We first use a piece-wise linear model inherited from . In addition, since the larger quantity of data (longer VLBI CPO time series) supports improved sampling capability, we allow the locations of the nodes (time points separating different pieces of the piece-wise linear function) to vary freely, except the first and the last ones that are set to be at the beginning and the end of the VLBI CPO time series used in the inversion. Experiments show that this method suffers from convergence issues for FCN-related parameters. This is easy to understand because multiple different arrangements of node positions may explain the data equally well, which inevitably leads to different plausible configurations during the MCMC sampling process. We address this issue by applying a clustering algorithm (k-mean) to the output samples of an initial run, in which the node positions are free to adjust, selecting the “most frequent” combination of node positions. The MCMC is then rerun with these fixed node positions.

In addition, since the amplitude variations between nodes is clearly not linear in time, we implement a cubic spline to replace the linear function in the inversion process. This approach allows us to capture the continuous and non-linear changes in the FCN amplitude over time, while still allowing the node positions to vary together with the BEPs. The results show that this modification significantly reduces the multimodality of the FCN-related parameters during MCMC sampling, to the extent that clustering and rerunning are no longer necessary. We discuss these results in more detail in the dedicated section.

The dynamical ellipticities e estimated from all series are in good agreement, lying at the edge of the 1σ range of the MHB2000 value. This parameter is related to the precession rate, which is correlated with the 18.6 year nutation. The CPO series used here cover about two periods of the 18.6 year nutation.

The dynamical ellipticity of the FOC (ef), the real part of the coupling constant at the CMB (ℜ(KCMB)), and the compliance β cannot be estimated independently, as they appear as a linear combination in the equation describing the nutations e.g., Eq. 29 in. The estimate of ef+ℜ(KCMB)-β is significantly larger than the MHB2000 value. The compliance β is fixed to the value calculated from the Preliminary Reference Earth Model PREM, by , which includes the effects of mantle anelasticity), since it is the parameter for which the theoretical value has the smallest uncertainty . Any departure of the real part of β from this value can be considered included in this term. A reevaluation and numerical separation of these correlated parameters will require careful theoretical consideration.

More notably, the estimated imaginary part of KCMB has a larger absolute value than MHB2000. This increase was already noted by , but it becomes more pronounced in the present work as the value approaches the 2σ boundary of MHB2000 value. Following the relationship illustrated in Fig. 1 of see also Fig. 2 of, for a conductive layer of 200 kilometers with a high conductivity of 5×105S m-1 (equal to that of the core) and a large skin depth (penetration of the time-varying magnetic field into the lower mantle), the matching with KCMB derived from VLBI observation would require an RMS radial magnetic field of approximately 0.75 mT through electromagnetic coupling alone. Such a field strength would require unrealistically high and uniform conductivity throughout the lower mantle , which is likely too optimistic given that the skin depth is of the order of a few hundred meters . Recent work by proposes an alternative mechanism involving topographic coupling through form drag and internal wave excitation at the CMB, which could explain the observed dissipation without requiring unrealistically high lower-mantle conductivity or magnetic field strength. The larger ℑ(KCMB) values we observe may therefore reflect contributions from multiple coupling mechanisms rather than purely electromagnetic effects.

This makes it difficult to interpret our estimate of ef, since the contribution of electromagnetic coupling to ℜ(KCMB) is uncertain. We can nevertheless provide constraints under different scenarios. estimated that ef exceeds the hydrostatic value (ef[hydrostatic]=0.002548) by about 3.8 %, corresponding to an increase of the core equatorial radius of 370 m. From our estimates, if ℜ(KCMB)=0, ef increases by about 4.9 % relative to the hydrostatic value. If we attribute all observed ℑ(KCMB) to electromagnetic coupling, leading to ℜ(KCMB)≈2.5×10-5, then ef increases about 4.0 % comparing to the hydrostatic value. In both cases, this implies a non-hydrostatic contribution corresponding to an increase in the core equatorial radius between about 370 and 500 m.

Our estimation of ℜ(KICB) appears to be almost half of the MHB2000 value. This decrease can already be seen in , but is again much more noticeable in this work. This parameter is related to the Free Inner Core Nutation (FICN) frequency (in cycle per mean solar day), as explained in and : 4σFICN≃-1+α2es+ν-KICB where α2=0.8294 is a constant defined in and related to properties of the Earth's interior such as the density jump at the CMB, and es=0.00242 is the dynamical ellipticity of the SIC. Both values are computed from PREM. KICB is the complex coupling constant at the ICB and ν is the compliance related to deformation of the ICB.

Similar to ℜ(KCMB), ℜ(KICB) appears as a linear combination with α2es +ν. In our inversion setup, the values of es and α2 are fixed to those calculated by from PREM, under the assumption that the inner core boundary corresponds to a surface of hydrostatic equilibrium, which was supported by studies in the late 1990 s, estimating dynamic topographies associated with mass anomalies in the inner core and in the mantle to be below 100 m at ICB . Note that the compliance ν is also fixed to the value calculated from PREM for the same reason as β. Considering electromagnetic coupling only, our value would require a magnetic field at the ICB lower than in MHB2000, and thus closer to geodynamo results (between 10 and 40 Gauss [0.1–0.4 T], ). Recent studies by and have revealed, from seismic data, possible viscous deformation at the inner core boundary (ICB) occurring on timescales of years. The hydrostatic assumption for the inner core in nutation computations could thus be revised. In addition, the density jump at the ICB could also be reconsidered in light of possible evidence for a mushy zone at the ICB .

Moreover, our estimation of the imaginary part of KICB is slightly larger in absolute value compared to MHB2000, which is, in a way, contradictory to the result for the real part, considering the relationship shown in Fig. 2 of see also Fig. 2 of for electromagnetic coupling at the ICB.

Compliances describe the deformability of the Earth or of specific regions of its interior, under a given forcing. The two compliances that are important to consider in our free parameter set are κ, which characterizes the deformability of the whole Earth under tidal forcing, and γ, which characterizes the deformability of the fluid core under tidal forcing.

These compliances reflect the anelasticity of Earth's interior, which is frequency dependent. Their values are initially derived from PREM , whose rheological parameters were constrained by high-frequency seismic data (with periods from several seconds to one hour) and subsequently extrapolated to the nutation band (diurnal frequencies) through a power-law scaling with frequency to account for anelasticity. The exponent of this power-law scaling, however, is poorly constrained.

In MHB2000, the compliance values estimated from VLBI data were consistent with the theoretical values obtained through this extrapolation. Our results break this consistency: our estimate of κ remains in complete agreement with MHB2000, but our estimate of γ is significantly smaller. Since both compliances are corrected from their elastic values through the same anelastic extrapolation procedure, a shift in γ without a corresponding shift in κ cannot be explained by simply adjusting the power-law exponent. This points to a limitation in the current anelastic correction framework, or in the PREM model from which the elastic compliances are derived, and calls for careful systematic revision.

The atmosphere contributes to nutation mainly at the prograde and retrograde annual, semi-annual, and ter-annual frequencies. There is also a contribution to the retrograde annual nutation due to the resonance effect with the FCN. Evaluating these effects using atmospheric angular momentum (AAM) data remains non-trivial, as it requires data with a high sampling rate to capture quasi-diurnal effects in a frame tied to the Earth. The largest effect occurs at the prograde annual frequency. Therefore, a prograde annual term is included among the estimated parameters to account for this atmospheric effect as it is the most important contribution . The results of our estimations of this term differ by about 20 µas among different solutions from various IVS analysis centers. It should be noted that this prograde annual term is empirical and will absorb not only the atmospheric contribution but also any other signals appearing in VLBI observations as prograde annual oscillations, which are typically associated with the Earth's revolution around the Sun and the resulting annual temperature variations.

Two CPO series are generated using Calc/nµSolve to test the effect of different mapping functions of the tropospheric delay in VLBI data processing. The considered mapping functions are VMF1 and NMF . The resulting differences appear primarily in the prograde annual term at the level of 2 µas, which corresponds to the uncertainty of this parameter in individual data series. The differences in KCMB and KICB are not significant.

We have considered the same set of additional parameters that are adjusted at the same time with the geophysical parameters as those in : the obliquity rate dϵ/dt, constant offsets cψ and cϵ, and the modeling uncertainty σM. Their presence in MHB2000 is not obvious, but they were actually evaluated by before the final BEP estimations for MHB2000 by .

The obliquity rate dϵ/dt was estimated together with 21 nutation frequencies by . Their result is also included in Table . also estimated the secular obliquity change with nutation amplitudes fixed to MHB2000 values, where they found dϵ/dt=-0.252±0.003masyr-1, which is closer to our result. Note that the FCN is also estimated in both cases with its time-varying amplitude represented as piece-wise linear functions, with 6 and 10 nodes for time series spanning about 20 years. However, their results for the constant offsets cψ and cϵ are not given.

The parameter σM, introduced by , incorporates both the modeling error and a constant correction to the standard deviation of the data see Eq. 28 in. Standard deviations are used to represent the uncertainties of CPO estimates from VLBI observations, but they are usually smaller than the true uncertainties. With σM, this possible underestimation of CPO uncertainties is taken into account. It thus partially serves the same purpose as the parameter introduced by to correct data errors.