Free Core Nutation Period Research Articles

Deep ensemble geophysics-informed neural networks for the prediction of celestial pole offsets

SUMMARYCelestial Pole Offsets (CPO), denoted by dX and dY, describe the differences in the observed position of the pole in the celestial frame with respect to a certain precession-nutation model. Precession and nutation components are part of the transformation matrix between terrestrial and celestial systems. Therefore, various applications in geodetic science such as high-precision spacecraft navigation require information regrading precession and nutation. For this purpose, CPO can be added to the precession-nutation model to precisely describe the motion of the celestial pole. However, as Very Long Baseline Interferometry (VLBI)—currently the only technique providing CPO—requires long data processing times resulting in several weeks of latency, predictions of CPO become necessary. Here we present a new methodology named Deep Ensemble Geophysics-Informed Neural Networks (DEGINNs) to provide accurate CPO predictions. The methodology has three main elements: (1) deep ensemble learning to provide the prediction uncertainty; (2) broad-band Liouville equation as a geophysical constraint connecting the rotational dynamics of CPO to the atmospheric and oceanic Effective Angular Momentum (EAM) functions and (3) coupled oscillatory recurrent neural networks to model the sequential characteristics of CPO time-series, also capable of handling irregularly sampled time-series. To test the methodology, we use the newest version of the final CPO time-series of International Earth Rotation and Reference Systems Service (IERS), namely IERS 20 C04. We focus on a forecasting horizon of 90 days, the practical forecasting horizon needed in space-geodetic applications. Furthermore, for validation purposes we generate an independent global VLBI solution for CPO since 1984 up to the end of 2022 and analyse the series. We draw the following conclusions. First, the prediction performance of DEGINNs demonstrates up to 25 and 33 percent improvement, respectively, for dX and dY, with respect to the rapid data provided by IERS. Secondly, predictions made with the help of EAM are more accurate compared to those without EAM, thus providing a clue to the role of atmosphere and ocean on the excitation of CPO. Finally, free core nutation period shows temporal variations with a dominant periodicity of around one year, partially excited by EAM.

Estimation of free core nutation parameters and availability of computing options

The Earth's Free Core Nutation (FCN) causes Earth tides and forced nutation with frequencies close to the FCN that exhibit resonance effects. High-precision superconducting gravimeter (SG) and very long baseline interferometry (VLBI) provide good observation techniques for detecting the FCN parameters. However, some choices in data processing and solution procedures increase the uncertainty of the FCN parameters. In this study, we analyzed the differences and the effectiveness of weight function and ocean tide corrections in the FCN parameter detection using synthetic data, SG data from thirty-one stations, and the 10 celestial pole offset (CPO) series. The results show that significant discrepancies are caused by different computing options for a single SG station. The stacking method, which results in a variation of 0.24–5 sidereal days (SDs) in the FCN period (T) and 103-104 in the quality factor (Q) due to the selection of the weighting function and the ocean tide model (OTM), can effectively suppress this influence. The statistical analysis results of synthetic data shows that although different weight choices, while adjusting the proportion of diurnal tidal waves involved, do not significantly improve the accuracy of fitted FCN parameters from gravity observations. The study evaluated a series of OTMs using the loading correction efficiency. The fitting of FCN parameters can be improved by selecting the mean of appropriate OTMs based on the evaluation results. Through the estimation of the FCN parameters based on the forced nutation, it was found that the weight function P1 is more suitable than others, and different CPO series (after 2009) resulted in a difference of 0.4 SDs in the T and of 103 in the Q. We estimated the FCN parameters for SG (T = 430.4 ± 1.5 SDs and Q = 1.52 × 104 ± 2.5 × 103) and for VLBI (T = 429.8 ± 0.7 SDs, Q = 1.88 × 104 ± 2.1 × 103).

A Spectral Element Method to Compute Earth’s Free Core Nutation

The Free Core Nutation (FCN) is a rotational mode caused by non-alignment of the rotation axis of the core and of the mantle. Its period observed by VLBI and superconducting gravimetry is around 430 sidereal days (Sd) with precision of better than 1 Sd, while its “theoretical” period calculated by traditional approaches and a given Earth model ranges from 450 to 470 Sd. Their gap of about 30 Sd is significant compared with its observation precision. We propose a spectral element method to compute the period of FCN and obtain a period of 434 Sd which is very close to the observed value.

Relationship between free core nutation and geomagnetic jerks

Recent studies have indicated a correlation between Earth’s free core nutation (FCN) and geomagnetic jerks (GMJs). However, some uncertainties still need to be resolved before their relationship can be confirmed. The variations in the amplitude and phase of the FCN result from the comprehensive influence of the surface fluid layer and core–mantle couplings, which makes its correlation with GMJs difficult to verify. The FCN period mainly depends on the inertia coupling and the dissipative couplings (such as viscous, electromagnetic and topographic couplings) at the core–mantle boundary according to the theory of Earth rotation. Whatever the GMJ mechanism, it is most likely to affect the FCN by changing the core–mantle couplings. This study was conducted to effectively determine variations in the FCN period by considering atmospheric and oceanic effects, investigate any correlation between the two phenomena, and analyze how the FCN relates to GMJs. Using the normal time–frequency transform, we extracted signals in the nutation band from the atmospheric and oceanic angular momentum functions. We used the broadband Liouville equations to estimate the atmospheric and oceanic effects on nutation terms. Using a sliding window of 2 years, we fitted five nutation terms most affected by FCN resonance from the celestial pole offsets with FCN model removed. The FCN period variation was estimated by using weighted least square method. The results indicated a correlation between the FCN and GMJ. Further, we analyzed the relationship between the geomagnetic fluctuations and FCN based on both the core–mantle couplings and the possible GMJ mechanism.

Frequency dependence of the polar motion resonance

SUMMARY The nutation of the Celestial Intermediate Pole can be considered as a retrograde diurnal polar motion. As the common polar motion, it presents a resonance, but with period TPM and quality factor QPM differing from the ones characterizing the Chandler wobble (TCW = 430.2−431.6 d, QCW in the interval (56 255) according to Nastula & Gross): according to the nutation analysis presented in a separate paper, this period is about TPM = 380 d and the quality factor becomes −10. In this study, we aim to revisit the geophysical interpretation of this result. Two complementary factors account for the observed values: the non-equilibrium response of the ocean to the pole tide potential in the diurnal band, and the resonance of the solid Earth tide at the free core nutation period. This leads to a resonance of TPM in the vicinity of the free core nutation period, confirmed by estimates derived from nutation analysis.

Estimation of free core resonance parameters based on long-term strain observations in the diurnal frequency band

The free core nutation (FCN) is one of the Earth’s rotational eigenmodes, which is caused by the retrograde motion of the liquid core relative to the mantle. The FCN period and Q-factor are determined by the elastic properties of the core/mantle boundary (CMB) and their electromagnetic interaction. In the celestial coordinate system, the FCN period is about 430 days; in the Earth-fixed reference frame this effect manifests itself in the form of the free core resonance (FCR) whose frequency falls in the diurnal tidal band. FCR observation requires highly accurate measurement of the amplitudes and phases of the near-diurnal tidal waves. In particular, the parameter estimates for minor waves K1, P1, Ψ1, and Φ1 are critically important for evaluating the FCR effect, i.e., the period and decay of this resonant mode. The progress in the experimental study of FCR is mainly due to the accumulation of the data from superconducting gravimeters and VLBI; at the same time, also the data of the precision laser strainmeters were used. In this work, the FCR effect is studied based on the long-term strain precision records by two European stations: Baksan, Russia (laser interferometer–strainmeter with a measuring armlength of 75 m [Milyukov et al., 2005; 2007] and Gran Sasso, Italy (two perpendicular laser interferometer–strainmeters, BA and BC, each with a measuring armlength of 90 m [Amoruso and Crescentini, 2009]).

Estimation of Free Core Resonance Parameters Based on Long-Term Strain Observations in the Diurnal Frequency Band

The free core nutation (FCN) is one of the Earth’s rotational eigenmodes, which is caused by the retrograde motion of the liquid core relative to the mantle. The FCN period and Q-factor are determined by the elastic properties of the core/mantle boundary (CMB) and their electromagnetic interaction. In the celestial coordinate system, the FCN period is about 430 days; in the Earth-fixed reference frame this effect manifests itself in the form of the free core resonance (FCR) whose frequency falls in the diurnal tidal band. FCR observation requires highly accurate measurement of the amplitudes and phases of the near-diurnal tidal waves. In particular, the parameter estimates for minor waves K1, P1, Ψ1, and Φ1 are critically important for evaluating the FCR effect, i.e., the period and decay of this resonant mode. The progress in the experimental study of FCR is mainly due to the accumulation of the data from superconducting gravimeters and VLBI; at the same time, also the data of the precision laser strainmeters were used. In this work, the FCR effect is studied based on the long-term strain precision records by two European stations: Baksan, Russia (laser interferometer–strainmeter with a measuring armlength of 75 m [Milyukov et al., 2005; 2007] and Gran Sasso, Italy (two perpendicular laser interferometer–strainmeters, BA and BC, each with a measuring armlength of 90 m [Amoruso and Crescentini, 2009]).

Detection of free core nutation resonance variation in Earth tide from global superconducting gravimeter observations

In order to verify the time variability of free core nutation (FCN) period, global superconducting gravimeter (SG) observations were analyzed based on synthetic test data. The gravity data series were synthesized to check the detectability of resonance variation caused by FCN period change. The tests indicate that the discrepancy between the FCN periods determined by SG and VLBI observations is caused by the high correlation between the FCN parameter and the amplitude factor of the ψ1 wave. The K1 wave is more sensitive to the FCN period change than other diurnal waves. The limit of the standard deviation of the K1 wave is found for more precisely observing the FCN period change. Tidal parameters of diurnal waves estimated from long series of 20 global SG stations were analyzed. A common variation trend is found in the amplitude factor of both K1 and ψ1 waves in all 8 stations above the limit, which indicates the FCN period may be not so stable in time. Furthermore, the variation in the K1 and ψ1 waves constrains the FCN period change to between 2.5 and 4 sidereal days, which also agrees with the possible variation from the current VLBI and SG observations.

Free core nutation period inferred from the gravity measurements at Józefosław

The Free Core Nutation (FCN) is an important eigenmode which affects both Earth rotation and body tide. The FCN parameters, the resonance period and the quality factor are important for understanding the dynamics of the Earth at nearly diurnal periods. Those parameters are usually estimated either from the Very Long Baseline Interferometry (VLBI) observations of nutation, or from the tidal gravity measurements. In this paper we investigate the determination of the FCN parameters from gravity records covering a period of more than three years, collected with the use of a LaCoste&Romberg Earth Tide no. 26 gravimeter, located at Jozefoslaw observatory near Warsaw. From the resonant enhancements of gravimetric factors and phases of diurnal tidal gravity waves, we could infer the FCN period to be equal to 430 sidereal days. This result is in very good agreement with previous gravimetric and VLBI nutation results, confirming the discrepancy in the dynamic flattening of the outer liquid core from its theoretical value based on the hydrostatic equilibrium assumption. The estimated FCN quality factor (Q ≈ 1300) is considerably smaller than the VLBI nutation result, which confirms that the local gravity measurements are more sensitive than VLBI global analyses to site-dependent phenomena (such as atmospheric and indirect ocean tidal effects). We also investigated the importance of gravimetric corrections in the FCN analysis, including numerical tests and simulations. This allowed us to estimate the uncertainty of FCN parameters due to improper or incomplete set of environmental corrections. We took also into account the impact of gravimetric factor errors and tidal wave selection on estimated FCN parameters. We demonstrated that despite relatively noisy measurements due to unfavorable gravimeter location, we were able to obtain very good results in case when proper correction and tidal wave selection were applied.

Observation of the Earth’s nutation by the VLBI: how accurate is the geophysical signal

We compare nutation time series determined by several International VLBI Service for geodesy and astrometry (IVS) analysis centers. These series were made available through the International Earth Rotation and Reference Systems Service (IERS). We adjust the amplitudes of the main nutations, including the free motion associated with the free core nutation (FCN). Then, we discuss the results in terms of physics of the Earth’s interior. We find consistent FCN signals in all of the time series, and we provide corrections to IAU 2000A series for a number of nutation terms with realistic errors. It appears that the analysis configuration or the software packages used by each analysis center introduce an error comparable to the amplitude of the prominent corrections. We show that the inconsistencies between series have significant consequences on our understanding of the Earth’s deep interior, especially for the free inner core resonance: they induce an uncertainty on the FCN period of about 0.5 day, and on the free inner core nutation (FICN) period of more than 1000 days, comparable to the estimated period itself. Though the FCN parameters are not so much affected, a 100 % error shows up for the FICN parameters and prevents from geophysical conclusions.

Estimation of the free core nutation period by the sliding-window complex least-squares fit method

Estimation of the free core nutation (FCN) period is a challenging prospect. Mostly, two methods, one direct and one indirect, have been applied in the past to address the problem by analyzing the Earth orientation parameters observed by the very long baseline interferometry. The indirect method estimates the FCN period from resonance effects of the FCN on forced nutation terms, whereas the direct method estimates the FCN period using the Fourier Transform (FT) approach. However, the FCN period estimated by the direct FT technique suffers from the non-stationary characteristics of celestial pole offsets (CPO). In this study, the FCN period is estimated by another direct method, i.e., the sliding-window complex least-squares fit method (SCLF). The estimated values of the FCN period for the full set of 1984.0–2014.0 and four subsets (1984.0–2000.0, 2000.0–2014.0, 1984.0–1990.0, 1990.0–2014.0) range from −428.8 to −434.3 mean solar days. From the FT to the SCLF method, the estimate uncertainty of the FCN period falls from several tens of days to several days. Thus, the SCLF method may serve as an independent direct way to estimate the FCN period, complementing and validating the indirect resonance method that has been frequently used before.

Free core nutation observed by VLBI

The signature of free core nutation (FCN) is found in the motion of the celestial intermediate pole in the celestial reference frame and in the resonance behaviour of the frequency-dependent Earth tidal displacement in its diurnal band. We focus on estimation of the FCN parameters, i.e. the period and amplitude. We run several global adjustments of 27 years of very long baseline interferometry (VLBI) data (1984.0 - 2011.0) to determine the FCN period from partial derivatives of the VLBI observables with respect to the FCN as contained in the nutation of the celestial intermediate pole and in the solid Earth tidal displacement in the diurnal band. Finally, we estimate the FCN period by a global adjustment from both phenomena simultaneously, which has not been done before. We find that our estimate of the FCN period of -431.18 +/- 0.10 sidereal days slightly deviates from the conventional value of -431.39 sidereal days. Additionally, we present our empirical model of the FCN with variable amplitude and phase compatible with the estimated period.

Application of superconductive gravity technique on the constraints of core-mantle coupling parameters

The parameters, i.e. the Period and the Quality factor, of the Earth’s free core nutation (FCN) are closely related to the dissipative coupling between the core and the mantle. Based on the FCN parameters obtained from the actual observations and theoretical simulation, significantly constrained in this study were several key parameters near the core-mantle boundary (CMB), related to the core and mantle coupling, including viscosity at the top of liquid core, conductivity at the bottom of the mantle, and dynamic ellipticity of the CMB. In order to choose high quality observations from global stations of the superconducting gravimeters (SG) on the Global Geodynamics Project (GGP) network, we adopted two criteria, the standard deviations of harmonic analysis on tidal observations and the quality of the FCN parameters calculated with the observations from single station. After the mean ocean tidal effects of the recent ocean tidal models were removed, the FCN parameters were retrieved by stacking the tidal gravity observations from the GGP network. The results were in a good agreement with those in the recent research by using the SG and/or the VLBI observations. Combined with an FCN theoretical model deduced by angular momentum method, the viscous and electromagnetic coupling parameters near the CMB were evaluated. Numerical results indicated that the viscosity at the top of the liquid core was in the range from 6.6×102 to 2.6×103 Pa·s, which was in good agreement with those obtained from the Earth’s nutation, the FCN and variations in the length of day (LOD). The conductivity at the bottom of the mantle should be as large as 2.6×106–1.0×107 S m−1 to match the FCN quality factors from the actual observations. The dissipative coupling had a little influence of 1–2 sidereal days for the FCN period.

Lander radio science experiment with a direct link between Mars and the Earth

Mars' Orientation and rotation Parameters (called here MOP): precession, nutation, polar motion and length of day (LOD) variations, are related to the interior of the planet as well as to the dynamics of its atmosphere. The MOP can be determined using the Doppler shift on radio signal data due to the motion of a probe landed on Mars relative to tracking stations on Earth. In this paper we perform numerical simulations for assessing the precision on the determination of the MOP using Direct-To-Earth (DTE) X-band Doppler measurements for a nearly equatorial lander. We then discuss how a better knowledge of the MOP could improve our understanding on the interior structure. The sensitivity of such a DTE radio link to these rotation parameters is first investigated. This shows that the latitude of the landing site must be higher than 20° to detect the Chandler Wobble component of the polar motion in DTE Doppler data and must be at least 40° to get tight constraints on it. It is found that the precision in the determination of the seasonal LOD variations will be significantly improved after about 350 days of operation, reaching the 5% level after 550 days, thereby better constraining the CO2 mass budget in the Martian atmosphere and ice caps. The current precision in the precession rate (25milliarcsecond (mas) per year) will be matched after 150 days of mission. An uncertainty of less than 5mas/year will be reached after 700 days, improving the precision on the polar moment of inertia by a factor of five. The precision on the determination of the amplitudes of nutation is estimated at a few mas after one Martian year of mission (about 12mas on the prograde/retrograde semi and terannual amplitudes) allowing for the detection of the contribution expected from the liquid core. Considering Mars with a liquid core in accordance with recent geodesic measurements, the Free-Core-Nutation (FCN) period is estimated with a precision of less than 10 days after 550 days of mission in case of an actual FCN period equal to −240 days and with a precision of 1 day after 300 days of observation for an FCN period at −230 days, which is close to the forcing terannual period of 228.9 days (resonance). It is shown that only 100 days of mission duration are needed to detect the liquid core contribution to the nutation near the resonance from the estimates of the amplitudes of nutation. Finally, the precisions on core physical parameters are inferred from the FCN period estimate. We show that a precision of 30% for the core moment of inertia and 5% for the dynamical flattening can be obtained after one Martian year of mission duration.

On the coupling between magnetic field and nutation in a numerical integration approach

Nutation amplitudes are computed in a displacement field approach that incorporates the influence of a prescribed magnetic field inside the Earth's core. The existence of relative nutational motions between the liquid core and its surrounding solid parts induces a shearing of the magnetic field. An incremental magnetic field is then created, which in return perturbs the nutations themselves. This problem has already been addressed within a nutation model computed from an angular momentum budget approach. Here we incorporate the magnetic field influence directly in the motion equation and in the boundary conditions used in precise nutation theory, and a new strategy to compute nutations is established. As in previous studies, we assume that the root-mean-square of the radial magnetic field amplitude at the core-mantle boundary is 6.9 Gauss, that the magnetic diffusivity at the bottom of the mantle and in the fluid outer core side is 1.6 m(2)/s, and that the thickness of the conductive layer at the bottom of the mantle is 200 m. The Coriolis force is included in this work. The results show that the free core nutation period decreases by 0.38 days, and that the out-of-phase (in-phase) amplitudes of the retrograde 18.6 year and the retrograde annual nutations increase (decrease) by 20 and 39 mu as, respectively. Comparisons of these results with previous studies are made, and discussions are also presented on the contribution of Coriolis force and the prescribed magnetic field on the coupling constants.

Temporal variations in free core nutation period

Based on the nearly diurnal resonance in the tidal gravity observations, the temporal variations in period of the Earth’s free core nutation (FCN) are investigated by using the tidal gravity observations of 18-year duration recorded continuously with a superconducting gravimeter (SG) at Brussels. The effects of the global oceanic tide loading and local barometric pressure on the SG observations have been removed by using eleven high-precision global digital models of oceanic tides and barometric pressure measurements recorded simultaneously at the same site. The results indicate that there exist decade-scale variations in the FCN period. The results should be further confirmed by the measurements using other space-based geodetic techniques (such as the very long baseline interferometry) and the SG observations from globally distributed stations.

Free core nutation resonance parameters from VLBI and superconducting gravimeter data

Context: Free core nutation (FCN) can be observed by its associated resonance effects on the forced nutations of the Earth's figure axis, as observed by very long baseline interferometry (VLBI), or on the diurnal tidal waves, retrieved from the time-varying surface gravity recorded by superconducting gravimeters (SG). Aims: In this paper, we study the sensitivity of both techniques to FCN parameters. Methods: We analyze surface gravity data from 15 SG stations and VLBI delays accumulated over the last 24 years. Results: We obtain estimates of the FCN period and quality factor that are consistent for both techniques. The inversion leads to a quality factor centered on ~16 600 with an uncertainty of ~3500 from SG and of ~900 from VLBI, and to a resonant period within [ -423.3, -430.5] days for SG and [ -427.8, -431.4] days for VLBI (3σ interval).

On models of Mars’ interior and amplitudes of forced nutations: 2. The effects of liquid core and mantle elasticity

The numerical modeling of Mars’ forced nutation of the real Mars’ model with a radially heterogeneous, compressible, self-gravitating liquid core and elastic mantle is performed. Effects of three-axiality of Mars’ figure are taken into account to compute values for the periods of free core nutation (FCN) and the amplitudes of forced nutations. The effects of the liquid core for different models are of the order of few percents for the prograde semi-annual component and of the order of 10–15% for the semi-annual retrograde component. The absolute values of these effects are of the order of 5–20 ms of arc (mas) and 1–3 mas, respectively. Taking into account, that the modern accuracy of VLBI measurements for the case of the Earth is of the order of 0.02 mas, we see, that important results about Mars’ internal structure may be obtained even if the accuracy of Mars’ nutational measurements 1–2 order of magnitude less than accuracy for the Earth. The effects of the shear stress relaxation during long time intervals are studied: the corrections to the nutational amplitudes for prograde semi-annual component are between 8.93 and 9.8, for retrograde semi-annual component are between −4.81 and −8.2 ms of arc, and the period of free core nutation is varied in the limits between 257.2 and 295.8 Earth's solar days. The differences exceed significantly the errors of Mars’ nutational components measurements, and, consequently, these measurements can give new important information about Martian mantle rheology. The possibilities of inferring properties of the interior structure of Mars based on the measurements of its rotation are discussed.

Lander radioscience for obtaining the rotation and orientation of Mars

The paper presents the concept, the objectives, the approach used, and the expected performances and accuracies of a radioscience experiment based on a radio link between the Earth and the surface of Mars. This experiment involves radioscience equipment installed on a lander at the surface of Mars. The experiment with the generic name lander radioscience (LaRa) consists of an X-band transponder that has been designed to obtain, over at least one Martian year, two-way Doppler measurements from the radio link between the ExoMars lander and the Earth (ExoMars is an ESA mission to Mars due to launch in 2013). These Doppler measurements will be used to obtain Mars’ orientation in space and rotation (precession and nutations, and length-of-day variations). More specifically, the relative position of the lander on the surface of Mars with respect to the Earth ground stations allows reconstructing Mars’ time varying orientation and rotation in space. Precession will be determined with an accuracy better by a factor of 4 (better than the 0.1% level) with respect to the present-day accuracy after only a few months at the Martian surface. This precession determination will, in turn, improve the determination of the moment of inertia of the whole planet (mantle plus core) and the radius of the core: for a specific interior composition or even for a range of possible compositions, the core radius is expected to be determined with a precision decreasing to a few tens of kilometers. A fairly precise measurement of variations in the orientation of Mars’ spin axis will enable, in addition to the determination of the moment of inertia of the core, an even better determination of the size of the core via the core resonance in the nutation amplitudes. When the core is liquid, the free core nutation (FCN) resonance induces a change in the nutation amplitudes, with respect to their values for a solid planet, at the percent level in the large semi-annual prograde nutation amplitude and even more (a few percent, a few tens of percent or more, depending on the FCN period) for the retrograde ter-annual nutation amplitude. The resonance amplification depends on the size, moment of inertia, and flattening of the core. For a large core, the amplification can be very large, ensuring the detection of the FCN, and determination of the core moment of inertia. The measurement of variations in Mars’ rotation also determines variations of the angular momentum due to seasonal mass transfer between the atmosphere and ice caps. Observations even for a short period of 180 days at the surface of Mars will decrease the uncertainty by a factor of two with respect to the present knowledge of these quantities (at the 10% level). The ultimate objectives of the proposed experiment are to obtain information on Mars’ interior and on the sublimation/condensation of CO 2 in Mars’ atmosphere. Improved knowledge of the interior will help us to better understand the formation and evolution of Mars. Improved knowledge of the CO 2 sublimation/condensation cycle will enable better understanding of the circulation and dynamics of Mars’ atmosphere.

Determination of the free core nutation period from tidal gravity observations of the GGP superconducting gravimeter network

This study is based on 25 long time-series of tidal gravity observations recorded with superconduct- ing gravimeters at 20 stations belonging to the Global Geodynamic Project (GGP). We investigate the diurnal waves around the liquid core resonance, i.e., K1, ψ1 and ϕ1, to determine the free core nutation (FCN) period, and compare these experimental results with models of the Earth response to the tidal forces. For this purpose, it is necessary to compute corrected amplitude factors andphasedifferencesbysubtractingtheoceantideload- ing (OTL) effect. To determine this loading effect for each wave, it was thus necessary to interpolate the con- tribution of the smaller oceanic constituents from the four well determined diurnal waves, i.e., Q1,O1,P1,K1. It was done for 11 different ocean tide models: SCW80, CSR3.0, CSR4.0, FES95.2, FES99, FES02, TPXO2, ORI96, AG95, NAO99 and GOT00. The numerical re- sults show that no model is decisively better than the others and that a mean tidal loading vector gives the most stable solution for a study of the liquid core res- onance. We compared solutions based on the mean of the 11 ocean models to subsets of six models used in a previous study and five more recent ones. The cali- bration errors put a limit on the accuracy of our global results at the level of ±0.1%, although the tidal factors of O1 and K1 are determined with an internal precision of close to 0.05%. The results for O1 more closely fit