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The results of numerical modeling of the values of the Chandler wobble period of Mars have been presented for a set of internal structure models that satisfy all currently available observable data: geodetic (average radius, mass, moment of inertia, tidal Love number k2) and crustal thickness and core radius values obtained from seismic data processing. Andrade rheology was used to take into account inelasticity when calculating model values of the tidal Love number k2 and the Chandler wobble period. It has been shown how the model values of the Love number k2 and the Chandler period depend on the Andrade rheological parameter and the adopted viscosity distribution. [ABSTRACT FROM AUTHOR]

A structural analysis of the time series of pole coordinate changes (version C01 IERS) for the period of 1975.0–2011.0 has been performed based on the nonlinear least squares method. Average estimates of the parameters of the main components of the pole movement—namely, Chandler, annual, and semiannual wobbles—are obtained for this period. The obtained values of periods T and amplitudes A of the main components are as follows: T = 433.49 ± 0.22 days and A = 160 ± 3 mas for the Chandler oscillations; T = 365.19 ± 0.37 days and A = 93 ± 5 mas for the annual oscillations; and T = 183.03 ± 0.34 days and A = 4 ± 2 mas for the semiannual oscillations. Changes in the pole coordinates are examined in the time series when focusing on the manifestation of Chandler oscillations. The dynamics of oscillation parameters (including amplitude, period, phase, and Q factor) is studied. Changes in the Chandler oscillation parameters show their interdependence. The correlation coefficient between phase and period variations is +0.94, and a similar relationship is observed between phase and amplitude variations with a correlation coefficient of +0.88. It is shown that the phase change precedes the changes in the amplitude and in the period. This behavior of the parameters of the Chandler wobble suggests that changes in the period and in the amplitude should be considered a consequence of the phase changes. It is revealed that an increase in the amplitude of Chandler oscillations correlates with a decrease in the attenuation decrement with a correlation coefficient of –0.98. These findings align with the statistical patterns articulated by Melchior, which are indicative of (a) inconstancy of the period of Chandler oscillations over time and (b) proportional changes between the period and the amplitude of oscillations. Thus, preference should be given to the one-component complicated model of the Chandler pole movement with a variable period for the studied period of time. [ABSTRACT FROM AUTHOR]

We reconstructed the Chandler Wobble (CW) from 1962 to 2022 by combining the Eigen-oscillator excited by geophysical fluids of atmospheric and oceanic angular momentums (AAM and OAM). The mass and motion terms for the AAM are further divided with respect to the land and ocean domains. Particular attention is placed on the time span from 2012 to 2022 in relation to the observable reduction in the amplitude of the CW. Our research indicates that the main contributor to the CW induced by AAM is the mass term (i.e., the pressure variations over land). Moreover, the phase of the AAM-induced CW remains relatively stable during the interval of 1962–2022. In contrast, the phase of the OAM-induced CW exhibits a periodic variation with a cycle of approximately 20 years. This cyclic variation would modulate the overall amplitude of the CW. The noticeable amplitude deduction over the past ten years can be attributed to the evolution of the CW driven by AAM and OAM, toward a state of cancellation. These findings further reveal that the variation in the phase difference between the CW forced by AAM and OAM, may be indicative of changes in the interaction between the solid Earth, atmosphere, and ocean. [ABSTRACT FROM AUTHOR]

The principal moments of inertia (PMIs) with the principal axes are usually taken as the dynamic figure parameters of Mars; they can be deduced from satellite-observed degree-two gravitational potentials in recent global gravity models and from the dynamic ellipticities resulting from precession observations. These PMIs are natural and significant for the geodetic, geophysical, and geodynamic problems of Mars, which are functions of internal density distributions. In this study, a closed and concise formula for determining the PMIs of the entire planet and its core was developed based on the second invariants of gravity and a multipole expansion. We deduced the polar oblateness J2 and the equatorial ellipticity J22 of Mars to be 1.9566 × 10−3 and 6.3106 × 10−5, respectively. The preferred principal moments of inertia of Mars are A = 2.66589 × 1036 kg·m2, B = 2.66775 × 1036 kg·m2, and C = 2.68125 × 1036 kg·m2. These values indicate that Mar is slightly triaxial. The equatorial principal moment of inertia of the Martian core is 1.46008 × 1035 kg·m2, accounting for ~5.47% of the planet’s PMI; this result is critical for investigating the density and size of the core of Mars, and the planet’s free core nutation.

The paper is devoted to the analysis of two components of the Earth's rotation, Chandler wobble (CW) and free core nutation (FCN). They are oscillations with near-constant periods but variable amplitude and phase. The variations of the amplitude and phase of the CW and FCN have already been considered in the literature, and both showed similar behavior such as a recent significant decrease of the amplitude and large phase change. However, the CW and FCN amplitude and phase variations are, to a large extent, predicted for the current epochs, and their today's variations need regular updates with obtaining new observations. In this work, the CW and FCN parameters have been re-computed using the latest data and compared with the data published earlier. It was found that the currently obtained amplitude and phase variations generally agreed with the data published earlier. The main difference is that the epochs of the current minimum of amplitude and phase jump or both CW and FCN happened somewhat later than was predicted in previous publications. The delay is about two years for the CW relative to the prediction made in 2010 and about one year for the FCN with respect to the prediction made in 2022. [ABSTRACT FROM AUTHOR]

The period and quality factor of the Chandler wobble (CW) are useful for constraining the Earth's internal structure properties, such as the mantle elasticity. It has been shown that the CW is mainly excited by a combination of atmospheric and oceanic processes; hence based on a deconvolution method, we can remove them from the excitation sequence of the CW to estimate its period P and quality factor Q. We finally re-estimate P = 432.3 ± 0.8 days and Q = 85 ± 15 for the CW. Based on those two estimates, we investigate the relationship between the geomagnetic jerks and the excitation sequences of the CW. The geomagnetic jerks or jerk bounds are well consistent with the sudden changes of the excitation sequences of the CW. This demonstrates that the geomagnetic jerks could be a possible excitation source of the CW. It is crucial for understanding the excitation of the CW and for deeper geophysical insights into the geomagnetic jerks.

The period TCW and quality factor QCW of the Chandler wobble (CW) as well as polar motion (PM) transfer functions are all determined by the Earth’s layered structure, mass distribution, elasticity, rheology and energy dissipation, via the Earth’s dynamic figure parameters and complex degree-2 Love numbers. However, most previous studies used geophysical excitations derived from real-valued PM transfer functions to invert for TCW and QCW, thus leading to results that are not self-consistent. By separating the observed PM into the freely decaying CW and the excited PM, a traverse-based method is proposed to search values of TCW and QCW that can fit both sides simultaneously, yielding the self-consistent estimates of TCW = 430.4 mean solar days and QCW = 130. This implies the degree-2 tidal Love number k = 0.35011 − 0.00226i and load Love number k' = − 0.36090 + 0.00233i, and the PM transfer functions TNL = 1.80001 − 0.00692i (non-loading) and TL = 1.15040 − 0.00023i (loading) valid at the Chandler period. [ABSTRACT FROM AUTHOR]

To analyze quasi‐periodic signals (with time‐varying complex amplitudes/frequencies) typically contained in geophysical observables is a quest that has seen continual advances in numerical techniques over the decades. In this study, based on transient z‐pole estimation (in Hankel matrices), a state‐space analysis referred to as Hankel Spectral Analysis (HSA) was developed. Based on the Hankel total least squares and incorporating truncated singular value decomposition and its shift‐invariant property, the HSA aims to decompose the closely spaced sinusoids robustly and orthogonally. Upon using a sliding windows process, the HSA can be used for decomposing and analyzing quasi‐periodic signals, in the support of consecutive parameter spectra {Ai, fi, θi}. Based on a series of experiments, we first confirmed the superiority of the HSA for decomposing different signal constituents (e.g., amplitude‐frequency modulation, mutation, and episodic signals). In real applications, as examples, we use HSA to analyze the polar motion (PM) and Earth's dynamic oblateness (ΔJ2). For the PM, we obtained the time‐varying Chandler wobble (CW) and Annual wobble, and first confirmed that there are four phase jumps in the CW since the 1900s; we find that all of those phase jumps are synchronized by the sharp decrease of Chandler intensity and period, and their random excitation mechanism was discussed. For the ΔJ2, the 18.6 and 10.5 years signals were re‐extracted, and we found that its interannual‐to‐decadal oscillations contribute to multiple global gravity anomalies. These results indicate the great potential of the HSA in decomposing and extracting the recorded periodic/quasi‐periodic signals from geophysical observations. Plain Language Summary: In signal analysis, the refined extracting of the periodic and quasi‐periodic signals remains a considerable challenge. In this study, we propose a powerful time‐series separator, the Hankel Spectral Analysis (HSA), which supports instant structured detection that traditional methods cannot afford. We have designed multiple sets of simulations to show the HSA's well‐restored ability for various and complex quasi‐periodic signals. Although the HSA has some significant advantages, it has two fundamental limitations: (a) it is applicable only to data series which are physically constituted by a sum of harmonic components; (b) the number of harmonic components needs to know beforehand. In the real application, we focus on the study of polar motion (PM) and the Earth's dynamic oblateness (ΔJ2). Here, we mainly attempted to resolve their recent disputes about the Chandler wobbles (CW) phase jumps in PM and the long periodic signal in ΔJ2. Subsequently, the origins of CW phase jumps are re‐examined; and our results show that there is a particular consistency between the four‐phase jump events and the valleys of amplitude and period. In addition, we re‐extracted the ∼18.6 and 10.5 years signals in ΔJ2 and found that the famous "1998 anomaly" related to the fluctuations on interannual‐decadal timescales. Key Points: Hankel Spectrum Analysis is proposed for more exactly decomposing and parameterizing signals with time‐varying amplitudes/frequenciesMultiple phase jumps of the Chandler wobble accompanied by sudden drops in instantaneous amplitude and period are detectedWe access the Earth's oblateness ΔJ2 of the residual ∼18.6 years tidal signal, ∼10.5 years signal, and multiple global gravity anomaly peaks [ABSTRACT FROM AUTHOR]

By taking into account the variable free polar motion (PM) known as the Chandler wobble (CW) and irregular forced PM excited by quasi-periodic changes in atmosphere, oceans and land water (described by the data of effective angular momenta EAM), we propose a short-term PM forecast method based on the Holt-Winters (HW) additive algorithm (termed as the HW-VCW method, with VCW denoting variable CW). In this method, the variable CW period is determined by minimizing the differences between PM observations and EAM-derived PM for every 8-year sliding timespan. Compared to the X- and Y-pole forecast errors (ΔPMX and ΔPMY) of the International Earth Rotation and Reference Systems Service (IERS) Bulletin A, our results derived from operational EAM can reduce ΔPMX by up to 38.4% and ΔPMY by up to 34.3% for forecasts ranging from 1 to 30 days. Further, we prove that using EAM forecast instead of operational EAM in the HW-VCW method can achieve similar accuracies. [ABSTRACT FROM AUTHOR]

SUMMARY: Accurate determination of the Chandler wobble (CW) period (T CW) and quality factor (Q CW) is of great significance to our understanding of the Earth's dynamic figure parameters, elasticity, rheology and energy dissipation. T CW and Q CW were typically determined in the time domain using the digital filter designed by Wilson; however, we developed an alternative method to estimate T CW in the frequency domain. We adopted the frequency domain expression solving the Liouville equation for polar motion (eq. 3 in the following) rather than the time domain to separate the free-damping CW and excited parts. Next, we substituted various excitation functions derived from the outputs of several general circulation models and selected monthly gravity models into the above frequency domain expression; hence we estimate T CW. The preferred T CW value using this method and the least difference combination mgm90 model is 430.4 ± 2.0 mean solar days. Comparing with previous studies within the error range, our results provide an independent way of estimating T CW. [ABSTRACT FROM AUTHOR]

The Earth's rotation is nonuniform: the position of the pole drifts and describes circles with the annual and Chandler (433 days) periods; the Earth's rotation velocity also varies. In the beginning of 2000s, the amplitude of the Chandler wobble began to decrease and in 2017–2020 reached a historical low comparable only with the low of late 1920s. We extract the Chandler time series by different methods (removal of the trend and annual oscillation, Panteleev filtering, and others) from series of EOP C04 and EOP C01 data on the polar motion and demonstrate that its phase also varies from early 2010s. Such variation in the phase occurred in the 1920s–1940s and equaled rad; at present, it approaches 2 rad. Possible interpretations of the phenomena, their manifestations in the polar motion spectrum, and probable causes are discussed. [ABSTRACT FROM AUTHOR]

Being a torque free motion of the rotating Earth, Chandler wobble is the major component in the Earth’s polar motion with amplitude about 0.05-0.2 arcsec and period about 430-435 days. Free core nutation, also called nearly diurnal free wobble, exists due to the elliptical core-mantle boundary in the Earth and takes almost the whole part of un-modelled variation of the Earth’s pole in the celestial sphere beside precession and nutation. We hereby present a brief summary of their theories and report their recent features acquired from updated datasets (EOP C04 and ECMWF) by using Fourier transform, modelling, and wavelet analysis. Our new findings include (1) period-instability of free core nutation between 420 and 450 days as well as its large amplitude-variation, (2) re-determined Chandler period and its quality factor, (3) fast decrease in Chandler amplitude after 2010.

Physics, chemistry, biology or finance are just some examples out of the many fields where complex Ornstein-Uhlenbeck (OU) processes have various applications in statistical modeling. They play role e.g. in the description of the motion of a charged test particle in a constant magnetic field or in the study of rotating waves in time-dependent reaction diffusion systems, whereas Kolmogorov used such a process to model the so-called Chandler wobble, the small deviation in the Earth's axis of rotation. A common problem in these applications is deciding how to choose a set of a sample locations in order to predict a random process in an optimal way. We study the optimal design problem for the prediction of a complex OU process on a compact interval with respect to integrated mean square prediction error (IMSPE) and entropy criteria. We derive the exact forms of both criteria, moreover, we show that optimal designs based on entropy criterion are equidistant, whereas the IMSPE based ones may differ from it. Finally, we present some numerical experiments to illustrate selected cases of optimal designs for small number of sampling locations. [ABSTRACT FROM AUTHOR]

In this paper we consider the statistical analysis of multivariate multiple nonlinear regression models with correlated errors, using Finite Fourier Transforms. Consistency and asymptotic normality of the weighted least squares estimates are established under various conditions on the regressor variables. These conditions involve different types of scalings, and the scaling factors are obtained explicitly for various types of nonlinear regression models including an interesting model which requires the estimation of unknown frequencies. The estimation of frequencies is a classical problem occurring in many areas like signal processing, environmental time series, astronomy and other areas of physical sciences. We illustrate our methodology using two real data sets taken from geophysics and environmental sciences. The data we consider from geophysics are polar motion (which is now widely known as "Chandlers Wobble"), where one has to estimate the drift parameters, the offset parameters and the two periodicities associated with elliptical motion. The data were first analyzed by Arato, Kolmogorov and Sinai who treat it as a bivariate time series satisfying a finite order time series model. They estimate the periodicities using the coefficients of the fitted models. Our analysis shows that the two dominant frequencies are 12 h and 410 days. The second example, we consider is the minimum/maximum monthly temperatures observed at the Antarctic Peninsula (Faraday/Vernadsky station). It is now widely believed that over the past 50 years there is a steady warming in this region, and if this is true, the warming has serious consequences on ecology, marine life, etc. as it can result in melting of ice shelves and glaciers. Our objective here is to estimate any existing temperature trend in the data, and we use the nonlinear regression methodology developed here to achieve that goal. © 2007 Elsevier B.V. All rights reserved.

We investigate a possible link between polar motion and (i) seismic energy release and (ii) timing and intensity of eruptions at Mount Etna (Italy) for which a dense observational database is available. Our study suggests that the seismicity around Mount Etna increases during the largest excursions of the Earth rotation pole, which result from the constructive interference of the climate‐driven seasonal and Chandler wobbles. To a lesser extent, a similar link is detected between the pole tide and the erupted volume of magma. We infer that, by creating periodic changes in the state of stress in the crust through a variable centrifugal force and a periodic vertical displacement (pole tide), variations in polar motion influence eruption timing and magnitude and the rate of seismic energy release from volcanoes. This study provides new evidences on the sensitivity of volcanoes to external forcing. Plain Language Summary: The Earth surface is continuously deformed by a wide range of phenomena including lunisolar tides and centrifugal forces due to changes in the Earth's rotation rate and in the direction of its rotation axis. The latter class of deformations is typically of a centimeter on interannual timescales. On another side, some volcanoes appear to be sensitive to small perturbations of the crustal stress. We detected similar variability in the centrifugal forces arising from the Earth's rotation axis direction variability and Mount Etna seismicity and erupted magma volume. This suggests that the variation of the Earth's rotation axis direction plays a significant role as modulator of the activity of Mount Etna. Key Points: Seismicity and eruption timing and magnitudes at Mount Etna show interannual variations in phase with the solid Earth pole tideEarth's Chandler and annual wobbles may alter the state of stress of the Earth's crust at interannual to decadal timescales [ABSTRACT FROM AUTHOR]

Complex Ornstein–Uhlenbeck (OU) processes have various applications in statistical modelling. They play role e.g. in the description of the motion of a charged test particle in a constant magnetic field or in the study of rotating waves in time-dependent reaction diffusion systems, whereas Kolmogorov used such a process to model the so-called Chandler wobble, small deviation in the Earth’s axis of rotation. In these applications parameter estimation and model fitting is based on discrete observations of the underlying stochastic process, however, the accuracy of the estimation strongly depend on the observation points. This paper studies the properties of D-optimal designs for estimating the parameters of a complex OU process with a trend. In special situations we show that in contrast with the case of the classical real OU process, a D-optimal design exists not only for the trend parameter, but also for joint estimation of the covariance parameters, moreover, these optimal designs are equidistant. [ABSTRACT FROM AUTHOR]

Sea level changes with a period of 14 months (P14) are usually associated with the pole tide, which is the oceanic response to the Chandler wobble in the Earth's axis of rotation. The amplitudes of these changes in the Baltic Sea are anomalously large, much larger than follows from the equilibrium pole tide theory. It appears that the oscillations are related to meteorological forcing rather than to pole motions. To examine the P14 properties, we used long-term (48-213 years) tide gauge records from 77 stations located in the Baltic Sea and adjacent area of the North Sea. High-resolution sea level spectra revealed a cluster of peaks with periods from 410 to 450 days. The temporal variations in the P14 sea level amplitudes and peak frequencies are found to be considerable and poorly correlated with the modulation of the Earth's pole motions. In contrast, our findings for 1871-2011 demonstrate strong resemblance between temporal (year-to-year) variations of the P14 peak and zonal wind, confirming earlier results of Ekman (1996) and O'Connor et al. (2000), who indicated the major role of the zonal wind in formation of the 14-month oscillations in the Baltic and North seas. The integral amplitudes of the P14 oscillations in the Baltic Sea gradually increase eastward from the entrance (the Danish straits) to the farthest ends of the sea with the largest amplitudes (up to 6.5 cm) observed at the very head of the Gulf of Finland. Similar P14 amplification toward the west coast of Denmark is also observed along the south coast of the North Sea. The geographical distribution of the P14 in these two regions are almost the same as of the seasonal sea level oscillations, indicating presumably similar formation mechanisms. Extensive shallow-water areas in the Baltic Sea and southeastern part of the North Sea, combined with intense meteorological forcing, appear to be two key factors responsible for the formation of anomalously strong 14-month oscillations in these regions. [ABSTRACT FROM AUTHOR]

Polar motion consists of both free (Chandler wobble, with approximately 14-month period) and forced components. The latter are caused by different excitations of geophysical origin. Very long-periodic (or secular) part is most probably due to post-glacial rebound, shorter periodic part (with dominant annual period) are caused mainly by motions of the atmosphere and oceans. Recently it was also proposed that impulse-like excitations due to geomagnetic jerks might be responsible for rapid changes of the amplitude and phase of Chandler wobble. In order to precisely determine the parameters of the free part, it is necessary to consider all these influences. We use the IERS combined solution C04 together with ERA atmospheric/oceanic excitations in the interval 1974.0–2014.0, and also additional excitations due to nine geomagnetic jerks, registered during this interval, to determine the period and quality factor of Chandler wobble, free from these geophysical effects. We obtained solutions for three different time intervals: 1974.0–1994.0, 1994.0–2014.0, and 1974.0–2014.0. The estimated values of Q -factor are much smaller if GMJ excitations are used in addition to atmospheric and oceanic ones, and they are determined with higher accuracy. Our preferred values, valid for the whole interval 1974.0–2014.0, are P = 432.86 ± 0.04 days and Q = 35.0 ± 0.3 . [ABSTRACT FROM AUTHOR]

The Chandler wobble (CW) and annual wobble (AW) are the two main components of polar motion, which are difficult to separate because of their very close periods. In the light of Fourier dictionary and basis pursuit method, a Fourier basis pursuit (FBP) spectrum is developed, which can reduce spectral smearing and leakage caused by the finite length of the time series. Further, a band-pass filtering method based on FBP spectrum (FBPBPF), which can effectively suppress the edge effect, is proposed in this paper. The simulation test results show that the FBPBPF method can effectively suppress the edge effect caused by spectral smearing and leakage and that its reconstruction accuracy at the boundary is approximately three times higher than the Fourier transform band-pass filtering method, which is based on Hamming windowed FFT spectrum, in extracting quasi-harmonic signals. The FBPBPF method is then applied to Earth's polar motion data during 1900-2015. Through analyzing the amplitude and period variations of CW and AW, and calculating the eccentricity variation of the AW, we found that: (1) the amplitude of the CW is currently at a historic minimum level, and it is even possible to diminish further until a complete stop; and (2) the eccentricity of the AW has a gradually decreased fluctuation during the last 116 years. [ABSTRACT FROM AUTHOR]

Observed polar motion consists of uniform circular motions at both positive (prograde) and negative (retrograde) frequencies. Generalized Euler-Liouville equations of Bizouard, taking into account Earth's triaxiality and asymmetry of the ocean tide, show that the corresponding retrograde and prograde circular excitations are coupled at any frequency. In this work, we reconstructed the polar motion excitation in the Chandler band (prograde and retrograde). Then we compared it with geophysical excitation, filtered out in the same way from the series of the Oceanic Angular Momentum (OAM) and Atmospheric Angular Momentum (AAM) for the period 1960-2000. The agreement was found to be better in the prograde band than in the retrograde one. [ABSTRACT FROM AUTHOR]

Satellite altimeter sea surface height observations include the geocentric displacements caused by the pole tide, namely the response of the solid Earth and oceans to polar motion. Most users of these data remove these effects using a model that was developed more than 20 years ago. We describe two improvements to the pole tide model for satellite altimeter measurements. Firstly, we recommend an approach that improves the model for the response of the oceans by including the effects of self-gravitation, loading, and mass conservation. Our recommended approach also specifically includes the previously ignored displacement of the solid Earth due to the load of the ocean response, and includes the effects of geocenter motion. Altogether, this improvement amplifies the modeled geocentric pole tide by 15 %, or up to 2 mm of sea surface height displacement. We validate this improvement using two decades of satellite altimeter measurements. Secondly, we recommend that the altimetry pole tide model exclude geocentric sea surface displacements resulting from the long-term drift in polar motion. The response to this particular component of polar motion requires a more rigorous approach than is used by conventional models. We show that erroneously including the response to this component of polar motion in the pole tide model impacts interpretation of regional sea level rise by $$\pm $$ 0.25 mm/year. [ABSTRACT FROM AUTHOR]

From the maps of regional contribution to atmospheric angular momentum (AAM) over the period 1948–2011 (NCEP/NCAR reanalysis data) time domain excitation in Chandler frequency band was extracted by Panteleev’s filtering method. This permits us to investigate the evolution of the regional atmospheric influence on Chandler wobble. It appears that the temperate latitudes bring the strongest inputs. For pressure term they are limited to continents, and highlight the role of Europe. For the wind term they mostly result from ocean area, encompassing in particular North Atlantic. A quasi-20 year cycle is found in the regional patterns of the atmospheric excitation. The integrated AAM is finally compared with the geodetic excitation reconstructed from the observed polar motion. [ABSTRACT FROM AUTHOR]