EARTH TIDES AND SUPERCONDUCTING
ZHU Yaozhong, SUN Heping and WU Bin
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China
The main progresses in the fields of the measurement and theoretical research of the Earth tides as well as the superconducing gravimeter observations and its application to geodynamics achieved by Chinese scholars in recent years (1999-2002) are briefly reviewed in this paper.
I. EARTH TIDE MEASUREMENT
By fitting the orbits of satellite to the highly precise measurement of satellite laser ranging (SLR) to Lageos-1 and combining the altimetry-derived ocean parameters, Wu et al. recovered the amplitudes and phases of the Love number k2 for different tidal waves. The obtained k2 values for M2, S2, K1 and Q1 waves are 0.3027, 0.3014, 0.2573 and 0.2968 respectively. The phase lag of k2 is 0.°12 0.°09 for M2, and 0.°14 0.°12 for K1. The amplitude and phase lag of k2 for the 18.6-year tide are 0.3154 0.0070 and 3.°1 2.°0 respectively.
Institute of Geodesy and Geophysics of Chinese Academy of Sciences discussed how to use two different methods, i.e. the length of day (LOD) method and satellite orbit perturbation (SOP) method to study the Love number k2 of the long period solid Earth tides Mf and Mm After taking the effect of non-equilibrium ocean tide in account, the real parts of k2 given by these two methods agree well for Mf and Mm waves, but the imaginary part of k2 obtained by the LOD method is negative. However the result determined by the SOP method is positive, which is consistent with the phase lag of k2 predicted by the anelastic Earth model.
Using the SLR and ground gravity observation data, the solid Earth tidal displacement Love numbers h2 and l2 have been determined. The obtained h2 values for M2, S2, K1 and Q2 waves are 0.6062, 0.6114, 0.5234 and 0.6024 respectively. The SLR solution indicates that the result of the vertical displacement Love number h2 is better than one of the horizontal displacement Love number l2.
Based on the comparison of the theoretical values predicted by two anelastic models with the recently observed ones of tides and the Earth rotation variations at M2, Mf, Mm, Chandler wobble and 18.6-year frequencies, Zhu et al. (2000a) analyzed the response of mantle anelasticity at different tidal frequencies. The observed results of space geodesy were used to constrain the theoretical models of anelasticity. The results show that Zschau’s model can interpret the observed amplitude of the anelastic Love number in the regime from the seismic frequency to the 18.6-year frequency, but at some frequencies, there are some discrepancies between the observed and theoretical phase lags. The anelastic Love numbers predicted by a single absorption band model do not accord well with the observed values at these frequencies.
Using the tidal solutions obtained from SLR and the ocean tidal solutions from satellite altimetry, Institute of Geodesy and Geophysics，CAS computed the secular changes in the Moon’s orbit elements and the Earth rotation rate in the ecliptic reference system. The SLR-derived secular change in the Moon’s mean motion caused by the total tidal dissipation is -24.78 arc sec/century2. The secular change in the Earth rotation rate caused by the solar and lunar tides is -5.25×10-22 rad/s2. Taking the non-tidal effect of the changes in the Earth rotation rate into account, the corresponding change in the length of day is 1.49 ms/century. Combining the tidal solutions determined by altimetry and SLR, the effects of the solid Earth and oceans on the secular variations of the Moon’s mean motion and Earth rotation rate are distinguished. The result shows that the tidal energy dissipation in the solid Earth is about 3%-4% of the tidal energy dissipation in oceans.
A new method to calculate the secular change in the Earth rotation rate caused by the tidal dissipation was developed by Institute of Geodesy and Geophysics, CAS. In this method the tidal torques exerted by the tidal potential from the Earth on the Moon and the Sun are directly calculated. Based on the SLR-determined tidal parameters, the tidal secular change in the Earth rotation is obtained with high level of accuracy, the value is -6.01´10-22 rad/s2. The tidal secular change in the Earth rotation derived from the astronomical records from 700 B.C. to. 1600 A.D. is -4.5´10-22 rad/s2. After removing the tide effect, the non-tidal acceleration in the Earth rotation rate is 1.5´10-22 rad/s2. So the secular change of geopotential coefficient is derived, that is =-3.2 ´10-11/year. This value agrees very well with the results of estimated directly from satellite tracking data.II. THEORETICAL STUDY OF EARTH TIDE
The computation of the Love numbers usually uses the numerical method to solve a system of ordinary differential equations (ODEs) that is singular at the Earth center. When using the traditional methods such as the Runge-Kutta method to solve such kinds of ODEs, the special treatment must be taken at the Earth center. Wuhan University applied a new method, the Chebyshev collocation method, to solve this problem. This method does not require the special treatment of the ODEs at the Earth center, and can give high precise results. The programs for computing the Love numbers and the loading Love numbers of the SNREI Earth model using the Chebyshev collocation method have been provided.
Wuhan University explained the fundamental concepts and theory of the Earth tides, including tidal generating force, tidal generating potential, tidal waves, Love numbers, various observable tidal phenomena, the theories of tidal and loading tidal deformation of the SNREI Earth model, etc.
Institute of Geodesy and Geophysics, CAS discussed the tidal motion equations of the anisotropic media in the upper mantle. Based on the parameters of the Earth’s model given by Dziewonski, the Love numbers and loading Love numbers were calculated by using the classic Runge-Kutta numerical integral method. The results show that the effect of considering the anisotropic property in the upper mantle or not on the Love numbers is relatively small (about 0.06%). However, the effect to the loading Love numbers is relatively large (about 2.5%).III. SUPERCONDUCTING GRAVIMETER OBSERVATION
Using the tidal observations obtained by the LCR-ET spring gravimeters during the cooperation of China-British and China-Germany, Institute of Geodesy and Geophysics, CAS determined the calibration factor of the superconducting gravimeter (SG) based on the method of the weighed sum of the main tidal wave amplitudes. Using the FG5 absolute gravimeter and the SG measurements at the same station during two periods in 1999 and 2000, Institute of Geodesy and Geophysics, CAS determined the calibration factor based on the method of the least square polynomial fit. It shows that the results obtained by these two methods are identical.
Using the international standard analysis methods, Institute of Geodesy and Geophysics, CAS filtered the data obtained in a period of 48 hours with a band-pass filter. According to the characteristics of different angular frequencies of tidal waves and the properties of the odd and even band filters, the tidal gravity waves, such as the diurnal, semidiurnal and ter-diurnal waves, are separated after the instrumental drift was eliminated from the observations. The tidal changing characteristics of the gravity field at different areas, such as China, Japan, Belgium, France and so on, are studied. The possible factors including choosing of the different tidal generating potentials and band filters, considering the air pressure correction or not, getting rid of error data, and so on, which may affect the accuracy of the analytic results, are discussed. The results show that the accuracy of the observed amplitude factors of the main waves is better than 0.04%. Based on the basic concept given by Venedikov, Institute of Geodesy and Geophysics, CAS developed “wavelet analysis method” in which a set of continuous wavelet filter functions to substitute three discrete odd and even filters were used. This method can be applied to analyze the data obtained by the different samplings.
Institute of Geodesy and Geophysics, CAS analyzed the observation data of 8 instruments, especially the long-term SG observations from 1988 to 1994, the improved values of the international gravity tidal reference (IGTR) at Wuhan Station were obtained with high precision. Comparing with the latest theoretical tidal model, the deviation of the amplitude factor decreased from the original one of 4.4‰ to the new one of 3.6‰. Institute of Geodesy and Geophysics, CAS calculated the synthesized gravity tide signals using Wuhan IGTR value, the latest gravity tidal theoretical model, the latest oceanic tidal model and the characteristic of nearly diurnal resonance in gravity tidal observations. The results show that the discrepancy between the synthesized tidal signals and the SG observations at Wuhan is 0.225 mGal.
Institute of Geodesy and Geophysics, CAS discussed the error estimation in the theoretical calculation of the atmospheric gravity signals with statistic technique. The effects of the uncertainty of Earth model, the air pressure data error and the lower density distribution of the station on the calculated results were also studied. Based on the observed gravity tidal residuals and the air pressure at Wuhan Station, and by using the techniques of the correlation and linear regression analysis, Institute of Geodesy and Geophysics, CAS studied the character of the atmospheric gravity in frequency and time domains respectively. After correcting air pressure, the observed gravity tidal residual amplitudes are reduced significantly at all frequency bands. It is shown that the correction using admittances in frequency domain is better than the one using admittances in time domain.
Based on the observation results of the SG at stations Wuhan and Tokyo, Institute of Geodesy and Geophysics, CAS investigated the suitability problem of the latest ocean tidal models (including Schwiderski, CSR3.0, FES95.2, ORI and ORI96). The residual vectors of the gravity tidal observations obtained by considering the loading correction with various ocean models or not were analyzed and compared. The results indicate that the oceanic loading effect is the main part of the residual vector of the gravity tidal observation. After the oceanic loading correction, the residual amplitude of the M2 wave at Wuhan Station is reduced about 70%, the residual amplitudes of the M2 and K1 waves at Tokyo Station are reduced about 75% and 90% respectively. Based on the Farrell loading theory, the influence of the ocean loading on the surface gravity tide was calculated.IV. APPLICATION OF SUPERCONDUCTING GRAVIMETER TO GEODYNAMICS
Institute of Geodesy and Geophysics, CAS determined precisely the period, quality factor and resonance intensity of the Earth’s free core nutation (FCN) using the SG observations. The result shows that the eigenperiod of the FCN is 429.0 sidereal days, which is accord with the former results and is about 30 sidereal days smaller than the theoretical value. It proves the conclusion that the real dynamics ellipticity of liquid core is larger 5% than that on the assumption of the hydrostatic equilibrium. The obtained quality factor is close to the one obtained with the VLBI observations. Meanwhile the influence of the ocean loading on the FCN resonant parameters was investigated. The result shows that the effects of different oceanic tidal models on the FCN eigenperiod and the real part of the resonant intensity are comparatively small, the main difference of the result is shown in the quality factor Q values, and the resonant intensity is strong dependent on the oceanic tide models. The influence of the uncertainty of different oceanic tidal models on the FCN parameters can be partly counteracted by staking of observations corrected with these models.REFERENCES
 Guo Junyi, Ning Jinsheng, Zhang Feipeng, 2001, Chebyshev-collocation method applied to solve ODEs in geophysics singular at the Earth center. Geophys. Res. Lett., 28(15): 3027-3030.
 Guo Junyi，2001, Chebyshev Collocation Method with Applications in Geophysics — Principle and Programs. Press of the Wuhan Technical University of Surveying and Mapping, Wuhan, 102-110.
 Guo Junyi, 2001, Introduction to Geophysics. Press of Surveying and Mapping, Beijing, 233-268 (in Chinese).
 Hsu Houze, Sun Heping, Xu Jianqiao, et al., 2000, Study on Wuhan international tidal gravity reference. Science in China (Series D), 30(5): 549–553 (in Chinese).
 Liu Lintao, Hsu Houze, Sun Heping, et al., 2000, New method of Precise Harmonic analysis of the tidal gravity observations. Science in China (Series D), 30(4): 442–448 (in Chinese).
 Luo Shaocong, Sun Heping, Xu Jianqiao, 2001, Theoretical computation in the precision estimation of the change in gravity and displacements due to atmospheric loading. Acta Geodaetica et Cartographic Sinica, 30(4): 309–315 (in Chinese).
 Peng Bibo, Wu Bin, Hsu Houze, 2000, Determination of long term variations of low degree geopotential field. Acta Geodaetica et Cartogrphica Sinica, 29(Sup.): 38-42 (in Chinese).
 Peng Bibo, Wu Bin, Hsu Houze, 2000, Determination of displacement Love numbers from SLR data. Acta Geodaetica et Cartogrphica Sinica, 29(2): 305-309 (in Chinese).
 Sun Heping, Chen Xiaodong, Hsu Houze, et al., 2001, Accurate determination of calibration factor for tidal gravity observation of a GWR-superconducting gravimeter. Acta Seismologica Sinica, 23(6): 651–658 (in Chinese).
 Sun Heping, Hsu Houze, Ducarme B, et al., 1999, Comprehensive comparison and analysis of tidal gravity observations obtained with superconducting gravimeters at stations in China, Belgium and France. Chinese Science Bulletin, 44(8): 750-755.
 Sun Heping, Takemoto S, Hsu Houze, et al., 2001, Precise tidal gravity recorded with superconducting gravimeters at stations Wuhan/China and Kyoto/Japan. Journal of Geodesy, 74: 720–729.
 Sun Heping, Hsu Houze, Xu Jianqiao, 2001, Determination of the new tidal parameters obtained with a superconduction gravimeter at station Wuhan/China. Journal of the Geodetic Society of Japan, 47(1): 347-352.
 Sun Heping, Takemoto Shuzo, Hsu Houze, et al., 2000, Observation results of the high precision tidal gravity with superconducting gravimeters at Stations Wuhan and Kyoto. Acta Geodaetica et Cartographic Sinica, 29(2): 181–187 (in Chinese).
 Sun Heping, Hsu Houze, Luo Shaocong, et al., 1999, Study of the ocean models using tidal gravity observations obtained with superconducting gravimeter. Acta Geodaetica et Cartographic Sinica, 28(2): 115–120 (in Chinese).
 Sun Heping, Zhou Jiangcun, 2002, Correction problem of the ocean loading signals on gravity measurements at fundamental stations in crust movement observation network in China. Advance in Earth Sciences, 17(1): 39–43 (in Chinese).
 Wu Bin, Peng Bibo, Zhu Yaozhong, Hsu Houze, 2001，Determination of Love numbers using satellite laser ranging. Journal of the Geodetic Society of Japan, 47(1): 174-180.
 Wu Bin, Zhu Yaozhong, Peng Bibo, 1999, Determination of solid Earth tide phase lag by satellite observation data. Chinese Science Bulletin, 44(5): 459-461.
 Wu Bin, Peng Bibo, Hao Xinghua, 2000, Determination of the Love number k2 of waves Mm and Mf in solid Earth tide — the results from space geodesy. Acta Astronomy Sinica, 41(3): 306-311 (in Chinese).
 Xu Jianqiao, Sun Heping, Luo Shaocong, 1999, Determination of the synthetic gravity tides at Wuhan Fundamental Station. Acta Seismologica Sinica, 19(3): 26-31 (in Chinese).
 Xu Jianqiao, Hao Xinghua, Sun Heping, 1999, Influence of atmospheric pressure on tidal gravity at Wuhan Station. Acta Geodaetica et Cartographic Sinica, 28(1): 21–27 (in Chinese).
 Xu Jianqiao, Hsu Houze, Sun Heping, et al., 1999, Investigation of the Earth’s nearly diurnal free wobble resonance using tidal gravity observations with superconducting gravimeters. Chinese Journal of Geophysics, 42(5): 599–608 (in Chinese).
 Xu Jianqiao, Sun Heping, Luo Shaocong, 2001, Investigation of the Earth free core nutation using the international observations obtained with superconducting gravimeters. Science in China (Series D), 31(9): 719–726 (in Chinese).
 Zhang Yonghong, Li Guoying, Sun Heping, 1999, Effects of anisotropic medium in the upper mantle on the Earth tide and loading tide. Chinese Journal of Geophysics, 42(3): 333-339 (in Chinese).
 Zhu Yaozhong, Wu Bin, Peng Bibo, 2000a, Constraints of the Earth rotation and tidal parameters on mantle anelasticity. Acta Astronomy Sinica, 41(1): 58-61 (in Chinese).
 Zhu Yaozhong, Wu Bin, Peng Bibo, 2000b, Secular changes in the Moon’s mean motion and Earth rotation rate due to tidal dissipation. Acta Geodaetica et Cartogrphica Sinica, 29(Suppl.): 1-4 (in Chinese).