Direct measurement of diurnal polar motion

by ring laser gyroscopes

K. U. Schreiber, A. Velikoseltsev, and M. RothacherForschungseinrichtung Satellitengeodasie, Technische Universitat Munchen, Kotzting, Germany

T. KlugelBundesamt fur Kartographie und Geodasie, Fundamentalstation Wettzell, Kotzting, Germany

G. E. Stedman and D. L. WiltshireDepartment of Physics and Astronomy, University of Canterbury, Christchurch, New Zealand

Received 14 September 2003; revised 29 March 2004; accepted 19 April 2004; published 23 June 2004.

[1] We report the first direct measurements of the very small effect of forced diurnal polarmotion, successfully observed on three of our large ring lasers, which now measure theinstantaneous direction of Earth’s rotation axis to a precision of 1 part in 108 whenaveraged over a time interval of several hours. Ring laser gyroscopes provide a new viabletechnique for directly and continuously measuring the position of the instantaneousrotation axis of the Earth and the amplitudes of the Oppolzer modes. In contrast, the spacegeodetic techniques (very long baseline interferometry, side looking radar, GPS, etc.)contain no information about the position of the instantaneous axis of rotation of the Earthbut are sensitive to the complete transformation matrix between the Earth-fixed andinertial reference frame. Further improvements of gyroscopes will provide a powerful newtool for studying the Earth’s interior. INDEX TERMS: 1210 Geodesy and Gravity: Diurnal and

subdiurnal rotational variations; 1294 Geodesy and Gravity: Instruments and techniques; 3210 Mathematical

Geophysics: Modeling; KEYWORDS: diurnal polar motion, ring laser, Sagnac effect

Citation: Schreiber, K. U., A. Velikoseltsev, M. Rothacher, T. Klugel, G. E. Stedman, and D. L. Wiltshire (2004), Direct

measurement of diurnal polar motion by ring laser gyroscopes, J. Geophys. Res., 109, B06405, doi:10.1029/2003JB002803.

1. Diurnal Polar Motion

[2] The Earth’s angular velocity vector varies in time,both in direction and magnitude. Changes in magnitudecorrespond to a variation of the length of day (LOD) withrespect to atomic clocks, amount up to a few millisecondsand consist mainly of various long-period and seasonalcomponents from solid Earth tides and the interaction withatmosphere and ocean and small diurnal and semidiurnalterms from ocean tides.[3] The direction of the Earth’s rotation axis also varies

with respect to both space- and Earth-fixed referencesystems. The principal component with respect to theEarth-fixed frame is the well-known Chandler wobble, withan amplitude of 4–6 m at the poles and a period of about432 days. This is a free mode of the Earth, i.e., it would stillbe present in the absence of the external gravitational forces.It is believed that the Chandler wobble would decay due todissipative effects in the Earth’s interior, were it not con-tinually excited by seismic activity and by random noise ofthe atmosphere.[4] The Chandler wobble is overlaid by daily variations

whose amplitudes are an order of magnitude smaller, some40–60 cm at the Earth’s surface (see Figure 1) [McClure,

1973; Frede and Dehant, 1999]. These Oppolzer terms arisefrom external torques due to the gravitational attraction ofthe Moon and Sun. Since the Earth is an oblate spheroidwith an equatorial bulge which is inclined to the plane of theecliptic, the net gravitational torque of the Moon and Sun ondifferent parts of the Earth’s surface does not exactly cancelout as it would if the Earth were a perfect sphere.[5] In an Earth-fixed reference system, such as that of a

ring laser fixed to the Earth’s surface, the forced retrogradediurnal polar motion is best viewed as a principal mode, theso-called ‘‘tilt-over mode’’ (K1), with the period of exactlyone sidereal day (23.93447 hours), whose amplitude ismodified as the angles and distances between the Earth,Moon, and Sun vary over the course of their orbits. Thecomplete spectrum of nutation modes can be understood asthe beat frequencies of the tilt-over mode with frequenciescorresponding to relevant orbital parameters: half a tropicalmonth, half a tropical year, the frequency of perigee, etc.The beat periods are clustered around one sidereal day.The O1 and P1 modes, with beat periods of 25.81934 and24.06589 hours, have the largest amplitudes after the K1

mode and arise from the change in angle between the Earth’sequatorial bulge and the Moon and Sun, respectively.(See Moritz and Mueller [1987] for basic theoreticaldetails.)[6] The spectrum of Oppolzer terms has the beauty that

since it arises from external gravitational torques, the

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, B06405, doi:10.1029/2003JB002803, 2004

Copyright 2004 by the American Geophysical Union.0148-0227/04/2003JB002803$09.00

B06405 1 of 5

frequencies are known precisely but the amplitudes,which depend on the properties of the Earth such as itsinertia tensor, elasticity, and liquid core are not soprecisely understood and could potentially reveal muchof geophysical interest. To date models of the Earth’sinterior have been built up which are broadly consistentwith very long baseline interferometry (VLBI) measure-ments of the instantaneous orientation of the Earth’srotation axis. However, direct routine measurements ofthe amplitude of the Oppolzer terms with ring laserswould open up a new field of geophysical studies. Inthis paper we report a first step in that direction.

2. Effect of Polar Motion on Ring LaserGyroscopes

[7] Ring lasers measure absolute rotation [Stedman,1997]. In the experimental setup, two laser beams propagatein opposite directions around a closed path. If the instru-ment rotates, the effective path length is slightly shorter forthe counterrotating beam. In an active laser cavity, as is thecase for our instruments, lasing is achieved when an integralnumber of wavelengths circumscribe the ring perimeter.Since the path length is slightly different for the corotatingand the counterrotating beams, the lasing frequencies arealso slightly different in each case, and the beat frequency ofthe two laser beams, the Sagnac frequency, is readilymeasurable. For an Earth-fixed rectangular cavity of perim-eter length P and area A, with normal n the Sagnacfrequency is given by

df ¼ 4A

lPn �6; ð1Þ

where 6 is the instantaneous angular velocity vector(referred to an inertial frame) and l is the wavelength ofthe laser beams in the absence of rotation. The Sagnacfrequency will vary with: (1) changes in the instantaneousposition of the Earth’s rotation axis and its angularvelocity, which alter 6; (2) tilts from solid Earth tidesand tidally induced ocean loading changes, which alterthe local normal, n; (3) any changes to the scaling factorA/P, which could result from local thermal or pressureinduced variations in the dimensions of the ring laser; and(4) any changes in the refractive index seen by eachbeam in the laser cavity, as might result from changes ingas composition.[8] Although different mechanisms contribute to

changes in df, their individual effects can be separated.First, unwanted effects from thermal expansion and thelike can be minimized by suitably isolating the ring lasersand by building ring lasers with the largest possible ratioA/P to increase their sensitivity. Likewise variation inrefractive index is an engineering design question, andwhile problems such as tiny cavity leaks may give slowdrifts in df, they would not have strongly periodicsignatures. Finally, tilts from solid Earth and ocean tidescan be readily distinguished from diurnal polar motion,by tiltmeter measurements on the ring laser.[9] Table 1 summarizes the performance of some of

the largest existing ring lasers. The gyroscope sensitivitydata represent the experimentally obtained instrumentalresolution (averaging time of 3 hours), while the maxi-mum amplitude of diurnal polar motion (DPM) is mea-sured at each ring laser site. While the longitude of aring laser location determines the phase of the polarmotion signal, the latitude defines the projection of therotation vector onto the ring laser normal and thereforedetermines the amount of the amplitude of the polarmotion signal that is mapped into the ring laser measure-ments by equation (1).

3. Details of a Diurnal Polar Motion Model

[10] McClure [1973] carried out a detailed investigationof diurnal polar motion, using a purely elastic deformableEarth as the basic theoretical model. The additionalcorrection for liquid core effects has been shown toproduce no significant differences for the motion ofthe rotation axis [Wahr, 1981; Brzezinski, 1986]. Adopt-ing body-fixed coordinates, the instantaneous position ofthe rotation axis is determined as a linear superpositionof sinusoidal Oppolzer modes. By convention, the modesare expressed in terms of fundamental arguments corres-ponding to particular orbital parameters. Specifically, the

Figure 1. Behavior of the Earth’s instantaneous axis in anEarth-fixed frame (with respect to the ConventionalInternational Origin) in the period of January to June2002. The large circle is the Chandler wobble. Super-imposed are the smaller circles of the Oppolzer terms:diurnal variations whose dominant amplitude varies overhalf a month.

Table 1. Sensitivity of Some of Our Ring Lasers and Theoretical

Maximum Amplitude of Diurnal Polar Motion Effect at These

Instruments

Ring Laser Area, m2 fSagnac, HzGyroscope

Sensitivity dW/W DPMmax dW/W

C-II 1 79.4 1 � 10�7 1 � 10�7

G 16 348.6 1 � 10�8 9.8 � 10�8

UG1 367 1512.8 3 � 10�8 1 � 10�7

B06405 SCHREIBER ET AL.: DIURNAL POLAR MOTION DETECTION BY RING LASERS

2 of 5

B06405

longitude diurnal variation, Dj, and the obliquity diurnalvariation, D�, are given by

Dj ¼Xi

�Ai sin fM þXj

NijFj

!ð2Þ

D� ¼Xi

Ai cos fM þXj

NijFi

!; ð3Þ

respectively, where fM is the Greenwich mean sidereal hourangle, Ai are amplitudes, and Nij are integer coefficients (asgiven by Brzezinski [1986]) which multiply the fundamentalarguments like, for example,

F1 � l ¼ 134:96340251 þ 171; 791; 592:217800t þ 31:879200t2

þ 0:05163500t3; ð4Þ

the mean anomaly of the Moon. Similar expressions areobtained for the mean anomaly of the Sun, the differencebetween the mean longitude of the Moon and the meanlongitude of the ascending node of the lunar orbit, the meanelongation of the Moon from the Sun, and, finally, themean longitude of the ascending node of the Moon. In allthese expressions, the time t is measured in Julian centuriesof 36,525 days of 86,400 s since J2000. Furthermore, theGreenwich mean sidereal hour angle fM relates to theGreenwich Mean Sidereal Time (GMST) as follows:

fM ¼ GMST2p

86400þ pþ 2pdu; ð5Þ

GMST ¼ 24110:548410 þ 8; 640; 184:8128660Tu þ 0:0931040T2u

� 6s:2� 10�6T3u ; ð6Þ

where Tu = du/36525, with du the number of days elapsedsince 1200 UT1 1 January 2000. Finally, the longitude l ofeach ring laser must be added to the Greenwich meansidereal angle (5) to synchronize observations.[11] The tables given by McClure [1973] and Brzezinski

[1986] provide up to 160 values of diurnal polar motionamplitudes and coefficients. We will keep only the largest 6coefficients, as all the remaining coefficients together con-tribute less than 1 milliarc second (mas) to the amplitude ofpolar motion and are not yet within the resolution obtainedwith a gyroscope. Calculations with these assumptionssuperimposed on the official International Earth RotationService (IERS) C04 polar motion series [Dick and Richter,2002] were used to compute Figure 1. This C04 series refersto the Celestial Intermediate Pole (CIP) [Capitaine et al.,2002] defined to be free of diurnal polar motion terms, asopposed to the instantaneous axis of rotation. One canclearly see the diurnal circles of the rotation axis as itprogresses through a larger cycle of the Chandler wobble.The resulting ‘‘latitude oscillations’’ have two main com-ponents with a periodicity of nearly 24 hours and approx-imately 14 days. The maximum amplitude represents 0.02 sof arc at the pole. We wish to point out that besides the herediscussed forced diurnal polar motion there are many more

effects that have subdaily or daily contributions to Earthrotation (e.g., from atmosphere, ocean). However, at thecurrent level of performance of our ring laser gyroscopethey are not yet observable. This also applies for length ofday variations.

4. Direct Observation of Polar Motion by LargeRing Lasers

[12] The effect of Earth tides and ocean loading as smallperiodic variations have already been successfully identifiedin the time series of the Sagnac frequency of the small C-IIring laser [Schreiber et al., 2003] located in an undergroundcavern in Christchurch (New Zealand). An unambiguousdetection of polar motion has now been provided by ourother ring lasers, G and UG1, the results from G being themost accurate and definitive. G is a semimonolithic squareHeNe ring laser of 16 m2 area constructed from Zerodur andlocated in a thermally stable underground laboratory at theFundamentalstation Wettzell (Germany) [Schreiber et al.,2001]. The G ring laser was operated for most of the time inthe year 2002. The most stable conditions ever obtainedfrom any of our large ring lasers were met between 24 Julyand 27 October, corresponding to the days 205 and 300 ofthe year 2002. The overall drift of the raw data over theentire time is as low as 2 mHz or expressed as an relativeerror Df/f < 6 � 10�6 over 95 days, to our knowledge aworld record for gyroscope stability.[13] A section of the observed time series, data taken

between days 270 and 290, is shown in Figure 2. A constantvalue of 348.635 Hz has been subtracted from each mea-surement. In addition to the raw ring laser data one can findthe corresponding tilt signatures from the body tides[Agnew, 1997] and the diurnal polar motion [McClure,1973; Brzezinski, 1986], both converted to the respectivevariations of the Sagnac frequency according to equation (1)and offset by 100 and 180 mHz for better illustration in thediagram. The residual data set is displayed offset by�100 mHz after the corrections were applied and shows agood agreement between the measurements and the models

Figure 2. Time series of a section of the Sagnac frequencytaken by the G ring laser. The computed contributions of theEarth body tides and diurnal polar motion to the Sagnacfrequency are shown offset by 100 and 180 mHz,respectively, at the top. The resultant time series with bothcorrections applied is shown offset at the bottom.

B06405 SCHREIBER ET AL.: DIURNAL POLAR MOTION DETECTION BY RING LASERS

3 of 5

B06405

adopted. What remains is the signature from instrumentaldrift and some transient perturbations of still unknowngeophysical origin.[14] We have computed power spectra of ring laser

measurements from various available time series of differentlengths of 3 different ring lasers, C-II, G, and UG1. Figure 3shows the result. There are two groups of signals evident inthe spectrum of the G ring, one at a period around 12 hours25 min and one at a period of around 1 day. In the lattergroup we can identify the K1 and O1 mode contributions todiurnal polar motion. Their periods of 23.9345 and25.8195 hours [Frede and Dehant, 1999] correspond tofrequencies of 11.606 and 10.758 mHz. Our measurementsyield frequencies of 11.665 and 10.754 mHz, which agreewell with theory within the spectral measurement resolutionof 129 nHz. The P1 mode also has an expected amplitudewell within current resolution, but since its period is veryclose to that of K1, longer time series are required toseparate out its peak in the spectrum.[15] The group of higher frequencies is due to variations

of the orientation of the ring laser on the Earth surfaceinduced by solid Earth tides. Again the frequencies of23.204 and 22.293 mHz are within the resolution of theexpected S2 and M2 tides signals. The Earth tides contributelittle to the diurnal terms because north directed diurnal tidaltilts are close to zero in midlatitudes.[16] The best available time series of UG1 and C-II are

much shorter. Therefore the resolution of their spectra is notso high. In addition, C-II shows a significant masking of thediurnal polar motion signal, probably due to backscatterinduced frequency pulling as an instrumental artefact[Schreiber et al., 1998]. Furthermore, the power spectraldensities obtained for daily polar motion and solid Earthtides as measured with the C-II ring laser are about oneorder of magnitude larger than expected. By contrast, forUG1 and G this is not the case. We believe that this is

related to the backscatter induced pulling of the Sagnacfrequency which is strong in C-II but negligible for UG1and G. Because of the long timescale, instrumental driftfrequencies below 5 mHz (G) or 8 mHz (UG1) were fully cutoff by the application of an 8th degree Butterworth high-pass filter. However, one can clearly identify diurnal polarmotion signals and solid Earth tides in all three instruments.The tidal signal is stronger at Christchurch due to localocean loading on Banks Peninsula.[17] The larger a ring laser in a given location is, the

higher is its sensitivity to such small signals. Figure 4 showsa time series from the 367 m2 UG1, which we believe is thelargest working ring laser gyroscope in the world [Dunn etal., 2002]. It dramatically shows one of the advantages ofring lasers, namely, the very high time resolution of themeasurements, making this technique very promising forthe study of subdaily variations in Earth rotation. Inaddition, continuous observations are possible over longtime intervals. In contrast, gyroscopes are local sensors, andthe data may contain contributions from local effects whichare not yet understood. Indications for this are apparent inFigure 4 as well.[18] Since the raw data of the UG1 ring laser show far more

drift than the G ring laser, the data were band-pass-filteredwith cutoff frequencies at 5 and 40 mHz. The computed polarmotion signal was treated in the same way in order to avoidartefacts from phase shifts caused by the filter process, and issuperimposed on the measurement data in Figure 4. The dataset has been corrected for Earth tides and ocean loading[Agnew, 1997]. Local disruptions of the ring laser setup havenot been reduced from the data set. Figure 4 shows theimpressive sensitivity of modern large ring laser gyroscopesapplied to the field of geophysics.

5. Summary

[19] The sensitivity and performance of large ring lasershave improved significantly in recent years with the con-struction of the C-II, G0, G, and UG1 gyroscopes. Today it

Figure 3. Power spectra of ring laser time series fromC-II, G, and UG1. The expected contributions from solidEarth tides and contributions to polar motion from the K1

and O1 mode are clearly present in all measurements. Thespectrum of G reveals the most details because the data setis much longer and the instrument is mostly free frombackscatter and drift.

Figure 4. An example of a time series of a measurementof diurnal polar motion carried out on the UG1 ring laser.Because of slow instrumental drift, the data were band-passfiltered. The computed theoretical polar motion signal wastreated similarly and is superimposed on the Sagnacfrequency measurements. The data set was also correctedfor Earth tides and ocean loading.

B06405 SCHREIBER ET AL.: DIURNAL POLAR MOTION DETECTION BY RING LASERS

4 of 5

B06405

is possible to measure variations in the location of therotational pole of the Earth to within a few centimeters.As our instruments progressively improve in stability weexpect evidence of further geophysical signals with periodsof well over a day in the future. Since both our ring laserlaboratories in Germany and New Zealand are nearly atantipodal points, we unfortunately cannot yet distinguishthe different effects of the components Dj and D� of thepolar motion. This will require at least one additional high-quality ring laser site displaced 90� in longitude from ourcurrent laboratories.[20] It is important to note that over the past 30 years the

theoretical model of forced diurnal polar motion has beendeveloped and is used to reduce these contributions fromVLBI measurements. There are still some uncertaintiesremaining, since the theoretical models use some simplifi-cations to account for a deformable Earth. Large ring lasergyroscopes provide a new and independent technologywhich can measure the amplitudes and frequencies of theforced modes of Earth rotation directly and monitor polarmotion to unprecedented high temporal resolution. Thiseffectively provides a direct probe of the Earth’s momentof inertia. Therefore we expect that future improvements inring laser technology will lead to quantitative improvementsin the nutation models themselves and provide a new toolfor studying aspects of the Earth’s interior.

[21] Acknowledgments. These combined ring laser results werepossible because of a collaboration of Forschungseinrichtung Satelliten-geodasie, Technische Universitat Munchen, Germany, University of Canter-bury, Christchurch, New Zealand, and Bundesamt fur Kartographie andGeodasie, Frankfurt, Germany. University of Canterbury research grants,contracts of the Marsden Fund of the Royal Society of New Zealand, andalso grants from the Deutsche Forschungsgemeinschaft (DFG) are grate-fully acknowledged.

ReferencesAgnew, D. C. (1997), NLOADF: A program for computing ocean-tideloading, J. Geophys. Res., 102, 5109–5110.

Brzezinski, A. (1986), Contribution to the theory of polar motion for anelastic earth with liquid core, Manuscr. Geod., 11, 226–241.

Capitaine, N., et al. (Eds.) (2002), Proceedings of the IERS Workshop onthe Implementation of the New IAU Resolutions, IERS Tech. Note 29,Int. Earth Rotation Serv., Cent. Bur., Frankfurt am Main, Germany.

Dick, W. R., and B. Richter (Eds.) (2002), IERS Annual Report 2001,123 pp., Int. Earth Rotation Serv., Cent. Bur., Frankfurt am Main,Germany.

Dunn, R. W., D. E. Shabalin, R. J. Thirkettle, G. J. MacDonald, G. E.Stedman, and K. U. Schreiber (2002), Design and initial operation of a367 m2 rectangular ring laser, Appl. Opt., 41(9), 1685–1688.

Frede, V., and V. Dehant (1999), Analytical versus semi-analytical determi-nations of the Oppolzer terms for a non-rigid Earth, J. Geod., 73, 94–104.

McClure, P. (1973), Diurnal polar motion, GSFC Rep. X-529-73-259, God-dard Space Flight Cent., Greenbelt, Md.

Moritz, H., and I. I. Mueller (1987), Earth Rotation: Theory and Observa-tion, Ungar, New York.

Schreiber, K. U., C. H. Rowe, D. N. Wright, S. J. Cooper, and G. E.Stedman (1998), Precision stabilization of the optical frequency in a largering laser gyroscope, Appl. Opt., 37(36), 8371–8381.

Schreiber, K. U., A. Velikoseltsev, T. Klugel, G. E. Stedman, andW. Schluter (2001), Advances in the stabilisation of large ring lasergyroscopes, paper presented at the Symposium Gyro Technology, Univ.of Stuttgart, Stuttgart, Germany.

Schreiber, K. U., T. Klugel, and G. E. Stedman (2003), Earth tide and tiltdetection by a ring laser gyroscope, J. Geophys. Res., 108(B2), 2132,doi:10.1029/2001JB000569.

Stedman, G. E. (1997), Ring-laser tests of fundamental physics and geo-physics, Rep. Prog. Phys., 60, 615–688.

Wahr, J. M. (1981), The forced nutations of an elliptical, rotating, elasticand oceanless earth, Geophys. J. R. Astron. Soc., 64, 705–727.

�����������������������T. Klugel, Bundesamt fur Kartographie und Geodasie, Fundamentalsta-

tion Wettzell, D-93444 Kotzting, Germany. (kluegel@wettzell.ifag.de)M. Rothacher, Technische Universitat Munchen, Forschungseinrichtung

Satellitengeodasie Arcisstr. 21, D-80333 Munchen, Germany. (markus.rothacher@bv.tu-muenchen.de)K. U. Schreiber and A. Velikoseltsev, Technische Universitat Munchen,

Forschungseinrichtung Satellitengeodasie Fundamentalstation Wettzell,D-93444 Kotzting, Germany. (schreiber@wettzell.ifag.de; alex@wettzell.ifag.de)G. E. Stedman and D. L. Wiltshire, Department of Physics and

Astronomy, University of Canterbury, Private Bag 4800, Christchurch,New Zealand. (geoffrey.stedman@canterbury.ac.nz; david.wiltshire@canterbury.ac.nz)

B06405 SCHREIBER ET AL.: DIURNAL POLAR MOTION DETECTION BY RING LASERS

5 of 5

B06405