r e x a m p l e , the o b s e r v e d s p l i t t i n g o f the C h a n d l e r p e a k .  B a n d R e j e c t F i l t e r i n g of Annual and N o i s e T h e o r e t i c a l l y , the p r e d i c t e d d a t a s e r i e s s h o u l d c o n s i s t o f two m a i n p e r i o d i c i t i e s - t h e 435 d a y C h a n d l e r a n d the 365 d a y a n n u a l - p l u s random noise.  So i n o r d e r to e x a m i n e the C h a n d l e r l a t i t u d e v a r i a t i o n , the  a n n u a l c o m p o n e n t a n d n o i s e m u s t b e e x t r a c t e d f r o m the l a t i t u d e w h i l e k e e p i n g i n m i n d the c r i t e r i a d i s c u s s e d a b o v e .  signal,  To accomplish  t a s k , a z e r o - p h a s e b a n d r e j e c t f i l t e r w a s a p p l i e d to the F o u r i e r  this  transform  o f the p r e d i c t e d s e r i e s .  G e n e r a l l y , t h r e e d i f f e r e n t r e g i o n s o f the F o u r i e r  spectrum were filtered.  N a r r o w b a n d f i l t e r s w e r e a p p l i e d to a r e g i o n  n e a r z e r o f r e q u e n c y a n d to the p e a k a t 365 d a y s .  B o t h of these w e r e a p p l i e d  so that the F o u r i e r a m p l i t u d e w a s r e d u c e d to the n o i s e l e v e l zero.  r a t h e r t h a n to  In a d d i t i o n , a w i d e b a n d f i l t e r w a s u n e d to r e m o v e t o t a l l y a l l c o n t r i b u -  t i o n s ( a s s u m e d to b e n o i s e ) a t f r e q u e n c i e s a b o v e a p e r i o d o f 200 d a y s . a c c o m m o d a t e the v a r i a b l e c o n d i t i o n s f r o m e p o c h to e p o c h a n d f r o m to o b s e r v a t o r y ,  To  observatory  the w i d t h a n d c u t o f f l e v e l o f the two n a r r o w b a n d r e j e c t  f i l t e r s w e r e c h o s e n to f i t the F o u r i e r s p e c t r u m i n q u e s t i o n .  What this  f i l t e r i n g a c c o m p l i s h e s i s to p r o d u c e a z e r o m e a n v a l u e , a n d to r e m o v e the  71  annual and all but the low frequency portion of random noise - which is left in order to accommodate phase and amplitude shifts in the Chandler component. To illustrate that this filtering method performs properly, the synthetic series Z(t) (Figure 22) was processed. The desired output of the band reject filtering will then be the original synthetic Chandler series of Figure 25. Figure 28 shows how the Fourier spectrum of the extended series is altered by the filtering.  There was no need to apply the filter at zero  frequency since the data series was already at zero mean value. The filtered result (inverse Fourier transform of Figure 28) is depicted by the solid line of Figure 29. The dotted line in the same diagram is the known input Chandler component.  Considering the fact that a large amount of the noise could not be  filtered out, the two signals are very much alike.  The main difference is  that the sharp breaks are not nearly so obvious in the filtered result as in the original, although the breaks are still visually detectable.  However, since  the sudden shifts of phase and amplitude in the actual latitude variation (if such shifts exist) might be a good deal smaller than these synthetic ones, it is desirable to have a method, other than visual scanning, for their detection.  Phase and Amplitude  Demodulation  In their paper that purports to correlate earthquakes with sudden events in the Chandler wobble, Smylie and Mansinha (1968) fit smooth arcs to the data by a least squares fit.  The point where a fitted arc passes outside  some circle of validity marks where a sudden event occurs and a new smooth  F i g u r e 28 : E f f e c t o f band-reject f i l t e r i n g o f annual on the spectrum o f the extended s y n t h e t i c l a t i t u d e v a r i a t i o n (see F i g u r e s 23 &24)  °~1  -1.9  1  48.1  1  98.1 TIME  Figure  29  1  148.1  1  T  198.1  ( J U L I A N DAYS)  248.1 (XlO  1  r  298.1  348.1  )  : Chandler v a r i a t i o n r e s u l t i n g from b a n d - r e j e c t f i l t e r i n g of the synthetic data. The d a s h e d l i n e r e p r e s e n t s t h e known s y n t h e t i c Chandler v a r i a t i o n .  74 arc begins. that a n y event; or  See  F i g u r e 6.  large deviation  This  from  method  the e x p e c t e d  w h e t h e r o r not that e v e n t  mean value.  dures somewhat, phase and harmonic,  To  suffers  from  smooth arc will produce  represents a real  a v o i d this p r o b l e m  a purely analytic  and  shift in phase,  that y i e l d s  f u n c t i o n was  amplitude  the t i m e - v a r i a b l e  developed.  By  quasi-  i t i s m e a n t that the p e r i o d i c s i g n a l h a s c o n s t a n t p e r i o d but that  the p h a s e a n d a m p l i t u d e  a r e p e r m i t t e d to v a r y w i t h t i m e .  C h a n d l e r latitude s e r i e s  o b e y s this c r i t e r i a .  d e n o t e d b y the f u n c t i o n  f, i t w i l l be o f the  f(t)  I f this  an  to s i m p l i f y the c o m p u t i n g p r o c e -  technique  amplitude of a quasi-harmonic  the m a j o r w e a k n e s s  =  A(t) e  M  w  the  So i f the C h a n d l e r s i g n a l i s  form:  A ( t ) cos ( w Q t  c a n be a l t e r e d to the c o m p l e x  Supposedly,  0 (t) )  +  (5.1)  form  t  o  +  0  (  t  )  >  - iw t then s i m p l e m u l t i p l i c a t i o n by e A ( t) a n d p h a s e 0 ( t ) .  °  w i l l y i e l d a c o m p l e x function of  Since  (9 -  cos  H_  2  )  =  sin 0  E q u a t i o n 5. 1 y i e l d s f(t  " J L  1 ww  o  )  =  A  ( t) s i n (w  t + °  0  (t) )  Therefore,  f ( t ) + i f ( t - - J2 L w- ) Q  and m u l t i p l y i n g by  e  1 W  o'  =  (t) A  i(w e  o  t+0(t  >>  amplitude  (f (t) + i f (t - __0 2 wQ  ) ) e" 1  w  ofc  = A  (t) e i 0 ( t )  (5. 2)  T h u s , t a k i n g the a m p l i t u d e a n d p h a s e o f the l e f t h a n d s i d e o f t h i s e q u a t i o n g i v e s the t i m e - v a r i a b l e a m p l i t u d e a n d p h a s e o f the s i g n a l  f.  s i m p l e c o m p u t a t i o n to t e s t d a t a y i e l d s e x c e l l e n t r e s u l t s .  A p p l i c a t i o n of this The  resultant phase  a n d a m p l i t u d e t i m e v a r i a t i o n s f o r the s i g n a l o f F i g u r e 25 a r e p l o t t e d a s the s o l i d l i n e s i n F i g u r e s 30 a n d 3 1 .  The  o n l y d e v i a t i o n s f r o m the a c t u a l p h a s e  a n d a m p l i t u d e o c c u r at the b r e a k s , w h i c h s h o u l d a p p e a r a s s t r a i g h t v e r t i c a l lines.  The  a c t u a l b r e a k c o r r e s p o n d s to the i n i t i a l d e v i a t i o n f r o m a h o r i z o n t a l  l i n e , a n d the l a r g e f l u c t u a t i o n s i m m e d i a t e l y f o l l o w i n g a r e c a u s e d b y the a v e r a g i n g c h a r a c t e r i s t i c o f the m e t h o d .  N e x t , the d e m o d u l a t o r  the f i l t e r e d s y n t h e t i c C h a n d l e r s e r i e s o f F i g u r e 2 9 . i n F i g u r e s 30 a n d 31) a r e s u r p r i s i n g l y g o o d . amplitude are immediately obvious.  The  The  was  a p p l i e d to  results (dashed lines  b r e a k s in both phase  and  A l s o , i t c a n be s e e n ( h e r e a n d i n a l l the  e x a m p l e s p r o c e s s e d ) that the p h a s e i s the m o r e r e l i a b l e i n d i c a t o r of a b r e a k . T h a t i s , the b a n d r e j e c t f i l t e r i n g c a u s e s u n p r e d i c t a b l e d i s t o r t i o n s o f the amplitude v a r i a t i o n i f large sudden shifts are p r e s e n t . s e e m s to b e f r e e o f t h e s e e f f e c t s .  The  phase variation  I  -1.9  1 48.1  1 98.1  TIME  1 148.1  1 198.1  ( J U L I A N DAYS}  1 ,248.1  (XlO3  1 298.1  )  l i r e 30 : P h a s e v a r i a t i o n o f t h e s y n t h e t i c C h a n d l e r . T h e  solid line is t h e phase o f the s y n t h e t i c C h a n d l e r m o t i o n r e c o v e r e d by b a n d - r e j e c t f i l t e r i n g and the dashed l i n e i s the phase o f t h e known o r i g i n a l input Chandler.  1 348  in  CJ  CO  O CM  LU CO  V 7-  M O  *  A •  _l Q_ cr  o  •3.9  Figure  31  48.1  9B.1  TIME  148.1  198.1  ( J U L I A N DAYS)  (X10 J  .248.1  )  ~1 298.1  348.1  J Amplitude v a r i a t i o n o f the s y n t h e t i c Chandler. S o l i d and dashed l i n e s r e p r e s e n t the f i l t e r e d and a c t u a l v e r s i o n s r e s p e c t i v e l y .  78  CHAPTER  THE  P H A S E AND  VI  A M P L I T U D E R E S U L T S FOR  5 I. L . S.  THE  OBSERVATORIES  A f t e r b a n d r e j e c t f i l t e r i n g to i s o l a t e the C h a n d l e r c o m p o n e n t o f l a t i t u d e v a r i a t i o n , one o b t a i n s a s e t o f o v e r l a p p i n g t e n y e a r t i m e s e r i e s f o r each observatory.  In o r d e r to p r o d u c e  lapping portions were averaged.  a c o n t i n u o u s C h a n d l e r s e r i e s , the o v e r -  F i g u r e s 32 to 36 a r e the r e s u l t i n g  l a t i t u d e v a r i a t i o n s f o r the i n d i c a t e d o b s e r v a t o r i e s .  Chandler  T h e r e a r e two m a i n f e a t u r e s  o f t h e s e r e s u l t s that a r e c o m m o n to e v e r y o b s e r v a t o r y : 1.  A  s h i f t i n m e a n v a l u e o c c u r r i n g at J u l i a n d a y 2 4 2 7 8 0 0 (1935.0)  w h i c h c o r r e s p o n d s to a c h a n g e i n s t a r c a t a l o g u e s u s e d b y the observatories.  The  s h i f t i s c l o s e to +.02"  f o r e a c h o f the  t h r e e o b s e r v a t o r i e s f u n c t i o n i n g at t h i s t i m e . 2.  A l a r g e i n c r e a s e i n a m p l i t u d e b e g i n n i n g at a p p r o x i m a t e l y J u l i a n d a y 2429300  (1939.0).  B o t h o f t h e s e f e a t u r e s i n d i c a t e that the f i l t e r i n g t e c h n i q u e i s at l e a s t c a p a b l e o f r e t a i n i n g i n f o r m a t i o n about f a i r l y r a p i d a m p l i t u d e s h i f t s .  An  even m o r e  en-  c o u r a g i n g f a c t o r i s that the o v e r l a p p i n g p o r t i o n s o f the s e v e r a l t e n y e a r  Chandler  s e r i e s at a p a r t i c u l a r s t a t i o n a r e v e r y s i m i l a r p r i o r to b e i n g a v e r a g e d .  That i s ,  two  c o m p l e t e l y i n d e p e n d e n t a p p l i c a t i o n s o f the f i l t e r i n g p r o c e s s to d i f f e r e n t d a t a  i n t e r v a l s y i e l d , i n the o v e r l a p p i n g r e g i o n , two are  almost identical.  C h a n d l e r v a r i a t i o n s e r i e s that  T h i s c o n s i s t e n c y i m p l i e s that the f i l t e r i n g t e c h n i q u e i s  p r o d u c i n g a r e s u l t c l o s e to s o m e d e f i n i t i v e C h a n d l e r  motion.  Figure  32  : Chandler  latitude  variation  for  Mislsawa.  to o  Figure  33  : Chandler  latitude  variation  for  Ukiah  U3  a  (=}  Figure  34  : Chandler  latitude  variation  for  Gaithersburg.  00  Figure  35  t  Chandler  latitude  variation  for  Kitab  (£3  O  Figure  36  : Chandler  latitude  variation  for  Carloforte.  84  In o r d e r to e x a m i n e the C h a n d l e r v a r i a t i o n i n m o r e d e t a i l , i t i s a d v a n t a g e o u s to l o o k at the p h a s e a n d a m p l i t u d e o f the p e r i o d i c m o t i o n as a function of t i m e . section.  The  T h i s i s a c c o m p l i s h e d b y the m e t h o d d e s c r i b e d i n the p r e v i o u s  a m p l i t u d e o f the C h a n d l e r m o t i o n f o r e a c h o f the f i v e o b s e r v a t o r i e s  i s p l o t t e d i n F i g u r e 3 7.  S i m i l a r l y , the p h a s e i s p l o t t e d i n F i g u r e 38.  It s h o u l d  be n o t e d that the p h a s e h a s b e e n a d j u s t e d to the G r e e n w i c h m e r i d i a n b y the l o n g i t u d e o f the o b s e r v a t o r y to the o u t p u t o f the d e m o d u l a t o r , z e r o point i n t i m e was  c h o s e n to be J u l i a n d a y 2 4 2 3 3 2 3 .  adding  a n d that the  T h i s m e a n s that a  l a t i t u d e s i g n a l at G r e e n w i c h w o u l d h a v e z e r o p h a s e i f the p e r i o d i c s i g n a l r e a c h e d a m a x i m u m o n d a y 2423323 o r o n a n y i n t e g e r n u m b e r o f p e r i o d s b e f o r e o r a f t e r wards.  The  C h a n d l e r p e r i o d was  c h o s e n to b e 4 3 7 . 0 d a y s b a s e d o n the v a l u e  that m a k e s the p h a s e c o n s t a n t i n the l a s t p o r t i o n o f the t i m e i n t e r v a l ( a f t e r 2429700).  J.D.  T h i s c r i t e r i o n f o r c h o o s i n g the e x a c t v a l u e o f the p e r i o d i s s o m e w h a t  a r b i t r a r y ( e x c e p t e d v a l u e = 435 d a y s ) , a n d i s a d o p t e d p r i m a r i l y b e c a u s e o f the r e l a t i v e c o n s i s t e n c y a n d f l a t n e s s o f the v a r i o u s p h a s e c u r v e s d u r i n g t h i s t i m e interval. It i s i m m e d i a t e l y o b v i o u s that s o m e m a j o r c h a n g e s i n b o t h the p h a s e a n d a m p l i t u d e o f the C h a n d l e r l a t i t u d e v a r i a t i o n o c c u r r e d i n the i n t e r v a l f r o m 1922. 7 to 1 9 4 9 . 0 .  P e r h a p s the m o s t e n c o u r a g i n g r e s u l t f r o m F i g u r e s 37  a n d 38 i s the c o n s i s t e n c y f r o m s t a t i o n to s t a t i o n , e s p e c i a l l y f o r p h a s e v a r i a t i o n . T o f u r t h e r s i m p l i f y i n t e r p r e t a t i o n , b o t h the p h a s e a n d v a r i a t i o n s w e r e v i s u a l l y a v e r a g e d o v e r the n u m b e r o f o b s e r v a t o r i e s . s a m e t i m e , a v i s u a l e s t i m a t e o f the m a x i m u m p o s s i b l e e r r o r i n the was  amplitude A t the average  o b t a i n e d b y i n c l u d i n g a l l the s i g n a l e x c e p t the e x t r e m i t i e s o f l a r g e p e a k s  242331.1  242431.1  242531.0  24263^1.0^ 242730.^ ^ ^ 0 . 8 ^ ^ ^ . 8  243030.7  Figure 3? : Chandler amplitude variation f o r a l l observatories.  243130.6  PHRSE VARIATION -0.5  98  1.0  2.5  (RADIANS) 4.0  5.5  7.0  87  that do not correlate with peaks at other stations. are shown in Figures 39 and 40.  The results of this process  The dotted line represents the average and  the solid line the error bounds.  Phase Results It is immediately apparent that there are several abrupt changes in the phase of the Chandler motion. days 2424511 (May,  The two most definite occur at Julian  1926) and 2425311 (March, 1928). The possible errors in  these dates are approximately plus or minus three months. Another very possible event occurs at Julian day 2426010 (February, 1930). After that, there is a more or less gradual increase in phase until approximately. 1941 (2430207) after which the phase is quite constant until the end of the record in 1949.0. Other possible (but quite uncertain) abrupt shifts occur at times 2426360 (January, 1931), 2427410 (December, 1933), 2429310 (February, 1939) and 2430207 (September, 1941). The most definite of these (based on the behavior of all four records) is the second to last.  Amplitude Results It is fairly difficult to make any meaningful conclusions about the amplitude variation because of the large error bounds.  However, it is obvious  that there is a large increase (from . 08" to . 20") beginning at approximately Julian day 2429300 (February, 1939). This starting point corresponds closely to a possible phase shift.  In the time at the beginning of the record where the  error bounds on the amplitude variation are relatively small, two of the minimum values of the averaged amplitude (at 2424500 and 2425300) correspond exactly to  IO CM  UJ CO  in  O < < >6 UJ Q  ZD  o  6 2423311  2424311  24253H  2426311  TIME Figure  39  : Amplitude  variation  2427311  2428311  (JULIAN averaged  over  2429311  2430311  2431311  DAYS) number  of  observatories.  CO  00  o  ^—. in CO in"  <  <  o  10  rr < >  (Nl  LU CO < X CL m o  i  2423311  2424311  2425311  T 2426311  TIME Figure  2427311  2428311  (JULIAN  40 : P h a s e v a r i a t i o n a v e r a g e d  T  2429311  i  2430311  — I  2431311  DAYS )  o v e r a number o f o b s e r v a t o r i e s .  oo so  90  the two major phase shifts.  Another relation between phase and amplitude is  that amplitude changes significantly only in regions where the phase is constant; that is, at the beginning and end sections of the record. Guinot (1972) has obtained phase and amplitude variation of the Chandler wobble for the time interval 1900.2 to 1970.0. His results (Figures 41 and 42) are based on a least squares fit of a sine function to each 1. 2 years of latitude data from up to fifteen latitude observatories. data point every .6 years.  This method yields a  His value for the phase is given in years instead of  in radians (1. 19 years = 2 TT radians), and he plots the variation of colatitude rather than latitude. After taking these differences into account, his results and the results illustrated in Figures 41 and 42, are, so far as the main features are concerned, in agreement.  Interpretation with Respect to Established Theories According to Smylie and Mansinha (1967) abrupt shifts like the two main shifts in phase observed should be coincidental with or just prior to major earthquakes. earthquakes ( > served shift (May, inactivity.  Table 3 is a list taken from Richter (1958) of the major  7.9) that occurred during the epoch of interest.  The first ob-  1926) marks the end of two years of complete earthquake  A series of major earthquakes begins only one month later.  The  second observed event (March, 1928) occurs at the end of nearly one year of inactivity and once again marks the beginning of an active time.  The only other  event that is considered likely is the one between February, 1930 and January, 1931 (the exact location is somewhat uncertain because there may  in fact be two  events). Once again, the shift occurs close to the end of a time of quiet (1. 5  T A B L E III  E A R T H Q U A K E S O F MAGNITUDE G R E A T E R T H A N 7. 9  Dates of Observed Chandler Events  Dates of Earthquakes  Magnitude  Nov. 11, 1922 Feb. 3, 1923 Sept. 1, 1923 Apr. 14, 1924 June 26, 1924  8.4 8.4 8.3 8.3 8. 3  June 26, 1926 Oct. 3, 1926 Oct. 26, 1926 Mar. 7, 1927 May 22, 1927  8. 3 7.9 7.9 7.9 8. 3  M a r c h 9, 1928 June 17, 1928 Dec. 1, 1928 M a r c h 7, 1929 . June 27, 1929  8. 1 7.9 8.3 8. 6 8. 3  May, 1926  March, 1928  may be one event  (Feb. 1930 ( (Jan. 16, 1931  Jan. 15, 1931  7.9  Feb. 2, 1931 Aug. 10, 1931 Oct. 13, 1931  7.9 7.9 7.9  91  92  y e a r s ) a n d m a r k s the b e g i n n i n g o f a n a c t i v e t i m e .  The most obvious conclusion  one c o u l d m a k e f r o m t h e s e o b s e r v a t i o n s i s that a b r u p t s h i f t s i n the C h a n d l e r m o t i o n a r e a s s o c i a t e d w i t h the o n s e t o f e a r t h q u a k e a c t i v i t y .  This  conclusion  o f c o u r s e i g n o r e s the c a s e s w h e r e a q u i e t e r a i s not e n d e d b y a n a b r u p t c h a n g e i n the C h a n d l e r m o t i o n , a n d a l s o the s h i f t s (though c o n s i d e r e d l e s s l i k e l y to b e r e a l ) that do n o t c o r r e s p o n d to the s a m e c i r c u m s t a n c e s . arises:  Another question  What m e c h a n i s m c o u l d p o s s i b l y cause s u c h l a r g e , r a p i d phase s h i f t s ?  A c c o r d i n g to G u i n o t (1972), t h e y a r e n o t l i n k e d to a n y g e o p h y s i c a l e v e n t .  Since  c a u t i o n i s the w a t c h w o r d , t h e r e won't b e a n y w i l d g u e s s e s h e r e a b o u t t h i s matter.  T h e c o r r e s p o n d e n c e between constant phase and r a p i d change i n a m p l i -  tude i s d i f f i c u l t to e x p l a i n - i t m i g h t b e s u r m i s e d that the e x c i t a t i o n m e c h a n i s m i s m o r e e f f i c i e n t w h e n the p h a s e i s a t a s t a b l e v a l u e . this w o r k : motion.  One thing i s c e r t a i n  from  e a r t h q u a k e s c o u l d n o t b e d e t e c t e d e v e n i f t h e y d i d e x c i t e the C h a n d l e r  T h a t i s , t h e n o i s e l e v e l i s a p p r o x i m a t e l y t h r e e t i m e s a s l a r g e a s the  l a r g e s t p o s s i b l e e f f e c t o f a m a j o r e a r t h q u a k e ( c a l c u l a t e d to b e a b o u t o n e foot o r . 0 1 " f o r the 1964 A l a s k a e a r t h q u a k e - S m y l i e a n d M a n s i n h a , An  1972).  i m p o r t a n t r e s u l t f r o m m o s t s t u d i e s o f the C h a n d l e r w o b b l e i s  a n e s t i m a t e o f the d a m p i n g t i m e o r a l t e r n a t e l y o f the Q f a c t o r .  E s t i m a t e s of  t h e s e q u a n t i t i e s a r e n o r m a l l y o b t a i n e d b y f i t t i n g the s m o o t h e d p o w e r d e n s i t y f u n c t i o n ( p r o p o r t i o n a l to the s q u a r e d F o u r i e r t r a n s f o r m o f the data) to f u n c t i o n s like  1 /  (  )  1 /  (w - w Q )  - (w2 - w  2 or  2  )  2  + 4w2  ?^ 2  (6. 1)  1 2 +  ( - )  (6.2)  93  depending on whether the excitation mechanism is thought to be random abrupt events or a continuous multi-frequency forcing function. It is unclear from most of this kind of curve fitting whether or not the effect of finite data length is considered in calculations.  The case of a multi-frequency excitation pro-  ducing a damped response as in Figure 9, can be ruled out by the fact that the phase demodulation indicates an almost single-valued Chandler frequency. It should be noted that the single frequency is not a result of the band-r eject filtering, since a specific aim of that filtering was to leave frequencies near the Chandler undisturbed. The noisiness of the amplitude variation makes it extremely difficult if not impossible to say anything about the existence of damped wavelets that would occur if the Chandler wobble was randomly excited. However, by merely modelling the observed phase and amplitude by simple straight lines (dotted line in Figures 39 and 40), and using these values as the phase and amplitude of an undamped sinusoid, one obtains the Fourier spectrum of Figure 43.  The dashed line is a line visually fitted in order to smooth the spectrum  and also to approximately model a damped response.  Part of the width of the  resulting peak is due to the sine function associated with the finite data length. However, a good portion of the width is due to the effect of phase and amplitude variation - similar to that observed in the previous chapter. Fitting very roughly the .parameters of equation 6. 1 to this curve, yields a damping time of approximately 7 years and a corresponding Q factor of 18.  That is, a perfect  sinusoidal signal (Q factor = °o) altered by simple phase and amplitude variation to emulate the observed values, yields a spectrum characteristic of a damped signal with a Q factor of approximately 20. It seems likely then, that the Q  Figure  kJ,  x Fourier  amplitude  spectrum  of  simplified  undamped  Chandler  motion  95  f a c t o r o f the a c t u a l w o b b l e i s s o m e w h a t h i g h e r t h a n i s c o m m o n l y t h o u g h t .  Conclusions 1.  The  o p t i m u m v a l u e f o r the p e r i o d o f the C h a n d l e r w o b b l e i s 437  b a s e d o n the c r i t e r i a that the t i m e v a r i a t i o n o f p h a s e be 2.  The  v a l u e o f the Q  vious estimates. 3.  minimized.  factor for wobble i s a l m o s t c e r t a i n l y h i g h e r than p r e Its e x a c t v a l u e i s e x t r e m e l y u n c e r t a i n .  A b r u p t s h i f t s i n the p h a s e o f the C h a n d l e r w o b b l e a r e p o s s i b l y a s s o c i a t e d w i t h the o n s e t o f e a r t h q u a k e  4.  days  The  activity.  c a u s e o f t h e s e p h a s e s h i f t s a n d a l s o o f l a r g e v a r i a t i o n s i n the C h a n d l e r  amplitude is unknown.  H o w e v e r , the v a r i a t i o n s a r e p r o b a b l y too l a r g e to  be c a u s e d b y the p r e s e n t l y e s t i m a t e d e f f e c t o f e a r t h q u a k e s . 5.  The  suggested c o r r e l a t i o n between i n d i v i d u a l earthquakes and changes i n  the C h a n d l e r w o b b l e c o u l d s t i l l b e t r u e t h o u g h t h e r e i s no e v i d e n c e to support such a r e l a t i o n .  96  BIBLIOGRAPHY  Bendat, J.S., and A. G. Piersol, Measurement and Analysis of Random Data, John Wiley and Sons, Inc., New York, 1966. Burg, J.P., Maximum entropy spectral analysis, preprint of paper presented at the 37th Meeting of the SEG, Oklahoma City, Okla. , 1967. Camera, L., Risultati del Servizio Internazionale Delle Latitudini dal 1935.0 al 1941.0. v. 9, 1957. Cecchini, G., II problema della variazione delle latitudini, Publ. Reale Obs. Astron. di B r e r a in Milano, v. 61, 7-96, 1928. Colombo, G., and I.I. Shapiro, Theoretical model for the Chandler Wobble, Nature, v. 217, 156-157, 1968. Dahlen, F. A., The excitation of the Chandler wobble by earthquakes, Geophys. J. R. Ast. S o c , v. 25, 157, 1971. Fedorov, F.P., and Y.S. Yatskiv, The cause of the apparent "bifurcation" of the free nutation period, Soviet Astron., v. 8, 608-611, 1965. Fellgett, P., unpublished manuscript referred to in Munk &t McDonald's Rotation of the Earth. 174, I960. Guinot, B., The Chandlerian Wobble from 1900 to 1970, Astron. & Astrophys., v. 19. 207-214, 1972. Haubrich, R. A., An examination of the data relating pole motion to earthquakes, in Earthquake Displacement Fields and the Rotation of the Earth, ed. by L. Mansinha, D.E. Smylie, and A. E. Beck, SpringerVerlag, New York, 1970. Iijimi, S., On the Chandler and annual ellipses in the polar motion as obtained from every 12 year period, Pub. Ast. S. J. v. 24, 109, 1972. Jeffreys, H. and Vicente, R., The theory of nutation and the variation in latitude: The Roche model core, Monthly Notices, Royal Astronomical Society, v. 117, 162, 1957. Kimura, H., Results of the International Latitude Service from 1922. 7 to 1931.0, v. 7, 1935.  Mansinha, L., and D.E. Smylie, Effect of earthquakes on the Chandler wobble and the secular polar shift, J. Geophys. Res., v. 72, 4731-4743, 1967. Melchior, P., Latitude variation, Progress in Physics and Chemistry of the Earth, v. 2, Pergamon Press, 1957. Munk, W., and E.S.M. Hassan, Atmospheric excitation of the Earth's wobble, Geophys. J., v. 4, 339-356, I960. Munk, W., and G. J. F. McDonald, The Rotation of the Earth, University Press, I960.  Cambridge  Pachenko, N.I., Discussion, Astron. J. v. 64, 94, 1959. Press, F., Displacements, strains, and tilts at teleseismic distances, J. Geophs. Res., v. 70, 2395-2412, 1965. ~~ Richter, Charles F., Elementary Seismology, Wilt Freeman and Company, San Francisco, 1958. Robinson, Enders A., Multichannel Time Series Analysis with Digital Computer Programs, Holden-Day, San Francisco, 1967. Rochester, M.G., Polar wobble and drift: A brief history, in Earthquake Displacement Fields and the Rotation of the Earth, ed. by L. Mansinha, D.E. Smylie, and A. E. Beck, Springer-Verlag, New York, 1970. Rochester, M.G. and M.G. Pedersen, Spectral Analyses of the Chandler Wobble, in Rotation of the Earth, I. A. U. publication, 33-38, 1972. Rochester, M. G., and D.E. Smylie, Geomagnetic core-mantle coupling and the Chandler wobble, Geophys. J., v. 10, 289-315, 1965. Rosenhead, L., The annual variation of latitude, Mon. Not. - Geophys. Suppl. 2, v. 140, 1929. Runcorn, S. C , A possible cause of the correlation between earthquakes and polar motions, in Earthquake Displacement Fields and the Rotation of the Earth, ed. by L. Mansinha et al.. Springer-Verlag, New York, 1970. Smylie, D.E., and L. Mansinha, Earthquakes and the observed motion of the rotation pole, J. Geophys. Res., v. 74, 7661-7673, 1968.  98  Smylie,  D . E . , G . K . C . C l a r k e , and L . M a n s i n h a , Deconvolution  o f the p o l e  p a t h , i n E a r t h q u a k e D i s p l a c e m e n t F i e l d s a n d the R o t a t i o n o f the E a r t h , e d . b y L,. M a n s i n h a , D . E . S m y l i e ,  and A . E .  Beck,  S p r i n g e r - V e r l a g , New Y o r k , 1970. Smylie,  D . E . a n d L . M a n s i n h a , T h e E l a s t i c i t y of D i s l o c a t i o n s i n R e a l E a r t h M o d e l s a n d C h a n g e s i n the R o t a t i o n o f the E a r t h , G e o p h y s . J . R . A s t r . S o c . . v. 23, 329-354, 1971.  Smylie,  D . E . , C l a r k e , G . K . C , a n d T . J . U l r y c h , A n a l y s i s o f the I r r e g u l a r i ties i n the E a r t h ' s R o t a t i o n , M e t h o d s i n C o m p u t a t i o n a l P h y s i c s , v. 13, 1 9 7 3 .  Stacey,  F.D., A re-examination  o f c o r e - m a n t l e c o u p l i n g a s the c a u s e o f the  w o b b l e , i n E a r t h q u a k e D i s p l a c e m e n t F i e l d s a n d the R o t a t i o n o f the E a r t h , e d . b y L . M a n s i n h a et a l . . S p r i n g e r - V e r l a g ,  New  Y o r k , 176-180, 1970. Takeuchi,  H . , O n the E a r t h t i d e o f the c o m p r e s s i b l e and e l a s t i c i t y . v. 3 1 , 6 5 1 ,  E a r t h of variable density  T r a n s a c t i o n s o f the A m e r i c a n G e o p h y s i c a l  Union,  195(T  U l r y c h , T . J . , S m y l i e , D . E . , J e n s e n , O. G . C l a r k e , G . K . C , P r e d i c t i v e f i l tering and smoothing of short r e c o r d s , J . Geophys. R e s , in press. W a l k e r , A . a n d A . Y o u n g , F u r t h e r n o t i c e s o n the a n a l y s i s o f the v a r i a t i o n i n l a t i t u d e , M o n t h l y N o t i c e s , R o y a l A s t r o n o m i c a l S o c i e t y , v . 117, 119-141, 1957. W e l l s , F . J . , P h . D . T h e s i s at B r o w n U n i v e r s i t y , 1 9 7 2 . Y a s h k o v , V . A . , S p e c t r u m o f the m o t i o n o f the E a r t h ' s p o l e s , S o v i e t A s t r o n o m y , v . 8, 6 0 5 - 6 0 7 , 1 9 6 5 . Y u m i , S. , P o l a r m o t i o n i n r e c e n t y e a r s , i n E a r t h q u a k e D i s p l a c e m e n t F i e l d s a n d the R o t a t i o n o f the E a r t h , e d . b y L>. M a n s i n h a et a l . , S p r i n g e r - V e r l a g , New Y o r k , 1970. * E d i t o r ' s note:  Added i n p r e s s .  R u d n i c k , P., The spectrum o f t h e v a r i a t i o n i n l a t i t u d e , T r a n s a c t i o n s American G e o p h y s i c a l U n i o n , 35» ^9t 1956. S p i t a l e r , R., Petermanns M i t t e i l u n g e n , Erganzungband, 29, 137» 1901.  of the  99  APPENDIX I  The Burg Algorithm (after Smylie, personal notes)  Let the time series f. be the input to a linear unit prediction filter.  Then the prediction of the process at the time k is /  N  and the error of the prediction is E  g,g,....,g 1 2 n  is minimized.  This produces the set of equations: 2  where  - f k  are chosen so that  gj 0 ff (0) + g 0 ff (-D +  gj 0  f k  ff  (N-1)  0££ ( \ )  +  +  g 0 (l-N) = 0 (l) n  f f  + g 0ff (°) n  ff  =%  <) N  represents the stationary autocorrelation of lag \  In addition, the error power  { for an N point predictor can be written  K l  }  100  r {  E  l  l J  P  ) =  N+ 1  =  ^ff  ( 0 )  N j=l  J  Combining this equation with the previous set yields the matrix equation: N 0 (O)  0 (N) ff  with  ^f^" )  ?> (-N)  0 (N-l)  0 (O)  1  ff  1  ff  ff  N+1 0 0 0 0 0  N  ff  - gN  N  1  The Burg algorithm concerns the iterative solution of this matrix equation for both the coefficients  a  ^ and the autocorrelations  0^ (i) simultaneously.  F i r s t consider the case N = 0. Then P  =  1  0  *ff  1  (0) = 1  —  M  M  V  ^ j=l  1  If  2  j  Since all the additional steps are identical in form, all that is required at this point is to show how to go from  N to N + 1.  For step N we have T. N  p *N+1 0  1 a  NI  a a  =  N2 N3 • •  a ' NN  0 0 0 0 0 _  where the first subscript on the  Ni  refers to the  order of the iteration  101  It is assumed that all the elements are known. The N + 1 equation is then written T  1  N+l  N+2  a  0  N+l, 1  =  a  N+l, 2  0 0  *  •  a  0  '  0  N+l, N+l This can be written 1  T N+l  0 *  a  a  N. 1  N, N N+l, N+l  N.N  a* N. 1 1  0  ' N+l 0 0 0 0  +  N+l 0 0 0 0  N+l, N+l  N+1  L  which implies the relations  a  N+l, 1  +  N, 1  a *  N,N  N+l.N  a  a  N+l, N+l  N, N  +  a  a  N+l, N+l  N+l and  0  ff  (N+l)  I j=l  N+l.j  0  ff( l-J> N +  N, 1  (A-l)  102  To  solve a l l these unknowns, it i s o n l y n e c e s s a r y  a  nvr. , „ , , • N+l, N+l  This is accomplished c  to c a l c u l a t e the v a l u e  b y m i n i m i z i n g the e s t i m a t o r f o r P j 6  N  +  2  M-N-l  p  1  = N+2  V  2(M-N-1)  i  ^ j=l  N+l  *=**  j  ^»  i+N+l'  a  w i t h r e s p e c t to the r e a l a n d i m a g i n a r y p a r t s o f s t i t u t e d f o r the  I  F  I  N+l.N+l  a f t e r h a v i n g6 s u b -  f r o m the e q u a t i o n s ( A - l ) .  Normally,  the e s t i m a t e o f e r r o r p o w e r f o r b a c k w a r d p r e d i c t i o n i s a l s o i n c l u d e d i n the minimization.  When this i s done, m i n i m i z a t i o n y i e l d s : M-N-l  J  2 a  N+l, N+l  h *  N,l  [VN+1+  N.NVJ  V N ' " " '  +  *  N.lW---  M-N-l  I j  [ I f L  = 1  1  +  j+N+1  a  '" " *  IN, IN  +  a  l~* , f  j  +  N, N  *  N, 1 fj+l  T h u s , a l l the e l e m e n t s o f s t e p  f  Q  +....+  a  f  I  N.Nj+11  N , 1 j+N  *j -1 HN J j +N  +  N+l  J  * "+  a N.N  * f  j+N>  I  C  1  J  a r e known, a n d one c a n p r o c e e d i n e x a c t l y  the s a m e m a n n e r to s t e p N+2, a n d so f o r t h u n t i l the d e s i r e d n u m b e r o f c o e f f i cients i s obtained. A p r i n t o u t o f the F o r t r a n s u b r o u t i n e to p e r f o r m  a single i t e r a -  t i o n o f the a b o v e p r o c e d u r e i s i n c l u d e d b e l o w . In t h i s p r o g r a m : M  =  n u m b e r o f data points  NN  =  o r d e r of iteration - produces r  F  =  data  series  • NN, l  , i = 1, . . .  NN  103  G  = prediction filter coefficients on output  P E F AND  PER  are storage arrays that simplify calculations in  the next iteration and must be declared in the main program.  12 1 1  2  4 6 3 10  SUBROUTINE 6 P E C ( M , N N , F , G , P E F , P E R ) REAL *8 G (M ) , P E F ( M ) , P E R I M ) , S N , S D , H ( 2 0 0 0 ) REAL F ( M ) N=NN-1 I F ( N . N E . O ) GO TO 11 DO. 12 J = 1» M . . PEF(J)=0D0 P E R U ) =0D0 SN=0D0 SD=0D0 JJ=M-N-1 DO ,2 J = l , J J _ SN = S N - 2 . D 0 * ( F ( J + N+ 1 ) + P E F ( j ) ) * ( F ( J ) + PER (" J ) j SD = SD + ( F (J +N+1) +PEF ( J ) ) **2+ (F ( J ) +PER ( J ) )**2 G(NN)=SN/SD I F ( N . E U . 0 ) G 0 TO 3 D O 4 J=1,N K=N-J+1 H(J)=G(J)+G(NN)*G(K) DO 6 J = 1 , N G ( J ) = H (J ) JJ=JJ-1 DO 10 J=1 , J J PER(J)=PER(J)+G(NN)*PEF(J)+G(NN)#F(J+NN) PEF(J)=PE F(J+1)+G(NN)*PER(J+ L)+G(NN)*F(J+l) RETURN END .  

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