On long-period polar motion

 Five separate polar motion series are examined in order to understand what portion of their variations at periods exceeding several years represents true polar motion. The data since the development of space-geodetic techniques (by themselves insufficient for study of long-period motion), and a variety of historical astrometric data sets, allow the following tentative conclusions: retrograde long-period polar motion below about −0.2 cpy (cycles per year) in pre-space-geodetic data (pre-1976) is dominantly noise. For 1976–1992, there is poor agreement between space-geodetic and astrometric series over the range −0.2 to +0.2 cpy, demonstrating that classical astrometry lacked the precision to monitor polar motion in this frequency range. It is concluded that all the pre-1976 astrometric polar motion data are likely to be dominated by noise at periods exceeding about 10 years. The exception to this is possibly a linear trend found in some astrometric and space geodetic series. At frequencies above prograde +0.2 cpy (periods shorter than about 5 years), historical astrometric data may be of sufficient quality for comparisons with geophysical excitation time series. Even in the era of space geodesy, significant differences are found in long-period variations in published polar motion time series. Received: 27 March 2001 / Accepted: 15 October 2001     

A study of gravity variations caused by polar motion using superconducting gravimeter data from the GGP network

Long-term continuous gravity observations, recorded at five superconducting gravimeter (SG) stations in the Global Geodynamic Project (GGP) network, as well as data on orientation variations in the Earths rotation axis (i.e. polar motion), have been used to investigate the characteristics of gravity variations on the Earths surface caused by polar motion. All the SG gravity data sets were pre-processed using identical techniques to remove the luni-solar gravity tides, the long-term trends of the instrumental drift, and the effects of atmospheric pressure. The analysis indicates that the spectral peaks, related to the Chandler and annual wobbles, were identified in both the power and product spectral density estimates. The magnitude of gravity variations, as well as the gravimetric amplitude factor associated with the Chandler wobble, changed significantly at different SG stations and during different observation periods. However, when all the SG observations at these five sites were combined, the gravimetric parameters of the Chandler wobble were retrieved accurately: 1.1613 ± 0.0737 for the amplitude factor and –1°.30 ± 1°.33 for the phase difference. The value of the estimated amplitude factor is in agreement with that predicted theoretically for the zonal tides of an elastic Earth model.     

Period variations of the Chandler wobble

Variations in the period of the Chandler wobble have been discussed since its discovery by Chandler in 1892. Various authors engaged in the investigation of polar motion time series suggest both a variable and an invariable period. It cannot be resolved by the analysis of time series whether the Chandler period is variable. By studying the influence of mass redistributions on the Chandler period it has been found that it is in fact variable, but the magnitude of such variation is much smaller than that found by polar motion time series analysis. For the currently available time series of polar motion, it is sufficient to assume an invariable Chandler period. AcknowledgmentsUseful discussions with Dr. F. Barthelmes and Dr. K. Fleming are gratefully appreciated.     

Possible improvement of Earth orientation forecast using autocovariance prediction procedures

 Autocovariance prediction has been applied to attempt to improve polar motion and UT1-UTC predictions. The predicted polar motion is the sum of the least-squares extrapolation model based on the Chandler circle, annual and semiannual ellipses, and a bias fit to the past 3 years of observations and the autocovariance prediction of these extrapolation residuals computed after subtraction of this model from pole coordinate data. This prediction method has been applied also to the UT1-UTC data, from which all known predictable effects were removed, but the prediction error has not been reduced with respect to the error of the current prediction model. However, the results show the possibility of decreasing polar motion prediction errors by about 50 for different prediction lengths from 50 to 200 days with respect to the errors of the current prediction model. Because of irregular variations in polar motion and UT1-UTC, the accuracy of the autocovariance prediction does depend on the epoch of the prediction. To explain irregular variations in x, y pole coordinate data, time-variable spectra of the equatorial components of the effective atmospheric angular momentum, determined by the National Center for Environmental Prediction, were computed. These time-variable spectra maxima for oscillations with periods of 100–140 days, which occurred in 1985, 1988, and 1990 could be responsible for excitation of the irregular short-period variations in pole coordinate data. Additionally, time-variable coherence between geodetic and atmospheric excitation function was computed, and the coherence maxima coincide also with the greatest irregular variations in polar motion extrapolation residuals. Received: 22 October 1996 / Accepted: 16 September 1997     

Predictability of the Earth’s polar motion

The numerical prediction of the Earth’s polar motion is of both theoretical and practical interest. The present paper is aimed at a comprehensive, experimental study of the predictability of polar motion using a homogeneous BIH (Bureau International de l’Heure) data set for the period 1967–1983. Based on our knowledge of the physics of the annual and the Chandler wobbles, we build the numerical model for the polar motion by allowing the wobble period to vary. Using an optimum base length of six years for prediction, this “floating-period” model, equipped with a nonlinear least-squares estimator, is found to yield polar motion predictions accurate to within 0″.012 to 0″.024 depending on the prediction length up to one year, corresponding to a predictability of 89–82%. This represents a considerable improvement over the conventional fixed-period predictor, which, by its nature, does not respond to variations in the apparent wobble periods (in particular, a dramatic decrease in the periods of both the annual and the Chandler wobbles after the year 1980). The superiority of the floating-period predictor to other predictors based on critically different numerical models is also demonstrated.     

Linear drift and periodic variations observed in long time series of polar motion

 Two long time series were analysed: the C01 series of the International Earth Rotation Service and the pole series obtained by re-analysis of the classical astronomical observations using the HIPPARCOS reference frame. The linear drift of the pole was determined to be 3.31 ± 0.05 milliarcseconds/year towards 76.1 ± 0.80° west longitude. For the least-squares fit the a priori correlations between simultaneous pole coordinates x _p , y _p were taken into account, and the weighting function was calculated by estimating empirical variance components. The decadal variations of the pole path were investigated by Fourier and wavelet analysis. Using sliding windows, the periods and amplitudes of the Chandler wobble and annual wobble were determined. Typical periods in the variable Chandler wobble and annual wobble parameters were obtained from wavelet analyses. Received: 21 January 2000 / Accepted: 28 August 2000     

Variability of polar motion oscillations with periods from 20 to 150 days in 1979–1991

Variability of short period oscillations of polar motion with periods ranging from 20 to 150 days were investigated in the period 1979–1991. The new computation method of time variable band pass filter spectra and the Wavelet Transform method were applied. These oscillations are elliptical with variable amplitudes. Modulation periods of amplitude variations of these oscillations of about two and three years were found. Correlations of short period oscillations of polar motion and of effective atmospheric angular momentum (EAAM) excitation functions show annual variations and connections of their increases with El Niño phenomena.     

Nonlinear mechanism for polar motion with period of 7 months

 The 7 month-period (sub-Chandler) wobble is considered with respect to the nonlinear dynamical equation of polar motion. Starting with the frequency modulation of Chandler wobble (CW) in the model developed by introduction of damping from perturbed visco-elastic deformation, the rotation equation of the CW becomes a resonance model with a time-dependent parameter. According to evolution calculation, the parameter resonance model is essentially identical to the reality of CW observations. If the frequency of CW is modulated about 3 by visco-elastic deformation, then the amplitude of CW will be modulated by greater than 70. On the other hand, bifurcation may occur according to the nonlinear dynamical system of the parameter resonance model, i.e. a pair of solutions splitting from the main CW. One is the 7-month-period wobble and the other is a motion with a period of about 28 months. Although the latter is very weak, the 7-month-period wobble will be observed as the stability condition is satisfied. The maximum amplitude is about 22.89 mas and the average 12.65 mas. This is identical to what is observed in reality. Received: 7 April 2000 / Accepted: 5 March 2001     

Short-term polar motion forecasts from earth system modeling data

Polar motion predictions for up to 10 days into the future are obtained from predicted states of the atmosphere, ocean and continental hydrosphere in a hind-cast experiment covering 2003–2008. High-frequency mass variations within the geophysical fluids are the main cause of wide-band stochastic signals not considered in the presently used statistical prediction approach of IERS bulletin A for polar motion. Taking EAM functions based on forecasted model states, derived from ECMWF medium-range forecasts and corresponding LSDM and OMCT simulations, into account the prediction errors are reduced by 26%. The effective forecast length of the model combination is found to be 7 days, primarily limited by the accuracy of the forecasted atmospheric wind fields. Highest improvements are found for forecast days 4–5 with prediction skill scores of the polar motion excitation functions improved by a factor up to 5. Whereas bulletin A forecasts can explain the observed variance within the first 10 days only by up to 40%, half of the model forecasts reach relative explained variances between 40 and 80%.     

Analysis of decade-long time series of GPS-based polar motion estimates at 15-min temporal resolution

We present results from the generation of 10-year-long continuous time series of the Earth’s polar motion at 15-min temporal resolution using Global Positioning System ground data. From our results, we infer an overall noise level in our high-rate polar motion time series of 60 \(\upmu \hbox {as}\) (RMS). However, a spectral decomposition of our estimates indicates a noise floor of 4 \(\upmu \hbox {as}\) at periods shorter than 2 days, which enables recovery of diurnal and semidiurnal tidally induced polar motion. We deliberately place no constraints on retrograde diurnal polar motion despite its inherent ambiguity with long-period nutation. With this approach, we are able to resolve damped manifestations of the effects of the diurnal ocean tides on retrograde polar motion. As such, our approach is at least capable of discriminating between a historical background nutation model that excludes the effects of the diurnal ocean tides and modern models that include those effects. To assess the quality of our polar motion solution outside of the retrograde diurnal frequency band, we focus on its capability to recover tidally driven and non-tidal variations manifesting at the ultra-rapid (intra-daily) and rapid (characterized by periods ranging from 2 to 20 days) periods. We find that our best estimates of diurnal and semidiurnal tidally induced polar motion result from an approach that adopts, at the observation level, a reasonable background model of these effects. We also demonstrate that our high-rate polar motion estimates yield similar results to daily-resolved polar motion estimates, and therefore do not compromise the ability to resolve polar motion at periods of 2–20 days.     

Wavelet Modeling of Regional and Temporal Variations of the Earth’s Gravitational Potential Observed by GRACE

This work is dedicated to the wavelet modeling of regional and temporal variations of the Earth’s gravitational potential observed by the GRACE (gravity recovery and climate experiment) satellite mission. In the first part, all required mathematical tools and methods involving spherical wavelets are provided. Then, we apply our method to monthly GRACE gravity fields. A strong seasonal signal can be identified which is restricted to areas where large-scale redistributions of continental water mass are expected. This assumption is analyzed and verified by comparing the time-series of regionally obtained wavelet coefficients of the gravitational signal originating from hydrology models and the gravitational potential observed by GRACE. The results are in good agreement with previous studies and illustrate that wavelets are an appropriate tool to investigate regional effects in the Earth’s gravitational field. Electronic Supplementary Material Supplementary material is available for this article at     

Assessment of periodic sub-diurnal Earth rotation variations at tidal frequencies through transformation of VLBI normal equation systems

We present an empirical model for periodic variations of diurnal and sub-diurnal Earth rotation parameters (ERPs) that was derived based on the transformation of normal equation (NEQ) systems of Very Long Baseline Interferometry (VLBI) observing sessions. NEQ systems that contain highly resolved polar motion and UT1-TAI with a temporal resolution of 15 min were generated and then transformed to the coefficients of the tidal ERP model to be solved for. To investigate the quality of this model, comparisons with empirical models from the Global Positioning System (GPS), another VLBI model and the model adopted by the conventions of the International Earth Rotation and Reference Systems Service (IERS) were performed. The absolute coefficients of these models agree almost completely within 7.5 μ as in polar motion and 0.5 μs in UT1-TAI. Several bigger differences exist, which are discussed in this paper. To be able to compare the model estimates with results of the continuous VLBI campaigns, where signals with periods of 8 and 6 h were detected, terms in the ter- and quarter-diurnal band were included in the tidal ERP model. Unfortunately, almost no common features with the results of continuous VLBI campaigns or ERP predictions in these tidal bands can be seen.     

Atmospheric and oceanic forcing of the rapid polar motion

The rapid polar motion for periods below 20 days is revisited in light of the most recent and accurate geodetic and geophysical data. Although its amplitude is smaller than 2 mas, it is excited mostly by powerful atmospheric processes, as large as the seasonal ones. The residual amplitude, representing about 20% of the total excitation, stems from the oceans. Rapid polar motion has an irregular nature that is well explained by the combined influence of the atmosphere and the oceans. An overall spectrum reveals cycles principally at 20, 13.6 (fortnightly tidal period) and 10 days (corresponding to the normal atmospheric mode Y₃¹{\Psi_3^1}), but this is only an averaged feature hiding its strong variability over seasonal time scales. This explains why it is so delicate to determine an empirical model of the tidal effect on polar motion. The variability in both amplitude and phase of the 13.6-day term is probably caused by a lunar barometric effect, modulated by some sub-seasonal thermal processes. The irregularities of the prominent cycles of the short-term polar motion are well explained by the atmospheric and oceanic excitations. The oceanic variability reinforces the atmospheric one, as they were triggered by the same agent, maybe seasonal and inter-annual thermal variations.     

New Solution for the Earth’s Free Wobble and Its Geophysical Implications

In this paper, the theory of the free wobble of the triaxial Earth is developed and new conclusions are drawn: the Euler period should be actually expressed by the first kind of complete elliptic integral; the trace of the free polar motion is elliptic and the orientations of its semi-minor and major axes are approximately parallel to the Earth’s principal axes A and B, respectively. In addition, the present theory shows that there is a mechanism of frequency-amplitude modulation in the Chandler wobble, whi...     

The temporal variation of the spherical and Cartesian multipoles of the gravity field: the generalized MacCullagh representation

 The Cartesian moments of the mass density of a gravitating body and the spherical harmonic coefficients of its gravitational field are related in a peculiar way. In particular, the products of inertia can be expressed by the spherical harmonic coefficients of the gravitational potential as was derived by MacCullagh for a rigid body. Here the MacCullagh formulae are extended to a deformable body which is restricted to radial symmetry in order to apply the Love–Shida hypothesis. The mass conservation law allows a representation of the incremental mass density by the respective excitation function. A representation of an arbitrary Cartesian monome is always possible by sums of solid spherical harmonics multiplied by powers of the radius. Introducing these representations into the definition of the Cartesian moments, an extension of the MacCullagh formulae is obtained. In particular, for excitation functions with a vanishing harmonic coefficient of degree zero, the (diagonal) incremental moments of inertia also can be represented by the excitation coefficients. Four types of excitation functions are considered, namely: (1) tidal excitation; (2) loading potential; (3) centrifugal potential; and (4) transverse surface stress. One application of the results could be model computation of the length-of-day variations and polar motion, which depend on the moments of inertia. Received: 27 July 1999 / Accepted: 24 May 2000     

Long-term polar motion prediction using normal time–frequency transform

This paper presents normal time–frequency transform (NTFT) application in harmonic/quasi-harmonic signal prediction. Particularly, we use the normal wavelet transform (a special NTFT) to make long-term polar motion prediction. Instantaneous frequency, phase and amplitude of Chandler wobble, prograde and retrograde annual wobbles of Earth’s polar motion are analyzed via the NTFT. Results show that the three main wobbles can be treated as quasi-harmonic processes. Current instantaneous harmonic information of the three wobbles can be acquired by the NTFT that has a kernel function constructed with a normal half-window function. Based on this information, we make the polar motion predictions with lead times of 1 year and 5 years. Results show that our prediction skills are very good with long lead time. An abnormality in the predictions occurs during the second half of 2005 and first half of 2006. Finally, we provide the future (starting from 2013) polar motion predictions with 1- and 5-year leads. These predictions will be used to verify the effectiveness of the method proposed in this paper.     

Prediction of polar motion

Based on an analysis of polar motion behavior, we found the possibility of predicting polar motion up to one year in advance. Comparing these predicted polar coordinates with the observed ones (smoothed), the rms of the differences is about 0".02. The differences of the relative polar motion are much smaller. For any time interval of 20–30 days throughout the whole year, the rms of the relative polar motion differences is about 0".01. It appears that 80–90% of the polar motion is composed of the stable, predictable Chandler and annual terms.     

Results of three year observation with a superconducting gravimeter at the GeoForschungsZentrum Potsdam

The superconducting gravimeter (SG) TT70 has been continuously recording gravity data at the GeoForschungsZentrum (GFZ) Potsdam absolute gravity site since July 1992. The recorded data are edited and preprocessed by spike and step detection and elimination and gap filling. An atmospheric pressure correction is carried out on gravity data in the time domain with a complex admittance before tidal fitting. The atmospheric pressure admittance is calculated from tide free output of SG data and local atmospheric pressure using the cross spectral method. The ground water level admittance is determined by a single coefficient. Improvements with these corrections are shown in analysis results. New tidal parameters for Potsdam site are determined and compared with recordings of an Askania spring gravimeter at a nearby site. Deviations against the Wahr-Dehant-model are shown. Polar motion data of the IERS (International Earth Rotation Service, Paris) are used to calculate variations of centrifugal acceleration caused by polar motion (pole tide). These are compared with the corrected tide free output of SG series. For drift determination the polar motion correction is applied on SG data. The nutation period equivalent to the Earth's Nearly Diurnal Free Wobble is calculated from the SG data with a value of T_FCN = (437.4 ± 1.5) sidereal days. This result is compared with those obtained from other SG stations. Received 19 December 1995; Accepted 13 September 1996     

由SLR观测的地心周期性运动(1993-2006年)

地球质心(CM)是整个地球的质量中心,地心运动是由地球系统的质量重新分布激励的,特别是流体圈层.利用SLR对LAGEOS 1/2卫星的距离观测,解算1993-2006年期间的地心运动时间序列,然后分别利用小波变换和最小二乘法分析该序列,发现地心运动存在长期和周期性变化.地心的长期运动表明地壳形状在变化.季节性变化是地心运动的主项,主要是由地球流体圈层的质量分布造成的,如海洋、大气和陆地水等.地心运动还存在其他周期性和准周期性变化.地心运动存在2～5年的长周期变化.许多周期都存在渐变,这表明整个地球系统质量和环境存在不规则的变化.

Abstract：

The center of mass of Earth (CM) is the center of total Earth's mass. The geocenter motion may be excit-ed by the mass redistribution of Earth system, especially the fluid layer. A time series of geocenter motion meas-ured with SLR on LAGEOS 1/2 is estimated and then analyzed with the wavelet transformation and the least squaresmethod, respectively. The secular and periodic variations of geocenter motion are detected. The long-term move-ment indicates that the crustal figure is changing, the north hemisphere and 180-degree hemisphere are shrinking, and the south hemisphere and O-degree hemisphere are swelling. The seasonal variations are the main componentswhich may be caused from the mass distribution of Earth fluid layer, e.g. ocean, atmosphere and continental wa-ter, There are many other periodic or quasi-periodic variations. There are long periodic variations through 2 to 5 years, Many periods gradually change, which indicates that there exist nonregular fluxes for the environment and mass of the whole Earth system.