Drift of the Earth’s Principal Axes of Inertia from GRACE and Satellite Laser Ranging Data

Round 1

Reviewer 1 Report

This is an interesting and scientifically relevant study into a determination of the motion of the Earth's principal axes of inertia using data sets giving time varying, second degree Stokes coefficients. The coefficients are derived from SLR solutions and from the GRACE mission.

Overall the results from the two time series agree sufficiently to support the authors' important conclusions that the Earth's principal axes of inertia are exhibiting hitherto-unknown, non-negligible drifts with magnitudes that exceed the GGOS goals of delivering a TRF at an accuracy level of 1mm and a stability of 0.1mm/yr.

I recommend that a sentence be included to confirm that the GRACE and SLR solutions are fully independent, as some GRACE solutions do use SLR results for low-degree SHCs.

Some specific suggestions to improve clarity/English in the text:

L50: of the knowledge of the Earth’s gravity field knowledge

L51: were already capable able of

L67: Earth’s elasticity yielding subsequent

L91: The deviations are matter to be small enough - needs a clearer explanation - too small to be of concern??

L155: and comparably - replace with 'correspondingly'

L194: more clear to find - replace with 'straightforward to determine'

L204: the signification level - replace with 'level of significance'

L214: accelerate at the particular epoch - what epoch, what does this mean?

L235: the overlapping of the solutions for each rotation is practically given across the entire time interval, - replace with 'the solutions for each of the three rotations Rx, Ry and Rz are very similar across the entire time interval'

L242 neither tight - replace with 'closely aligned'

L252: For instance, if the observed - replace with 'whether or not'

L253: maybe one among - replace with 'concerning' ?

L255: is if 'whether' these variations of the Earth’s inertia tensor might affect

 

Author Response

This is an interesting and scientifically relevant study into a determination of the motion of the Earth's principal axes of inertia using data sets giving time varying, second degree Stokes coefficients. The coefficients are derived from SLR solutions and from the GRACE mission.

Overall the results from the two time series agree sufficiently to support the authors' important conclusions that the Earth's principal axes of inertia are exhibiting hitherto-unknown, non-negligible drifts with magnitudes that exceed the GGOS goals of delivering a TRF at an accuracy level of 1mm and a stability of 0.1mm/yr.

I recommend that a sentence be included to confirm that the GRACE and SLR solutions are fully independent, as some GRACE solutions do use SLR results for low-degree SHCs.

 

Response:

We greatly appreciate the time and effort you took to read our manuscript in detail that helped improve the quality of the paper. We made our best to meet all of the points and comments that you raised in the revised version. Concerning SHC solutions, we agree with your point. We modified introduction section (see page.2 line.71-72)

“Among the different data series available we have chosen two independent solutions to present the results derived from the GRACE-based SHC monthly values provided by the Centre for Space Research (CSR), …"

 

Some specific suggestions to improve clarity/English in the text:

L50: of the knowledge of the Earth’s gravity field knowledge

Response:

It is done!

L51: were already capable able of

Response:

It is done!

L67: Earth’s elasticity yielding subsequent

Response:

It is done!

L91: The deviations are matter to be small enough - needs a clearer explanation - too small to be of concern??

Response:

It is done!

L.92: The deviations are matter to be small enough to obtain the rotation matrix by deriving an analytical approximation up to the first order.

L155: and comparably - replace with 'correspondingly'

Response:

It is done!

L194: more clear to find - replace with 'straightforward to determine'

Response:

It is done!

L204: the signification level - replace with 'level of significance'

Response:

It is done!

L214: accelerate at the particular epoch - what epoch, what does this mean?

Response:

It is done!

L.220: “around the years 2005-2006”

L235: the overlapping of the solutions for each rotation is practically given across the entire time interval, - replace with 'the solutions for each of the three rotations Rx, Ry and Rz are very similar across the entire time interval'

Response:

It is done!

L242 neither tight - replace with 'closely aligned'

Response:

It is done!

L252: For instance, if the observed - replace with 'whether or not'

Response:

It is done!

L253: maybe one among - replace with 'concerning' ?

Response:

It is done!

L255: is if 'whether' these variations of the Earth’s inertia tensor might affect

Response:

It is done!

Author Response File: Author Response.pdf

Reviewer 2 Report

Review of manuscript 684167

Drift of the Earth's Principal Axes of Inertia from GRACE and Satellite Laser Ranging Data

by Ferrándiz et al.

submitted to Remotesensing

 

The manuscript focuses on a significant consequence of the very precise measurements of the Earth's gravitational field provided by the two GRACE twin satellites as well as by geodetic satellites by means of the SLR technique. In particular, the time-varying components of the low degree coefficients allow for the measurement of the variation of the moments of inertia of the Earth produced by the complex mass redistribution of our planet due to various physical phenomena in relationship with the Earth’s elasticity and rotation and Moon and Sun attraction.

The results are expressed in terms of the components of the infinitesimal antisymmetric rotation matrix from the ATRF frame to the PAI frame, as defined in the manuscript.

This work is new and original and based not only on the GRACE measurements and the SLR data of the geodetic satellites, but also on an analytical approach previously developed in the literature by some of the authors of the current manuscript.

This Reviewer is in favor of publication of this work after minor revision.

 

Remarks

The manuscript is well written and it is clear the methodology followed by the authors and the fact that their results are preliminary and that further analyzes are still needed, as for the determination of the spectral content of the data.

However, some improvements of the current results are already possible on the basis of the following aspects.

In section 3, the authors analyzed the two different data sets — UT-CSR for GRACE data and DGFI-TUM for SLR data — and compared the results obtained with a preliminary discussion in section 4. Their conclusion is that the differences obtained are little, the data sets are consistent and, consequently, the results are robust enough to be significant and can be considered a physical feature of the Earth change, not an artifact.

However, their comparison is based on data characterized by a different spatial resolution (monthly vs weekly) and on different time scales (January 2002 to August 2018 and January 2000 to February 2018). Of course, the longer the time interval of the data the greater the physical content, but, in view of a comparison of data obtained with different techniques, the possible differences in the final results should be evaluated with two different data sets determined exactly on the same time span, for instance from January 2002 to February 2018.

I suggest

to anticipate in their section 3 the comparison between the GRACE and SLR data, that is the current figure 4 of section 4, with the general considerations already introduced analyze independently the two sets of data over the same time span, as already done, with the new results for the various linear fits

Moreover, the SLR data may be further analyzed by smoothing the data in such a way to reduce some of the high frequencies and compare again the results on a monthly basis with those of GRACE. This may help in regard to the discussion about a possible influence of the different modeling of the “mean pole” in the two data sets as discussed in section 4: CSR-GRACE with a linear model and DGFI-TUM with a cubic model.

About this point, why not introduce in the analysis also the GRACE RL06 data from GFZ? In this data sets the model for the mean pole is the same of the CSR data, as well as the spatial resolution. Moreover, the SLR data may be compared with GRACE GFZ RL05, whose data are characterized by a weekly resolution alongside a monthly one.

Obviously, this possible extension of the analysis to other GRACE measurement data is not mandatory, but pleasant to have in order to possibly reinforce the results.

Finally, please check the final expressions for Rx, Ry and Rz at the end of section 2.3, there are a few typos compared with Eqs. (7).

Best regards

Author Response

Answer to Reviewer #2:

 

The manuscript focuses on a significant consequence of the very precise measurements of the Earth's gravitational field provided by the two GRACE twin satellites as well as by geodetic satellites by means of the SLR technique. In particular, the time-varying components of the low degree coefficients allow for the measurement of the variation of the moments of inertia of the Earth produced by the complex mass redistribution of our planet due to various physical phenomena in relationship with the Earth’s elasticity and rotation and Moon and Sun attraction.

The results are expressed in terms of the components of the infinitesimal antisymmetric rotation matrix from the ATRF frame to the PAI frame, as defined in the manuscript.

This work is new and original and based not only on the GRACE measurements and the SLR data of the geodetic satellites, but also on an analytical approach previously developed in the literature by some of the authors of the current manuscript.

This Reviewer is in favor of publication of this work after minor revision.

Response:

Thank you for your comment. We greatly appreciate the time and effort you took to read our manuscript in detail and for your very detailed and constructive reviews that greatly helped improve the quality of the paper.

 

 

Remarks

The manuscript is well written and it is clear the methodology followed by the authors and the fact that their results are preliminary and that further analyzes are still needed, as for the determination of the spectral content of the data. However, some improvements of the current results are already possible on the basis of the following aspects.

In section 3, the authors analyzed the two different data sets — UT-CSR for GRACE data and DGFI-TUM for SLR data — and compared the results obtained with a preliminary discussion in section 4. Their conclusion is that the differences obtained are little, the data sets are consistent and, consequently, the results are robust enough to be significant and can be considered a physical feature of the Earth change, not an artifact.

However, their comparison is based on data characterized by a different spatial resolution (monthly vs weekly) and on different time scales (January 2002 to August 2018 and January 2000 to February 2018). Of course, the longer the time interval of the data the greater the physical content, but, in view of a comparison of data obtained with different techniques, the possible differences in the final results should be evaluated with two different data sets determined exactly on the same time span, for instance from January 2002 to February 2018.

I suggest

to anticipate in their section 3 the comparison between the GRACE and SLR data, that is the current figure 4 of section 4, with the general considerations already introduced analyze independently the two sets of data over the same time span, as already done, with the new results for the various linear fits

Moreover, the SLR data may be further analyzed by smoothing the data in such a way to reduce some of the high frequencies and compare again the results on a monthly basis with those of GRACE. This may help in regard to the discussion about a possible influence of the different modeling of the “mean pole” in the two data sets as discussed in section 4: CSR-GRACE with a linear model and DGFI-TUM with a cubic model.

 

Response:

Following your suggestion, we analyzed the SLR DGFI-TUM Stokes coefficients filtered to have a time series with monthly resolution concerning this specific comment. The results are reported to you on the table below, which corresponds to table 3 of the paper - with different resolution. The results for weekly and monthly DGFI data are very close; besides, the results for the smoothed DFGI data are closer to the ones for the CSR data (table 2) than the original. Therefore, the fitted coefficients look sensitive to smoothing, but in a small percent. Taking into account that we cannot ensure which time resolution is better a priori, we decided to keep in the paper the original fits, thus analyzing each data set with the same temporal resolution used to derive it.

We have included in the paper a short comment on that at the end of section 3 (line 180-184):

To discard that the higher temporal resolution of the DGFI input data might cause some distortion of the results, we have also analyzed a smoothed version of them with monthly resolution like the CSR data, and found that there is no relevant difference in the results. The resulting drifts are R’_x = -1.96±0.60, R’_y = 11.42± 0.39, R’_z = 120.55±71.46 cm/y, very close to the weekly ones

It is complemented with another one in the discussion – end of para 2 (line 208-209):

In fact, when the CSR based results are compared to the smoothed DGFI ones, the corresponding CI95 overlap.

 

Table: Results derived from the SLR DGFI-TUM Stokes coefficients with monthly resolution. WRMS after fitting a linear function to each, biases, and drifts of the equivalent rotations Rx, Ry, and Rz from ATRF to PAI. Units are [cm] and [cm/year], time origin is 2000.0.

	WRMS [cm]	Bias [cm]	Drift [cm/year]
Rx	45.83	1002.65±5.77	-1.96±0.60
Ry	29.88	452.51±3.81	11.42± 0.39
Rz	5385.23	-7384.78±672.93	120.55±71.46

 

 

 

About this point, why not introduce in the analysis also the GRACE RL06 data from GFZ? In this data sets the model for the mean pole is the same of the CSR data, as well as the spatial resolution. Moreover, the SLR data may be compared with GRACE GFZ RL05, whose data are characterized by a weekly resolution alongside a monthly one.

Obviously, this possible extension of the analysis to other GRACE measurement data is not mandatory, but pleasant to have in order to possibly reinforce the results.

Of course, we would like to analyze the GFZ GRACE data, but the time allowed for revision is very short (few days) and there are also other candidate series that would deserve to be analyzed. The main purpose of the paper is to verify a hypothesis mentioned in the manuscript already as a conclusion: That the observed principal axes of inertia of the Earth are clearly neither tight to the ITRF axes nor oscillating around some specific “mean” equilibrium position but exhibit a non-negligible drift, whose magnitude is above the accuracy of GGOS goal. We analyzed first the CSR series because its release started in the early years of GRACE and thus it is well known. Then, we chose an SLR series, based on input data from other technique and computed with independent software, which also had better temporal resolution, weekly. The NASA/GSFC SLR series, another “classic” product, is not yet provided with such resolution higher than monthly, although expected soon.

Taking into account that the two analyses performed show a good agreement, in spite of the different temporal resolution and time intervals of the corresponding input data, then that you pointed that the extension to other input data was not mandatory, and finally that the journal policy urges prompt revisions, we have not analyzed any additional data set of Stokes coefficients. However, we think the suggested discussion of analyzing other GRACE solutions is of high interest and we will consider your suggestion for sure, but in future work.

 

Finally, please check the final expressions for Rx, Ry and Rz at the end of section 2.3, there are a few typos compared with Eqs. (7).

Thank you and sorry for not having noticed them before submission. We have found and corrected the typos.

There is one in the 2^nd eq. (10):, the sign + in the denominator must be changed to –

There are also two typos in the last two equalities just before 2.4. The Greek subindices are wrong; they have been changed to “xi”

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Very well done report. An important contribution to geophysical geodesy. In the future we will have a better representation of the Barycenter (the COM of the Two-Body field and harmonic expansion) with the Lunar GRAIL coefficients and improvements to COM, LOD, CWh (Chandler Wobble harmonics), Moments of Inertia and terrestrial rotation and reference frames.

Author Response

We would like to thank you for the positive feedback. We greatly appreciate the time and effort you took to read our manuscript in detail.