Characterizing Stalagmites’ Eigenfrequencies by Combining In Situ Vibration Measurements and Finite Element Modeling Based on 3D Scans

Round 1

Reviewer 1 Report

I think that the manuscript might be accepted in the current format.

It is well written and everything is well explained.

I only have some small notes:

1) Line 210. In the second equation, the variable I (moment of inertia) is missed

2) Line 325. It would be "... the natural frequency of structures ..."

3) Lines 404-408. I suggest to rewrite this paragraph.

You mention:

The use of ambient seismic noise recording over long periods of time has the advantage that also transient events (near or far earthquakes; quarry blasts, etc.) or variations in external conditions  (seasons, floods, soil saturation, etc.) are covered.

First, in your case, you only analyzed 22 days, so it can not be considered a long period (e.g. for including seasons, floods, etc effects)

Anyway, for characterizing the natural frequency of the stalagmite you do not need to be recording days. It would be only required if you were interested in measuring the possible evolution of the natural frequency along the time. In this case, the measurements should be taken in continuous mode, maybe years.

About the transients, they does not affect to the estimation of the natural frequency of the stalagmite.

Parolai, S., & Galiana-Merino, J. J. (2006). Effect of transient seismic noise on estimates of H/V spectral ratios. Bulletin of the Seismological Society of America, 96(1), 228-236.

You also mention in this paragraph:

These external events or conditions could bring variations of the natural frequency over time [51].

This is true for the case analyzed in the paper [51], i.e. buildings, which are subject to different external factors that can affect their structure. But I do not think that is the case inside the cave.

4) Line 419. I would be " ... ambient noise makes possible to ..."

5) Line 428. I would write "... quarry blasts as a first step ..."

Author Response

Dear editors and reviewers,

We would like to thank the two reviewers for their constructive comments. We took them into account and it greatly improved the manuscript. In the attachment, we reply "in line" to all the comments of Reviewer 1.

I remain at your disposal should you have more questions and comments,

Aurélie Martin, for the authors.

Author Response File: Author Response.docx

Reviewer 2 Report

The authors present an analysis of very low, ambient vibrations of a stalagmite followed by finite element analysis of the same stalagmite.
This is an interesting contribution to the difficult problem of the dynamic properties of speleothems.
However, before the publication, some important clarifications must be introduced to the submission.

1. The Authors effectively identified two natural frequencies, yet such low ambient excitations (less than 1x10^-13 m) may not be able to excite some modes due to perhaps residual nonlinearities in the response. This problem is known in the applications of very low ambient excitations. The Authors should somehow comment on this reservation.

2. The Authors put measuring sensors at the base of the stalagmite and at the height of 2.52m. Why this height was chosen? In case only one sensor must be used, to avoid putting it in a vibration node of an important eigenform, the higher it is mounted, the better. Perhaps this height is the reason that the lower mode is shown smaller than the higher one in Fig. 5. Please confirm that the sensor at the height of 2.52 m had horizontal axes exactly parallel to the axes of the sensor at the base.
a) the stalagmite represent different properties in each of the horizontal axes of the sensors. Mixing the responses in two perpendicular axes is thus not correct. Please provide two separate figures for the plot shown in the lower left corner of Fig. 4. The responses separated along the perpendicular axes (of two sensors) should provide more information, including different responses of the stalagmite in separate directions.
b) Did the Authors check eigenproblem FEM solutions, including and excluding the mass of the relatively heavy sensor (1.7kg), which would affect a natural frequency and its directivity.

2. The Authors have obtained eigenproblem solution by using very approximate formulas and the Finite Element Method (FEM).
In the case of the information about solving eigenproblem by FEM it is very poorly reported (just four lines: 219-222).
a) it is not written how the elliptical elements are modeled. How many separate solid elements per elliptical cylinder were used? Are they at all divided into smaller FEM elements? The Authors applied this method to better understand a phenomenon of "splitting eigenmodes" in different directions. This reviewer doubts if such knowledge is so important. Anyway the more the stalagmite is complicated the more the "splitting" of frequencies occur. The Authors could see it clearly by solving eigenproblem of a cantilever with rectangular cross-section a x a with a double first frequency. When the cross-section becomes rectangular (a x b) the split of frequencies simply increases with 'b' deviating from 'a'. Similar phenomenon occurs with stalagmite differing with dynamic properties along perpendicular horizontal directions Thus anyway the full-size mesh model of the whole stalagmite, for obvious reasons, gives much better results than any approximate models, which makes the elipsoidal analysis unnecessary after a well developed FEM analysis is carried out.
b) what are the solid elements applied in FEM. Please describe number of nodes/faces for each element and/or give precise reference to literature where this element is described (including perhaps page number of the software manual - if necessary).
c) What was the total number of finite elements used in each analysis?
d) what was the number of respective dynamic degrees od freedom?
e) information shown in figure 10 is misleading. The first three are approximate models of cantilever beam, while the last two models are finite element models - not elliptical cross-section or 3d scan as written there. They must be described as FEM models with respective numbers of finite elements or dynamic degrees of freedom involved. Ellipses with internal division onto finite elements (how many per elliptical cylinder, how many in total). So called '3d scan' is just FEM mesh with unknown (?) dynamic degrees of freedom. How many?

3. The results of the eigenproblem strongly depend on key parameters used in FEM eigenproblem solution. Please comment on the effect of the accuracy of the rather rounded values of density rho=2500kg/m^3 and Young moduls E=22GPa.
Suppose for example it is +/-5% accuracy for Young modulus. It would be interesting for the readers to know how the +/-5% difference of E or rho would affect the computed natural frequencies?

Conclusion.
In order to properly address the reported problems, it is advised to rearrange the solutions of the FEM eignenproblem into separate points, particularly the 'elliptical slice' approach and full FEM analysis (called wrongly '3d scan' here) with exhausting explanations as described above.
When eigenmodes are sorted the usual approach in the dynamics of mechanical systems is to put them with increasing natural frequencies. So if one writes mode no.1, the next mode with higher nat. frq. is mode number 2 etc. If the Authors decide to number the modes in pairs please include number like 1a, 1b and then 2a and 2b with all the modes with f1a<f1b<f2a

Author Response

Dear editors and reviewers,

We would like to thank the two reviewers for their constructive comments. We took them into account and it greatly improved the manuscript. In the attachment, we reply "in line" to all the comments of Reviewer 2.

I remain at your disposal should you have more questions and comments,

Aurélie Martin, for the authors.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The Authors implemented all but one suggestion of this reviewer and the paper is substantially improved. In the letter/rebuttal on page 2, the Authors presented modified Fig. 4 with the measured responses separately measured along x & y directions. Yet in the corrected manuscript the old versions of the three figures of the lower part of Fig. 4 can be seen.

The separate responses along x & y (in different colors green & gray) bring totally new interesting information. Please introduce these 3 figures also for the readers of Geosciences. These figures can either replace existing 3 figures or can be additionally added depending on what is easier for the Authors.

Apart from this, the paper is ready for publication

Author Response

Dear Reviewers and Editors,

We would like to thank the reviewers one more time for the constructive comments. We took them into account and it greatly improved the manuscript. Following Reviewer 2’s remark, we have decided to highlight the observed amplification results. We have therefore added a few words on this matter in the abstract, the introduction, and the conclusion in addition to what has been added in the text (results and discussion). Below (in docx file) you will find our responses to the Reviewer.

I remain at your disposal should you have more questions and comments

A. Martin for the authors

Author Response File: Author Response.docx