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

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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 C H A N D L E R W O B B L E b y J o h n A l e x a n d e r L i n t o n B . A . S c . , 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 i n 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 for the D e g r e e of M a s t e r o f S c i e n c e i n the D e p a r t m e n t o f G e o p h y s i c s W e a c c e p t this t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a M a y , 1973 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be gran t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed w ithout my w r i t t e n p e r m i s s i o n . Department o f G>e<?£?/^cj.-r/c 5 The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date Z7o^7<e f , i A B S T R A C T N o r m a l l y , the wobble of the e a r t h has been dealt with i n a manner that a s s u m e s that the two m a i n p e r i o d i c components have constant phase and amplitude. A n i n i t i a l a ssumption of this thesis i s that both these p a r a m e t e r s can v a r y with time. A technique of p r e d i c t i v e f i l t e r i n g is u s e d to determine the C h a n d l e r component of the wobble f r o m b a s i c latitude m e a s u r e m e n t s at the five I. L. S. o b s e r v a t o r i e s . A s i m p l e an a l y t i c p r o c e -dure i s employed to obtain the phase and amplitude v a r i a t i o n of the p e r i o d i c Chandler motion. The r e s u l t s indicate that m a j o r changes i n both phase and amplitude o c c u r in the p e r i o d 1922. 7 to 1949. T h e s e changes a r e p o s s i b l y a s s o c i a t e d with earthquake a c t i v i t y , although there is nothing to indicate that there is a c o r r e l a t i o n between i n d i v i d u a l earthquakes and events in the C h a n d l e r motion. The c a l c u l a t e d p e r i o d of the Chandler wobble i s 437 days and the damping time i s so u n c e r t a i n that a value approaching i n f i n i t y i s not u n l i k e l y . i i TABLE OF CONTENTS Chapter Page I INTRODUCTION 1 EARLY HISTORY OF WOBBLE RESEARCH 2 The Annual Wobble 3 The Chandler Wobble 7 Chandler Damping 10 Chandler Excitation 12 SUMMARY 16 II THEORY OF EARTH WOBBLE 17 Choice of Axes 19 Perturbation Equations 19 Rotational Deformation 23 Special Solutions to Equation (2.9) 24 III THE LATITUDE SERVICES 30 Method of Observation 32 Reduction of Data by the I. L. S. . . . 33 Instantaneous Pole Path 34 The Problem , . 35 IV PREPARATION OF EQUALLY SPACED LATITUDE MEANS AT EACH OBSERVATORY . . 37 Time Variable Noise Removal 45 V FREQUENCY DOMAIN FILTERING TO OBTAIN CHANDLER LATITUDE VARIATION 56 Motivation 56 Predictive Filtering 58 The Effect of Phase and Amplitude Shifts on the Fourier Spectrum 62 Band Reject Filtering of Annual and Noise 70 i i i T A B LE OF CONTENTS (Continued) Chapter Page VI THE PHASE AND AMPLITUDE RESULTS FOR THE 5 I. L. S. OBSERVATORIES 78 Phase Results 87 Amplitude Results 87 Interpretation with Respect to Established Theories 90 Conclusions 95 BIBLIOGRAPHY 96 APPENDIX I 99 iv LIST OF TABLES Table Page I Annual Components of the Rotation Pole in Units of ". 01 5 II Estimates of Chandler Period 5a III Earthquakes of magnitude greater than 7. 9 (after Richter, 1958) 9Qa V LIST OF FIGURES Figure Page 1 Power spectrum of Polar Motion and Latitude Data (taken from Munk and McDonald, 1960) 4 2 Power spectrum of Polar Motion (taken from Rudnick, 1956) 4 3 Chandler polar motion amplitude characteristics (taken from Yumi, 1970) 9 4 Chandler period variation (taken from Iijimi, 1971) 9 5 Power spectrum showing split Chandler peak (taken from Rochester and Pedersen, 1972) 11 6 Smylie and Mansinha's fitting of smooth arcs to observed polar motion (taken from Scientific American, Dec. 1971) 15 7 Orientation of reference axes . 20 8 Polar motion corresponding to damped free wobble of the earth 26 9 Response of damped earth as a function of frequency 27 10 Longitude of I. L. S. observatories 31 11 Configuration of star pair with respect to zenith and celestial pole 32 12 Filter to Remove Random Noise 46 13 Output from variable noise filter (solid line) and constant noise filter (solid line with circles). The dotted line is the original series .... 49 14-18 Plots of basic smoothed data from each of the observatories 51-55 vi LIST OF FIGURES (Continued) Figure Page 19 Fourier transform of 10 years of a perfect sinusoid (period = 435 days). The dotted vertical line represents transform of an infinite sinusoid of the same frequency 57 20 Fourier transform of synthetic wobble data of Figure 22 59 21 Method for Predictive Extension of a Data Series 61 22 Synthetic latitude variation composed of 2 main frequency components (annual and Chandler) plus random noise 63 23 Prediction of time series in Figure 22 to five times its original length 64 24 The Fourier amplitude spectrum of the extended series (Figure 23) 65 25 Chandler variation used to create the series of Figure 22. Composed of sine wavelets all of the same frequency (1/435 days"*) but each varying in phase and amplitude 66 26 Fourier amplitude spectrum of Figure 25. The dashed line is the spectrum of a sinusoid with constant phase and amplitude 67 27 Synthesis of the Fourier transform of a truncated sinusoid 69 28 Effect of band-reject filtering of annual on the spectrum of the extended synthetic latitude variation (Figure 23 and Figure 24) 72 29 Chandler variation resulting from band-reject filtering of the synthetic data. The dashed line represents the known synthetic Chandler variation 73 vii LIST OF FIGURES (Continued) Figure Page 30 Phase variation of the synthetic Chandler. The solid line is the phase of the synthetic Chandler motion recovered by band-reject filtering and the dashed line is the phase of the known original input Chandler 76 31 Amplitude variation of the synthetic Chandler. Solid and dashed lines represent the filtered and actual versions respectively . . 77 32-36 Chandler latitude variation for each of the observatories 79-83 37 Chandler amplitude variation for all observatories 85 38 Chandler phase variation for all observatories 86 39 Amplitude variation averaged over number of observatories 88 40 Phase variation averaged over number of observatories 89 41 Chandler amplitude variation after Guinot (1972) 91 42 Chandler phase variation after Guinot (1972) 91 43 Fourier amplitude spectrum of simplified undamped Chandler motion 94 viii ACKNOWLEDGEMENT I wish to thank Doug Smylie and Tad Ulrych for their generous support and encouragement. 1 I N T R O D U C T I O N At f i r s t glance, the d e s c r i p t i o n of the earth's rotation about its a xis would s eem a s i m p l e task. That i s . i f the e a r t h i s t r e a t e d as a r i g i d , rotating sphere, one could s i m p l y apply the r e l e v a n t c l a s s i c a l laws of p h y s i c s as developed by E u l e r . However, the r e a l e a r t h i s neither r i g i d nor s p h e r i c a l . T h e s e deviations f r o m the 'ideal' body r e s u l t i n complex p e r t u r b a t i o n s i n the r o t a -tion of the earth. The bulging of the e a r t h at the equator produces the v a r i a -tion i n the d i r e c t i o n of the axis of angular momentum i n s p a c e ( p r e c e s s i o n and nutation). Another effect i s the v a r i a t i o n i n the length of the day. The most important p e r t u r b a t i o n to geophysicists, and the subject of this thesis, 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 of the instantaneous r o t a t i o n axis with r e s p e c t to the geographic f r a m e of the e a r t h - or alternately, the movement of the r o t a t i o n a l pole (point near N o r t h pole where instantaneous r o t a t i o n axis p i e r c e s surface) over a h o r i z o n t a l plane p a r a l l e l to the equator. T h i s 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 of wobble ar e i n t e r m s of the motion of the r o t a t i o n a l pole in units of distance - the complete planar motion being d e s c r i b e d by two d i r e c t i o n components. Succeeding chapters of this thesis w i l l deal with the h i s t o r y of e a r t h wobble r e s e a r c h , the c l a s s i c a l theory, and the pole path data, a l l with a view of j u s t i f y i n g a new a p p roach to the a n a l y s i s of wobble data. 2 E A R L Y HISTORY OF WOBBLE R E S E A R C H Since the history of research involving the earth's wobble is thoroughly documented by Munk and McDonald (I960) in their book on the rotation of the earth, only the highlights are discussed here. It was Euler (1765) who first predicted that a rigid sphere would, in addition to its normal rotation, wobble freely about its axis of greatest rotational inertia. He pre-dicted that the period of the wobble should be 305 days. Pe'ters, Bessel, Kelvin and Newcomb all searched without success (or with false success) for a motion of the predicted period. In 1888, Kdstner discovered a variation in the constant of aberration that he attributed to a . 2 second annual variation in latitude. To show that this was in fact a variation due to wobble, measure-ments were made at stations in B e r l i n and Waikiki - roughly 180 degrees apart - and the measurements showed the expected opposite change in latitude. S. C. Chandler in 1891 announced that the wobble, in addition to an annual component, also contains a motion of period 428 days. Newcomb in the same year showed that the 428 day period could, in fact, be the free wobble period lengthened from 305 to 428 days by the non-rigidity of the earth. These dis-coveries and the lack of confidence in the observational evidence to that time led to the establishment of the International Latitude Service (I. L. S. ) which functioned from 1899. 0 to 1962. 0 when it was replaced by the International Polar Motion Service. Since the time the I. L. S. was established, the major thrust 3 of r e s e a r c h has been centered on two aspects of the wobble p r o b l e m : 1. A n a l y s i s of the data and d e t e r m i n a t i o n i n d e t a i l of the c h a r a c t e r i s t -i c s of the two m a i n frequency components (Chandler and annual). 2. E x p l a n a t i o n of these c h a r a c t e r i s t i c s by a p l a u s i b l e m odel for the r e a l earth. The annual wobble The annual wobble has, for convenience's sake, been u n i v e r s a l l y t r e a t e d 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, i s just s i m p l e c i r c u l a r or e l l i p t i c a l m otion of the r o t a t i o n a l pole. On f i r s t glance at the power spe c t r u m s depicted i n F i g . 1 & 2 (both a r e taken f r o m Munk and McDonald), this a ssumption would s eem to be j u s t i f i e d by the n a r r o w n e s s of the annual peak. The power s p e c t r a for p e r f e c t c i r c u l a r motion of p e r i o d one y e a r would have a peak width of 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 peak width i n these plots of . 03 to . 06 c y c l e s / y e a r . Unfortunately, the s i m p l i c i t y this i m p l i e s i s not quite achieved. Walker and Young (1957) point out. i n their exhaustive a n a l y s i s of I. L. S. data that, "the components of the annual motion v a r y a c c o r d i n g to type of s e r i e s and i n t e r v a l , there being noticeable d i f f e r e n c e s in both the amplitudes and phases of the s e v e r a l compo-nents. " T a b l e I gives v a r i o u s e s t i m a t e s of the amplitude for each of the two d i r e c t i o n a l components of annual p o l a r motion - rrij towards Greenwich, r r ^ towards ninety degrees east of Greenwich, The amplitude v a r i e s by a l m o s t a factor of two between m i n i m u m and maximum. T h i s v a r i a t i o n means that i f the annual wobble i s to be m o d e l l e d as s i m p l e c i r c u l a r motion, the m odel 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 # Source Interval m i m2 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 Walker & Young 1899-1954 -6.4 -7. 1 7. 0 -4.6 1957 1900-1934 -5. 5 -7. 0 7. 5 -4.6 1900-1920 -4.8 -6. 0 6.6 -3. 7 Jeffreys 1940 1912-1935) and ) -3. 2 -7. 8 5. 6 -1.6 Markowitz 1916-1940) 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 1890-1890-1890-1890-1922-1892-1908-1900-1891-1900-1890-1929-1897-1897-1930-1891 -1915 1924 1924 1922 1938 1933 1921 1940 1945 1920 1924 1953 1957 1922 1957 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 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) * after Wells, 1972. 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 observa-tory (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 deter-mined 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 attri-buted 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 compo-nent. 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 Chandler motion enables one to draw con c l u s i o n s about the r i g i d i t y , shape and n o n - e l a s t i c i t y of the r e a l earth. The e s s e n t i a l feature of the Chandler wobble i s that its s p e c t r a l r e p r e s e n t a t i o n r e s e m b l e s that of a damped o s c i l l a t o r - that i s , the s p e c t r a l peak i s s p r e a d over f r e q u e n c i e s near the peak value (see F i g . 1 & 2). T a b l e II (taken f r o m Wells, 1972) shows the Chandler p e r i o d (period at peak) during v a r i o u s epochs as c a l c u l a t e d by the authors in d i c a t e d . T h i s table i l l u s t r a t e s that the Chandler wobble might have t i m e - v a r i a b l e amplitude and frequency. P l o t s by Y u m i (1970) and I i j i m i (1971) ( F i g , 3 & 4), indicate how the p e r i o d and amplitude of the two C h a n d l e r components v a r y with time. The amplitude v a r i a t i o n i s e s p e c i a l l y r e m a r k a b l e - between ten and t h i r t y feet; It should be pointed out that these authors have u s e d v a r i o u s f o r m s of h a r m o n i c a n a l y s i s -sometimes-on data lengths as short as ten y e a r s (Yumi) - and t h e r e f o r e r e s u l t s that indicate a shift i n frequency a r e subject 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 of the technique. E s t i m a t e s of the damping time (time for signal to decay to 1/e of its o r i g i n a l value) v a r y f r o m two y e a r s (Walker and Young, 1957) to ninety y e a r s (Pachenko). T h e s e estimates of damping time l e a d to a Q factor ( d i m e nsionless m e a s u r e of energy dissipation) of t h i r t y to fifty, which i s s e v e r a l times s m a l l e r than Q values c a l c u l a t e d f r o m s e i s m i c data. F e l l g e t t (I960) c a l c u l a t e d a damping time of 12.4 y e a r s but concluded that the value is " u n c e r t a i n by a factor of at l e a s t ten. and damping times as short as 2. 5 y e a r s o r as long as s e v e r a l hundred y e a r s a r e not excluded. " The 9 « g -io\ a. E o aoL—i l i L j i L _j 1 i 1993 ISOS 1317 1929 19*/ 19S3 I96ST F i g u r e 3 : C h a n d l e r p o l a r m o t i o n a m p l i t u d e c h a r a c t e r i s t i c s ( t a k e n f r o m Yuml, 1970) a a> T3 o **Z <u a 1M\ l.20\ 1.16 1.14-i — i — r T — i — i — i — i — r L rn. J JL i i i J_ 190072 2 4 3 6 F i g u r e 4 : C h a n d l e r p e r i o d v a r i a t i o n ( t a k e n f r o m I i j i m l , 1971) 10 apparent disparity between Q factors still remains. One solution (Wells. 1972) would be a frequency dependent Q. 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. This is not apparent in the observations. Yashkov (1965) carried out harmonic analysis on polar motion data and produced a split Chandler peak. Fig. 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 c t 7 0 o •"60 O x 50 o cc 40 UJ * ? 30 20 10 .5 BAND WIDTH L A .7 .8 .9 FREQUENCY (CPY) • /V. VY. 1.0 I.I Figure 5 s Power spectrum showing s p l i t Chandler peak (taken from Rochester and Pedersen, 1972) TOWARD 90 DEGREES Figure 6 : Smylle and Manslnha's f i t t i n g of smooth arcs to observed polar 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 factor of 100 to 200 - as opposed to a Q factor from wobble data of 30 to 50. 2. 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 com-pletely. 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 preces-sion 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 dis-location theory to earth models and produced much larger displacement fields than had been anticipated. Smylie and Mansinha (1967) used Press's tech-niques to calculate theoretical displacement fields of earthquakes and the associated change in the moment of inertia. They concluded that the pertur-bations 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. For 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 affect the Chandler wobble. 16 SUMMARY The most important facts that can be gleaned f r o m this b r i e f h i s t o r y a r e that: 1. both the annual and C h a n d l e r components a r e time v a r i a b l e i n amplitude and perhaps in frequency. 2. no assumption should be made about the damping time of the C h a n d l e r motion. 3. the Chandler may have r a n d o m sudden changes i n phase and amplitude. 17 CHAPTER II THEORY OF EARTH 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 « • — i i ~~~ = ~~~ 4- :w x L L L inertial 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 = /{{ t Y i [ r x fvv"x r 4- v (r)) p (7)] dV earth 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 2 &.. - x. x. "1 w. 1 L i j 1 J J J when the standard summation convention (repeated subscripts indicate summa-tion over that subscript) is used. The e"- are unit vectors in the coordinate directions and c$ij is the Kronecker delta. The angular momentum is now L = earth w. p fr) dV  1 L ij 1 J J J 4- fff r x v ( r ) p ( r ) dV earth ' e. I., w. 4- J 1 i J J where I.. = fff \ r 2 cf'.. - x. x. 1 p ( r ) dV (Z. 2) XJ earth L *J 1 J J ~ is the inertia tensor of the earth, and = JyT 7 x v ( r ) p ( r ) dV earth ' 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 coordin-ate system is attached to the observatories - relative motion of the observa-tories 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 Fig. 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 1 1 = I 2 2 = A A N D l33 = 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 ( 1 9 6 0 ) . 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 (geographic North pole) instantaneous rotation axis — fixed in space except for effects of precession (to Greenwich) Iml earth s centre of mass Y ( 9 0 ° E of Greenwich) F i g u r e 7 : O r i e n t a t i o n o f r e f e r e n c e a x e s 2 1 w = f\ m 1 4- JX 4- e^f\. (1 4- m 3) J\ - angular velocity of earth; r r i | , m^. < < 1 Il l5 s A + Cl l+ bl l hi- A + C 2 2 + B 2 2 X 3 3 * C + C 3 3 + B 3 3 X 1 3 = ! 3 1 = B 1 3 + C 1 3 lZ3 = I 3 2 = B 2 3 + C 2 3 I 1 2 = I 2 1 = b 1 2 + c 1 2 b is the contribution to I., from rotational deformation above that due to ij XJ steady, diurnal rotation, and c-j is deformation due to other causes. Then, the angular momentum components are Ll = hi W 1 + I 1 2 W 2 + I 1 3 W 3 + A = A - a m i + j\ ( b 1 3 + c 1 3 ) +• x1 L 2 = hi w l + * 2 2 W 2 + J 2 3 W 3 + ^ 2 = A A m 2 + A ( b 2 3 + c 2 3 ) + Xz L 3 = I 3 1 wj + I 3 2 w2 4- I 3 3 w3 4- Xz = C A (1 + m 3) 4- A ( c 3 3 4- b 3 3 ) 4- / 3 where terms like (b.j 4- c.j) m k J\ are neglected in comparison to those like (b.j 4- C . J) J TL . Substituting into the Liouville equation ( 2 . 3 ) yields 22 P ~ £ + A J\ m 4- _TL (b , 4- c ) 4- C yL2m_ 1 1 1 13 13 -jT_ 2Am 2- J l ? ( b 2 3 4- c 2 3) - y i _ i 2 (2.4) p = A jT.m 4- JT (b 4- c ) 4- i + A A m 2 ^ 23 2 3 ^ * + SL ( b 1 3 4- c 1 3) 4- /I i i - C J l 2 m (2.5) = C J l m 3 4- y i ( c 3 3 4- b 3 3) 4- / 3 (2.6) where, in addition to the previous approximation, we now neglect products of lj and mj. Now, let m = rrij 4- i r = p + i p — 1 2 i - ix± az By multiplying equation 2. 5 by i = (V-l ), and adding the result to equation 2. 4. we obtain m 71 A - i m -A.2 ( C - A.) = _Q -- i j / l - ( c l 3 4- i c 2 3)_fL - ( b u 4- i b'23 )_/! - i ( c 1 3 4- i c 2 3 ) / l 2 - i ( b ^ + i b 2 3 ) y i _ 2 (2.7) 23 T h i s i s the equation of motion for the wobble - m gives the complex angular displacement of the axis of rotation away f r o m "e^. o r . i f one wishes, the motion of the instantaneous pole of r o t a t i o n over the sur f a c e of the earth , with . 1 second of angular motion equivalent to ten feet of polar motion. The r e -maining equation m3 = 77T [ P 3 " ^ 3 - < '33 + b33 >^ -] < 2' 8 ) governs the change in length of day. Rotational D e f o r m a t i o n Since the t e r m s b.. are i n fact functions of m, it is u s e f u l at !J — this point to obtain the exact functional r e l a t i o n s h i p . It can be shown that k_ d m m , _TL b e. I c. 12 =  l* 3 G k d m 7 J\ b. 2 23 " 3 G B = k2 d 5 ml ^2 13 3~G where d equals the r a d i u s of the earth , G i s the g r a v i t a t i o n a l constant and k i s the Love number-defined such that i f the s y m m e t r i c e a r t h is subjected 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 as the gradient of a second o r d e r s p h e r i c a l h a r m o n i c potential U^. the additional potential V at the d e f o r m e d s u r f a c e i s V = k 2 U-,. T h e s e r e s u l t s a r e b a s e d on the assumption of l i n e a r str e s s - s t r a i n r e l a t i o n s , and a r e d e r i v e d on page 25 of Munk and M c D o n a l d 24 (1960). Substituting for b.^  in equation 2.7 gives m/L(A+ — ) - i rn j \ ( C - A - _2 ) - 3G - 3 G = H - 1 " " ( ^13 + 1 ' 1 ( c13 + 1 c 2 3 ) ^ If we neglect k^d _ / l /3G compared to A, and allow (C-A)/A to be replaced c 2 5 2 by (C-A)/C and k2<i A /3GA by k,,d .A. /3GC (introduces an error of only one part in 300 in each case), then 2 - i < r o 2 . = X A [JZ - 1 - i i ^ - ( C 1 3 + i C 2 3 ) y L - i (C 1 3 4- i C 2 3) J1 2J (2.9) ^ A 5 Z k ?d y, where <TQ = (H - ) C - A H = (3 The quantity H 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 m = 0 i <r t from which m = A e ° (A = constant) 25 This represents a circular motion of the instantaneous pole of rotation with angular velocity (T0 and radius A. For a rigid earth with the same preces-sional constant H as the real earth, the angular velocity is given by i CTQ = 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 ? d 5 i l 2 cr = A H - — 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 H - (k 2d 5 JT.2/3GC) 435 days 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. To allow for damping of the motion, 0s is allowed to be complex: cr = w. + i/t o ° 26 w is the angular frequency of the free wobble as c a l c u l a t e d p r e v i o u s l y , and o f is the damping time. E q u a t i o n 2.9 then y i e l d s the solution m . - t/Z i w Qt A e e and the motion of the pole i s a west to east s p i r a l as shown i n F i g . 8. F i g . 8. P o l a r motion c o r r e s p o n d i n g to damped fre e wobble of 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 Damping A l l o w <P Q to be complex as above, and assume the right hand side of equation 2.9 (which r e p r e s e n t s the e x c i t a t i o n or f o r c i n g function of the d i f f e r e n t i a l equation) i s of the f o r m i <rt o < e where o( and <T a r e the amplitude and frequency of the e x c i t a t i o n . T h e n equation 2 . 3 becomes 27. m - i CT rn_ = <=< e C T t The steady-state solution of this equation i s m i ( CT- CT0) (Tt It follows that m V ( cr w Q ) 2 + ( i / r ) 2 A s s u m i n g that the amplitude of the e x c i t a t i o n , o £ , i s the same for a l l f r e q u e n c i e s , a plot of /m { v e r s u s frequency would look l i k e F i g . 9'. B y fitt i n g a c u r v e of this type to the actual data, one obtains an estimate of the wo t p a r a m e t e r C and thus of the damping factor Q = — £ — . 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) = heaviside function = O t < O = 1 t > O J = a complex constant (b) F(t) = Jcf(t) cf (t) is the dirac delta function. The general solution of equation 2.9 is given by i Q" t ' m(t) = e [ m Q - j F ( t ) e~ 1 d "tJ - oo Upon substituting for F(t) from case (a), a simple integration gives m(t) = e . t < 0 J i c r n t m(t) = (m Q - ~ ) e 4- : . t > 0 — i <rn' i cr -o o The spiral motion after t = O has modified equilibrium pole position J U x . new amplitude m 0 - i c j - o and new phase 0 given C a s e (b) gives m(t) = m Q e t < 0 m (t) = ( m 0 - J ) e t > 0 H e r e , the e q u i l i b r i u m pole p o s i t i o n i s u n a l t e r e d for t > 0, but both the phase and amplitude a r e changed by the addition of the quantity J . F r o m these examples, one would expect the Chandler motion due to earthquake e x c i t a t i o n to be smooth s p i r a l s i n t e r r u p t e d by sudden shifts in phase and amplitude and p o s s i b l y , also i n e q u i l i b r i u m pole p o s i t i o n . 30 CHAPTER III THE LATITUDE 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 orienta-tion 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 Kitab, Russia 140° 0 8 ' East / 60° 2 9 ' East / " " m to Greenwich \ Gaithersburg, Maryland Ukioh, California \ 77° 12' West 123° 13' Wes t 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. Kitab was formerly located at Tschardjin - some 3° to the west. Carloforte was close for nearly four years during World War II. For 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 Fig. 11. P = Z = E = S. = l Z. = l <fi = celestial pole zenith celestial equator star i zenith angle to declination of S. Fig. 11. Configuration of star pair with respect to zenith and celestial pole. The latitude 0 is given by or upon addition by 0 = 0 = (fl - Zl and 0 = cf2 + Z 2 cfl + cT2 Z 2 - Z l (3. 1) The difference Z^ - Zj 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 values must be found i n star catalogues. The important part of this p r o c e s s is maintaining the telescope l e v e l (or a constant zenith). T h i s was originally-done v i s u a l l y , but most m o d e r n m e a s u r e m e n t s a r e made with the Photographic Zenith Tube ( P Z T ) . In this i nstrument, the l e v e l l i n g device i s a f r e e s u r f a c e of m e r c u r y with r e f l e c t s a s e r i e s of four star images onto a photographic plate f r o m which the zenith distance z i s m e a s u r e d d i r e c t l y . R eduction of Data by the I. L.'S. Unfortunately, the data published by the I. L , S. a r e not the b a s i c 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 equation 3. 1 which i n t e r m s of m e a s u r e d quantities b ecomes 0 = d" + R ' M + I. S. (3.2) where cf = adopted value of mean d e c l i n a t i o n of the star p a i r ( ^ ( <$i + <^2 ) a t 'he time of mea s u r e m e n t - time v a r i a b l e because of star 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 = angle of telescope shift per one hal f r e v o l u t i o n of the m i c r o m e t e r - includ i n g a t e r m for temperature dependence. ° I. S. = a t e r m to account for i n e q u a l i t i e s of the m i c r o m e t e r s c r e w . The I. L. S. under K i m u r a added to this equation s e v e r a l c o r r e c t i o n s b a s e d on the data i t s e l f . A c o r r e c t i o n for d e c l i n a t i o n is i n c l u d e d in o r d e r to make the 34 mean latitude given by each star p a i r equal to the mean value for the star p a i r group of which it is a p a r t . The group mean is obtained by weighting a c c o r d i n g to the trend of the data i n a manner totally i n c o m p r e h e n s i b l e . A c o r r e c t i o n for the time v a r i a b l e part of M is also obtained. See K i m u r a ( 1 9 3 5 ) . F o r t u n a t e l y , only the latter c o r r e c t i o n has a l o c a l effect - the c o r r e c t i o n to dec l i n a t i o n produces the same effect at e v e r y station. L u i g i C a m e r a operated the I. L.S. f r o m 1 9 3 5 . 0 to 1 9 4 8 , and happily his method of red u c t i o n is s t r a i g h t - f o r w a r d . He introduces 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 and l a s t d e s c r i b e d above for K i m u r a - the di f f e r e n c e being the s i m p l i c i t y of his assumptions and subsequent c a l c u l a t i o n s . See C a m e r a ( 1 9 5 7 , page 2 5 6 ) . Instantaneous P o l e P a t h M o s t of the a c c e s s i b l e data p u b l i s h e d by the I. L . S. c o n s i s t s of monthly c a l c u l a t i o n s of the two d i r e c t i o n a l components of the p o l a r m o t ion. Latitude v a r i a t i o n at station i , & 0 j , due to p o l a r m o tion i s just ^ 0^  = r r i j cos + s i n f • where m^ and m^ 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 components of angular p o l a r displacement away f r o m the x^ a x i s . t . is the longitude of station i . 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 . Since latitude v a r i a t i o n can be caused by e r r o r s i n star p o s i t i o n s , t e m p e r a t u r e fluctuations and other s i m i l a r v a r i a t i o n s that do not depend on p o l a r motion, latitude v a r i a t i o n is written 0. = m. cos "C*. + m_ si n 7j. f- Z X A 1 Cd 1 35 z is a term added to account for non-polar latitude variation. This equation is fitted to the observed ^ 0- at each station by choosing m m_ and obs \ c z which make 5 a minimum. The Problem To this point, we have been describing other people's accomp-lishments. From these are drawn the major objectives of this thesis: • 1. To achieve a clear separation of the two spectral components from relative-ly 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 C H A P T E R IV P R E P A R A T I O N O F E Q U A L L Y S P A C E D L A T I T U D E M E A N S A T E A C H O B S E R V A T O R Y Our 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 i n d i v i d u a l 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 and m i c r o m e t e r v a r i a t i o n d e s c r i b e d above. A m a x i m u m of sixteen such o b s e r v a t i o n s are made each night at each of the five o b s e r v a t o r i e s . Since o b s e r v a t i o n s cannot be made at a l l times due to weather conditions, many nights have no readings at a l l and some have v e r y few. In o r d e r to do time s e r i e s a n a l y s i s , it 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 , it is d e s i r a b l e to attach g r e a t e r weight to nights where many obs e r v a t i o n s were made. L o o k i n g f i r s t at nights when at l e a s t one m e a s u r e m e n t was made at a p a r t i c u l a r station, the mean value estimate of latitude on night j i s given by N (4-1) i=l where N = number of o b s e r v a t i o n s on night j i .th , th x - 1 latitude o b s e r v a t i o n on j night j Note that the symbol s\ i n d i c a t e s that the quantity i s an e s t i m a t o r o n l y . The unbiased v a r i a n c e e s t i m a t o r for any o b s e r v a t i o n x1 i s j 38 N i=l (Bendat and Piersol, 1966, . • pg 126) Assuming each x is independent, the variance estimator of the mean j value s . is J N var ft] = Nirhr 2 ['r'H i=l (Bendat and Piersol, 1966, Pg 135) For nights when N=l, f ^ l 2 var Is J = .1 (second) j 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 Z = I * J ^i,. (4-3) k * k k+j J=N where the summation is only over nights with observations and k refers to a night with no observations. Taking expected values of Z^ .: N . N EA Z. J- = >. a J E S s f = L a j s j=-N *" "'J j=-N k k + J s. , . represents the actual latitude variation at time k+j. Let s, . = s, + E3 k+j * J k+j k k 39 where E'' is a term to account for the small variation in actual latitude from k time k until time k+j. N N Now r ~ . . • E ( z j = Y aJ s + £ a J E J j = -N J=-N N and, if in addition the a? are constrained by £ s? - 1, then k j=-N k f 1 N j E i Z k l = S k + 2 \ \ (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 k is unnecessary. 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'k <4-5> j=-N which is quite unwieldy. However, if var(z^) is expanded in the same manner, the solution is simplified somewhat. That is, 2 v a r ( Z k ) = E { [ z k - E ( Z k ) ] }' „ ? N N . = E [ Z |- s - 2 s I aJ E [ - £ aJE?l I k J k k k k L ^ k k J j =-N j=-N 40 Combining this result with equation (4-5) yields N 2 D = var (Z, ) + [ £ J K 3 1 (4-6) k k j=-N k k J N 2 For convenience, let I V a'' E I = A . Then D. = var (Z. ) +• A, . L j=-N k k J k k k' The value of var (z^) can be determined directly from the identity N _ N a s ) = 2- (a ) v a r (s ) j=-N k k + J j=-N. k k + J (Bendat and Piersol, 1966 Pg 67) which is valid if the Is^  are independent - as is assumed. Thus, we have finally 2 N . 2 N 2 °k • e{ K - \] }• i <%>v-«v,1 + [ £ < K\  (4- 7) J=-N j=_N j j The unknowns in the above include the coefficients a and the terms E . k k The coefficients a are chosen so that, in addition to satis-N j k fying the condition 51 a =1, they minimize var (z ) . Normally, one would j=-N k k choose to minimize D^. However, this leads to a set of up to 2 N+1 linear equations, which makes solving for the coefficients a tedious and expensive business. In addition, sample calculations indicate that << var (Zk) for most cases - which means that Dn « var (Z, ). Minimizing var (z, ) with k k k respect to the a? yields k 41 a' = v a r (\ + j > where C i s a constant d e t e r m i n e d by the c r i t e r i a N j aJ = 1 j=-N k 2 T o m i n i m i z e the effect of the t e r m A, = Y a E 3 \ let k j=-N L k k J i w.C a = J ( 4 _ 8 ) k v a r ( ? k + . ) where w. i s a triangle function with equation w. = w - I J I w J o N+1 o N • Once again C is d e t e r m i n e d by £ a = 1. The effect of in c l u d i n g the j=-N . k factor w. i n the d e t e r m i n a t i o n of a-' i s to give l e s s emphasis to t e r m s J k j of Z where E is l i k e l y to be l a r g e . T h i s amounts to a rough m i n i m i z a t i o n k k of A ^ and thus of D^. The t r i a n g l e shape is a l m o s t c e r t a i n l y not the o p t i m u m one. However, it is unquestionably better than the box c a r function which is a s s u m e d i f j C \ ' v a r f t ' ) k+j The only r e m a i n i n g unknowns f r o m equation (4-7) are the t e r m s E"' . Since E^ is a function of the actual latitude v a r i a t i o n , it cannot be evaluated without k knowing the latitude s i g n a l e x a c t l y . However, it is p o s s i b l e to define an expec-ted value of A = \Y a? E | i n t e r m s of a u t o - c o r r e l a t i o n s c a l c u l a t e d on k L k k J j the b a s i s that the actual latitude s e r i e s i s only one d e t e r m i n a t i o n of a sto c h a s t i c p r o c e s s that can be s i m p l y r e p r e s e n t e d as s = x cos (wk + 6) xC where x = constant amplitude of latitude v a r i a t i o n w = angular frequency of the m a i n p e r i o d i c v a r i a t i o n 9 = a random v a r i a b l e with u n i f o r m p r o b a b i l i t y d i s t r i b u t i o n 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 ) = E j s k s k + ^ J = J p(9) cos (wk + 0) cos (wk + w\ 0 + 0) dO Since p(0) = , integration yields 2 0(X) = I . cosw\ To simplify the calculations, it is desirable to expand the above equation for the case where w\ is small: e u > * f 2 [ i - 4^ ] This is further simplified by assuming a linear 0 such that (X) = [ l - n|X|] (4-9) where n is chosen so that 0(X) < 0 (X) over the range - N < 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 w 2N n = It remains now to write in terms of the above auto-correlation function: N N A, = E { V V a j a m E j E m ) k lj=-Nm=-T* k k k k i Subsituting E1* = s, . . - s yields 6 k k+j k ' N N A = E { 2 £ a J a m (s . - s ) (s - s )} k k+i k k+m k J j=-N m=-N which, after taking expected values inside the summation reduces to N N ^ = J* m=ZN * k \ ^ ^ + ' ^ ^ V j S ^ - E { s k s k + m } + E ( s k 2 }] But. E { s k + j s k + m } = 0 ( k - m ) and similarly for other terms. Therefore N N A k = ^ I a j a m [ 0 (k -m ) - 0 (j) . 0 ( m) j=-N m = -N k k 0 ( 0 ) ] N Remembering that Z a J = 1 , and taking non-indexed terms outside the j=-N k summation: N N N = 0 (0) - 2 £ a j 0 (j) + I I a j a m 0 (m - j) j=-N k j = _ N m = _ N k k 44 Substituting in the l i n e a r a p p r o x i m a t i o n for the a u t o c o r r e l a t i o n (equation 4-9), gives 2 N N N A k = If- [z I a k ( j , - I I a k a k | j - m | 2 L j=-N j = - N m = - N K (4-10) F o r the values x and w given above, and for N=9 days, 2 2 2 .00003 sec /day Thus in summary: The latitude at time k i s e s t i m a t e d by N I a J t . (4-3) z k j=-N k k +J and the v a r i a n c e of z k i s es t i m a t e d by > 2 N 2 E { [ z k - \] )- £ N [<] ™ r « V , > + \ ( 4 - 7 ) It should be r e m e m b e r e d that a l l the summations above are only o v e r nights with data. The value of N was chosen to be nine days - bas e d on the m i n i -m um f i l t e r length r e q u i r e d to b r i d g e a l l but the most extreme data gaps. B e c a u s e of the summation 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 z^, the v a r i a n c e of z, with r e s p e c t to s, i s m u c h s m a l l e r than the v a r i a n c e of the latitude k K estimate "s^ for nights with data. T o r e m e d y this s i t u a t i o n , the interpolate tion f i l t e r was applied a l s o to o b s e r v a t i o n nights - in t r o d u c i n g an additional c o e f f i c i e n t a.^  so that the mean value of latitude, "3 , on the night i n question, k k is i n c l u d e d i n the c a l c u l a t i o n of z^. C a r r y i n g out the ca l c u l a t i o n s d e s c r i b e d i n the s u m m a r y above y i e l d latitude and latitude v a r i a n c e estimates for each night. Since only the v a r i a t i o n in latitude (as opposed to the absolute latitude given by z^), i s r e q u i r e d , the mean latitude of the station as de t e r m i n e d by C a m e r a (1957, pg Z13) was subtracted f r o m each z^, k = l , , N. The 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 have zero mean value because the wobble motion contains a slowly v a r y i n g component that i n effect makes the mean value v a r y with t i m e . T i m e V a r i a b l e N o i s e R e m o v a l The latitude 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 p r o c e s s p r o v i d e an estimate of the m e a s u r e m e n t noise i n the r e d u c e d o b s e r v a t i o n s . T h e r e f o r e , allow z k = s k + n k (4-11) > where n k i s a white noise p r o c e s s with c h a r a c t e r i s t i c s such that = n k i f k = j 0 i f k 4 j A f i l t e r designed to extract the noise f r o m the signal z i s shown sc h e m a t i -c a l l y i n F i g u r e 12. The output y k of such a f i l t e r i s given by y* = X h* "k-i (4"12> The optimum f i l t e r c o e f f i c i e n t s h^ a r e d e t e r m i n e d by the leas t s quares fit of the f i l t e r output y, to the d e s i r e d output s . T h i s r e s u l t s i n the k k equation 46 Figure 12. Filter to Remove Random Noise N dp (k. k-m) = sz j=-N £ hj $ (k - j , k - m) m = - N 0, . N (4-13) where <I> ( X, t) = E-( z z J zz 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): $ (k, j) = 3> (k, j) 4- <J> (k, j) (4-14) zz ss nn * z (k. j) = % s (k. j) (4-15) The cross autocorrelations <p (k. j) and $ (k, j) are zero because of sn x * •" ns the independence of s^ and n. and because E n. =0. Since J *nn<k' J) = E { n k n l } - n s.J ^ k c a n be w r i t t e n n = * (k. k) - * (k. k) k zz v ' ' ss E { [Zk - ^ k ]2} Dk The quantity D 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 . The l a t i -tude v a r i a t i o n is a s s u m e d to be a stationary p r o c e s s . T h i s means that * s (k. j) = 0 (k - j) where 0 i s the n o r m a l time independent a u t o c o r r e l a t i o n of the latitude s e r i e s . Equations (4-14)and (4-15) can now be w r i t t e n (k, j) = 0 (k - j) + D k 0 j (4-16) zz sz (k, j) = 0 (k - j) (4-17) W r i t i n g equation (4-13) out in f u l l for v a r i o u s values of m and i n c o r p o r a t i n g the above substitutions f o r the a u t o c o r r e l a t i o n s , p roduces the following set of equations: -N 0 N _ N h 0 (0) + + h 0 ( - N) + + h, 0 (- 2 N) + h D, „ k k k k k+N = 0 ( - N) - N 0 +N n h 0 (N) + . . . + h 0 (0) + . . . + h . 0 (N) + h D, = 0 (0) k k • k k K • * • h" kN 0 (2N) + . . . + hg 0 (N) + . . . + h™ 0 (0) + h * D k N = 0 (N) 48 T o s i m p l i f y , let 0(0) . 0(2N) 0 (-2N) 0 (0) H -N ^k 0 i k •N B 0 (-N) 0/0) 0 (N) and n D k+N 0 D 0 0 0 0 0 k+N-1 0 0 0 *0 D k-N T h e n and ( T + JJ ) H = B -1 H = ( T + U) B (4-18) -1 where ( T + E ) ( T + H) = identity m a t r i x . E q u a t i o n (4-18) was u s e d to obtain the noise f i l t e r c o e f f i c i e n t s h^ for a given point k. A f t e r the f i l t e r was a p p l i e d at that point, the f i l t e r was moved ten days along the data and 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 means with ten day sampling i n t e r v a l i s obtained. T e n days was chosen i n o r d e r to be compatible with the data of the B u r e a u Internationale de l'Heure. A u t o c o r r e -lations were obtained f r o m the unsmoothed data by using the B u r g a l g o r i t h m as d e s c r i b e d i n Appendix I, and the m a t r i x i n v e r s i o n was a c c o m p l i s h e d by standard I B M p r o g r a m SL.E. T y p i c a l r e s u l t s a r e shown in F i g u r e 13. A l s o plotted i n F i g u r e 13 a r e the r e s u l t s of another technique whereby the same 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 (equal to the average v a r i a n c e for the complete s e r i e s ) and u s i n g the L e v i n s o n i n v e r s i o n technique (Robinson, 1967, page 43). F i g u r e 13 : O u t p u t f r o m v a r i a b l e n o i s e f i l t e r ( s o l i d l i n e ) 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 c i r c l e s ) . The d o t t e d l i n e i s t h e o r i g i n a l s e r i e s . 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 indicate that the l a t t e r t e c h -nique i s somewhat i n f e r i o r to the method used. T h i s is perhaps due to n u m e r i c a l 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 for a m a t r i x so l a r g e (37 x 37). In a p p l i c a t i o n , the smoothing f i l t e r length was chosen a r b i t r a r i l y to be 37 days (N=18) - the only c r i t e r i a being that the f i l t e r must be short enough not to cause h a r m f u l high-frequency cutoff. It should be kept i n m i n d that only the m e a s u r e m e n t noise has been f i l t e r e d . A s can be seen f r o m the r e s u l t i n g time s e r i e s for 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), there i s s t i l l a f a i r l y l a r g e noise factor which can p o s s i b l y be attributed 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 . The ten day mean values that a r e output f r o m Eq u a t i o n (4-12) and that a r e plotted i n F i g u r e s 14 to 18, r e p r e s e n t the b a s i c data for s p e c t r a l a n a l y s i s . CO o Figure lit- : Basic smoothed data for Misisawa. Figure 15 : Basic smoothed data f o r Ukiah. U l ro F i g u r e 16 : B a s i c smoothed d a t a f o r C a r l o f o r t e . F i g u r e 17 : B a s i c smoothed data f o r K i t a b . Figure 18 : B a s i c smoothed data f o r Gaithersburg. 56 C H A P T E R V F R E Q U E N C Y D O M A I N F I L T E R I N G T O O B T A I N 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 Mo t i v a t i o n Two of the o r i g i n a l o bjectives of this thesis were: 1. T o separate the two m a i n frequency components f r o m short r e c o r d s . 2. T o remove the annual component without 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 shifts i n the Chandler motion and without f o r c i n g the annual component to be a constant s p e c t r a l delta function. T r a d i t i o n a l methods of h a r m o n i c a n a l y s i s include p r i o r assumptions about the frequency components being dealt with. E a c h component i s a s s u m e d to be a pe r f e c t h a r m o n i c that does not v a r y i n phase, amplitude or mean value with t i m e . In addition, a l l the t r a d i t i o n a l methods assume that outside the known i n t e r v a l , the data s e r i e s i s either z e ro or c y c l i c . B e s i d e s the obvious i n a c c u r a c y of these extensions, they cause s e r i o u s i n t e r f e r e n c e effects i n the frequency domain that make the s e p a r a t i o n of cl o s e f r e q u e n c i e s v e r y d i f f i c u l t . 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 of a p e r f e c t s i n u s o i d of p e r i o d 435 days - bas e d on a p p r o x i m a t e l y ten y e a r s of data. If this s i n u s o i d was continued to i n f i n i t y i n both 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 would be the v e r t i c a l line i n the m a i n peak of the same plot. The di f f e r e n c e between the F o u r i e r t r a n s f o r m s of the truncated s i n u s o i d and the continuous s i n u s o i d i l l u s t r a t e s quite d r a m a t i c a l l y the u n d e s i r a b l e effect of extending the known F i g u r e 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 ( p e r l o d = ^ 3 5 d a y s ) 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 i n f i n i t e s i n u s o i d o f t h e same f r e q u e n c y . data with z e r o e s . F o r the case of the C h a n d l e r and annual components of latitude v a r i a t i o n , a p p r o x i m a t e l y fif t y y e a r s of data a r e r e q u i r e d to obtain enough s e p a r a t i o n of the frequency components so that the t r u n c a t i o n effects on the r e s p e c t i v e peaks do not i n t e r f e r e too s e v e r e l y . F o r a r e c o r d length of ten y e a r s , what should be two d i s t i n c t f r e q u e n c y peaks become m i r e d into one v e r y b r o a d peak. F i g u r e 20, which 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 ten y e a r s of synthetic wobble data (see F i g u r e 22), e x e m p l i f i e s this u n d e s i r a b l e phenomenon. The dotted l i n e s r e p r e s e n t the actual peak l o c a t i o n s . It seems obvious that i f the o b j e c t i v e s d e s c r i b e d above are to be f u l f i l l e d , some new s p e c t r a l technique i s r e q u i r e d . P r e d i c t i v e F i l t e r i n g r A method of p r e d i c t i v e f i l t e r i n g developed by U l r y c h et a l (1973) reduces the mutual i n t e r f e r e n c e p r o b l e m s of h a r m o n i c a n a l y s i s . E s s e n -t i a l l y , the method i s just the p r e d i c t i o n of the data s e r i e s s e v e r a l times i n both the b a c k w a r d and f o r w a r d d i r e c t i o n in such a manner that, 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 d e t e r m i n a t i o n of a s t a t i s t i c a l p r o c e s s , the 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 . T h i s means that no new i n f o r m a t i o n i s added by extending the o r i g i n a l s e r i e s . The advantage of the technique i s that the i n c r e a s e d data length causes the i n t e r f e r e n c e effects to be g r e a t l y r e d u c e d . ot The p r e d i c t i o n c o e f f i c i e n t s g., j = 1, 2, N, for p r e -J dieting the s e r i e s z at the time of i n t e r v a l s ahead of the data point k are defined by the equation I I 1 1—: 1 1 1 i 0.0 0.DB3 0.125 0.188 0.25 U„3]3 0.375 0.438 0.5 FREQ U/DflY) CX30"2 ) F i g u r e 20 : F o u r i e r t r a n s f o r m o f s y n t h e t i c wobble d a t a o f F i g u r e 22. <•£ 60' N Zk+a = A Zk+1 - j N is l i m i t e d to values l e s s than the total number of data points 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 for backwards p r e d i c t i o n , b. , are defined J by N Z a b. Z. , . . K- a ^ J k+j - 1 U l r y c h et a l (1973) has shown that c a l c u l a t i n g these c o e f f i c i e n t s i s equivalent to applying the unit p r e d i c t i o n f i l t e r ( a = 1) or t i m e s , each time i n c o r p o r a t i n g the p r e d i c t e d point as the new la s t point of data. A proof i s given i n Smylie et a l (1973). T h e r e f o r e , only one c a l c u l a t i o n of the c o e f f i c i e n t s g? and b: is n e c e s s a r y to obtain a p r e d i c t i o n for a l l values of o . Computation 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 somewhat by the fact that for a r e a l time s e r i e s 1 1 g = b T h i s equality r i s e s out of the method of computation - which i s j j the m i n i m i z a t i o n of N N E l ( z , x ! - I g 1 z ) 2 + (Z - X b Z ) 2 J I k+1 f- & i k + l - j ' k-1 j k - l + j J J = l J J j=i J J 1 1 with r e s p e c t to the c o e f f i c i e n t s g . and b . T h i s m i n i m i z a t i o n y i e l d s N + 1 •> j l i n e a r equations. F o r t u n a t e l y , J . B u r g (1967) has developed an i t e r a t i v e method ( c a l l e d the B u r g al g o r i t h m ) that s i m u l t a n e o u s l y produces a u t o c o r r e l a -tions and unit p r e d i c t i o n c o e f f i c i e n t s for a given data s e r i e s . T h i s a l g o r i t h m is d e s c r i b e d i n Appendix I. Once the unit p r e d i c t i o n c o e f f i c i e n t s have been obtained, the s e r i e s is extended one point at a time i n both f o r w a r d and back-wards d i r e c t i o n s . The extension p r o c e d u r e i s i l l u s t r a t e d s c h e m a t i c a l l y in .61 F i g u r e 21. Continuing in this manner, the s e r i e s can be p r e d i c t e d to i n f i n i t y 1 Step 1 1 1 1 g3 g2 g l X + X 4- x 4 x 4-ZN - 3 ZN-2 ZN-1 ZN ZN+1 V o r i g i n a l data move f i l t e r f o r w a r d one i n t e r v a l 1 g5 1 1 g4 g3 1 g2 l gi Step 2 x 4- x 4- x 4- x 4- x • Z N - 3 ZN - 2 ZN - 1 ZN ZN4-1 ZN4-2 o r i g i n a l data F i g u r e 21. Method f o r P r e d i c t i v e E x t e n s i o n of a Data S e r i e s . in both d i r e c t i o n s (if one i s r i c h enough). A c c o r d i n g to U l r y c h et a l (1973) in the l i m i t , as the s e r i e s is 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 density f u n c -tion of the s e r i e s approaches the m a x i m u m entropy s p e c t r a l e s t i m a t o r ( M E S E ) . T h i s i m p l i e s that the extension adds no s t a t i s t i c a l i n f o r m a t i o n to that inherent i n the data. T o apply this method, the data r e c o r d was di v i d e d into o v e r l a p p i n g epochs of n e a r l y ten y e a r s (360 points or 3600 days). The l e n g t h of p r e d i c t i o n f i l t e r was chosen to be 250 points (2500 days) on the b a s i s that the r e s o l u t i o n of the annual and Ch a n d l e r frequency components r e q u i r e s at leas 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 of the M E S E technique - and that i f many m o r e than one half as many co e f f i c i e n t s as data points a r e used, then the B u r g a l g o r i t h m i s not st a b l e . T h e s e ten y e a r data r e c o r d s were extended two t i m e s in each d i r e c t i o n to create a time s e r i e s of n e a r l y fifty y e a r s . T o i l l u s t r a t e the value of this technique, the s y n -thetic signal of F i g u r e 22 (composed to produce a signal roughly comparable to the o b s e r v e d one by u s i n g the time s e r i e s of F i g u r e 25 to m o d e l the Chandler v a r i a t i o n and a p e r f e c t s i n u s o i d of amplitude . 15 seconds to r e p r e s e n t the annual and r a n d o m l y generated noise of standard deviation .02 seconds to r e p r e s e n t m e a s u r e m e n t noise) was extended in the p r e s c r i b e d manner to p r o -duce the s e r i e s of F i g u r e 23. 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 of the p r e d i c -ted p o r t i o n of this plot are 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 being that the extension appears to become s i n g l e - f r e q u e n c y valued towards the e x t r e -m i t i e s . The true worth of the technique i s i m m e d i a t e l y obvious when the fast F o u r i e r t r a n s f o r m S ( F F T ) of the o r i g i n a l and extended s e r i e s are c o m p a r e d . See F i g u r e s 20 and 24. The t r a n s f o r m of the o r i g i n a l s i g n a l has one m a i n peak: the only i n d i c a t i o n that there might be two r e a l components i s a slight bulging on the high frequency side of the m a i n peak. B y c o m p a r i s o n , the t r a n s f o r m of the extended s e r i e s has two v e r y w e l l separated m a i n peaks c o r r e s p o n d i n g to the known input frequency components. T h e r e f o r e , having a method that achieves the objective of c l e a r r e s o l u t i o n , it r e m a i n s to develop a b a s i s for f i l t e r i n g out the annual component and unwanted noise such that the second objective above is met. The E f f e c t of P h a se and Amplitude Shifts on the F o u r i e r S p e c t r u m Synthetic wavelets c o n s i s t i n g of truncated sinusoids of constant p e r i o d equal to 435 days, but v a r y i n g i n amplitude and absolute phase were added end to end to produce the function X(t) shown in F i g u r e 25. C o m p a r i n g the F o u r i e r t r a n s f o r m of a continuous s i n u s o i d (as in F i g u r e 19) and the t r a n s f o r m of X(t) (which are s u p e r i m p o s e d in F i g u r e 26) demonstrates the t y p i c a l effect of i n t r o d u c i n g sudden phase and/or amplitude s h i f t s . 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 of 2 main f requency components (annual and Chandler) p lus random n o i s e . F i g u r e 23 : P r e d i c t i o n o f t i m e s e r i e s i n F i g u r e 22 t o f i v e t i m e s i t s o r i g i n a l l e n g t h . F i g u r e 2k : The F o u r i e r a m p l i t u d e s p e c t r u m o f t h e e x t e n d e d s e r i e s ( F i g u r e 23). us o 0 f N UJo" 5 = H ct: CE UJ ZDo. I— • cr i ~ I 1 ~~r 1 1 98.1 148.1 198.1 ,248.1 298.1 TIME ( J U L I A N DAYS) ( X l O 1 ) -1.9 46.1 I 348.1 Figure 25 : Chandler v a r i a t i o n used to create the series of Figure 22 . Composed of sine wavelets a l l of the same frequency (1/^35 days ) but each varying i n phase and amplitude. F i g u r e 26 : F o u r i e r amplitude spectrum o f F i g u r e 2 5 . The dashed l i n e i s the spectrum of a s i n u s o i d with constant phase and amplitude. 0 68 the F o u r i e r amplitude becomes much m o r e spre a d over f r e q u e n c i e s near the known c e n t r a l f r equency. In addition, the peak shape tends to be a s y m m e t r i c and the peak frequency i s shifted away f r o m the known value (1/435 days T h i s behaviour can be explained in g e n e r a l t e r m s by v i s u a l i z i n g the synthetic data in t e rms of the wavelets that compose i t . E a c h of the wavelets can be r e p r e s e n t e d as a continuous sine function m u l t i p l i e d by the 'boxcar' function of a p p r o p r i a t e width. See left hand side of F i g u r e 27. The F o u r i e r t r a n s f o r m of the product of two functions i s the convolution of 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 as seen on the right hand side of F i g u r e 27. The b r e a d t h of the s i n e function ( F o u r i e r t r a n s f o r m of b o x c a r function) 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 length of the b o x c a r function (or length of wavelet). T h e r e f o r e , the data s e r i e s c omposed of short wavelets w i l l be a complex sum of broadened sine functions. T h i s r e s u l t s in a F o u r i e r amplitude s p e c t r u m that i s some-* what 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 of the continuous s i n u s o i d . The F o u r i e r t r a n s f o r m of such a s e r i e s of wavelets can be d e t e r m i n e d a n a l y t i c a l l y . However, except for the case of two wavelets, the number of p a r a m e t e r s makes the solution e n o r m o u s l y c o m p l i c a t e d . The case of two wavelets has been examined by F e d o r o v and Y a t s k i v (1965) and they show that equal amplitude wavelets 180 degrees out of phase w i l l produce a split s p e c t r a l peak. T h u s , the apparent s p l i t t i n g of the Chandler component mentioned p r e v i o u s l y i s now attributed to a l a r g e phase shift (Wells 1972). It i s apparent then, that in o r d e r to p r e s e r v e phase or amplitude shift i n f o r m a t i o n , f r e -quencies near the c e n t r a l C h a n d l e r frequency must be left u n a l t e r e d (if p o s s i b l e ) by the p r o c e s s of f i l t e r i n g out the annual component. A l s o , it i s T I M E DOMAIN F R E Q U E N C Y D O M A I N co F i g u r e 27 : S y n t h e s i s o f the F o u r i e r transform of a t r u n c a t e d s i n u s o i d . 70 obvious that by m e r e l y int r o d u c i n g sudden phase and amplitude shifts in a pe r f e c t h a r m o n i c , that it is p o s s i b l e to cause: (1) the F o u r i e r s p e c t r u m of such a s i g n a l to be tota l l y devoid of meaning so far as h a r m o n i c amplitude and frequency i n f o r m a t i o n i s c o n c e r n e d . (2) spurious a r t i f i c i a l effects that could be i n t e r p r e t e d as being r e a l - for example, the o b s e r v e d splitting of the Chandler peak. Band 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 data s e r i e s should c o n s i s t of two m a i n p e r i o d i c i t i e s - the 435 day Ch a n d l e r and the 365 day annual - plus r a n d o m n o i s e . So i n o r d e r to examine the Chandler latitude v a r i a t i o n , the annual component and noise must be e x t r a c t e d f r o m the latitude s i g n a l , while keeping in m i n d the c r i t e r i a d i s c u s s e d above. T o a c c o m p l i s h this task, a zero-phase band r e j e c t f i l t e r was a p p l i e d to the F o u r i e r t r a n s f o r m of 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 , three di f f e r e n t regions of the F o u r i e r s p e c t r u m were f i l t e r e d . N a r r o w band f i l t e r s were a p p l i e d to a r e g i o n near zero frequency and to the peak at 365 da y s . Both of these were a p p l i e d so that the F o u r i e r amplitude was r e d u c e d to the noise l e v e l r a t h e r than to z e r o . In addition, a wide band f i l t e r was uned to r e m o v e totally a l l c o n t r i b u -tions (assumed to be noise) at f r e q u e n c i e s above a p e r i o d of 200 da y s . T o accommodate the v a r i a b l e conditions f r o m epoch to epoch and f r o m o b s e r v a t o r y to o b s e r v a t o r y , the width and cutoff l e v e l of the two n a r r o w band r e j e c t f i l t e r s were chosen to fit the F o u r i e r s p e c t r u m in question. What this f i l t e r i n g a c c o m p l i s h e s is to produce a z e r o mean value, and to r e m o v e the 7 1 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 compo-nent. 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 Figure 28 : E f f e c t of band-reject f i l t e r i n g of annual on the spectrum of the extended synthetic latitude v a r i a t i o n (see Figures 23 &24) °~1 1 1 1 T 1 r -1.9 48.1 98.1 148.1 198.1 248.1 298.1 348.1 TIME ( J U L I A N DAYS) ( X l O 1 ) F i g u r e 29 : C h a n d l e r v a r i a t i o n r e s u l t i n g f r o m b a n d - r e j e c t f i l t e r i n g o f t h e s y n t h e t i c d a t a . 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 C h a n d l e r v a r i a t i o n . 74 a r c b e g i n s . See F i g u r e 6. T h i s method su f f e r s f r o m the m a j or weakness that any l a r g e deviation f r o m the expected smooth a r c w i l l produce an event; whether o r not that event r e p r e s e n t s a r e a l shift in phase, amplitude or mean value. T o avoid this p r o b l e m and to s i m p l i f y the computing p r o c e -dures somewhat, a p u r e l y anal y t i c technique that y i e l d s the t i m e - v a r i a b l e phase and amplitude of a q u a s i - h a r m o n i c function was developed. B y quasi-h a r m o n i c , it i s meant that the p e r i o d i c s i g n a l has constant p e r i o d but that the phase and amplitude a r e p e r m i t t e d to v a r y with t i m e . Supposedly, the Chandler latitude s e r i e s obeys this c r i t e r i a . So i f the Chandler s i g n a l is denoted by the function f, it w i l l be of the f o r m : f(t) = A ( t ) cos ( w Q t + 0 (t) ) (5.1) I f this can be a l t e r e d to the complex f o r m A ( t ) e M wot + 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 ° w i l l y i e l d a complex function of amplitude A( t) and phase 0 (t) . Since cos ( 9 - H_ ) = s i n 0 2 Equation 5. 1 y i e l d s f(t " J 1 w ) = A ( t) s i n (w t + 0 (t) ) L wo ° T h e r e f o r e , f ( t ) + i f ( t - - J L - ) = A( t ) ei ( wot + 0 ( t> > 2 w Q and m u l t i p l y i n g by e 1 Wo ' (f (t) + i f (t - __0 ) ) e" 1 w o f c = A (t) e i 0 ( t ) (5. 2) 2 w Q T h u s , taking the amplitude and phase of the left hand side of this equation gives the t i m e - v a r i a b l e amplitude and phase of the signal f. A p p l i c a t i o n of this s i m p l e computation to test data y i e l d s excellent r e s u l t s . The resultant phase and amplitude time v a r i a t i o n s for the signal of F i g u r e 25 are plotted as the s o l i d l i n e s i n F i g u r e s 30 and 31. The only deviations f r o m the actual phase and amplitude o c c u r at the b r e a k s , which should appear as straight v e r t i c a l l i n e s . The actual 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 deviation f r o m a h o r i z o n t a l l i n e , and the l a r g e fluctuations i m m e d i a t e l y following are caused by the a v e r a g i n g c h a r a c t e r i s t i c of the method. Next, the demodulator was a p p l i e d to the f i l t e r e d synthetic C h a n d ler s e r i e s of F i g u r e 29. The r e s u l t s (dashed l i n e s in F i g u r e s 30 and 31) are s u r p r i s i n g l y good. The b r e a k s in both phase and amplitude a r e i m m e d i a t e l y o b v i o u s . A l s o , it can be seen (here and i n a l l the examples p r o c e s s e d ) that the phase is 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 . That i s , the band r e j e c t f i l t e r i n g causes unpredictable d i s t o r t i o n s of the amplitude v a r i a t i o n i f l a r g e sudden shifts a r e p r e s e n t . The phase v a r i a t i o n seems to be free of these e f f e c t s . I 1 1 1 1 1 1 1 - 1 . 9 48.1 98.1 148.1 198.1 ,248.1 298.1 348 TIME ( J U L I A N DAYS} (XlO 3 ) l i r e 30 : Phase 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 . The s o l i d l i n e i s t h e phase o f t h e 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 t h e d a s h e d l i n e i s t h e phase o f t h e known o r i g i n a l i n p u t C h a n d l e r . in CJ CO O CM LU CO M O _ l Q_ c r V 7- * A • o ~1 298.1 •3.9 48.1 9B.1 148.1 198.1 .248.1 TIME ( J U L I A N DAYS) (X10 J ) 348.1 Figure 31 J Amplitude v a r i a t i o n of the synthetic Chandler. S o l i d and dashed l i n e s represent the f i l t e r e d and actual versions r e s p e c t i v e l y . 78 C H A P T E R VI T H E P H A S E A N D A M P L I T U D E R E S U L T S F O R T H E 5 I. L . S. O B S E R V A T O R I E S A f t e r band r e j e c t f i l t e r i n g to is o l a t e the C h andler component of latitude v a r i a t i o n , one obtains a set of o v e r l a p p i n g ten y e a r time s e r i e s for each o b s e r v a t o r y . In o r d e r to produce a continuous Chandler s e r i e s , the o v e r -lapping p o r t i o n s were a v e r a g e d . F i g u r e s 32 to 36 a r e the r e s u l t i n g C h a n d l e r latitude v a r i a t i o n s for the i n d i c a t e d o b s e r v a t o r i e s . T h e r e a r e two m a i n features of these r e s u l t s that are common to e v e r y o b s e r v a t o r y : 1. A shift i n mean value o c c u r r i n g at J u l i a n day 2427800 (1935.0) which c o r r e s p o n d s to a change i n star catalogues used by the o b s e r v a t o r i e s . The shift i s close to +.02" for each of the three o b s e r v a t o r i e s functioning at this t i m e . 2. A l a r g e i n c r e a s e i n amplitude beginning at a p p r o x i m a t e l y J u l i a n day 2429300 (1939.0). Both of these features indicate that the f i l t e r i n g technique i s at l e a s t capable of 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 amplitude s h i f t s . A n even m o r e en-couraging factor i s that the o v e r l a p p i n g p o r t i o n s of the s e v e r a l ten y e a r C h a n d l e r s e r i e s at a p a r t i c u l a r station are v e r y s i m i l a r p r i o r to being a v e r a g e d . That i s , two completely independent ap p l i c a t i o n s of the f i l t e r i n g p r o c e s s to dif f e r e n t data 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 Chandler v a r i a t i o n s e r i e s that are a lmost i d e n t i c a l . 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 technique i s p r o d u c i n g a r e s u l t c l o s e to some defi n i t i v e C h a n d l er m o t i o n . F i g u r e 32 : 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 f o r M i s l s a w a . to o F i g u r e 33 : 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 f o r U k i a h U3 a (=} F i g u r e 34 : 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 f o r G a i t h e r s b u r g . 00 F i g u r e 35 t 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 f o r K i t a b (£3 O F i g u r e 36 : 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 f o r C a r l o f o r t e . 84 In o r d e r to examine the Chandler v a r i a t i o n in m o r e d e t a i l , it is advantageous to look at the phase and amplitude of the p e r i o d i c motion as a function of t i m e . T h i s i s a c c o m p l i s h e d by the method d e s c r i b e d in the p r e v i o u s s e c t i o n . The amplitude of the Chandler motion for each of the five o b s e r v a t o r i e s is plotted in F i g u r e 3 7. S i m i l a r l y , the phase i s plotted in F i g u r e 38. It should be noted that the phase has been adjusted to the G r e e n w i c h m e r i d i a n by adding the longitude of the o b s e r v a t o r y to the output of the demodulator, and that the zero point in time was chosen to be J u l i a n day 2423323. T h i s means that a latitude signal at G r e e n w i c h would have zero phase 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 on day 2423323 or on any integer number of p e r i o d s before or a f t e r -w a r d s . The Chandler p e r i o d was chosen to be 437. 0 days ba s e d on the value that makes the phase constant i n the l a s t p o r t i o n of the time i n t e r v a l (after J.D. 2429700). T h i s c r i t e r i o n for choosing the exact value of the p e r i o d i s somewhat a r b i t r a r y (excepted value = 435 days), and i s adopted p r i m a r i l y because of the r e l a t i v e c o n s i s t e n c y and flatness of the v a r i o u s phase c u r v e s during this time i n t e r v a l . It i s i m m e d i a t e l y obvious that some m a j o r changes i n both the phase and amplitude of the Chandler latitude 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 1949.0. P e r h a p s the most encouraging r e s u l t f r o m F i g u r e s 37 and 38 is the c o n s i s t e n c y f r o m station to station, e s p e c i a l l y f or phase 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 , both the phase and amplitude v a r i a t i o n s were v i s u a l l y a v e r a g e d o v e r the number of o b s e r v a t o r i e s . At the same time, a v i s u a l estimate of the m a x i m u m p o s s i b l e e r r o r in the average was obtained by i n c l u d i n g a l l the signal except the e x t r e m i t i e s of l a r g e peaks 242331.1 242431.1 242531.0 24263^ 1.0^  242730.^ ^ ^ 0 . 8 ^ ^ ^ . 8 243030.7 243130.6 Figure 3? : Chandler amplitude variation for a l l observatories. PHRSE VARIATION (RADIANS) -0.5 1.0 2.5 4.0 5.5 7.0 98 87 that do not correlate with peaks at other stations. The results of this process are shown in Figures 39 and 40. 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. The two most definite occur at Julian days 2424511 (May, 1926) and 2425311 (March, 1928). The possible errors in these dates are approximately plus or minus three months. Another very pos-sible 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 2427311 2428311 2429311 2430311 T I M E ( J U L I A N D A Y S ) 2431311 F i g u r e 3 9 : A m p l i t u d e v a r i a t i o n a v e r a g e d o v e r number o f o b s e r v a t o r i e s . CO 00 o ^—. in CO in" < < o rr < > LU CO < X CL 1 0 (Nl m o i 2423311 2424311 T 2425311 T 2426311 2427311 2428311 2429311 T I M E (JULIAN DAYS ) i 2430311 — I 2431311 F i g u r e 40 : Phase v a r i a t i o n averaged over 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 con-stant; 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. This method yields a data point every .6 years. 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. Table 3 is a list taken from Richter (1958) of the major earthquakes ( > 7.9) that occurred during the epoch of interest. The first ob-served shift (May, 1926) marks the end of two years of complete earthquake inactivity. 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 TABLE III EARTHQUAKES OF MAGNITUDE GREATER THAN 7. 9 Dates of Observed Dates of Chandler Events Earthquakes Magnitude Nov. 11, 1922 8.4 Feb. 3, 1923 8.4 Sept. 1, 1923 8.3 Apr. 14, 1924 8.3 June 26, 1924 8. 3 May, 1926 June 26, 1926 8. 3 Oct. 3, 1926 7.9 Oct. 26, 1926 7.9 Mar. 7, 1927 7.9 May 22, 1927 8. 3 March 9, 1928 8. 1 June 17, 1928 7.9 Dec. 1, 1928 8.3 March 7, 1929 8. 6 . June 27, 1929 8. 3 may be (Feb. 1930 one ( Jan. 15, 1931 7.9 March, 1928 event (Jan. 16, 1931 Feb. 2, 1931 7.9 Aug. 10, 1931 7.9 Oct. 13, 1931 7.9 9 1 92 y e a r s ) and m a r k s the beginning of an active t i m e . The most obvious c o n c l u s i o n one could make f r o m these o b s e r v a t i o n s i s that abrupt shifts i n the Chandler motion a r e a s s o c i a t e d with the onset of earthquake a c t i v i t y . T h i s c o n c l u s i o n of c o u r s e i g n o r e s the cases where a quiet e r a i s not ended by an abrupt change in the Chandler motion, and also the shifts (though c o n s i d e r e d l e s s l i k e l y to be real) that do not c o r r e s p o n d to the same c i r c u m s t a n c e s . Another question a r i s e s : What m e c h a n i s m could p o s s i b l y cause such 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 Guinot (1972), they are not l i n k e d to any ge o p h y s i c a l event. Since caution i s the watchword, there won't be any w i l d guesses here about this m a t t e r . The co r r e s p o n d e n c e between constant phase and r a p i d change in a m p l i -tude i s d i f f i c u l t to explain - it might be s u r m i s e d that the excit a t i o n m e c h a n i s m is m o r e e f f i c i e n t when the phase i s at a stable v a l u e . One thing i s c e r t a i n f r o m this work: earthquakes could not be detected even i f they did excite the Chandler m o t i o n . That i s , the noise l e v e l i s a p p r o x i m a t e l y three t i m e s as l a r g e as the l a r g e s t p o s s i b l e effect of a m a j o r earthquake ( c a l c u l a t e d to be about one foot or .01" for the 1964 A l a s k a earthquake - S m y l i e and M a n s i n h a , 1972). A n important r e s u l t f r o m most studies of the Chandler wobble i s an estimate of the damping time or a l t e r n a t e l y of the Q f a c t o r . E s t i m a t e s of these quantities a r e n o r m a l l y obtained by fitting the smoothed power density function ( p r o p o r t i o n a l to the squared F o u r i e r t r a n s f o r m of the data) to functions l i k e 1 / ( ) - (w2 - w 2) 2 + 4w2 ?^2 (6. 1) 2 1 2 or 1 / (w - w Q) + ( - ) (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 ex-tremely difficult if not impossible to say anything about the existence of damped wavelets that would occur if the Chandler wobble was randomly excited. How-ever, 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 F i g u r e kJ, x F o u r i e r a m p l i t u d e s p e c t r u m o f s i m p l i f i e d undamped C h a n d l e r m o t i o n 95 factor of the actual wobble is somewhat higher than i s commonly thought. C o n c l u s i o n s 1. The optimum value for the p e r i o d of the Chandler wobble i s 437 days bas e d on the c r i t e r i a that the time v a r i a t i o n of phase be m i n i m i z e d . 2. The value of the Q factor for wobble i s a l m o s t c e r t a i n l y higher than p r e -vious e s t i m a t e s . Its exact value i s e x t r e m e l y u n c e r t a i n . 3. A b r u p t shifts i n the phase of the Chandler wobble ar e p o s s i b l y a s s o c i a t e d with the onset of earthquake a c t i v i t y . 4. The cause of these phase shifts and also of l a r g e v a r i a t i o n s i n the Chandler amplitude i s unknown. However, 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 caused by the p r e s e n t l y e s t i m a t e d effect of earthquakes. 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 Chandler wobble could s t i l l be true though there is no evidence 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 Brera 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. Soc, 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 Rota- tion 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, Springer-Verlag, 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 lati-tude: The Roche model core, Monthly Notices, Royal Astrono- mical 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, Cambridge University Press, I960. 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 Dis- placement Fields and the Rotation of the Earth, ed. by L. Man-sinha, 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 Rota-tion 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 S m y l i e , D.E., G.K.C. C l a r k e , and L . M a n s i n h a , Deconvolution of the pole path, 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 and the Rotation of the  E a r t h , ed. by 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. S m y l i e , D . E . and L . M a n s i n h a , The 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 and Changes in the Rotation of the E a r t h , Geophys. J.R.  A s t r . Soc. . v. 23, 329-354, 1971. S m y l i e , D.E., C l a r k e , G . K . C , and T . J . U l r y c h , A n a l y s i s of the I r r e g u l a r i -ties i n the Ear t h ' s Rotation, Methods in Computational P h y s i c s , v. 13, 1973. Stacey, F.D., A r e - e x a m i n a t i o n of co r e - m a n t l e coupling as the cause of the wobble, in Earthquake D i s p l a c e m e n t F i e l d s and the Rotation of  the E a r t h , ed. by 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. T a k e u c h i , H., On the E a r t h tide of the c o m p r e s s i b l e E a r t h of v a r i a b l e density and e l a s t i c i t y . T r a n s a c t i o n s of the A m e r i c a n G e o p h y s i c a l Union, v. 31, 651, 195(T U l r y c h , T . J . , S m y l i e , D.E., Jensen, O. G. C l a r k e , G . K . C , P r e d i c t i v e f i l -t e r i n g and smoothing of short r e c o r d s , J . Geophys. R e s , in p r e s s . W a l k e r , A. and A . Young, F u r t h e r notices on the a n a l y s i s of the v a r i a t i o n in latitude, Monthly N o t i c e s , R o y a l A s t r o n o m i c a l Society, v. 117, 119-141, 1957. W e l l s , F . J . , Ph.D. T h e s i s at B r o w n U n i v e r s i t y , 1972. Yashkov, V.A., Sp e c t r u m of the m o tion of the Earth's p o l e s , Soviet A s t r o n o m y , v. 8, 605-607, 1965. Y u m i , S. , P o l a r motion i n recent y e a r s , i n Earthquake D i s p l a c e m e n t F i e l d s  and the Rotation of the E a r t h , ed. by 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 press. Rudnick, P., The spectrum of the v a r i a t i o n i n l a t i t u d e , Transactions of the  American Geophysical Union, 35» ^9t 1956. S p i t a l e r , R., Petermanns Mitteilungen, Erganzungband, 29, 137» 1901. 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 f - f k k g , g , . . . . , g are chosen so that 1 2 n is minimized. This produces the set of equations: gj 0ff (0) + g 2 0ff (-D + + g n0 f f ( l-N) = 0 f f ( l) gj 0ff (N-1) + + g n 0ff (°) = % <N) where 0££ ( \ ) represents the stationary autocorrelation of lag \ In addition, the error power { K l } for an N point predictor can be written 100 r l { lEJ ) = PN+ 1 = ^ f f ( 0 ) N j=l J Combining this equation with the previous set yields the matrix equation: N 0ff(O) ^ f ^ " 1 ) ?>ff(-N) with 0ff(N) 0 f f(N-l) 0ff(O) 1 N N+1 0 0 0 0 0 1 N - g N The Burg algorithm concerns the iterative solution of this matrix equation for both the coefficients a^ and the autocorrelations 0^ (i) simultaneously. First consider the case N = 0. Then M 2 P = 0 (0) = — V If 1 *ff 1 1 M ^ 1 j j=l 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 1 p *N+1 a NI 0 a = 0 N2 a 0 N3 • 0 • 0 a ' NN 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 N+l 1 N+2 a N+l, 1 a N+l, 2 = 0 0 * 0 • a ' 0 0 N+l, N+l This can be written T N+l 1 a N. 1 N.N 0 N+l, N+l ' N+l 0 0 0 0 LN+1 + N+l, N+l 0 * a N, N a* N. 1 1 N+l 0 0 0 0 which implies the relations a N+l, 1 N, 1 + a N+l, N+l a N, N and a * N+l.N 0ff (N+l) N,N N+l I j = l + a a N+l, N+l N, 1 N+l.j 0 f f ( N + l - J > (A-l) 102 To solve a l l these unknowns, it i s only n e c e s s a r y to c a l c u l a t e the value a nvr. , „ , , • T h i s i s a c c o m p l i s h e d by m i n i m i z i n g the e s t i m a t o r for P N+l, N+l c j 6 N + 2 M - N - l p = 1 V i F I N+2 2(M-N-1) ^ I i+N+l' j=l with r e s p e c t to the r e a l and i m a g i n a r y p a r t s of a after having sub-N+l.N+l 6 stituted for the N + l j *=* * ^» f r o m the equations ( A - l ) . N o r m a l l y , the estimate of e r r o r power for b a c k w a r d p r e d i c t i o n is also i n c l u d e d i n the m i n i m i z a t i o n . When this is done, m i n i m i z a t i o n y i e l d s : M - N - l * * a 2 J [VN+1+ N,l V N ' " " ' N .NVJ h + N.lW---N + l , N+l M - N - l I [ I f + Q f +....+ a f I j = 1 L 1 j+N+1 N, 1 j+N N . N j + 11 a * 1 N, N j +N J ' " " * + IN, IN j -HN J l ~ * a * a * I C 1 , fj + N, 1 fj+l + * " + N.N fj+N> J T h u s , a l l the elements of step N+l are known, and one can p r o c e e d i n exactly the same manner to step N+2, and so f o r t h until the d e s i r e d number of c o e f f i -cients is obtained. A p r i n t out of the F o r t r a n subroutine to p e r f o r m a single i t e r a -tion of the above p r o c e d u r e i s incl u d e d below. In this p r o g r a m : M = number of data points N N = o r d e r of i t e r a t i o n - produces • , i = 1, . . . N N r NN, l F = data s e r i e s 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. SUBROUTINE 6 P E C ( M , N N , F , G , P E F , P E R ) REAL *8 G (M ) , P E F ( M ) , PERIM) , 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 . . P E F ( J ) = 0 D 0 12 P E R U ) =0D0 1 1 SN=0D0 SD=0D0 J J = 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 2 SD = SD + ( F (J +N+1) +PEF ( J ) ) **2+ (F ( J ) +PER ( J ) )**2 G ( N N ) = S N / S D I F ( N . E U . 0 ) G 0 TO 3 D O 4 J = 1 , N K = N - J + 1 4 H ( J ) = G ( J ) + G ( N N ) * G ( K ) DO 6 J = 1 , N 6 G ( J ) = H (J ) J J = J J - 1 3 DO 10 J=1 , J J P E R ( J ) = P E R ( J ) + G ( N N ) * P E F ( J ) + G ( N N ) # F ( J + N N ) 10 P E F ( J ) = P E F ( J + 1 ) + G ( N N ) * P E R ( J + L ) + G ( N N ) * F ( J + l ) RETURN END . 

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