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”?

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?