Polar Motion Time Series Research Articles

Advancing polar motion prediction with derivative information

Abstract Earth Orientation Parameters (EOP) are essential for monitoring Earth’s rotational irregularities, impacting satellite navigation, space exploration, and climate forecasting. This study introduces a hybrid prediction model combining least-squares (LS) and vector autoregression (VAR) to improve Earth’s Pole Coordinates (x, y) forecast accuracy. Using daily sampled IERS EOP 20 C04 data from 2013 to 2023, we conducted 1,000 yearly random trials, performing 48 forecasts per year. Our method evaluates six data combinations, including primary variables (x, y) and their derivatives ( x ̇ , y ̇ $\dot{x},\dot{y}$ ). Results show a systematic improvement in prediction accuracy, especially for ultra-short-term forecasts (10 days into future), with derivative information stabilizing the solutions. The best-performing combination ( x , y , x ̇ , y ̇ $x,y,\dot{x},\dot{y}$ ) achieved a mean absolute prediction error (MAPE) reduction (with respect to the reference data combination – x, y) of up to 8 % for the y and 7 % for the x over a whole 30-day forecast horizon. These findings highlight the effectiveness of incorporating derivatives of polar motion time series into prediction procedure.

Estimation of Earth Rotation Parameters and Prediction of Polar Motion Using Hybrid CNN–LSTM Model

The Earth rotation parameters (ERPs), including polar motion (PMX and PMY) and universal time (UT1-UTC), play a central role in functions such as monitoring the Earth’s rotation and high-precision navigation and positioning. Variations in ERPs reflect not only the overall state of movement of the Earth, but also the interactions among the atmosphere, ocean, and land on the spatial and temporal scales. In this paper, we estimated ERP series based on very long baseline interferometry (VLBI) observations between 2011–2020. The results show that the average root mean square errors (RMSEs) are 0.187 mas for PMX, 0.205 mas for PMY, and 0.022 ms for UT1-UTC. Furthermore, to explore the high-frequency variations in more detail, we analyzed the polar motion time series spectrum based on fast Fourier transform (FFT), and our findings show that the Chandler motion was approximately 426 days and that the annual motion was about 360 days. In addition, the results also validate the presence of a weaker retrograde oscillation with an amplitude of about 3.5 mas. This paper proposes a hybrid prediction model that combines convolutional neural network (CNN) and long short-term memory (LSTM) neural network: the CNN–LSTM model. The advantages can be attributed to the CNN’s ability to extract and optimize features related to polar motion series, and the LSTM’s ability to make medium- to long-term predictions based on historical time series. Compared with Bulletin A, the prediction accuracies of PMX and PMY are improved by 42% and 13%, respectively. Notably, the hybrid CNN–LSTM model can effectively improve the accuracy of medium- and long-term polar motion prediction.

Neural ODE Differential Learning and Its Application in Polar Motion Prediction

AbstractThis paper introduces a new learning algorithm for accurate, physically driven time series prediction. The fundamental assumption behind the method is that the phenomena follow Ordinary Differential Equations. We investigate the general case where the time series follows an ODE of degree . The resulting method is a learning algorithm based on the finite differences between the values of time series. We present the application of the method in the field of geodesy for polar motion prediction, the main objective of the present paper. We show that in this application, the linear form of the method is sufficient and offers competitive predictive performance. We present a baseline solution, in which we use historical polar motion time series from 1976 to predict up to the year 2020. The prediction horizon in this case is short‐term (up to 10 days into the future). In addition, we compare the prediction accuracy in the short‐term horizon with the results of the best performing model in the first Earth Orientation Prediction Comparison Campaign. On average, a 53% improvement in prediction performance is achieved. In further analyses, we compare the prediction accuracy for both short‐term and long‐term against the results of state‐of‐the‐art methods, namely Multichannel Singular Spectrum Analysis, and a combination of Singular Spectrum Analysis and Copula sampling. We show that the proposed method in this paper can outperform the mentioned two methods in both short and long‐term horizons, with an average improvement of the prediction performance of 54% and 52%, respectively.

Analysis of Earth’s polar motion and length of day trends in comparison with estimates using second degree stokes coefficients from satellite gravimetry

We analyzed the impacts of data span on trend estimates using Earth’s long-term polar motion time series, 1846-present, and using methodologies including singular spectrum analysis, and Panteleev’s filter to mitigate the time series containing transient signals. Our results show that the fluctuations of the mean rotational pole position, the Markowitz wobble, cannot be fully explained by the oceanic and atmospheric excitations. However, there exists plausible similarity with the variations of amplitudes of the Chandler wobble. To explain the abrupt deviation of the mean pole from the previous state after year 2000, we first compute Earth rotation excitations, using the temporal variations of the second-degree Stokes coefficients, C 21 , S 21 , estimated from GRACE, GRACE Follow-On and Satellite Laser Ranging (SLR), 2002–2021. We then compare their trend estimates with that of the Earth’s polar motion, and conclude that the drift of the pole is consistent with the climate-induced mass redistributions within the Earth system during the past two decades. However, the observed trend is not in exact agreement with the prediction values using contemporary glacial isostatic adjustment (GIA) process forward models. The analysis of the J 2 variations since 1976 from SLR and the corresponding length of day (LOD) changes, reveals a clear trend reversal around the year 2000. However, the observed J 2 variations can only explain ∼ 5 % of the long-term LOD changes. The remaining decadal signal in the LOD, usually accounted for by the angular momentum exchange at the core-mantle boundary, is observed to be anti-correlated with the Earth surface temperature anomaly. The geophysical explanation on these relationships remains elusive, and necessitates future studies.

Analysis of 25 Years of Polar Motion Derived from the DORIS Space Geodetic Technique Using FFT and SSA Methods.

Polar motion (PM) has a close relation to the Earth’s structure and composition, seasonal changes of the atmosphere and oceans, storage of waters, etc. As one of the four major space geodetic techniques, doppler orbitography and radiopositioning integrated by satellite (DORIS) is a mature technique that can monitor PM through precise ground station positioning. There are few articles that have analyzed the PM series derived by the DORIS solution in detail. The aim of this research was to assess the PM time-series based on the DORIS solution, to better capture the time-series. In this paper, Fourier fast transform (FFT) and singular spectrum analysis (SSA) were applied to analyze the 25 years of PM time-series solved by DORIS observation from January 1993 to January 2018, then accurately separate the trend terms and periodic signals, and finally precisely reconstruct the main components. To evaluate the PM time-series derived from DORIS, they were compared with those obtained from EOP 14 C04 (IAU2000). The results showed that the RMSs of the differences in PM between them were 1.594 mas and 1.465 mas in the X and Y directions, respectively. Spectrum analysis using FFT showed that the period of annual wobble was 0.998 years and that of the Chandler wobble was 1.181 years. During the SSA process, after singular value decomposition (SVD), the time-series was reconstructed using the eigenvalues and corresponding eigenvectors, and the results indicated that the trend term, annual wobble, and Chandler wobble components were accurately decomposed and reconstructed, and the component reconstruction results had a precision of 3.858 and 2.387 mas in the X and Y directions, respectively. In addition, the tests also gave reasonable explanations of the phenomena of peaks of differences between the PM parameters derived from DORIS and EOP 14 C04, trend terms, the Chandler wobble, and other signals detected by the SSA and FFT. This research will help the assessment and explanation of PM time-series and will offer a good method for the prediction of pole shifts.

Application of the AR‐z Spectrum to Polar Motion: A Possible First Detection of the Inner Core Wobble and Its Implications for the Density of Earth's Core

AbstractThe density jump at the inner core boundary of the Earth is poorly constrained by seismic data. One sensitive measure of the jump is the eigenperiod of the inner core wobble (ICW). Although theoretical studies on the ICW have been ongoing for decades, no observed ICW signal has been reported. To increase the detection level, here we use a recently proposed AR‐z method to analyze the 1960–2017 polar motion time series. We finally identify a +8.7 ± 0.2 year prograde motion with an ~2.67 ± 0.04 milliarcsecond amplitude. After confirming that this signal is almost a stationary oscillation and that atmospheric/oceanic/hydrological effects cannot seem to excite this signal, we carefully suggest that this ~8.7 year signal is possibly the ICW. According to this suggestion, we reconstruct a new 1‐D density model with a 507 ± 15 kg/m3 inner core boundary density jump. This new model can also provide a better fit than Preliminary Reference Earth model to the seismic overtone 2S1.

Possible enhancement of Earth’s polar motion predictions using a wavelet-based preprocessing procedure

Earth’s polar motion predictions are essential in near real-time applications including spacecraft navigation and satellite orbit determinations and are also important for geophysics studies. It has previously been demonstrated that the basic problem in predicting a polar motion time-series is to treat separately its low- and high-frequency components. In this paper, this problem is solved by the combination of wavelet transform with the least-squares (LS) extrapolation and autoregressive (AR) method. Wavelet transform is first employed to separate the deterministic low-frequency components including the Chandler and annual wobbles and irregular short-period high-frequency components of a polar motion time-series. Then the deterministic parts are predicted using LS extrapolation of models for the linear trend, Chandler and annual wobbles, while the remaining irregular parts together with LS residuals are forecasted using the AR stochastic method. The polar motion predictions are computed as a combination of the LS extrapolation and AR prediction. The results show that the polar motion predictions up to 360 days in the future obtained by this approach can be remarkably enhanced due to the use of the wavelet-based preprocessing procedure. We conclude that the separate treatment of low- and high-frequency components is of use to enhance polar motion predictions.

Polar motion prediction using the combination of SSA and Copula-based analysis

The real-time estimation of polar motion (PM) is needed for the navigation of Earth satellite and interplanetary spacecraft. However, it is impossible to have real-time information due to the complexity of the measurement model and data processing. Various prediction methods have been developed. However, the accuracy of PM prediction is still not satisfactory even for a few days in the future. Therefore, new techniques or a combination of the existing methods need to be investigated for improving the accuracy of the predicted PM. There is a well-introduced method called Copula, and we want to combine it with singular spectrum analysis (SSA) method for PM prediction. In this study, first, we model the predominant trend of PM time series using SSA. Then, the difference between PM time series and its SSA estimation is modeled using Copula-based analysis. Multiple sets of PM predictions which range between 1 and 365 days have been performed based on an IERS 08 C04 time series to assess the capability of our hybrid model. Our results illustrate that the proposed method can efficiently predict PM. The improvement in PM prediction accuracy up to 365 days in the future is found to be around 40% on average and up to 65 and 46% in terms of success rate for the {hbox{PM}}_{x} and {hbox{PM}}_{y}, respectively.

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.

One hybrid model combining singular spectrum analysis and LS + ARMA for polar motion prediction

Accurate real-time polar motion parameters play an important role in satellite navigation and positioning and spacecraft tracking. To meet the needs for real-time and high-accuracy polar motion prediction, a hybrid model that integrated singular spectrum analysis (SSA), least-squares (LS) extrapolation and an autoregressive moving average (ARMA) model was proposed. SSA was applied to separate the trend, the annual and the Chandler components from a given polar motion time series. LS extrapolation models were constructed for the separated trend, annual and Chandler components. An ARMA model was established for a synthetic sequence that contained the remaining SSA component and the residual series of LS fitting. In applying this hybrid model, multiple sets of polar motion predictions with lead times of 360days were made based on an IERS 08 C04 series. The results showed that the proposed method could effectively predict the polar motion parameters.

지구의 극운동에 대한 최소제곱법 분석: 스마일리의 추측에 상반됨

Abstract: From the Earth’s polar motion time series (IERS 08 C04, since 1981), after removal of seasonal variation byband-pass filtering, we acquired Earth’s free Eulerian motion (Chandler wobble) time series. By successive least squareerror fittings on it, we analyzed amplitude and phase variation of Chandler wobble. We attempted to identify anyprecursory behavior of the pole before large earthquakes but only to fail. Unlike Smylie’s conjecture there was noappreciable motion of the Earth’s pole detected at around the each times of recent six largest earthquakes of magnitudeover 8.5. Keywords: Polar Motion, Earthquake Precursor요약: 지구의 극운동 자료(IERS 08 C04, 1981년 이후)로부터 1 년 주기 성분을 제거한 뒤 얻어진 시계열에 대하여 최소제곱법을 적용하여 지구의 자유-오일러 운동(찬들러 워블) 시계열을 얻었으며, 이의 진폭과 위상 등의 변화를 조사하였다. 한편 대지진 발생 전의 전조현상으로서의 극의 움직임을 찾아보았으나 발견하지 못하였다. 스마일리의 추측과는 달리,규모 8.5 이상의 최근 지진 여섯 개의 발생시점 부근에서 극운동의 특별한 징후가 없었다. 주요어: 극운동, 지진전조현상 Introduction The Earth’s spin rotation is quite stable: however, the poleshows ceaseless and complicated movement. The precessionof the Earth, which is caused by luni-solar torque exerted onthe Earth’s equatorial bulge, exists in large amount. In fact, themagnitude of precession (50 arcsec per year) surpasses allother kinds of variations in the Earth’s spin rotation. Wehereby are concerned with the polar motion, which is torque-free motion aside from the forced precession and nutation, andrefers to the slow relative change of the Earth’s rotational polewith respect to the observer on the Earth itself. Chandlerwobble and annual wobble, both of a few hundred milliarcsecradii, are the dominant periodic components of the Earth’spolar motion, and they together show conspicuous beating of6.4 year period (Fig. 1a-b). While annual wobble is a drivenmotion, Chandler wobble can be regarded as authenticoscillating mode of the spinning Earth, and its period is about433 days. Aside from these periodic variations, there existssteady pole drift about 10 centimeters per year along 70W(Fig. 1a-b) due to ‘glacial isostatic adjustment’, which is alsoknown as ‘post glacial rebound.’ It is obvious that seasonalvariations of diverse processes of the Earth should beassociated with the annual periodic components of the polarmotion. Likewise there must be certain energy source to excite

Search for the 531-day-period wobble signal in the polar motion based on EEMD

Abstract. In this study, we use a nonlinear and non-stationary time series analysis method, the ensemble empirical mode decomposition method (EEMD), to analyze the polar motion (PM) time series (EOP C04 series from 1962 to 2013) to find a 531-day-period wobble (531 dW) signal. The 531 dW signal has been found in the early PM series (1962–1977), but cannot be found in the recent PM series (1978–2013) using conventional analysis approaches. By virtue of the demodulation feature of EEMD, the 531 dW can be confirmed to be present in PM based on the differences of the amplitudes and phases between different intrinsic mode functions. Results from three sub-series divided from the EOP C04 series show that the period of the 531 dW is subject to variations, in the range of 530.9–524 days, and its amplitude is also time-dependent (about 2–11 mas). Synthetic tests are carried out to explain why the 531 dW can only be observed in recent 30-year PM time series after using EEMD. The 531 dW is also detected in the two longest available superconducting gravimeter (SG) records, which further confirms the presence of the 531 dW. The confirmation of the 531 dW existence could be significant in establishing a more reasonable Earth rotation model and may effectively contribute to the prediction of the PM and its mechanism interpretation.

Separation of atmospheric, oceanic and hydrological polar motion excitation mechanisms based on a combination of geometric and gravimetric space observations

The goal of our study is to determine accurate time series of geophysical Earth rotation excitations to learn more about global dynamic processes in the Earth system. For this purpose, we developed an adjustment model which allows to combine precise observations from space geodetic observation systems, such as Satellite Laser Ranging (SLR), Global Navigation Satellite Systems, Very Long Baseline Interferometry, Doppler Orbit determination and Radiopositioning Integrated on Satellite, satellite altimetry and satellite gravimetry in order to separate geophysical excitation mechanisms of Earth rotation. Three polar motion time series are applied to derive the polar motion excitation functions (integral effect). Furthermore we use five time variable gravity field solutions from Gravity Recovery and Climate Experiment to determine not only the integral mass effect but also the oceanic and hydrological mass effects by applying suitable filter techniques and a land–ocean mask. For comparison the integral mass effect is also derived from degree 2 potential coefficients that are estimated from SLR observations. The oceanic mass effect is also determined from sea level anomalies observed by satellite altimetry by reducing the steric sea level anomalies derived from temperature and salinity fields of the oceans. Due to the combination of all geodetic estimated excitations the weaknesses of the individual processing strategies can be reduced and the technique-specific strengths can be accounted for. The formal errors of the adjusted geodetic solutions are smaller than the RMS differences of the geophysical model solutions. The improved excitation time series can be used to improve the geophysical modeling.

Methodology for the combination of sub-daily Earth rotation from GPS and VLBI observations

A combination procedure of Earth orientation parameters from Global Positioning System (GPS) and Very Long Baseline Interferometry (VLBI) observations was developed on the basis of homogeneous normal equation systems. The emphasis and purpose of the combination was the determination of sub-daily polar motion (PM) and universal time (UT1) for a long time-span of 13 years. Time series with an hourly resolution and a model for tidal variations of PM and UT1-TAI (dUT1) were estimated. In both cases, 14-day nutation corrections were estimated simultaneously with the ERPs. Due to the combination procedure, it was warranted that the strengths of both techniques were preserved. At the same time, only a minimum of de-correlating or stabilizing constraints were necessary. Hereby, a PM time series was determined, whose precision is mainly dominated by GPS observations. However, this setup benefits from the fact that VLBI delivered nutation and dUT1 estimates at the same time. An even bigger enhancement can be seen for the dUT1 estimation, where the high-frequency variations are provided by GPS, while the long term trend is defined by VLBI. The estimated combined tidal PM and dUT1 model was predominantly determined from the GPS observations. Overall, the combined tidal model for the first time completely comprises the geometrical benefits of VLBI and GPS observations. In terms of root mean squared (RMS) differences, the tidal amplitudes agree with other empirical single-technique tidal models below 4 μas in PM and 0.25 μs in dUT1. The noise floor of the tidal ERP model was investigated in three ways resulting in about 1 μas for diurnal PM and 0.07 μs for diurnal dUT1 while the semi-diurnal components have a slightly better accuracy.

GGOS-D: homogeneous reprocessing and rigorous combination of space geodetic observations

In preparation of activities planned for the realization of the Global Geodetic Observing System (GGOS), a group of German scientists has carried out a study under the acronym GGOS-D which closely resembles the ideas behind the GGOS initiative. The objective of the GGOS-D project was the investigation of the methodological and information-technological realization of a global geodetic-geophysical observing system and especially the integration and combination of the space geodetic observations. In the course of this project, highly consistent time series of GPS, VLBI, and SLR results were generated based on common state-of-the-art standards for modeling and parameterization. These series were then combined to consistently and accurately compute a Terrestrial Reference Frame (TRF). This TRF was subsequently used as the basis to produce time series of station coordinates, Earth orientation, and troposphere parameters. In this publication, we present results of processing algorithms and strategies for the integration of the space-geodetic observations which had been developed in the project GGOS-D serving as a prototype or a small and limited version of the data handling and processing part of a global geodetic observing system. From a comparison of the GGOS-D terrestrial reference frame results and the ITRF2005, the accuracy of the datum parameters is about 5–7 mm for the positions and 1.0–1.5 mm/year for the rates. The residuals of the station positions are about 3 mm and between 0.5 and 1.0 mm/year for the station velocities. Applying the GGOS-D TRF, the offset of the polar motion time series from GPS and VLBI is reduced to 50 μas (equivalent to 1.5 mm at the Earth’s surface). With respect to troposphere parameter time series, the offset of the estimates of total zenith delays from co-located VLBI and GPS observations for most stations in this study is smaller than 1.5 mm. The combined polar motion components show a significantly better WRMS agreement with the IERS 05C04 series (96.0/96.0 μas) than VLBI (109.0/100.7 μas) or GPS (98.0/99.5 μas) alone. The time series of the estimated parameters have not yet been combined and exploited to the extent that would be possible. However, the results presented here demonstrate that the experiences made by the GGOS-D project are very valuable for similar developments on an international level as part of the GGOS development.

Polar motion modeling, analysis, and prediction with time dependent harmonic coefficients

A time dependent amplitude model was proposed for the analysis and prediction of polar motion time series. The formulation was implemented to analyze part of the new combined solution, EOP (IERS) C 04, daily polar motion time series of 14 years length using a statistical model with first order autoregressive disturbances. A new solution approach, where the serial correlations of the disturbances are eliminated by sequentially differencing the measurements, was used to estimate the model parameters using weighted least squares. The new model parsimoniously represents the 14-year time series with 0.5 mas rms fit, close to the reported 0.1 mas observed pole position precisions for the x and y components. The model can also predict 6 months into the future with less than 4 mas rms prediction error for both polar motion components, and down to sub mas for one-step ahead prediction as validated using a set of daily time series data that are not used in the estimation.

The need for a new definition of a conventional international origin

The conventional international origin (CIO), established from observations made a century ago, is not directly related to observations by modern space-geodetic techniques. Both the greater precision of these techniques and improved knowledge of the structure of the Earth justify the need for a new CIO. We analyze recent polar motion time-series (VLBI, SLR, and GPS) to test estimators that might be used to establish such a new conventional origin. This new origin would be defined as the barycenter of the motion of the pole for a specific epoch. Consistency among the series examined is of the order of 2 milli-arc-seconds. A drift model can be employed in the analysis of specific series to establish an origin as the barycenter at a specific epoch, rather than the midpoint of the series. As an example, we estimate a “Conventional International Reference Origin” for the year 2000.0, using polar motion series that began in 1984.

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.

Autocovariance prediction of complex-valued polar motion time series

Autocovariance prediction formulae of complex-valued time series are derived and applied to predict pole coordinate data. The pole coordinate data are transformed into the polar motion radius and angular distance and afterwards their predictions are computed. The predictions of the radius and angular distance are transformed into the pole coordinate predictions. The radius and its prediction refer to the 6-year running mean of polar motion data and its autocovariance predictions. The prediction errors depend mostly on the starting epochs of prediction due to irregular variations of polar motion. The mean prediction errors of x and y pole coordinate data are slightly greater than those computed by the current method of polar motion prediction carried out in the IERS Sub-Bureau for Rapid Service and Prediction.

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.