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PHASE and AMPLITUDE VARIATION of CHANDLER WOBBLE by John

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PHASE and AMPLITUDE VARIATION of CHANDLER WOBBLE by John PHASE AND AMPLITUDE VARIATION OF CHANDLER WOBBLE by John Alexander Linton B.A. Sc., University of British Columbia, 1970 A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Master of Science in the Department of Geophysics We accept this thesis as conforming to the required standard The University of British Columbia May, 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of G>e<?£?/^cj.-r/c 5 The University of British Columbia Vancouver 8, Canada Date Z7o^7<e f , i ABSTRACT Normally, the wobble of the earth has been dealt with in a manner that assumes that the two main periodic components have constant phase and amplitude. An initial assumption of this thesis is that both these parameters can vary with time. A technique of predictive filtering is used to determine the Chandler component of the wobble from basic latitude measurements at the five I. L. S. observatories. A simple analytic proce• dure is employed to obtain the phase and amplitude variation of the periodic Chandler motion. The results indicate that major changes in both phase and amplitude occur in the period 1922. 7 to 1949. These changes are possibly associated with earthquake activity, although there is nothing to indicate that there is a correlation between individual earthquakes and events in the Chandler motion. The calculated period of the Chandler wobble is 437 days and the damping time is so uncertain that a value approaching infinity is not unlikely. ii 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 iii TABLE 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 INTRODUCTION At first glance, the description of the earth's rotation about its axis would seem a simple task. That is. if the earth is treated as a rigid, rotating sphere, one could simply apply the relevant classical laws of physics as developed by Euler. However, the real earth is neither rigid nor spherical. These deviations from the 'ideal' body result in complex perturbations in the rota• tion of the earth. The bulging of the earth at the equator produces the varia• tion in the direction of the axis of angular momentum in space(precession and nutation). Another effect is the variation in the length of the day. The most important perturbation to geophysicists, and the subject of this thesis, is the periodic variation in the orientation of the instantaneous rotation axis with respect to the geographic frame of the earth - or alternately, the movement of the rotational pole (point near North pole where instantaneous rotation axis pierces surface) over a horizontal plane parallel to the equator. This motion is called wobble. Most measurements of wobble are in terms of the motion of the rotational pole in units of distance - the complete planar motion being described by two direction components. Succeeding chapters of this thesis will deal with the history of earth wobble research, the classical theory, and the pole path data, all with a view of justifying a new approach to the analysis of wobble data. 2 EARLY HISTORY OF WOBBLE RESEARCH 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 Berlin 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 research has been centered on two aspects of the wobble problem: 1. Analysis of the data and determination in detail of the characterist• ics of the two main frequency components (Chandler and annual).
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