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

The University of British Columbia Vancouver 8, Canada

Date Z7o^7

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).

2. Explanation of these characteristics by a plausible model for the

real earth.

The annual wobble

The annual wobble has, for convenience's sake, been universally treated as a spectral delta function which, in more familiar terms, is just

simple circular or elliptical motion of the rotational pole. On first glance at the power spectrums depicted in Fig. 1 & 2 (both are taken from Munk and

McDonald), this assumption would seem to be justified by the narrowness of the annual peak. The power spectra for perfect circular motion of period one year would have a peak width of approximately . 04 cycles/year compared to the peak width in these plots of . 03 to . 06 cycles/year. Unfortunately, the

simplicity this implies is not quite achieved. Walker and Young (1957) point out. in their exhaustive analysis of I. L. S. data that, "the components of the annual motion vary according to type of series and interval, there being noticeable differences in both the amplitudes and phases of the several compo• nents. " Table I gives various estimates of the amplitude for each of the two directional components of annual polar motion - rrij towards Greenwich, rr^ towards ninety degrees east of Greenwich, The amplitude varies by almost a factor of two between minimum and maximum. This variation means that if the annual wobble is to be modelled as simple circular motion, the model 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 mi 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 Values of the Period, Years Source

1890-1915 1.13, 1.20, 1.27 Witting (1915) 1890-1924 1.20 Pollak (1927) 1890-1924 1.19 Stumpff (1927) 1890-1922 1.08, 1.14, 1.19, 1.27 Wahl (1938) 1922-1938 1.13 II 1892-1933 1. 223 Jeffreys (1940) 1908-1921 1. 202 n 1900-1940 1.108, 1.170, 1.208, 1.250 Labrouste (1946) 1891-1945 1. 196 Rudnick (1956) 1900-1920 1. 193 Walker & Young (1957) 1890-1924 1. 191 Danjon & Guinot (1954) 1929-1953 1. 172 11 1897-1957 1. 186 Pachenko (I960) 1897-1922 1.190 it 1930-1957 1.180 n 1891 -1952 1. 193 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 conclusions about the rigidity, shape and non-elasticity of the real earth.

The essential feature of the Chandler wobble is that its spectral representation resembles that of a damped oscillator - that is, the spectral peak is spread over frequencies near the peak value (see Fig. 1 & 2). Table

II (taken from Wells, 1972) shows the Chandler period (period at peak) during various epochs as calculated by the authors indicated. This table illustrates that the Chandler wobble might have time-variable amplitude and frequency.

Plots by Yumi (1970) and Iijimi (1971) (Fig, 3 & 4), indicate how the period and amplitude of the two Chandler components vary with time. The amplitude variation is especially remarkable - between ten and thirty feet; It should be pointed out that these authors have used various forms of harmonic analysis - sometimes-on data lengths as short as ten years (Yumi) - and therefore results that indicate a shift in frequency are subject to large errors due to resolution problems of the technique.

Estimates of the damping time (time for signal to decay to 1/e of its original value) vary from two years (Walker and Young, 1957) to ninety years (Pachenko). These estimates of damping time lead to a Q factor

(dimensionless measure of energy dissipation) of thirty to fifty, which is several times smaller than Q values calculated from seismic data.

Fellgett (I960) calculated a damping time of 12.4 years but concluded that the value is "uncertain by a factor of at least ten. and damping times as short as 2. 5 years or as long as several hundred years are not excluded. " The 9

«

g -io\ a. E o —i aoL l i L j i L _j 1 i 1993 ISOS 1317 1929 19*/ 19S3 I96ST

Figure 3 : Chandler polar motion amplitude characteristics (taken from Yuml, 1970)

1M\ i—i—i i r i—i—r T— — —

l.20\

a> 1.16

T3 o 1.14- **

L rn.

J JL i i i J_ 190072 2436

Figure 4 : Chandler period variation (taken from Iijiml, 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

ct70 o •"60 O x 50 o cc 40 UJ * ? 30 BAND WIDTH 20

10

LA • /V. VY. .5 .7 .8 .9 1.0 I.I FREQUENCY (CPY)

Figure 5 s Power spectrum showing split Chandler peak (taken from Rochester and Pedersen, 1972)

TOWARD 90 DEGREES

Figure 6 : Smylle and Manslnha's fitting of smooth arcs to observed polar motion (taken from Scientific 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 insignificant, to affect the Chandler wobble. 16

SUMMARY

The most important facts that can be gleaned from this brief history are that:

1. both the annual and Chandler components are time variable in

amplitude and perhaps in frequency.

2. no assumption should be made about the damping time of the

Chandler motion.

3. the Chandler may have random sudden changes in 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

= (7)] L /{{tYi [ r x fvv"x r 4- v (r)) p 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

rx(wxr) = w(r.r)- r(w.r)

which becomes

r x ( w x r ) = e. [r2 &.. - x. x. "1 w. 1 L ij 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

p L = w. fr) dV 1 L 1 J eartarth ij J J

4- fff r x v ( r ) p ( r ) dV earth '

e. I., w. 4- J 1 iJ J

where I.. = fff \ r2 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 7xv(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

l C I11 = I22 = A AND 33 = where two of the principal moments are equal because of cylindrical symmetry.

Perturbation Equations

The development of the equations governing the case where the axis of rotation is perturbed from its equilibrium position follows the method of Munk and McDonald (1960). The departures from uniform, diurnal rotation are almost undetectable in practice, so that first order perturbation theory should work extremely well. Therefore, let 20

Z (geographic North pole)

instantaneous rotation axis — fixed in space except for effects of Iml precession

earth s centre of mass Y (90° E of Greenwich)

(to Greenwich)

Figure 7 : Orientation of reference axes 21

m m ) w = f\ 1 4- JX 4- e^f\. (1 4- 3

J\ - angular velocity of earth; rri|, m^. << 1

Ill5sA+Cll+bll hi- A+C22 + B22 X33 * C + C33 + B33

X ! =B + C l = I = B + C 13 = 31 13 13 Z3 32 23 23

I12 = I21 = b12 + c12

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

L l = hi W1 + I12W2 + I13W3+ A

= A-ami+ j\ (b13 + c13) +• x1

wl + L2 = hi *22 W2 + J23 W3 + ^ 2

= AAm2 + A(b23+c23)+ Xz

wj L3 = I31 + I32 w2 4- I33 w3 4- Xz

A A = C (1 + m3) 4- (c33 4- b33 ) 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)JTL . Substituting into the Liouville equation (2. 3) yields 22

P ~ £ + A J\ m 4- _TL (b , 4- c ) 4- C yL2m_ 11 1 13 13

2 _i2 -jT_ Am2- Jl?(b23 4- c23) - yi (2.4)

p = A jT.m 4- JT (b 4- c ) 4- i +AAm 2 ^ 23 2 3 ^ *

+ SL /I ii 2 (b13 4- c13) 4- - C Jl m (2.5)

= CJl m3 4- yi (c33 4- b33) 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 - i ± a x z

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 -

-ij/l - ( cl3 4- i c23)_fL - ( bu 4- i b'23 )_/!

2 2 - i ( c13 4- i c23)/l - i ( b^ + i b23)yi_ (2.7) 23

This is the equation of motion for the wobble - m gives the complex angular

displacement of the axis of rotation away from "e^. or. if one wishes, the

motion of the instantaneous pole of rotation over the surface of the earth, with

. 1 second of angular motion equivalent to ten feet of polar motion. The re•

maining equation

m = P 3 77T [ 3 " ^3 - < '33+ b33 >^-] <2'8)

governs the change in length of day.

Rotational Deformation

Since the terms b.. are in fact functions of m, it is useful at !J — this point to obtain the exact functional relationship. It can be shown that

k_ d m m, _TL e. c. b I 12 = l* 3 G

k d m 7 J\

2 b.23 " 3 G

B = k2 d5ml^2 13 3~G

where d equals the radius of the earth, G is the gravitational constant and k is the Love number-defined such that if the symmetric earth is subjected

to a disturbing force field expressible as the gradient of a second order

spherical harmonic potential U^. the additional potential V at the deformed

surface is V = k2 U-,. These results are based on the assumption of linear

str ess-strain relations, and are derived on page 25 of Munk and McDonald 24

(1960). Substituting for b.^ in equation 2.7 gives

m/L(A+ — ) - i rn j\ ( C - A - _2 )

- 3G - 3G

= H - ( + 1 1 (c + 1 c ) 1 " " ^13 ' 13 23 ^

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 [JZ -1 -ii^ 2

2J - i (C13 4- i C23) J1 (2.9)

^ A5 Z k?d y,

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

This represents a circular motion of the instantaneous pole of rotation with

(T angular velocity 0 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

5 2 k?d il cr = AH- — o 3GC

The only unknown in this equation is k^. and Takeuchi (1950) has computed k^ = . 281 for one of Bullen's earth models. Using this value, the period of the free wobble is

2 TT 435 days 5 2 H - (k2d JT. /3GC) 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 calculated previously, and o

f is the damping time. Equation 2.9 then yields the solution

i w t . - t/Z Q m A e e

and the motion of the pole is a west to east spiral as shown in Fig. 8.

Fig. 8. Polar motion corresponding to damped free wobble of the earth.

C. Harmonic Excitation with Damping

Allow to be complex as above, and assume the right

i

where o( and

Then equation 2.3 becomes

CTt m - i CT rn_ = <=< e

The steady-state solution of this equation is

(Tt m i ( CT- CT0)

It follows that

m w )2 + (i/r )2 V ( cr Q

Assuming that the amplitude of the excitation, o£ , is the same for all frequencies, a plot of /m { versus frequency would look like Fig. 9'. By fitting a curve of this type to the actual data, one obtains an estimate of the

wo t parameter C and thus of the damping factor Q = —£—.

t I ml

Fig. 9. Response of damped earth as a function of frequency. 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 ' 1 J m(t) = e [mQ - j F (t) e~ d "t

- oo

Upon substituting for F(t) from case (a), a simple integration gives

m(t) = e . t < 0

J i crn t m(t) = (mQ - ~ ) e 4- : . t > 0 — i

The spiral motion after t = O has modified equilibrium pole position J Ux . new amplitude m0 - i cj-o and new phase 0 given Case (b) gives

m(t) = m Q e t < 0

m (t) = (m0 - J ) e t > 0

Here, the equilibrium pole position is unaltered for t > 0, but both the phase and amplitude are changed by the addition of the quantity J. From these examples, one would expect the Chandler motion due to earthquake excitation to be smooth spirals interrupted by sudden shifts in phase and amplitude and possibly, also in equilibrium pole position. 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° 08' East / 60° 29' East

/ " " m to Greenwich

\ Gaithersburg, Maryland

Ukioh, California \ 77° 12' West

123° 13' West

Fig. 10. Longitude of I. L. S. observatories

The stations have operated somewhat intermittently since the inception in 1899. Gaithersburg was closed from 1914 to 1931. 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 = celestial pole Z = zenith E = celestial equator S. = star i l Z. = l zenith angle to

Fig. 11. Configuration of star pair with respect to zenith and celestial pole.

(fl - = cf2 + The latitude 0 is given by 0 = Zl and 0 Z2

cfl + cT Z Z 2 2 - l or upon addition by 0 = (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 in star catalogues. The important part of this process is maintaining the telescope level (or a constant zenith). This was originally-

done visually, but most modern measurements are made with the Photographic

Zenith Tube (PZT). In this instrument, the levelling device is a free surface of mercury with reflects a series of four star images onto a photographic plate from which the zenith distance z is measured directly.

Reduction of Data by the I. L.'S.

Unfortunately, the data published by the I. L, S. are not the basic measurements of latitude. Basic latitude values 0 are given by equation 3. 1 which in terms of measured quantities becomes

0 = d" + R ' M + I. S. (3.2)

where cf = adopted value of mean declination of the star pair

+ ) at 'he time of measurement - time ( ^ ( <$i <^2

variable because of star pair proper motion.

R = micrometer measurement (turns).

M = angle of telescope shift per one half revolution of the

micrometer - including a term for temperature

dependence. °

I. S. = a term to account for inequalities of the micrometer

screw.

The I. L. S. under Kimura added to this equation several corrections based on the data itself. A correction for declination is included in order to make the 34

mean latitude given by each star pair equal to the mean value for the star pair group of which it is a part. The group mean is obtained by weighting according to the trend of the data in a manner totally incomprehensible. A

correction for the time variable part of M is also obtained. See Kimura

(1935). Fortunately, only the latter correction has a local effect - the correction to declination produces the same effect at every station. Luigi

Camera operated the I. L.S. from 1935. 0 to 1948, and happily his method of reduction is straight-forward. He introduces two corrections similar to the first and last described above for Kimura - the difference being the simplicity of his assumptions and subsequent calculations. See Camera (1957, page 256).

Instantaneous Pole Path

Most of the accessible data published by the I. L. S. consists of monthly calculations of the two directional components of the polar motion.

Latitude variation at station i, & 0 j , due to polar motion is just

^ 0^ = rrij cos + sin f •

where m^ and m^ are the respective directional components of angular polar displacement away from the x^ axis. t. is the longitude of station i. measured positively in the easterly direction. Since latitude variation can be caused by errors in star positions, temperature fluctuations and other similar variations that do not depend on polar motion, latitude variation is written

0. = m. cos "C*. + m_ sin 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

CHAPTER IV

PREPARATION OF EQUALLY SPACED LATITUDE

MEANS AT EACH OBSERVATORY

Our basic data consist of latitude observations from individual star pairs - including the corrections to declination and micrometer variation described above. A maximum of sixteen such observations are made each night at each of the five observatories. Since observations cannot be made at all times due to weather conditions, many nights have no readings at all and some have very few. In order to do time series analysis, it is essential to have uniformly spaced data. Also, it is desirable to attach greater weight to nights where many observations were made. Looking first at nights when at least one measurement was made at a particular station, the mean value estimate of latitude on night j is given by

N

(4-1) i=l

where

N = number of observations on night j

i .th , th x - 1 latitude observation on j night j

Note that the symbol s\ indicates that the quantity is an estimator only.

The unbiased variance estimator for any observation x1 is 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

J Z = I * ^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- = >. aJ E S s f = L aj 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 (zj = Y aJ s + £ aJ EJ 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

(4 4 E Z = S + 2 \ \ " ) i kl k 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

E 4 5 D E{zl}-\ -z\ k • I \ 'k <-> j=-N which is quite unwieldy. However, if var(z^) is expanded in the same manner,

the solution is simplified somewhat. That is,

2 {[z var(Zk) = E k-E(Zk)] }'

„ ? N N . = E [ Z |- s -2s I aJ E[- £ aJE?l I k J k k kkL^kkJ j =-N j=-N 40

Combining this result with equation (4-5) yields

N 2

3 D = var (Z, ) + [ £ J K 1 (4-6) k k j=-N k kJ

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 as ) = 2- (a ) var (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 2 N .2 N °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' = var (\+j> where C is a constant determined by the criteria

N j aJ = 1

j=-N k 2

To minimize the effect of the term A, = Y a E 3 \ let k j=-N L k kJ i w.C

a = J (4_8)

k var (?k+.) where w. is a triangle function with equation

w. = w - I J I w J o N+1 o N • Once again C is determined by £ a = 1. The effect of including the j=-N . k factor w. in the determination of a-' is to give less emphasis to terms J k j of Z where E is likely to be large. This amounts to a rough minimization k k of A^ and thus of D^. The triangle shape is almost certainly not the optimum

one. However, it is unquestionably better than the box car function which is

assumed if j C \ ' varft' ) k+j

The only remaining unknowns from equation (4-7) are the terms E"' . Since

E^ is a function of the actual latitude variation, it cannot be evaluated without k knowing the latitude signal exactly. However, it is possible to define an expec-

ted value of A = \Y a? E | in terms of auto-correlations calculated on k L k kJ j

the basis that the actual latitude series is only one determination of a stochastic

process that can be simply represented as

s = x cos (wk + 6) xC where x = constant amplitude of latitude variation w = angular frequency of the main periodic variation

9 = a random variable with uniform probability distribution from 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 jsksk+^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:

eu>*f2[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

w2N n =

It remains now to write in terms of the above auto-correlation function: N N

j m j m A, = E{ V V a a E E ) k lj=-Nm=-T* k k k ki

Subsituting E1* = s, . . - s yields 6 k k+j k ' N N A = E{ 2 £ aJ am (s . - s ) (s - s )} j=-N m=-N k k+i k k+m k J

which, after taking expected values inside the summation reduces to

N N

^ = J* m=ZN *k \ ^ ^ + ' ^ ^Vj S^

-E{sksk+m} +E(sk2 }]

But. E {sk+jsk+m} = 0(k-m )

and similarly for other terms. Therefore N N

j m Ak = ^ I a a [ 0 (k -m ) - 0 (j) . 0 (m) j=-N m = -N k k

0 (0)]

N Remembering that Z aJ =1, and taking non-indexed terms outside the j=-N k summation: N N N

= 0 (0) - 2 £ aj 0 (j) + I I aj am 0 (m - j)

j=-N k j=_N m= _N k k 44

Substituting in the linear approximation for the autocorrelation (equation 4-9), gives

2 N N N A a (j, - a a |j k = If- [z I k I I k k -m| 2 L j=-N j=-Nm=-N K (4-10)

For the values x and w given above, and for N=9 days,

2 2 .00003 sec /day 2

Thus in summary: The latitude at time k is estimated by N z aJ (4-3) k I t . j=-N k k+J

and the variance of zk is estimated by >

2 N 2

r + (4 7) E z £ [<] ™ «V,> \ - {[ k - \] )- N

It should be remembered that all the summations above are only over nights with data. The value of N was chosen to be nine days - based on the mini• mum filter length required to bridge all but the most extreme data gaps.

Because of the summation process involved in calculating z^, the variance of z, with respect to is much smaller than the variance of the latitude k sK, estimate "s^ for nights with data. To remedy this situation, the interpolate tion filter was applied also to observation nights - introducing an additional coefficient a.^ so that the mean value of latitude, , on the night in question, k "3k is included in the calculation of z^. Carrying out the calculations described in the summary above yield latitude and latitude variance estimates for each night. Since only the variation in latitude (as opposed to the absolute latitude given by z^), is required, the mean latitude of the station as determined by Camera (1957, pg Z13) was subtracted from each z^, k=l, , N. The resulting series will not necessarily have zero mean value because the wobble motion contains a slowly varying component that in effect makes the mean value vary with time.

Time Variable Noise Removal

The latitude variance estimates from the interpolation process provide an estimate of the measurement noise in the reduced observations.

Therefore, allow

zk = sk + nk (4-11)

> where nk is a white noise process with characteristics such that

if = nk k = j

0 if k 4 j

A filter designed to extract the noise from the signal z is shown schemati-

cally in Figure 12. The output yk of such a filter is given by

y* = X h* "k-i (4"12> The optimum filter coefficients h^ are determined by the least squares fit of the filter output y, to the desired output s . This results in the k k equation 46

Figure 12. Filter to Remove Random Noise

N

dp (k. k-m) = £ hj $ (k - j , k - m) sz j=-N

m = - N 0, . N (4-13)

where ( 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): 4- $ (k, j) = 3> (k, j) (k, j) (4-14) zz ss nn

* z (k. j) = %s (k. j) (4-15)

The cross autocorrelations

*nn

n = * (k. k) - * (k. k) k zz v ' ' ss

E { [Zk - ^k]2}

Dk

The quantity Dk is known directly from the interpolation process. The lati•

tude variation is assumed to be a stationary process. This means that

* (k. j) = (k - j) s 0

where 0 is the normal time independent autocorrelation of the latitude series.

Equations (4-14)and (4-15) can now be written

j (k, j) = (k - j) + Dk 0 (4-16) zz 0

(k, j) = 0 (k - j) (4-17) sz

Writing equation (4-13) out in full for various values of m and incorporating

the above substitutions for the autocorrelations, produces 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) + . . . + h . (N) + h D, = (0) k k 0• k 0 k K 0

• * • h" N (2N) + . . . + hg (N) + . . . + h™ (0) + h* D = (N) k 0 0 0 kN 0 48

To simplify, let -N 0(0) . 0 (-2N) ^k H 0

0(2N) 0 (0) ik •N

D 0 0 0 0 (-N) k+N

0 D 0 0 B 0/0) and n k+N-1 0 0 0 *0 D k-N 0 (N)

Then ( T + JJ ) H = B

-1 and H = ( T + U) B (4-18)

-1 where ( T + E ) ( T + H) = identity matrix.

Equation (4-18) was used to obtain the noise filter coefficients h^ for a given point k. After the filter was applied at that point, the filter was moved ten days along the data and re-calculated. Thus, a series of latitude means with ten day sampling interval is obtained. Ten days was chosen in order to be compatible with the data of the Bureau Internationale de l'Heure. Autocorre• lations were obtained from the unsmoothed data by using the Burg algorithm as described in Appendix I, and the matrix inversion was accomplished by standard IBM program SL.E. Typical results are shown in Figure 13. Also plotted in Figure 13 are the results of another technique whereby the same series is smoothed assuming a constant noise factor (equal to the average variance for the complete series) and using the Levinson inversion technique

(Robinson, 1967, page 43). Figure 13 : Output from variable noise filter (solid line) and constant noise filter (solid line with circles). The dotted line is the original series.

sO 50

This, and other non-plotted comparisons clearly indicate that the latter tech• nique is somewhat inferior to the method used. This is perhaps due to numerical instability in the Levinson inversion for a matrix so large (37 x 37).

In application, the smoothing filter length was chosen arbitrarily to be 37 days (N=18) - the only criteria being that the filter must be short enough not to cause harmful high-frequency cutoff. It should be kept in mind that only

the measurement noise has been filtered. As can be seen from the resulting time series for the various observatories (Figures 14 to 18), there is still a fairly large noise factor which can possibly be attributed to variations in air pressure, temperature, visibility and local vertical. The ten day mean values that are output from Equation (4-12) and that are plotted in Figures 14 to 18, represent the basic data for spectral analysis. CO o

Figure lit- : Basic smoothed data for Misisawa. Figure 15 : Basic smoothed data for Ukiah.

Ul ro Figure 16 : Basic smoothed data for Carloforte. Figure 17 : Basic smoothed data for Kitab. Figure 18 : Basic smoothed data for Gaithersburg. 56

CHAPTER V

FREQUENCY DOMAIN FILTERING TO OBTAIN

CHANDLER LATITUDE VARIATION

Motivation

Two of the original objectives of this thesis were:

1. To separate the two main frequency components from short records.

2. To remove the annual component without destroying information associated

with sudden phase and amplitude shifts in the Chandler motion and without

forcing the annual component to be a constant spectral delta function.

Traditional methods of harmonic analysis include prior assumptions about the frequency components being dealt with. Each component is assumed to be a perfect harmonic that does not vary in phase, amplitude or mean value with time. In addition, all the traditional methods assume that outside the known interval, the data series is either zero or cyclic. Besides the obvious inaccuracy of these extensions, they cause serious interference effects in the frequency domain that make the separation of close frequencies very difficult. Figure 19 is the Fourier transform of a perfect sinusoid of period

435 days - based on approximately ten years of data. If this sinusoid was continued to infinity in both directions, the Fourier transform would be the vertical line in the main peak of the same plot. The difference between the

Fourier transforms of the truncated sinusoid and the continuous sinusoid illustrates quite dramatically the undesirable effect of extending the known Figure 19 : Fourier transform of 10 years of a perfect sinusoid (perlod=^35 days) The dotted vertical line represents transform of an infinite sinusoid of the same frequency. data with zeroes. For the case of the Chandler and annual components of latitude variation, approximately fifty years of data are required to obtain enough separation of the frequency components so that the truncation effects on the respective peaks do not interfere too severely. For a record length of ten years, what should be two distinct frequency peaks become mired into one very broad peak. Figure 20, which represents the Fourier transform of ten years of synthetic wobble data (see Figure 22), exemplifies this undesirable phenomenon. The dotted lines represent the actual peak locations. It seems obvious that if the objectives described above are to be fulfilled, some new spectral technique is required.

Predictive Filtering

r

A method of predictive filtering developed by Ulrych et al (1973)

reduces the mutual interference problems of harmonic analysis. Essen• tially, the method is just the prediction of the data series several times in both the backward and forward direction in such a manner that, if the original series is considered to be one determination of a statistical process, the original and extended series are statistically identical. This means that no new information is added by extending the original series. The advantage of the technique is that the increased data length causes the interference effects to be greatly reduced.

ot The prediction coefficients g., j = 1, 2, N, for pre- J dieting the series z at the time of intervals 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 )

Figure 20 : Fourier transform of synthetic wobble data of Figure 22. <•£ 60'

N

Z Z k+a = A k+1 - j

N is limited to values less than the total number of data points available.

Similarly, the filter coefficients for backwards prediction, b. , are defined J N by a Z b. Z. , . . K- a ^ J k+j - 1

Ulrych et al (1973) has shown that calculating these coefficients is equivalent to applying the unit prediction filter ( a = 1) or times, each time incorporating the predicted point as the new last point of data. A proof is given in Smylie et al (1973). Therefore, only one calculation of the coefficients g? and b: is necessary to obtain a prediction for all values of o . Computation of the coefficients is simplified somewhat by the fact that for a real time series

1 1 g = b This equality rises out of the method of computation - which is j j the minimization of

N N

E 1 z 2 2 z g + (Z - Z l ( ,x! - I & ) X b ) J J I k+1 f- i k+l-j' k-1 j k-l+j l J J J J J= j=i

1 1 with respect to the coefficients g . and b . This minimization yields N + 1 •> j linear equations. Fortunately, J. Burg (1967) has developed an iterative method (called the Burg algorithm) that simultaneously produces autocorrela• tions and unit prediction coefficients for a given data series. This algorithm is described in Appendix I. Once the unit prediction coefficients have been obtained, the series is extended one point at a time in both forward and back• wards directions. The extension procedure is illustrated schematically in .61

Figure 21. Continuing in this manner, the series can be predicted to infinity

1 1 1 1 4 g3 g2 gl Step 1 x x 4- X + X 4- ZN+1

ZN-3 ZN-2 ZN-1 ZN

V original data

move filter forward one interval

1 1 1 1 l g5 g4 g3 g2 gi Step 2 x 4- x 4- x 4- x 4- x

Z Z Z Z Z N4-2 • ZN-3 N-2 N- 1 N N4-1

original data

Figure 21. Method for Predictive Extension of a Data Series.

in both directions (if one is rich enough). According to Ulrych et al (1973)

in the limit, as the series is predicted to infinity, the spectral density func•

tion of the series approaches the maximum entropy spectral estimator

(MESE). This implies that the extension adds no statistical information to

that inherent in the data. To apply this method, the data record was divided

into overlapping epochs of nearly ten years (360 points or 3600 days). The

length of prediction filter was chosen to be 250 points (2500 days) on the basis that the resolution of the annual and Chandler frequency components

requires at least 225 coefficients - according to the theoretical resolution of

the MESE technique - and that if many more than one half as many coefficients

as data points are used, then the Burg algorithm is not stable. These ten year data records were extended two times in each direction to create a time

series of nearly fifty years. To illustrate the value of this technique, the syn•

thetic signal of Figure 22 (composed to produce a signal roughly comparable to the observed one by using the time series of Figure 25 to model the Chandler variation and a perfect sinusoid of amplitude . 15 seconds to represent the annual and randomly generated noise of standard deviation .02 seconds to represent measurement noise) was extended in the prescribed manner to pro• duce the series of Figure 23. It is clear that the characteristics of the predic• ted portion of this plot are similar to the original - the main difference being that the extension appears to become single-frequency valued towards the extre• mities. The true worth of the technique is immediately obvious when the fast

Fourier transformS(FFT) of the original and extended series are compared.

See Figures 20 and 24. The transform of the original signal has one main peak: the only indication that there might be two real components is a slight bulging on the high frequency side of the main peak. By comparison, the transform of the extended series has two very well separated main peaks corresponding to the known input frequency components. Therefore, having a method that achieves the objective of clear resolution, it remains to develop a basis for filtering out the annual component and unwanted noise such that the second objective above is met.

The Effect of Phase and Amplitude Shifts on the Fourier Spectrum

Synthetic wavelets consisting of truncated sinusoids of constant period equal to 435 days, but varying in amplitude and absolute phase were added end to end to produce the function X(t) shown in Figure 25. Comparing the Fourier transform of a continuous sinusoid (as in Figure 19) and the transform of X(t) (which are superimposed in Figure 26) demonstrates the typical effect of introducing sudden phase and/or amplitude shifts. That is. Figure 22 : Synthetic latitude variation composed of 2 main frequency components (annual and Chandler) plus random noise. Figure 23 : Prediction of time series in Figure 22 to five times its original length. Figure 2k : The Fourier amplitude spectrum of the extended series (Figure 23). us o

0fN UJo"

5=H ct: CE

UJ ZDo. I— •

cr

i ~I 1 ~~r 1 1 I -1.9 46.1 98.1 148.1 198.1 ,248.1 298.1 348.1 TIME (JULIAN DAYS) (XlO1 ) Figure 25 : Chandler variation used to create the series of Figure 22. Composed of sine wavelets all of the same frequency (1/^35 days ) but each varying in phase and amplitude. Figure 26 : Fourier amplitude spectrum of Figure 25. The dashed line is the spectrum of a sinusoid with constant phase and amplitude. 0 68

the Fourier amplitude becomes much more spread over frequencies near the known central frequency. In addition, the peak shape tends to be asymmetric and the peak frequency is shifted away from the known value (1/435 days

This behaviour can be explained in general terms by visualizing the synthetic data in terms of the wavelets that compose it. Each of the wavelets can be represented as a continuous sine function multiplied by the 'boxcar' function of appropriate width. See left hand side of Figure 27. The Fourier transform of the product of two functions is the convolution of the individual Fourier transforms as seen on the right hand side of Figure 27. The breadth of the sine function (Fourier transform of boxcar function) is inversely proportional to the length of the boxcar function (or length of wavelet). Therefore, the data series composed of short wavelets will be a complex sum of broadened sine functions. This results in a Fourier amplitude spectrum that is some-* what smeared out compared to the spectrum of the continuous sinusoid. The

Fourier transform of such a series of wavelets can be determined analytically.

However, except for the case of two wavelets, the number of parameters makes the solution enormously complicated. The case of two wavelets has been examined by Fedorov and Yatskiv (1965) and they show that equal amplitude wavelets 180 degrees out of phase will produce a split spectral peak. Thus, the apparent splitting of the Chandler component mentioned previously is now attributed to a large phase shift (Wells 1972). It is apparent then, that in order to preserve phase or amplitude shift information, fre• quencies near the central Chandler frequency must be left unaltered (if possible) by the process of filtering out the annual component. Also, it is TIME DOMAIN FREQUENCY DOMAIN

co

Figure 27 : Synthesis of the Fourier transform of a truncated sinusoid. 70

obvious that by merely introducing sudden phase and amplitude shifts in a

perfect harmonic, that it is possible to cause:

(1) the Fourier spectrum of such a signal to be totally devoid of meaning

so far as harmonic amplitude and frequency information is concerned.

(2) spurious artificial effects that could be interpreted as being real - for

example, the observed splitting of the Chandler peak.

Band Reject Filtering of Annual and Noise

Theoretically, the predicted data series should consist of two main periodicities - the 435 day Chandler and the 365 day annual - plus random noise. So in order to examine the Chandler latitude variation, the annual component and noise must be extracted from the latitude signal, while keeping in mind the criteria discussed above. To accomplish this task, a zero-phase band reject filter was applied to the Fourier transform of the predicted series. Generally, three different regions of the Fourier spectrum were filtered. Narrow band filters were applied to a region near zero frequency and to the peak at 365 days. Both of these were applied so that the Fourier amplitude was reduced to the noise level rather than to zero. In addition, a wide band filter was uned to remove totally all contribu• tions (assumed to be noise) at frequencies above a period of 200 days. To accommodate the variable conditions from epoch to epoch and from observatory to observatory, the width and cutoff level of the two narrow band reject filters were chosen to fit the Fourier spectrum in question. What this filtering accomplishes is to produce a zero mean value, and to remove 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 : Effect of band-reject filtering of annual on the spectrum of the extended synthetic latitude variation (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 (JULIAN DAYS) (XlO1 )

Figure 29 : Chandler variation resulting from band-reject filtering of the synthetic data. The dashed line represents the known synthetic Chandler variation. 74 arc begins. See Figure 6. This method suffers from the major weakness that any large deviation from the expected smooth arc will produce an event; whether or not that event represents a real shift in phase, amplitude or mean value. To avoid this problem and to simplify the computing proce• dures somewhat, a purely analytic technique that yields the time-variable phase and amplitude of a quasi-harmonic function was developed. By quasi- harmonic, it is meant that the periodic signal has constant period but that the phase and amplitude are permitted to vary with time. Supposedly, the

Chandler latitude series obeys this criteria. So if the Chandler signal is denoted by the function f, it will be of the form:

f(t) = A(t) cos (wQt + 0 (t) ) (5.1)

If this can be altered to the complex form

A(t) e Mwot+ 0(t) >

- iw t then simple multiplication by e ° will yield a complex function of amplitude

A( t) and phase 0 (t) . Since

cos ( 9 - H_ ) = sin 0 2

Equation 5. 1 yields

f(t 1 ) = A ( t) sin (w t + 0 (t) ) " JL w wo °

Therefore,

i(w t+0(t f(t) + if(t- -JL- ) (t) o > > 2 wQ = A e and multiplying by e 1Wo' 1 w fc i0(t) (f (t) + i f (t - __0 ) ) e" o = A (t) e (5. 2) 2 wQ

Thus, taking the amplitude and phase of the left hand side of this equation gives the time-variable amplitude and phase of the signal f. Application of this simple computation to test data yields excellent results. The resultant phase and amplitude time variations for the signal of Figure 25 are plotted as the solid lines in Figures 30 and 31. The only deviations from the actual phase and amplitude occur at the breaks, which should appear as straight vertical lines. The actual break corresponds to the initial deviation from a horizontal line, and the large fluctuations immediately following are caused by the averaging characteristic of the method. Next, the demodulator was applied to the filtered synthetic Chandler series of Figure 29. The results (dashed lines in Figures 30 and 31) are surprisingly good. The breaks in both phase and amplitude are immediately obvious. Also, it can be seen (here and in all the examples processed) that the phase is the more reliable indicator of a break.

That is, the band reject filtering causes unpredictable distortions of the amplitude variation if large sudden shifts are present. The phase variation seems to be free of these effects. I 1 1 1 1 1 1 1 -1.9 48.1 98.1 148.1 198.1 ,248.1 298.1 348 3 TIME (JULIAN DAYS} (XlO ) lire 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. in

CJ CO

O CM LU CO

V

7- MO * A • _l Q_ cr

o ~1 •3.9 48.1 9B.1 148.1 198.1 .248.1 348.1 J 298.1 TIME (JULIAN DAYS) (X10 )

Figure 31 J Amplitude variation of the synthetic Chandler. Solid and dashed lines represent the filtered and actual versions respectively. 78

CHAPTER VI

THE PHASE AND AMPLITUDE RESULTS FOR THE 5 I. L. S. OBSERVATORIES

After band reject filtering to isolate the Chandler component of

latitude variation, one obtains a set of overlapping ten year time series for

each observatory. In order to produce a continuous Chandler series, the over•

lapping portions were averaged. Figures 32 to 36 are the resulting Chandler latitude variations for the indicated observatories. There are two main features of these results that are common to every observatory:

1. A shift in mean value occurring at Julian day 2427800 (1935.0)

which corresponds to a change in star catalogues used by the

observatories. The shift is close to +.02" for each of the

three observatories functioning at this time.

2. A large increase in amplitude beginning at approximately Julian

day 2429300 (1939.0).

Both of these features indicate that the filtering technique is at least capable of retaining information about fairly rapid amplitude shifts. An even more en• couraging factor is that the overlapping portions of the several ten year Chandler series at a particular station are very similar prior to being averaged. That is, two completely independent applications of the filtering process to different data intervals yield, in the overlapping region, two Chandler variation series that are almost identical. This consistency implies that the filtering technique is producing a result close to some definitive Chandler motion. Figure 32 : Chandler latitude variation for Mislsawa. to o

Figure 33 : Chandler latitude variation for Ukiah U3 a

(=}

Figure 34 : Chandler latitude variation for Gaithersburg. 00 Figure 35 t Chandler latitude variation for Kitab (£3 O

Figure 36 : Chandler latitude variation for Carloforte. 84

In order to examine the Chandler variation in more detail, it is

advantageous to look at the phase and amplitude of the periodic motion as a

function of time. This is accomplished by the method described in the previous

section. The amplitude of the Chandler motion for each of the five observatories

is plotted in Figure 3 7. Similarly, the phase is plotted in Figure 38. It should

be noted that the phase has been adjusted to the Greenwich meridian by adding

the longitude of the observatory to the output of the demodulator, and that the

zero point in time was chosen to be Julian day 2423323. This means that a

latitude signal at Greenwich would have zero phase if the periodic signal reached

a maximum on day 2423323 or on any integer number of periods before or after•

wards. The Chandler period was chosen to be 437. 0 days based on the value

that makes the phase constant in the last portion of the time interval (after J.D.

2429700). This criterion for choosing the exact value of the period is somewhat

arbitrary (excepted value = 435 days), and is adopted primarily because of the

relative consistency and flatness of the various phase curves during this time

interval.

It is immediately obvious that some major changes in both the

phase and amplitude of the Chandler latitude variation occurred in the interval

from 1922. 7 to 1949.0. Perhaps the most encouraging result from Figures 37

and 38 is the consistency from station to station, especially for phase variation.

To further simplify interpretation, both the phase and amplitude

variations were visually averaged over the number of observatories. At the

same time, a visual estimate of the maximum possible error in the average was obtained by including all the signal except the extremities of large peaks 242331.1 242431.1 242531.0 24263^1.0^ 242730.^ ^^0.8^^^.8 243030.7 243130.6

Figure 3? : Chandler amplitude variation for all 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 2431311 TIME (JULIAN DAYS)

39 CO Figure : Amplitude variation averaged over number of observatories. 00 o

^—. in CO in" <

< o

10 rr (Nl < >

LU CO < X CL

m o T i i T —I 2423311 2424311 2425311 2426311 2427311 2428311 2429311 2430311 2431311 TIME (JULIAN DAYS )

Figure 40 : Phase variation averaged over a number of observatories. sooo 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, 1928 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 event (Jan. 16, 1931 Feb. 2, 1931 7.9 Aug. 10, 1931 7.9 Oct. 13, 1931 7.9 91 92

years) and marks the beginning of an active time. The most obvious conclusion

one could make from these observations is that abrupt shifts in the Chandler

motion are associated with the onset of earthquake activity. This conclusion

of course ignores the cases where a quiet era is not ended by an abrupt change

in the Chandler motion, and also the shifts (though considered less likely to be

real) that do not correspond to the same circumstances. Another question

arises: What mechanism could possibly cause such large, rapid phase shifts?

According to Guinot (1972), they are not linked to any geophysical event. Since

caution is the watchword, there won't be any wild guesses here about this

matter. The correspondence between constant phase and rapid change in ampli•

tude is difficult to explain - it might be surmised that the excitation mechanism

is more efficient when the phase is at a stable value. One thing is certain from

this work: earthquakes could not be detected even if they did excite the Chandler

motion. That is, the noise level is approximately three times as large as the

largest possible effect of a major earthquake (calculated to be about one foot or

.01" for the 1964 Alaska earthquake - Smylie and Mansinha, 1972).

An important result from most studies of the Chandler wobble is

an estimate of the damping time or alternately of the Q factor. Estimates of

these quantities are normally obtained by fitting the smoothed power density

function (proportional to the squared Fourier transform of the data) to functions

like

1 / ( ) - (w2 - w 2) 2 + 4w2 ?^2 (6. 1)

2 12 or 1 / (w - wQ) + ( - ) (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 Figure kJ, x Fourier amplitude spectrum of simplified undamped Chandler motion 95

factor of the actual wobble is somewhat higher than is commonly thought.

Conclusions

1. The optimum value for the period of the Chandler wobble is 437 days

based on the criteria that the time variation of phase be minimized.

2. The value of the Q factor for wobble is almost certainly higher than pre•

vious estimates. Its exact value is extremely uncertain.

3. Abrupt shifts in the phase of the Chandler wobble are possibly associated

with the onset of earthquake activity.

4. The cause of these phase shifts and also of large variations in the Chandler

amplitude is unknown. However, the variations are probably too large to

be caused by the presently estimated effect of earthquakes.

5. The suggested correlation between individual earthquakes and changes in

the Chandler wobble could still be true though there is no evidence to

support such a relation. 96

BIBLIOGRAPHY

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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.

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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

Smylie, D.E., G.K.C. Clarke, and L. Mansinha, Deconvolution of the pole path, 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.

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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 12 n

is minimized. This produces the set of equations:

0ff 0ff gj (0) + g2 (-D ++ gn0ff(l-N) = 0ff(l)

N gj 0ff (N-1) + + gn 0ff (°) = % < ) where 0££ ( \ ) represents the stationary autocorrelation of lag \

In addition, the error power

{ Kl } for an N point predictor can be written 100 r l E { l J P (0) )= N+ 1 = ^ff N

j=l J

Combining this equation with the previous set yields the matrix equation:

N

0 1 1 ff(O) ^f^" ) ?>ff(-N) N+1 0 0 0 0 0 0 0 ff(N) ff(N-l) ff(O) N 0 with - g 1 N 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 11 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

p T. 1 where the first subscript N a *N+1 NI 0 on the Ni refers to the a = 0 N2 order of the iteration a 0 N3 • 0 • 0 a ' NN 0 _ 101

It is assumed that all the elements are known. The N + 1 equation is then written

T 1 N+l N+2 a 0 aN+l , 1 = 0 N+l, 2

* 0 • 0 a ' 0 N+l, N+l

This can be written

T 1 0 a* N+l a N. 1 N, N N+l, N+l

N.N a* N. 1 0 1

' N+l N+l 0 0 0 + 0 N+l, N+l 0 0 0 0 LN+1 which implies the relations a a + a N+l, 1 N, 1 N+l, N+l N, N

a * N,N + a a N+l.N N+l, N+l N, 1 (A-l) N+l and 0 N+ 0ff (N+l) I N+l.j ff( l-J> j = l 102

To solve all these unknowns, it is only necessary to calculate the value a , „,,• This is accomplished by minimizing the estimator for P N+lnvr. , N+l c j 6 N+2

M-N-l 1 p = V iF I N+2 2(M-N-1) ^ I i+N+l' j=l with respect to the real and imaginary parts of a after having sub- N+l.N+l 6

stituted for the N+l j *=** ^» from the equations (A-l). Normally, the estimate of error power for backward prediction is also included in the minimization. When this is done, minimization yields: M-N-l * * 2 + [VN+1+ ,l VN'""' N. VJ a J N N h N.lW--- N+l, N+l M-N-l

I [If + Q f +....+ a f I 1 1 j=1 L j+N+1 N, 1 j+N N.Nj+1

a ' " " *+ IN, IN * j -H1N J N, N j +N J a * a * IC l~* 1 ,f + f + + f j N, 1 j+l *" N.N j+N> J

Thus, all the elements of step N+l are known, and one can proceed in exactly the same manner to step N+2, and so forth until the desired number of coeffi• cients is obtained.

A print out of the Fortran subroutine to perform a single itera• tion of the above procedure is included below.

In this program:

M = number of data points

NN = order of iteration - produces • , i = 1, . . . NN r NN, l F = data series 103

G = prediction filter coefficients on output

PEF AND PER are storage arrays that simplify calculations in

the next iteration and must be declared in the main program.

SUBROUTINE 6PEC(M,NN,F,G,PEF,PER ) REAL *8 G (M ) ,PEF(M ), PERIM) ,SN,SD,H(2000) REAL F(M) N=NN-1 IF(N.NE.O) GO TO 11 DO. 12 J = 1» M . . PEF(J)=0D0 12 PERU ) =0D0 1 1 SN=0D0 SD=0D0 JJ=M-N-1 DO ,2 J = l ,JJ _ 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(NN)=SN/SD IF(N.EU.0)G0 TO 3 DO 4 J=1,N K=N-J+1 4 H(J)=G(J)+G(NN)*G(K) DO 6 J=1,N 6 G(J)=H (J ) JJ=JJ-1 3 DO 10 J=1 , JJ PER(J)=PER(J)+G(NN)*PEF(J)+G(NN)#F(J+NN) 10 PEF(J)=PE F(J+1)+G(NN)*PER(J+ L)+G(NN)*F(J+l) RETURN END .