Pure Appl. Geophys. Ó 2017 The Author(s) This article is an open access publication DOI 10.1007/s00024-017-1552-8 Pure and Applied Geophysics

On the Influence of Known Diurnal and Subdiurnal Signals in Polar Motion and UT1 on Ring Laser Gyroscope Observations

1 1,2 MONIKA TERCJAK and ALEKSANDER BRZEZIN´ SKI

Abstract—The ring laser gyroscope (RLG) technique has been Nevertheless, it is of great importance to study all of investigated for about 20 years as a potential complement to space them in detail, as knowledge about variations in Earth geodetic techniques in measuring variations of Earth rotation. The technique is of great interest, especially in the context of moni- rotation gives information about the phenomena toring rapid changes of rotation with sub-daily resolution. In this causing them and helps in studying Earth’s interior. paper, we review how the known high frequency signals in Earth Moreover, parameters of Earth rotation define the rotation parameters, including the so-called diurnal polar motion, diurnal and semidiurnal ocean tide effects in polar motion and UT1 instantaneous orientation of the Earth in space, which and librations, prograde diurnal in polar motion and semidiurnal in is necessary for practical purposes, like transforma- UT1, contribute to the RLG observable, the Sagnac frequency. Our tion between terrestrial and celestial reference results suggest that at the current accuracy level of the technique, the signals coming from diurnal polar motion and ocean tides frames, space navigation, or point positioning. should be taken into account in analysis while the influence of Variations in Earth rotation can be observed by libration can be neglected. We also point out that the contributing space geodetic techniques, like very long baseline signals are superimposed upon each other and can hardly be sep- interferometry (VLBI), global navigation satellite arated from the data from a single instrument. Our computations are done taking parameters of the RLG at the Wettzell Observatory systems (GNSS) or satellite laser ranging (SLR), but in Germany. However, we also consider how the strength of a recently also by a strictly ground-based technique particular signal depends on the geographic location of a hori- which is the ring laser gyroscope (RLG). The RLG has zontally mounted instrument. Finally, we discuss the relationship between the geographical location and terrestrial orientation of been investigated for about 20 years as a potential RLG and its consequence for the observed Sagnac signal. complement to the space geodetic techniques in the determination of Earth rotation parameters (ERP). Its Key words: Earth rotation, ring laser gyroscope, Sagnac fre- quency, diurnal polar motion, ocean tides, libration. potential has been noticed by Chao (1991) and studied by, for instance, Schreiber et al. (2004), Stedman et al. (2007). Additionally, Mendes Cerveira et al. (2009) and Nilsson et al. (2012) reported the advantages of 1. Introduction combining RLG data with VLBI observations. Although RLG data are considered mainly for inves- The Earth rotation vector continuously undergoes tigation of high-frequency variations in ERP, variations due to, e.g. the gravitational attraction of Schreiber et al. (2011) demonstrated the feasibility of the Moon and the Sun or processes in the liquid detecting the Chandler and annual wobbles from these envelopes of the solid Earth, including the atmo- observations. Nevertheless, the technique constantly sphere, the oceans and the outer core. These undergoes improvement so that today the RLG fluctuations differ in time-scales and magnitudes. observations are more accurate than a few years ago. Consequently, its potential in the monitoring of Earth rotation increases and requires further investigation. For a correct interpretation of the RLG data, it is 1 Department of Geodesy and Geodetic Astronomy, Warsaw necessary to properly model the expected signals. University of Technology, Pl. Politechniki 1, 00-661 Warsaw, The main factors that affect the observable of an Poland. E-mail: [email protected] 2 Space Research Centre, CBK PAN, Bartycka 18A, 00-716 RLG, i.e. the Sagnac frequency, refer to changes in Warsaw, Poland. M. Tercjak and A. Brzezin´ski Pure Appl. Geophys. the geometry of an instrument due to thermal and gyroscopes used for navigation are usually not bigger pressure variations, to laser performance (e.g. than 0.02 m2 and have a sensitivity of about 5  10À7 pffiffiffiffiffiffi backscatter coupling, dispersion), to variations of the rad=s Hz (Schreiber and Wells 2013), which is an normal vector of an instrument with respect to the insufficient level for geodetic aims. Nevertheless, the terrestrial reference frame, and to variations of the sensitivity is directly proportional to the scale factor Earth rotation vector. In this paper, we deal with the (a constant depending on the size of the instrument last effect, i.e. changes in ERP. We review the and the laser wavelength), which, in turn, increases known, predictable diurnal and subdiurnal signals in with the area of a gyroscope. Consequently, large ring polar motion (PM) and the universal time (UT1), laser gyroscopes having areas between 1 and about which influence the observed Sagnac frequency. We 834 m2 have been constructed to improve the sensi- consider the following three phenomena: the so- tivity by increasing the scale factor. However, large called diurnal polar motion of the instantaneous instruments bring about other issues, like increasing rotation axis, diurnal and semidiurnal signals in PM difficulty in constructing monolithic instrument and, and UT1 caused by ocean tides, and the so-called consequently, problems with mechanical stability. libration in PM and UT1. As we consider here the Taking into account all of those technical aspects, the impact of three-dimensional effects on the one-di- 4m 4 m large quasi-monolithic G-ring at the mensional signal, we study whether components of Wettzell Observatory is nowadays the most those effects amplify or cancel out each other. We stable and sensitive gyroscope, with the highest also verify how the impact of perturbations in ERP on potential to be used in geodesy (Schreiber and Wells the RLG observations varies with the change of 2013). location or orientation of an instrument. The Wettzel gyroscope has been operating since In Sect. 2 we present the mathematical back- 2001. It is located in a purpose-built underground ground and a brief description of the RLG apparatus laboratory, effectively isolated from external atmo- at the Wettzell Observatory, on which we conducted spheric conditions. A detailed description of the our investigation. In Sect. 3 we provide a description G-ring is given in Schreiber and Wells (2013). The of the investigated phenomena and their contribution whole monumentation minimizes external influences to observations of the G-ring at the Wettzell Obser- and protects the instrument from extensive tilting. vatory. In Sect. 4 we discuss the issue of orientation Nevertheless, six high precision tiltmeters, allowing of the normal vector of an instrument with respect to detection of 1 nrad tilts, are placed on the top of the the axes of the terrestrial system. We present the RLG to monitor orientation variations with respect to influence of ERP signals on observations from an the plumb line, e.g. due to the solid Earth tides or instrument like the Wettzell one, but depending on its local ground movements. However, taking into geographical location and other than horizontal ori- account the horizontal orientation of the RLG, only entation of the instrument. Conclusions are outlined tilts in the North-South direction must be monitored, in Sect. 5. This work represents a preliminary study as they cause variations in observations which are 3 of the use of RLG data for monitoring diurnal and orders of magnitude higher than the sensor resolution. subdiurnal signals (especially non-harmonic of geo- The East-West directed tilts have a negligible influ- physical origin) in Earth rotation parameters. ence on the observed RLG signal. The observable of the RLG is the so-called Sag- nac effect. In physical terms, the observed effect is a 2. The G-Ring Laser and the Sagnac Effect phase shift of two light beams propagating in oppo- site directions around a circuit of a rotating In general, gyroscopes are rotation sensors mostly gyroscope. Although the two beams travel the same used in navigation (aircraft, marine), but also in path in the same conditions, due to the sensor rotation consumer electronics (mobile phones, tablets) or in they traverse different distances. As a result the phase robotic guidance. Sensors used for these purposes are shift is observed which is directly proportional to the relatively small and not very sensitive. For instance, dot product of the area vector, normal to the On the Influence of Known Diurnal and Subdiurnal Signals gyroscope plane, and the vector of its rotation (Chao i ÀÁ m ¼ p À p_ À Pe_ Àih ; ð3Þ 1991). However, as the transition from phase to fre- X0 quency measurement greatly improves the gyroscope where the complex parameter m ¼ m þ im , with sensitivity (Schreiber and Wells 2013), instead of the x y the imaginary unit i, denotes the polar motion of the phase shift the beat frequency, called the Sagnac IRP, p ¼ x À iy describes the polar motion of the frequency, is used to describe the effect (e.g. Nilsson p p CIP, h denotes the Earth rotation angle, P ¼ dX þ et al. 2012): idY defines position of the CIP in the geocentric fsagn ¼ KX Á n þ finstr; with the scale factor celestial reference system (GCRS). Parameters dX 4 Á A and dY are the celestial pole offsets (nutation) of the K ¼ ; ð1Þ kL CIP (observed by VLBI) with respect to the position predicted by the conventional precession-nutation where L, A and k are the laser beam path length, the model. The dot denotes the time derivative. By area enclosed by the path and the wavelength of the taking the real and imaginary parts of Eq. (3) we get laser, respectively. They are assumed to be constant for the equivalent representation by 2 real-valued a particular instrument. In practice, these geometric equations: quantities are subject to fluctuations due to tempera- ture- and pressure-induced deformations, which is out 1 _ _ mx ¼ ðX0xp À y_p þ dX sin h À dY cos hÞ; of the scope of the paper. Vector X ¼ X0½mx my 1 þ X0 T ð4Þ mzŠ is the instantaneous rotation vector of the Earth- 1 my ¼ ðÀX0yp À x_p þ dX_ cos h þ dY_ sin hÞ: fixed system in space, with the dimensionless param- X0 eters m ; m ; m defining perturbations, and X x y z 0 See Mendes Cerveira et al. (2009) for extensive dis- denoting the mean angular speed of rotation. Vector n cussion of the equations. Note, that the polar motion is the normal vector of the instrument. For the hori- components mx, my in Eq. (4) do not contain this part zontally mounted RLG it is defined by of the lunisolar effect which is expressed by the n cos u cos k cos u sin k sin u T , where u and k ¼½ Š conventional precession–nutation model, the retro- are the geodetic latitude and longitude of an instru- grade diurnal signals, but the effect might be taken ment’s location. Finally, f is an instrumental offset. instr into account using a model reffered directly to the Consequently, (neglecting the instrumental errors, as IRP (e.g. Brzezin´ski 1986). Moreover, as we showed we do not deal with them within this paper) the basic in Tercjak et al. (2015) and as it was discussed by Sagnac frequency equation can be expressed by: Brzezin´ski (2009), the part involving the time _ _ fsagn ¼ KX0ðsin u þ mx cos k cos u derivatives of the nutation parameters (dX, dY) as yet ð2Þ can be neglected, and a simpler form of Eq. (3) can be þ my sin k cos u þ mz sin uÞ: used: Its unit is [Hz], and the variations in Earth rotation we i study here cause variations in the observed Sagnac m ¼ p À p_; ð5Þ X frequency at the level of 10À5 [Hz], i.e tens of ½lHz], 0 which strictly depends on the scale factor K. More- and consequently: over, to use the relation (2) for modeling the impact 1 1 of the known high frequency signals in ERP on the mx ¼ ðX0xp À y_pÞ; my ¼ ðÀX0yp À x_pÞ: X0 X0 RLG observations, we must take into account the fact ð6Þ that the available models of Earth rotation refer to the conventional celestial intermediate pole (CIP), and Variations of the axial component of rotation are not to the instantaneous rotation pole (IRP). Conse- expressed as the time derivative of quently, the relationship between CIP and IRP must dUT1 ¼ UT1 À UTC, where UTC denotes so-called be considered. For polar motion the first-order rela- coordinated universal time (see Petit and Luzum tionship is given by Brzezin´ski and Capitaine (1993): 2010 for a definition): M. Tercjak and A. Brzezin´ski Pure Appl. Geophys.

odUT1 Conventions 2010 (Petit and Luzum 2010). Based on m ¼ : ð7Þ z ot the above-mentioned models we compute compo- nents of the instantaneous rotation vector, then Although Eqs. 6 and 7 are adequate for reflecting the substitute it into Eq. (2) and derive the corresponding influence of signals in PM and dUT1 in the RLG variations of the Sagnac frequency, using parameters observations, we can also take advantage of the of the G-ring and the location of Wettzell. We take relationship (5) expressed in the frequency domain:  1 lHz as the resolution limit of the Sagnac frequency r signal (Prof. Ulrich Schreiber, personal communica- mðrÞ¼ 1 þ pðrÞ; ð8Þ X0 tion). The absolute value of 1 lHz refers to a relative precision of about 3  10À9. The time span of our where r ¼ 2p is the angular frequency of a particular T computation is 118 days, from 19.09.2014 to harmonic component with period T, positive for 15.01.2015. Below, we present a short description of prograde (counterclockwise) motion and negative for the models and the corresponding influence on the retrograde (clockwise) motion. This form of the observed Sagnac frequency. relationship between CIP and IRP is usable when considering splitting up PM signals into prograde and retrograde terms. Following e.g. Tian (2013) we can 3.1. Diurnal polar motion derive the relationship for the polar motion of the IRP The diurnal polar motion (DPM) refers to the as a function of prograde and retrograde amplitudes variations of the IRP in a body-fixed frame, due to a ap and ar, respectively, and the argument argðtÞ torque exerted by the tesseral components of the which is a linear combination of the five fundamental lunisolar tidal force. The perturbation is a near- arguments (Petit and Luzum 2010, section 5.7): circular retrograde motion with a period of approx- m ¼ apðcos½argðtފ þ i sin½argðtÞŠÞ þ arðcos½argðtފ imately one sidereal day and an amplitude varying in À i sin½argðtފÞ: ð9Þ time (Brzezin´ski 1986). The terrestrial perturbation of the IRP is associated with a celestial perturbation Consequently, the components m and m of m are x y which is not seen by the RLG. In case of the CIP the obtained as real and imaginary parts of the above whole perturbation appears as a celestial motion, i.e. relation. The corresponding expressions for a and a p r precession–nutation. See Brzezin´ski and Capitaine are presented below. (1993) for an extensive discussion about the rela- tionship between the motions of the instantaneous and conventional rotation poles. 3. Signals in Polar Motion and dUT1 The issue of the DPM has been thoroughly investigated for different Earth models, for instance The long-term aim of our research is to find out Woolard (1953) presented results for a rigid Earth; whether it is possible to use the RLG data for mon- McClure (1973) corrected those results for a purely itoring high-frequency variations in ERP, especially elastic Earth. Brzezin´ski (1986) corrected the results the non-harmonic signals of geophysical origin. of McClure (1973) for liquid-core effects using the However, before we do that, we first want to review dynamical theory of Wahr (1981a, b) and tabulated all known and modeled diurnal and subdiurnal signals numerical values of about 160 amplitudes for the affecting such observations. We start our studies with three axes: the instantaneous rotation axis, the signals in PM and dUT1. angular momentum axis, and the axis of figure. The As we have already mentioned, we consider here DPM model refers to the IRP, not the CIP, so the three phenomena: diurnal polar motion according to corresponding variations might be directly substituted the model developed by Brzezin´ski (1986), the into Eq. (2). A time series of the theoretical Sagnac influence of diurnal and semidiurnal ocean tides on frequency based on the Brzezin´ski (1986) DPM PM and dUT1 and librations in PM and dUT1, both model is presented in Fig. 1a. according to the models recommended in the IERS On the Influence of Known Diurnal and Subdiurnal Signals

(a) (b)

30 1 Diurnal Polar Motion diurnal OT in PM semidiurnal OT in PM 20 0.5

Hz] 10 μ

0 0

-10

Amplitude [ -0.5 -20

-30 -1 0 20 40 60 80 100 120 0 20 40 60 80 100 120

(c) (d) 0.4 4 diurnal OT in dUT1 semidiurnal OT in dUT1 Libration in PM Libration in dUT1 0.3 3

Hz] 0.2 μ 2

0.1 1

0 0 Amplitude [ -0.1 -1

-0.2 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Number of days since the initial epoch Number of days since the initial epoch

Figure 1 Influence of the modeled diurnal and semidiurnal signals in Earth rotation parameters on the Sagnac frequency as could be observed by the G-ring at the Wettzell Observatory; a diurnal polar motion, b diurnal (greater amplitude) and semidiurnal ocean tides effect in polar motion, c diurnal (upper) and semidiurnal (lower) ocean tides effect in dUT1, d libration effect in polar motion (upper) and in dUT1 (lower). For better visibility of the libration and OT effect in dUT1 the curves have been shifted vertically. For OT effect in PM the curves are superimposed upon each other to show the difference in the amplitudes

3.2. Ocean tides The currently recommended model of the diurnal and semidiurnal variations in polar motion and UT1 Gravitationally forced ocean tides (OT) contribute provided by the IERS Conventions 2010 is based on to variations in Earth rotation via the angular the work of Ray et al. (1994). It enables the estimation momentum exchange and, when considering polar of semidiurnal and prograde diurnal PM variations, as motion and dUT1, it is a dominant effect in the well as semidiurnal and diurnal dUT1 perturbations. diurnal and semidiurnal frequency bands. The phe- The retrograde diurnal equatorial component of the nomenon is quite regular and predictable, the OT contribution is included in the IAU 2000/2006 frequencies are well known, while the amplitudes precession–nutation model (Mathews et al. 2002), but and phases can be derived from ocean tide models regarding IRP it should be included in a retrograde and/or from observations of Earth rotation (Brze- DPM model. The DPM model, which we use here, zin´ski 2009). There have been several investigations does not take into account the OT effects. However, of the subject. Some of them (e.g. Ray et al. 1994) the inclusion of the OT effect in the DPM model resulted in analytical, geophysical models derived causes differences in amplitudes not bigger than with the use of the TOPEX/Poseidon altimetry data, 10 las (compare e.g. Brzezin´ski 1986 and Tian others (e.g. Watkins and Eanes 1994) derived empir- 2013). This corresponds to a Sagnac frequency signal ical models with parameters estimated from space at the level of 0.20 lHz, which is negligible at the geodetic techniques. current level of observation accuracy. M. Tercjak and A. Brzezin´ski Pure Appl. Geophys.

As the conventional model of OT effect in polar (Chao et al. 1991). Because of the axial asymmetry motion refers to the CIP, we performed and compared and the fact that the equator and ecliptic are not two ways of computation. In the first method we coplanar, the lunisolar torque gives rise to variations evaluated variations of all components of the CIP and in both polar motion (diurnal signal in the prograde then computed perturbations of the IRP by applying direction) and UT1 (semidiurnal perturbation). The relationships (6) and (7). In the second approach we phenomenon was named ‘‘libration’’, because of its further converted Eqs. (7) and (9) by taking into generic similarity to the libration of the Moon, by account the parameterization applied in the IERS Chao et al. (1991), who also estimated and then Conventions model. In the case of polar motion, we corrected its amplitudes (Chao et al. 1996). Further took coefficients xs, xc, ys, yc from Tables 8.2a and contribution to the topic came from, e.g. Brzezin´ski 8.2b of the IERS Conventions 2010 (xs, xc are the sin and Capitaine (2003, 2009) whose work constitutes a and the cos coefficients of xp and ys, yc are the basis for models provided by the IERS Conventions corresponding amplitudes of yp) and computed the 2010. prograde and retrograde amplitudes ap and ar as Although the libration has relatively small ampli- follows (Tian 2013): tudes and we expect that the signal might be not visible in the RLG observations, we decided to r xc À ys xs þ yc ap ¼ 1 þ À i ; investigate the signal, as the ring laser technique is X0 2 2 ð10Þ still under development. For this purpose we used r x þ y x À y a ¼ 1 À c s þ i s c ; coefficients given in Tables 5.1a and 5.1b of the IERS r X 2 2 0 Conventions 2010 and applied the same methods of where 1 Æ r are transfer coefficients taken from computations as in the case of ocean tides. The X0 Eq. (8). For the mz component we started with the Sagnac frequency variations based on the libration expression: model are presented together for PM and dUT1 in Fig. 1d. dUT1 ¼ us sinðargðtÞÞ þ uc cosðargðtÞÞ; ð11Þ where us and uc are the sine and cosine amplitudes from Tables 8.3a and 8.3b of the IERS Conventions 3.4. Results 2010. Then we used Eq. (11)inEq.(7) assuming From Fig. 1a–d, one can see that the main effect is additionally that arg(t) is a linear function of time. the DPM with a maximum amplitude of about That led us to the following expression for mz: 26 lHz. The signal is clearly visible in the RLG observations, which has been reported even in early mz ¼ rus cosðargðtÞÞ À ruc sinðargðtÞÞ ð12Þ works dealing with RLG data (e.g. Schreiber and ¼ uà cosðargðtÞÞ þ uà sinðargðtÞÞ: s c Wells 2013). This is not surprising, as amplitudes of As expected, both approaches gave same results, and the DPM are much larger than those from other therefore, they might be used interchangeably. The effects. The next largest signals are the semidiurnal Sagnac frequency based on the OT model is pre- OT effect in dUT1 and the diurnal OT effect in dUT1 sented separately for PM (Fig. 1b) and dUT1 and PM. The maximum amplitudes are 1.5, 1.1 and (Fig. 1c). 1.0 lHz, respectively. They are up to 30 times smaller than the DPM influence. However, they are above the current detection limit of the G-ring and 3.3. Libration should, therefore, be taken into account in the Assuming a triaxial Earth with the principal analysis of the signal. The smallest signals are the moments of inertia A, B, C (and A\B\C) the libration, both in PM and dUT1, and the semidiurnal libration is the direct effect of external (lunisolar) OT effect in PM with the maximum amplitudes of torque exerted on B-A, in the same way as precession 0.09, 0.17 and 0.13 lHz, respectively. All of them are and nutation arise from the torque exerted on C-A by one order of magnitude below the detection On the Influence of Known Diurnal and Subdiurnal Signals

2.5 threshold, thus at the current level of accuracy of the prograde retrograde technique they might be omitted in the analysis. 2

Hz] 1.5 Apart from the separate influence of each signal μ 1 we also compared the aggregated contribution of the 0.5 ocean tides and librations to check whether they 0 Amplitude [ amplify or cancel each other out. It can be seen from -0.5 Fig. 2 that although the libration signals amplify each -1 0 20 40 60 80 100 120 other, they are still much lower than the contribution Number of days since the initial epoch of OT and are still below the sensitivity threshold of Figure 3 the observations. Similarly, diurnal and semidiurnal Variation of the Sagnac frequency due to the retrograde and ocean tides effect in PM and dUT1 also amplify each prograde semidiurnal signals in polar motion excited by the ocean other. This suggests that their combined contribution tides will be visible, but there might be a problem to separate them. Although libration is negligibly small in Fig. 3 consists of the influence of all components for now, we added the signal to the influence of OT of prograde (top) and retrograde (bottom) semidiurnal and it turned out that the sum is slightly smaller than ocean tides over the entire period of analysis. It can the sum of OT only. This might be helpful in future be seen that the Sagnac frequency variations, due to analysis, when the RLG technique will be more both prograde and retrograde signals, are visible as accurate. modulated sinusoids. Both sine waves are of similar Although most of the results were quite expected, amplitude, however, they are almost out of phase, so the semidiurnal ocean tides effect in PM yielded a the sum of both is strongly reduced. To see that more surprisingly small signal, especially in comparison clearly, we separated the two largest semidiurnal tidal with the original amplitudes reported in the IERS components, M2 and S2, and plotted the first two days Conventions 2010. This suggests that the influence of in Fig. 4. (for better visibility only the first two days the prograde signals might be counterbalanced by the are depicted there). The comparison shows that the retrograde ones. To study this case in detail, we used prograde amplitudes are only slightly smaller than the Eq. (8) and split up the semidiurnal tidal signals into retrograde, and the two signals are roughly shifted by  prograde and retrograde components. The Sagnac 180 in phase. Moreover, it is visible that the M2 and frequency variations based on those signals are S2 signals are also shifted relative to each other, and shown in Figs. 3 and 4. Each of the curves shown consequently the retrograde signal of the M2 cancel out the retrograde signal of the S2, and the same with

10 the prograde counterparts. However, when studying a longer time period it turns out that their amplitudes 8 do not vary much, but the shift between the M2 and S2 6 Hz]

μ varies, and there are places where the signals coincide 4 and amplify each other, and then the whole semid- 2 iurnal OT signal in PM reaches its maximum.

Amplitude [ 0

-2

Sum of libration in PM and dUT1 Sum of OT in PM Sum of OT in dUT1 Sum of all "small" signals -4 4. Simulation of Changing the Location 0 20 40 60 80 100 120 or Orientation of an RLG Number of days since the initial epoch

Figure 2 It can be seen from Eq. (1) that the RLG obser- Variation of the Sagnac frequency due to the aggregated contri- vations depend on the orientation of the normal bution of different signals in Earth rotation (from top to bottom): libration in dUT1 and PM (offset 9 lHz), diurnal and semidiurnal vector of the instrument with respect to the instan- ocean tides effect in PM (offset 8 lHz), diurnal and semidiurnal taneous Earth rotation vector. The normal vector of ocean tides effect in dUT1 (offset 5 lHz), the sum of all ‘‘small’’ the horizontally mounted instrument can be expressed effects in PM and dUT1 M. Tercjak and A. Brzezin´ski Pure Appl. Geophys.

0.5 retrograde prograde 0.25 retrograde prograde 0.4 0.2 0.3 0.15 0.2 0.1

Hz] 0.1 μ 0.05 0 0 -0.1 -0.05 Amplitude [ -0.2 -0.1 -0.3 -0.15 -0.4 -0.2

-0.5 -0.25 0 0.4 0.8 1.2 1.6 2 0 0.4 0.8 1.2 1.6 2 Number of days since the initial epoch Number of days since the initial epoch

Figure 4 Same as in Fig. 3 but only the largest tidal components M2 (left plot) and S2 (right plot) are plotted over 2 days

00:00:00 06:00:00 12:00:00 18:00:00

50 50 50 50 ] °

[ 0 0 0 0 φ

-50 -50 -50 -50 Diurnal Polar Motion -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 λ [°] λ [°] λ [°] λ [°]

-30 -20 -10 0 10 20 30

Figure 5 Variation of the Sagnac frequency based on the model of the DPM depending on the location of a horizontally mounted instrument, given by latitude u and longitude k. Figures depict results for the day with the strongest influence of the effects, i.e. 22.12.2014. Units are [lHz] by geodetic latitude u and longitude k, as in Eq. (2). semidiurnal effects. A chosen day is a day with the We can easily conclude that the influence of a par- highest amplitudes in all signals. ticular phenomena influencing Earth rotation on the The influence of the libration is not considered observed Sagnac frequency will be different in hori- here since the total libration signal does not exceed zontally mounted instruments at different locations. the level of 0.40 lHz, independently of location. Therefore, we also performed computations for the However, it also turned out that the impact of the whole time span, with one hour resolution, using semidiurnal ocean tides effect in polar motion, during parameters (size and wavelength) of the G-Ring, but the whole investigated period, may reach 1.8 lHz, introducing other locations. We prepared a grid and it is comparable to the other OT components, and having a resolution of 3 by 6 (latitude and longitude, therefore, the signal cannot be ignored in further respectively) and at each point of the grid we com- analysis. Nevertheless, at the location of the Wettzell puted the Sagnac frequency based on all analyzed Observatory this signal is always as small, as visible models. In Figs. 5 and 6 we present results for the in Fig. 1b. In fact, there are two areas where the DPM and diurnal ocean tides for every 6 h of the influence of the semidiurnal ocean tides effect on 94th day of our time span (22.12.2014), and for every polar motion have a negligibly small contribution to 3 h of the first half of the same day for the the observed RLG signal between about 5 and 25 On the Influence of Known Diurnal and Subdiurnal Signals

00:00:00 06:00:00 12:00:00 18:00:00

50 50 50 50 ] °

[ 0 0 0 0 φ

-50 -50 -50 -50 diurnal OT in PM

-150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150

50 50 50 50 ] °

[ 0 0 0 0 φ

-50 -50 -50 -50 diurnal OT in dUT1

-150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150

00:00:00 03:00:00 06:00:00 12:00:00

50 50 50 50 ] °

[ 0 0 0 0 φ

-50 -50 -50 -50 semi-diurnal OT in PM -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150

50 50 50 50 ] °

[ 0 0 0 0 φ

-50 -50 -50 -50

semi-diurnal OT in dUT1 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 λ [°] λ [°] λ [°] λ [°]

-1.5 -1 -0.5 0 0.5 1 1.5

Figure 6 Variation of the Sagnac frequency due to the diurnal and semidiurnal signals in polar motion and dUT1 excited by the ocean tides, depending on the geographical location of a horizontally mounted instrument. The location is given by latitude u and longitude k. Figures depict results for the day with the strongest influence of the effects, i.e. 22.12.2014. Please note that for diurnal effects the time step is 6 h and for semidiurnal 3 h. Units are [lHz] and between À155 and À17 in longitude (where the However, there is no such dependence for the sign ‘-’ indicates the Western hemisphere). In con- DPM and the diurnal OT effect in PM. Their maxima trast, the strongest signal is expected Æ10 around the and minima (of the absolute value 39.9 lHz for the meridians À75 and 102 at the equator. The reason DPM and 1.6 lHz for the OT) move due to the is a strong polarization of the semidiurnal polar spinning Earth, and they are always in two points at motion excited by the ocean tides, shown in Fig. 7. the equator, distant from each other by about 180 One can see that the signal is strongly polarized in the (which is not surprising as we consider here diurnal direction being perpendicular to the Wettzell merid- effects). As also expected, the diurnal and semidiur- ian plane. Therefore, the polar motion excited by the nal ocean tides on dUT1 varies with latitude with the semidiurnal ocean tide is not visible in the G-ring strongest impact at the poles and the weakest at the signal. In general its magnitude on the observed equator. The maximum amplitudes at the poles are Sagnac frequency strongly depends on the longitude 1.5 lHz for the diurnal and 2.0 lHz for the semidi- of an RLG. urnal OT effect in dUT1. However, from the M. Tercjak and A. Brzezin´ski Pure Appl. Geophys.

° 90 Table 1 120° 60° 0.8 The most desirable locations of an RLG, where the maximum amplitude of the investigated signals is expected 0.6 150° 30° 0.4 No. Signals uk

0.2 1 Diurnal and semidiurnal OT in 90 or – dUT1 À90 ± ° ° 180 0 2 DPM and diurnal OT in PM 0 Any    3 Signals in mx component 0 0 or 180    4 Signals in my component 0 90 or À90 5 Semidiurnal OT in PM 0 102 or -150° -30° À75 Please note that we conduct only a theoretical discussion here, and ° ° -120 -60 do not consider whether locations are accessible or not -90°

Figure 7 visibility of the investigated signals. They are tabu- Polar motion of IRP caused by semidiurnal ocean tides effect. lated in Table 1. Minus sign of the longitude refers to the Western hemisphere and units are [mas] Although we do not consider here the real con- struction of a new instrument from Table 1 it is investigation of the aggregated influence of all ocean obvious that some of the locations, e.g. the polar tide signals over the whole time period, we can point regions, are not accessible. Moreover, locations out that at some epochs the signals may have either desirable for monitoring signals in PM and dUT1 the same or opposite signs, and thus their superpo- exclude each other. However, regarding Earth rota- sition may be constructive or destructive. This is tion monitoring any change of the geographical visible in Fig. 8, where the sum of all OT components location of the ring laser is equivalent to a change of is depicted for the day of the strongest influence of its orientation with respect to the crust, while keeping the signals. the location unchanged. Therefore, we decided to Hence, our analysis shows that the geographical discuss this problem in a more detailed way. location of a horizontally mounted RLG instrument is an important factor in the context of observations of 4.1. Location vs. orientation of an RLG Earth rotation variations. Looking at Figs. 5, 6, 7, 8 we can point out several locations which are of cru- As has already been expressed by Eq. (1), RLG cial importance from the point of view of the observations depend on the orientation of a plane

00:00:00 06:00:00 12:00:00 18:00:00

50 50 50 50 ] ° 0 0 0 0 [ φ

-50 -50 -50 -50

-150-100 -50 0 50 100 150 -150-100 -50 0 50 100 150 -150-100 -50 0 50 100 150 -150-100 -50 0 50 100 150

sum of OT in PM and dUT1 λ [°] λ [°] λ [°] λ [°]

-4 -2 0 2 4

Figure 8 Variation of the Sagnac frequency due to the aggregated influence of all ocean tide signals in Earth rotation depending on the geographical location of an instrument, given by latitude u and longitude k. Figures depict results for the day with the strongest influence of the effects, i.e. 22.12.2014. Units are [lHz] On the Influence of Known Diurnal and Subdiurnal Signals representing the optical cavity of an RLG with Locations 2–5 from the table refer to different respect to the terrestrial reference system. If the points at the equator. The normal vector of a planes of two instruments are parallel, the Sagnac horizontally mounted instrument located at the equa- signal, due to change of Earth rotation axis with tor (u ¼ 0) takes the form of T respect to the solid Earth, should be the same in both n2 ¼½cos k2 sin k2 0Š . After substituting the vector RLG, independently of their locations. Hence, instead into Eq. (13) and making further derivations we get: of considering other locations we can consider the tan Dk same place but a different orientation of the RLG. In cos z ¼ cos u cos Dk and tan A ¼À : 1 sin u other words, we can orient an instrument at location 1 1 ð15Þ (given by u1, k1) to make the same register obser- vations as a horizontally mounted RLG from location Additionally, if we want the instrument to be placed   2 (given by u2, k2). All we have to do is to introduce in the plane k2 equals 0 or 90 then z ¼ cos u1 cos k1 the normal vector of an instrument mounted hori- and A ¼ tan k1= sin u1 or z ¼ cos u1 sin k1 and zontally at location 2 (n2) into location 1 and derive A ¼Àtan k1 sin u1, respectively. the azimuth (A) and the zenith angle (z) of the vector We should emphasize here that we are conducting n2 at location 1 (v1). From a mathematical point of only a theoretical discussion, not taking into account view, we can simply use the transformation from the any technical aspects of the construction of ring global terrestrial system to the local astronomical lasers. However, as regards the orientation of existing system (e.g. Seeber 2003): instruments, most of them are mounted horizontally 2 3 and some of them vertically. Therefore, finally we sin z cos A 6 7 also take a quick look at signals observed by a v ¼DÀ1n ) 4 sin z sin A 5 1 2 vertically mounted instrument. In such an instrument cos z  2 3 the zenith angle of the normal vector is always 90 À sin u cos k1 À sin u sin k1 cos u and the observed signal will differ with the change of 6 1 1 1 7 ¼4 À sin k1 cos k1 0 5 the azimuth. To determine what location would be ‘‘observed’’ we can use the formula cos u cos k1 cos u sin k1 sin u 2 1 3 1 1 T n2 ¼ D½cos A sin A0Š , which leads to the relation: cos u cos k2 6 2 7 2 3  4 cos u sin k 5: ð13Þ cos u cos k2 2 2 6 2 7 4 5 sin u2 cos u2 sin k2 sin u If we want to find out what location would be ‘‘ob- 2 2 3 ð16Þ À sin u cos k1 cos A À sin k1 sin A served’’ by an instrument oriented in a particular way, 6 1 7 4 5 we can use the formula n2 ¼ Dv1. Using relation (13) ¼ À sin u1 sin k1 cos A þ cos k1 sin A : we can derive the orientation of an instrument at cos u1 cos A location 1 being equivalent to any location 2. How- Additionally, if the azimuth were 0 or 90 (North or ever, for the sake of completeness, we discuss below East direction), the observed location would be the locations which are pointed out in Table 1. ju j¼90 Àju j, k ¼ k or ju j¼0, Location 1 from Table 1 refers to the poles. The 2 1 2 1 2 k ¼ 90 þ k , respectively. latitude equals 90 (the North Pole) or À90 (the 2 1 South Pole) and consequently in the terrestrial reference frame n ¼½00 Æ 1ŠT . That leads to: 2 5. Concluding Remarks T v2 ¼Æ½Šcos u1 0 Æ sin u1 ; ð14Þ Our investigation concerns the contribution of the therefore, the normal vector of an instrument at known modeled high frequency components of Earth location 1 should have the zenith angle z ¼ 90 À rotation to the signal observed by ring laser gyro- ju j and the azimuth A ¼ 0 or 180. 1 scopes. For this purpose we computed, based on M. Tercjak and A. Brzezin´ski Pure Appl. Geophys. available models, the theoretical Sagnac frequency up, north and east directions, it would be possible to observed by the G ring at the Wettzell Observatory. observe signals at points u  40:9S, k  12:9E and Then we introduced new locations for a ring laser like u ¼ 0, k  102:9E. Hence, the mid-latitude point the Wettzell one to study how the signals vary with becomes an appropriate location for observing signals longitude and latitude. Lastly, we discussed equiva- in polar motion. Also, subdiurnal OT signals, although lence between the change of location and change of strongly polarized in the direction perpendicular to the orientation of an instrument in the context of regis- Wettzell meridian plane, could be then observed. tered signals. Based on the derived results we can Moreover, though it is beyond the scope of this work, point out a few conclusions and remarks. we also should remember that the normal vector of an First of all, we can point out that the lunisolar instrument undergoes some perturbations, and having diurnal polar motion is definitely the dominant signal, several instruments at the same place makes it easier from all of the Earth rotation variations at diurnal and to identify and model local effects affecting obser- subdiurnal frequencies, affecting the RLG observa- vations. In fact, such variations of the normal vector tions. However, the influence of diurnal and might be assumed the same for all instruments at the semidiurnal ocean tides on both polar motion and location, which limits the number of degrees of free- dUT1 should also be taken into account in analysis, dom in a set of observations. The number of degrees but they are rather visible as a sum of harmonics with of freedom has been discussed by, e.g. Chao (1991), the same frequencies, and there might be a problem and we are going to deal with the issue in further to separate them. The influence of the libration can analysis. hardly be detected at the current accuracy of the Summarizing, we reviewed the main diurnal and RLG. However, as the total influence of libration (in subdiurnal signals in PM and dUT1, which influence PM and dUT1) reaches about 0.4 lHz corresponding gyroscopic observations. This study can easily be to a relative accuracy level of 1  10À9, with the extended for other high frequency effects in Earth increasing accuracy of the RLG measurements the rotation, like the contributions of the atmospheric S1 signal should be considered. tide. The next step of our investigation will be a study Moreover, it is important that the high frequency of the contribution of local phenomena (like Earth signal observed by a horizontally mounted RLG tides or non-tidal loading effects) to the observed depends considerably on the geographical location of Sagnac frequency. the instrument. For instance, there are areas where the semidiurnal OT effect in PM are hardly noticeable, the signals in dUT1 are negligibly small at the equa- Acknowledgements tor, and the influence of signals in PM decreases when moving towards the poles. Also, the desired location We wish to thank Ulrich Schreiber and Andre´ for monitoring signals in PM and dUT1 exclude each Gebauer for helpful discussions and information other. Consequently, the component of interest is about the Wettzell ring laser. The first author is crucial for the choice of the location, and if all the indebted to Johannes Bo¨hm and Michael Schinde- components of Earth rotation should be monitored legger for turning her on to the RLG subject. Final using the RLG technique, several instruments located version of the paper benefited from helpful comments at appropriate places worldwide would be required. and suggestions of Benjamin F. Chao and the two However, any change of geographical location of a anonymous reviewers. horizontally mounted instrument is equivalent to an appropriate change of its orientation with respect to Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// the crust. This theoretically means that instead of creativecommons.org/licenses/by/4.0/), which permits unrestricted considering several locations, only one location with use, distribution, and reproduction in any medium, provided you several instruments can be considered. For example, give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if imagining that at the Wettzell location there were changes were made. three instruments with the normal vectors oriented in On the Influence of Known Diurnal and Subdiurnal Signals

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(Received December 5, 2016, revised April 8, 2017, accepted April 12, 2017)