the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Sep 2025
Basic Earth Parameters from VLBI observations using Bayesian inversions in the time domain: updated insights of the Earth's interior
Abstract. We present updated estimates of Basic Earth Parameters (BEP) from VLBI Celestial Pole Offset (CPO) time series spanning 1980-2025 using ensemble Markov Chain Monte-Carlo Bayesian inversion. Building upon Koot et al. (2008), we employ enhanced sampling algorithms and incorporate recent advances in ocean tidal modeling (Cheng and Bizouard, 2025). Key improvements include: (1) implementation of piece-wise cubic spline modeling for Free Core Nutation (FCN) amplitude variations, which significantly reduces multimodality in MCMC sampling compared to linear modeling; (2) integration of updated Ocean Tidal Angular Momentum (OTAM) values from FES 2014 ocean tidal atlas (Lyard et al., 2021) without the empirical 0.7 scaling factor previously applied; and (3) utilization of five diverse CPO series from different analysis centers spanning up to 45 years of observations.
Our results show good consistency across different CPO series, with estimated dynamical ellipticity values at the edge of the 1σ range of MHB 2000. Notable findings include a larger absolute value for the imaginary part of the core-mantle boundary coupling constant (Im(KCMB)), approaching the 2σ boundary of Mathews et al. (2002), which may reflect contributions from multiple coupling mechanisms, including topographic coupling through the "form drag" effect caused by wave interactions with irregular boundaries (Rekier et al., 2025). The real part of the inner core boundary coupling constant (Re(KICB)) is approximately half the MHB 2000 value, potentially indicating the need to revisit hydrostatic assumptions for the inner core given recent seismic evidence of viscous deformation. Compliance estimates suggest that frequency extrapolation methods from seismic to nutation bands require revision. The enhanced FCN free mode modeling successfully captures amplitude variations that differ from empirical models, particularly after 2000, though the physical interpretation of these differences requires further investigation.
The systematic discrepancies across multiple parameters suggest that the current nutation theory needs substantial updates to incorporate more realistic models of core-mantle coupling and inner core behavior.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2025-4428', Anonymous Referee #1, 13 Nov 2025
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AC1: 'Reply on RC1', Yuting Cheng, 06 Mar 2026
We thank Reviewer 1 for the careful reading and constructive comments, which have helped us improve the clarity of the manuscript. Below we address each comment individually.
General revisions (Comments 1–6, 10–12): We have addressed the reviewer's suggestions
- regarding missing references in the introduction (Comment 1),
- the typographical error on line 59 (Comment 2),
- the clarification of the data timespan which has been moved earlier in the manuscript (Comment 3),
- the addition of references for the sampling algorithms and the emcee software package (Comment 4),
- the phrasing on line 100 (Comment 5),
- the comparison between MCMC and Metropolis-Hastings, which has been clarified to reflect that the difference is about single-walker and ensemble but not the MCMC algorithm itself (Comment 6),
- the inclusion of the anelastic contribution in the calculation of beta (Comment 10),
- the assumptions on electrical conductivity and the rms radial field (Comment 11),
- and the specification that the "halved values" refer to the MHB 2000 model in the concluding remarks (Comment 12).
Comment 7 — On the 0.7 scaling factor: The matter with this scaling factor was a bit of a detective's work. When calculating ocean tidal effects on nutation, the angular momentum of only several "principal waves" gets calculated from the tidal maps; the rest of the tidal terms are interpolated by means of "admittance." In the paper of MHB2000, they applied a constant scaling factor of 0.7, determined by test fittings, to account for the inaccuracy of the motion terms. What we found in Cheng & Bizouard (2025) was that for these terms, the residual amplitudes estimated by least-squares from the CPO series are consistently about 0.3 of the theoretical values, which we suspect were what remained from the "incomplete subtraction" because of the 0.7 scaling factor. So in this work we decided not to apply it. This realisation could be attributed to the improvement of both ocean tide modelling and VLBI data quality since the early 2000s.Comment 8 — On "matter" and "motion" terms: The "matter" term arises from the pressure effect associated with the ocean height, and the "motion" from the ocean current. This clarification has been added to the text.
Comment 9 — On the different VLBI solutions: EOP products are produced following the same guidelines given by IERS or IVS, meaning the models used in the data processing are the same, but they can differ in terms of implementation in the different software packages used by different analysis centres. For solutions produced with the same software, the processing strategy and products can still differ (e.g., outlier detection, a priori values such as in-house EOP series instead of C04, etc.). These operational differences could cause small differences in the results of our BEP estimation, and could serve as a way to assess more realistic uncertainties apart from pure formal errors.
All these points have been corrected or clarified in the revised manuscript.Citation: https://doi.org/10.5194/egusphere-2025-4428-AC1
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AC1: 'Reply on RC1', Yuting Cheng, 06 Mar 2026
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RC2: 'Comment on egusphere-2025-4428', Thomas Herring, 02 Feb 2026
This paper is well written and clearly presented. It is a comprehensive update of estimates of Earth parameters from VLBI nutation time series. The work builds on and compares to earlier studies. It presents results with uncertainties that seem realistic. One small comment is that there are no errors in the cubic-spline estimates of the FCN amplitudes. The rapid increase in the FCN amplitude after 2000 is not seen in all analyses. Would error bars show that the differences are insignificant? I also wonder if the numering of Figures 2 and 3 should be reversed? Figure 3 seems to be the logical next figure after Figure 1. (Alternatively, changing Figure 2 to Figure 1 would put the two FCN figures next to each other.
I also wonder about the availability of software used for this analysis?
Citation: https://doi.org/10.5194/egusphere-2025-4428-RC2 -
AC2: 'Reply on RC2', Yuting Cheng, 06 Mar 2026
We thank Reviewer 2 for the positive assessment and the helpful suggestions.
Regarding the absence of error estimates on the cubic-spline FCN amplitudes: we have produced new plots for the FCN amplitude variations based on percentile calculations of the posterior ensembles. The two figures now include shaded regions representing an equivalence of 3σ.
Regarding the figure ordering: we have changed the order of figures as suggested.
Regarding the availability of the software: the program can be made available on request, but we do not have plans to make it publicly accessible in the near future.
Citation: https://doi.org/10.5194/egusphere-2025-4428-AC2
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AC2: 'Reply on RC2', Yuting Cheng, 06 Mar 2026
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- 1
The authors use a Bayesian inversion to recover estimates of Earth parameters from VLBI observations of the Earth’s nutation. The main innovations include the use of a better sampler for the Bayesian inversion, a better ocean tide model and a more flexible recovery of motion to the free core nutation. Comparisons of the revised Earth parameters with previous estimates offer new insights into the structure and dynamics of the Earth’s interior. The results are very interesting, although the manuscript is aimed at an expert audience. I had trouble following parts of the manuscript. These parts could benefit from clarification or more precise descriptions. Most of my comment deal with points of clarification.
Specific Comments
1. The introduction has no references before line 47. Statements are often made without documentation or support. For example, “… the precision (of VLBI) had a major improvement in the late 1980s”. What is “a conventional model” in the definition of the Celestial Pole Offsets?
2. line 59: lager -> larger
3. line 60: “We now have almost 25 years of data with better quality…”. The abstract suggests that 45 years of data are used in the inversion. It appears that “25 years” refers to the additional data available since the MHB 2000 model. This point is clarified in the next paragraph. It might be helpful to move this clarification earlier in the manuscript.
4. The sampling algorithm is updated from a single Metropolis-Hastings sampler to an ensemble Markov Chain Monte Carlo method. Are there references for the original and updated methods? Who is the author of the software package “emcee”?
5. line 100: repeated or nested use of “thanks” makes for an odd sentence.
6. line 118: “the MCMC has more sampling capabilities than the Metropolis-Hastings method”. MCMC is often viewed as a general class of methods, whereas the Metropolis-Hastings is a specific algorithm within that class. A comparison between MCMC and the Metropolis-Hastings methods is confusing.
7. Point of clarification on line 146 - “This suggests that the 0.7 scaling factor applied in the MHB model to reduce ocean tidal effects may have been unnecessary”. Is the idea that updated ocean tidal models have smaller amplitudes, so there is no need to apply a scaling factor to the new models? Presumably some reduction of the ocean tidal model used in MHB 2000 was “necessary”? Could the authors clarify?
8. Table 3: “matter” and “motion” terms refer to changes in moment of inertia and local momentum due to the ocean?
9. Several solutions are used as “observations” in the Bayesian inversion (see Table 3). Readers are given the names of the solutions and the processing software, but few other details are stated. For example, usn2024b.eoxy and gsf2023a.eoxy use the same processing software. What is the difference between these solutions? The authors note that these different solutions give similar results in the inversion. How should readers assess this statement without information about the input solutions?
10. line 186: Does the calculation of “beta” from PREM include the anelastic contribution?
11. line 191: What value of electrical conductivity is assumed in the calculation of the rms radial field (=0.75 mT)? How is a “realistic” conductivity assessed? How is a “realistic” rms radial field assessed?
12. line 276: “The halved Re(K^ICB) values,…..” . The earlier text makes it clear that the “halved values” refer to the MHB 2000 model. Why not be specific in the “Concluding remarks”?