the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 04 Sep 2024
Array-based ambient vibration modal analysis describes fracture-controlled mode shapes at a natural rock arch (Utah, USA)
Abstract. Fracture generation and propagation are primary mechanisms of structural degradation in natural rock arches and other freestanding rock landforms. However, methods to detect structural changes arising from fracturing are limited, particularly at sites with difficult access and high cultural value. Here we show how ambient vibration modal analysis can be used to identify fracture-controlled resonance modes at a sandstone arch in Utah (USA) aiding the selection of relevant modes for structural health monitoring. We characterized modal properties of Hunter Canyon Arch (i.e., resonance frequencies, damping ratios, and mode shapes) using spectral and cross-correlation analyses of data generated from an array of nodal geophones. Results revealed properties of nine resonance modes with frequencies between 1 and 12 Hz, damping ratios between 0.6 and 4.3 %, and an assortment of 3D mode shapes. Experimental data were then compared to numerical models implementing both homogeneous media and heterogeneous configurations generated through discretization of compliant zones in areas of mapped fractures. Results showed that all numerical solutions replicated the first two resonance modes of the arch, indicating these are insensitive to structural complexity derived from fractures and thus may be poor targets for monitoring. Meanwhile, heterogenous models with implemented fracture zones succeeded in matching the frequency and shape of one additional higher mode, indicating this mode is sensitive to fracture properties and thus most likely to respond to structural change from fracture propagation. Evolutionary crack damage modelling confirmed the sensitivity of this mode, and conversely the relative insensitivity of other modes, to simulated fracture propagation. While examination of fundamental modes is common in structural health monitoring studies, our results suggest that identifying changes in higher-order modes, i.e., those determined to be affected by fractured areas, may be more informative for characterizing structural damage in monitoring applications.
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RC1: 'Comment on egusphere-2024-1894', Anonymous Referee #1, 07 Oct 2024
This paper presents a seismic response analysis of the Hunter Canyon Arch (Utah, US). Both field observations and numerical modeling are adopted in the study. An array measurement of the arch’s ambient vibrations was performed with nine simultaneously recording seismic stations. A polarization analysis was performed on the recorded data, as well as a modal analysis using the cross-correlation technique. Modal frequencies, shapes, and corresponding damping ratios were estimated. Based on photogrammetry, a 3D geometrical arch model was developed, and present joint sets were characterized. This model was implemented into a finite element solver. The numerical model analysis was focused on the effects of fractures and their propagation on the modal frequencies and shapes. The numerical results were compared with the observations (modal frequencies, shapes). The authors demonstrate that the fundamental and the first higher modes are sensitive only to the overall shape and bulk modulus of the arch, while the third higher mode is sensitive to the presence of fracture and its extent. They conclude that the higher modes of the slender structures are more sensitive to the fractures (if present). Therefore, higher modes could be better suited for the monitoring of the localized damage compared to fundamental modes. The manuscript's topic is original, interesting, and suitable for the Earth Surface Dynamics journal. The presented analysis is in-depth, and the discussions are focused. The manuscript is well-written and comprehensible. In conclusion, I recommend only minor revisions.Specific comments:The addressing of the observed modes as presented in the text is not very convincing in some cases. This might be due to poor visibility of the weak modal motions in Fig. 4 (that is, the small vectors are too small):• Mode 2 (Fig. 4b) is addressed as a second-order transverse bending mode, but no node is visible in the observed shape, all points seem to be in phase. This, in my opinion, contradicts the simulation’s result (Fig 5c).• Mode 3 (Fig. 4c) looks very similar to Mode 1. The comparison with the simulation is also not very clear. The direct comparison of the motions at the observation points might be helpful.The cross-correlation technique is not very well introduced. A brief introduction would be helpful, outlining the suitability compared to other methods (frequency-domain decomposition, stochastic subspace identification)The relative modal mass (RMM) parameter is not well introduced in the text, although it is used as a criterion for the mode selection presented in Fig 5. A brief description should be included in the text, or RMM can be omitted in the text and the figure.The damping was not considered in the numerical simulations, although the observed values are discussed in the text. The possible implementation of damping in the simulations would give more insight. This could be stressed in the manuscript.The discussion of the damping ratios of different modes is not very clear. The damping ratio of the fundamental mode is found to be small, and it is stated in the manuscript: “These low damping values indicate that seismic energy is trapped within the structure and unable to propagate to the surrounding rock mass (Häusler et al., 2021b).” In contrast, in the explanation of the stronger damping of the higher modes, it is stated: “We hypothesize this change could arise from increasing participation of the abutment fracture area, and corresponding fracture shear and normal compliance, in modal deflection of the arch. The hypothesis of increasing energy transmissivity at the fracture scale is also supported by our numerical modeling results.” This is not clear. In general, seismic energy could leave the structure through the base and side of the arch which are assumed transparent. Indeed, the compliant fractures should result in lower transparency and, thus, even lower damping. However, the energy dissipation in the fracture might increase the damping. Please reformulate this part of the discussion.The angle difference presented in Fig. 6 is not defined in the text. Is the difference for the observed modes rather in horizontal or vertical direction?The name of the arch should be mentioned in the manuscript's title. Other arches are not studied and not much discussed in the text in the context of the study (higher modes and fractures).Different symbols should be used for the simulated modal frequencies (for example, in Fig. 5). You can use caret or tilde.Citation: https://doi.org/
10.5194/egusphere-2024-1894-RC1 -
AC1: 'Reply on RC1', Guglielmo Grechi, 30 Oct 2024
We thank the reviewer for the insightful comments and suggestions. We appreciate the time and effort invested in the review process and have carefully considered each point raised. Below, we provide responses to each specific comment:
SC1.1) The addressing of the observed modes as presented in the text is not very convincing in some cases. This might be due to poor visibility of the weak modal motions in Fig. 4 (that is, the small vectors are too small):
R1.1) We agree that the normalized modal vector amplitude at some monitoring stations is in some cases very small, especially for the first three modes in correspondence with the stations closest to the arch’s abutment. However, these reduced amplitudes indicate small participation of those points in the modal deformation pattern at those resonance modes, hence we believe those vectors may enhance rather than hide the visualization and interpretation of modal shapes.
SC1.2) Mode 2 (Fig. 4b) is addressed as a second-order transverse bending mode, but no node is visible in the observed shape, all points seem to be in phase. This, in my opinion, contradicts the simulation’s result (Fig 5c).
R1.2) It is indeed true that no nodal point can be observed from experimental results in Figure 4b, but this is because the transverse or out-of-plane bending takes place on the Radial-Vertical (YZ) plane, thus mostly involving the pier/pillar. Due to the impossibility to measure ambient vibrations anywhere else than on the arch lintel, we could only interpret this mode of vibration based on our experience and numerical modeling results. This modal deflection pattern is very well reproduced by our numerical modeling results. However, we understand that Figure 5c may not represent this second-order bending mode correctly, hence we have decided to modify that panel to enhance the visualization of mode 2. To better clarify this point I’ve attached a sketch created using our numerical modeling results for mode 1 and 2, that should highlight how these two modes are indeed a first- and second-order transverse bending modes (see the supplementary file).
SC1.3) Mode 3 (Fig. 4c) looks very similar to Mode 1. The comparison with the simulation is also not very clear. The direct comparison of the motions at the observation points might be helpful.
R1.3) This is an example of how reduced participations in modal deformation patterns at specific locations can highlight differences between the measured resonance modes. In particular, the main differences in the experimental results between Mode 1 and Mode 3 can be derived by observing normalized modal vectors at stations H06 and H07, both in terms of magnitude and direction. Considering Mode 1, station H06 features a small amplitude, transverse-oriented modal vector that is in-phase with all the other vectors measured on the arch (i.e., from H01 to H05). For the same resonance mode, station H07 shows no significant participation, and this is well-described by the absence of a visible modal vector at that specific location. Considering Mode 3, the modal vectors for stations H06 and H07 describe a different configuration from Mode 1. Here, despite their reduced amplitude, those small amplitude vectors align toward a radial-oriented direction with respect to the arch. The direct comparison between experimental and numerical modeling results for each resonance mode, both in terms of normalized modal displacement amplitudes and angle differences between modal vector directions, is provided in Figure 6.
SC2) The cross-correlation technique is not very well introduced. A brief introduction would be helpful, outlining the suitability compared to other methods (frequency-domain decomposition, stochastic subspace identification).
R2) We added a new paragraph to the manuscript:
“[…] Cross-correlation modal analysis, also referred to as Natural Excitation Technique (NExT) (Farrar and James III, 1997), is a time-domain, output-only method used to estimate modal properties of vibrating structures by analyzing the cross-correlation functions between output response measurements under ambient excitation. We selected this analytical method over Frequency Domain Decomposition (FDD) and Stochastic Subspace Identification (SSI) techniques (Brincker et al., 2001; Van Overschee and De Moor, 1996) on the basis of computational efficiency. The robustness of the cross-correlation modal technique, compared to FDD and SSI, has been previously demonstrated in comparative studies (Bessette-Kirton et al., 2022; Häusler et al., 2021a), establishing it as a reliable approach for accurate modal property estimation of natural structures under ambient excitation.”
SC3) The relative modal mass (RMM) parameter is not well introduced in the text, although it is used as a criterion for the mode selection presented in Fig 5. A brief description should be included in the text, or RMM can be omitted in the text and the figure.
R3) We have updated the corresponding paragraph, adding more detail and references to introduce the meaning of the relative modal mass parameter and its implications in modal analysis:
“[…] For this reason, we elected to analyze only those modes characterized by the highest relative modal mass (RMM) values (Fig. 5b). This direction-dependent parameter is frequently employed in engineering studies as it measures the extent of mass participation at each resonance mode (Table 1), thus allowing for assessment of the significance of specific eigenmodes in describing the dynamic behavior of a structure (Aenlle et al., 2021; Mayes et al., 2015).”
SC4) The damping was not considered in the numerical simulations, although the observed values are discussed in the text. The possible implementation of damping in the simulations would give more insight. This could be stressed in the manuscript.
R4) We appreciate the suggestion to include damping in our numerical simulations. This is indeed an interesting line of research that could further refine our understanding of the system’s dynamic behavior, particularly by providing a more detailed representation of energy dissipation during free vibrations. However, we believe that incorporating damping, while valuable, is beyond the scope of this study. The main objective of our research is to investigate the sensitivity of resonance frequencies and mode shapes to discrete rock mass fractures, and our numerical models were specifically calibrated to match these resonance modes, and the insights we derive are based on the comparison of modal frequencies and shapes. Furthermore, while we have experimentally measured damping ratios and discussed them in the context of the observed resonance modes, incorporating these into the numerical modeling would require careful validation steps. It would necessitate an additional layer of complexity in our modeling, including time-consuming efforts to develop, calibrate, and validate models that realistically capture energy dissipation mechanisms. Given the scope of our current work, we believe it is appropriate to leave this task for future research.
SC5) The discussion of the damping ratios of different modes is not very clear. The damping ratio of the fundamental mode is found to be small, and it is stated in the manuscript:
Line 287–288 “[…] These low damping values indicate that seismic energy is trapped within the structure and unable to propagate to the surrounding rock mass (Häusler et al., 2021b).”
In contrast, in the explanation of the stronger damping of the higher modes, it is stated:
Line 292–2945 “[…]. We hypothesize this change could arise from increasing participation of the abutment fracture area, and corresponding fracture shear and normal compliance, in modal deflection of the arch. The hypothesis of increasing energy transmissivity at the fracture scale is also supported by our numerical modeling results.”
This is not clear. In general, seismic energy could leave the structure through the base and side of the arch which are assumed transparent. Indeed, the compliant fractures should result in lower transparency and, thus, even lower damping. However, the energy dissipation in the fracture might increase the damping. Please reformulate this part of the discussion.
R5) Thank you for these insightful comments regarding the discussion of damping ratios across different modes. We have tried to improve the abovementioned paragraph by adding a more in depth discussion. We interpreted the very low damping values observed for Modes 1 and 2 (≤1%) as primarily resulting from an internal friction mechanism (i.e., material damping) and the minimal contact area of the abutment and base surfaces relative to the overall size of the arch. This limited contact area restricts pathways for seismic energy to dissipate into the surrounding rock mass, effectively trapping energy within the structure (radiation damping). The significant increase in the modal damping ratio observed at Mode 3 suggests the involvement of additional energy dissipation mechanisms.
Below is the revised paragraph incorporating the new discussion:
[…] “We hypothesize this change could arise from increasing participation of the abutment fracture area, and corresponding fracture shear and normal compliance, in modal deflection of the arch. In structural engineering, it is well-documented that frictional interfaces, such as joints and connections, contribute to energy dissipation due to micro-slip and frictional losses under dynamic loading (Cimellaro, 2023; Lazan, 1968). This phenomenon, often referred to as interface or slip damping, is critical in dynamic analysis and structural design. In engineering geology, the role of fractures in modifying energy dissipation mechanisms is less extensively studied, and there is a higher degree of uncertainty due to the complex nature of rock fractures (i.e., geometry, properties, contact areas). Despite this, it is reasonable that frictional interactions at fracture interfaces, as well as surface roughness, can contribute to increased energy dissipation within rock masses (Bandis et al., 1983; Goodman, 1980; Habaraduwa Peellage et al., 2024). The hypothesis of increasing energy dissipation at the fracture scale is also supported by our numerical modelling results.”
Our hypothesis is that, although we cannot precisely characterize the fracture behavior at the scale of our study, frictional mechanisms at the fracture scale—or even at the surface roughness scale—may enhance the damping of higher-order modes. This would lead to greater energy dissipation as seismic energy is partially transmitted and dissipated through these discontinuities.
SC6) The angle difference presented in Fig. 6 is not defined in the text. Is the difference for the observed modes rather in horizontal or vertical direction?
The angle difference presented in Figure 6 refers to the angular separation between the observed experimental and numerical modal vectors, regardless of orientation in horizontal or vertical directions. We retrieved the angle (θ) by evaluating the arccosine of the normalized dot product between vector pairs (i.e., experimental modal vectors were always used as reference). The equation was added to the text.
SC7) The name of the arch should be mentioned in the manuscript's title. Other arches are not studied and not much discussed in the text in the context of the study (higher modes and fractures).
R7) The title has been updated following the reviewer’s suggestion:
“Array-based ambient vibration modal analysis describes fracture-controlled mode shapes at the Hunter Canyon Arch (Utah, USA)”
SC8) Different symbols should be used for the simulated modal frequencies (for example, in Fig. 5). You can use caret or tilde.
R8) We have changed the labels for modeled and measured resonance modes in Figure 5 (see the supplementary file) as we agree that it could have been confusing. However, we believe it is not necessary to change the adopted symbols in other parts of the manuscript.
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AC1: 'Reply on RC1', Guglielmo Grechi, 30 Oct 2024
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RC2: 'Comment on egusphere-2024-1894', Anonymous Referee #2, 03 Nov 2024
The manuscript “Array-based ambient vibration modal analysis describes fracture-controlled mode shapes at a natural rock arch (Utah, USA)” by Grechi et al. shows how modal analysis of ambient vibrations can inform about the structural state of free-standing landforms. The approach is novel. The results exciting and promising. And this is also the flaw. The reader needs to be well versed in both fracture mechanics and stability, as well as in ambient seismic methods to be able to see the contribution this manuscript can make. The manuscript also lacks a bit in clarity of the structure and scope, and explanation of the major concepts and methods.
However, with some minor revision, this manuscript will be a great contribution to our understanding of the rate of fate of rock arches.
Detailed comments:
Title
The title reads quite complicated and not well descriptive of the main finding and scope.
- As far as I understand this is specific to the Hunter Canyon Arch and could maybe be also more general to natural rock arches in Utah. So, I would either make the title general to that “…at natural rock arches (Utah, USA)” or specific “at Hunter Canyon Arch (Utah, USA)”
- What is really the key finding here? – The title describes more of the method “Array-based ambient vibration modal analysis….” – isn’t the key finding rather that there is a mode shape that is indicative of fractures, which can be determined by ambient vibration analysis?
Abstract
The abstract, if read before the manuscript, reads quite confusing. This is partially due to unnecessary complicated sentence structure, as well as jargon and concepts that are not introduced.
- What is the motivation? What is the scope? What is the research question? How did you address this? If you could be a bit more straight forward/explicit on that, that would enhance the readability. E.g., the motivation is the degradation of free-standing rock arches by (progressive) damage evolution, the scope is to determine the structural state including the fractures, the research question/hypothesis is that the damage is affecting the mode shapes, your approach is to do an array-based ambient vibration modal analysis.
- Can you really show that it is the “Fracture generation and propagation” that leads to damage or is it disintegration and weakening of bonds, or buckling of force chains in the sandstone, too?
- These fractures lead to structural degradation but they are also the reason for the shape of the arch in the first place. What is the difference here (e.g., time-dependent, environmentally controlled material property changes)?
- Why do natural arches have a “high cultural value” and why does it matter here – you are not suggesting any way to preserve them? Or is it rather that not only the access (physically getting there) but also the accessibility (permission to get there) is limited. What you might want to point out is that only non-destructive methods are eventually allowed.
- Line 9ff: The sentence reads repetitive. Please simplify and here it would be great to get your hypothesis, so it is easier to follow why you are doing what.
- Some of the wording seems off: results “revealed”, and “assortment of 3D mode shapes”.
- Lines14-24: While this is exiting to read as a summary, it is hard to follow if you did not read the whole manuscript before. Please narrow down and ideally be more quantitative of the key finding and its implications.
1 Introduction
Imagine the reader skipped reading the abstract, they will not get what the motivation is. Starting with “Modal analysis” and in the following using jargon and very specific concepts without defining or introducing them makes it quite ambiguous, especially if the reader is not familiar with seismic methods including the deployment and analysis.
2 Study site…
The section reads quite unclear, and some of the methods could go into the supplements. For example, what do you mean by (line 82) “…appearance of precarious stability”? What is a (line 94) “fracture-bounded compartment”? How does the fracture sets link to your analysis or do you map them but then assume a continuum of the material and transmission of the arching stress? What is the difference between the mapped fracture sets and the fractures that degrade the stability (scale, openness, persistence…)? If the fractures (line 95f) have aperture how are your vibrations transmitted?
Figure 1 is rather confusing than providing a good overview. Only S1 and S2 fractures are mentioned later in the manuscript. The photos are very red and hard to see where the arch is and where the background rocks are. Can you provide one that just depicts the arch? Similar to figure 2a? Indicating fracture planes in red dashed lines is nearly invisible, even if not colorblind.
3 Modal analysis methods
3.1 This reads a bit confusing and could be linked to the fracture sets described before. For example, take Figure 2a and make it Figure 1. What do you mean by (line109) “but still encompassing the fracture-isolated rock volume”? What do you mean with (line 113) “Winds where calm”? -remember some readers might not understand why that piece of information is important and why you don’t report on other environmental conditions, e.g., T or RH. Please be clear on what this means, e.g., you mounted the samples with putty to provide good coupling, calm winds allowed for low environmental noise, and thus a high(er) signal to noise ratio of your measurements.
3.2 It is hard to follow what you did exactly. Especially, how you identified the “prominent peaks“ and why those are interpreted as the “resonance modes of the arch”. I see two options, one would be to share the code in the supplement, the other would be to detail the methods by which you picked the peaks.
In Figure 2b, please label the peaks you identified. They should match in all panels. To better compare it might be good to have the same y-axis for all directions. The x-axis differs between b panels and c and d, why? Can you combine c and d, and plot them under b, making the x axis not the mode but the frequency and just indicating the modes in all plots with a dashed/dotted line? – The dotted lines in b seem not to correspond to those in c and d.
How does the amplitude of cross-correlation (Figure 3a) enable “visualization of mode shapes along the linear array”? - which is not fully linear, as H08 is offset. Please be clear on what it shows. You have not defined what “mode shapes” are by now, and what do you mean by “relative modal displacement” (line 47). Please use the figure to better explain what the modal shape and changes thereof show, as well as what the composite vector and its inclination can tell. In this section you are describing the methods, not the results. Where can I find the cross correlograms for the other frequencies?
You are not referring to Figure 3b in the text at all. Is it necessary? Why is the direction of the ZZ component not indicated? There seem to be also some artefacts in the figure which are distracting. Maybe consider simplifying the 3D shaded model to outlines, e.g., sometimes less is more.
3.3 This section starts with a logical jump. 3.1 and 3.2 are about the physical arch and the measurement of the ambient vibrations. This section takes a step further by using the information from 3.1 on the structure, turning it into a numerical model arch. Then you use the measured vibrational modes to simulate the dynamics in the model arch. The aspect of how you set up the model geometry, what you include and what not reads very confusing. Please clarify between model setup (mesh, smoothing, spatial data, ...), material properties and structural features and with which assumption they have been implemented in the model, please also provide the boundary conditions. You seem to have different models you tried, please make their difference explicit and why you chose those. For example, what do you mean by (lines 182) “…generate these compliant mechanical zones.”, and “…a 1 m wide zone cutting…”, or (line 189) “…fractures were modelled as open zones”.
The dynamic modeling part is also not easy to understand as it is mixed in with structural aspects, without saying how the one affects the other. Please be clear on how you represent the dynamics in your models and how you get “data” from your model which is then compared with the field measurements. Please state how you compared the modeling and field data. It seems also that you are not showing the comparison of the first 4 resonance modes, but 1,2,3, and 6 (Figure 6).
4 Results
Here it would be good to link the figures better with the text – show and tell. For example, what does the Figure 2b really show? I can count and try to detect where the 9 peaks are, or you could label them and mark them throughout Figure 2 (see comment above).
Please present the results first and then interpret (and discuss, in the Discussion). For example, to highlight what the result of the spectral analysis is you need to make clear how this was done. Then state the results, you found 9 peaks in the arch which were not present in the reference station. The interpretation of those peaks is part of the discussion (strictly speaking). This is a pattern that is found throughout the results and makes it hard to discern what the result really are and what your interpretation is. This also leads to a repetition in the Discussion.
It would also be good if you could link your results to your research question and hypothesis, e.g., what was tested/looked for. For example, starting in line 225, it becomes unclear what you are aiming at in this manuscript: Are you interested in structural controls of the arch or fracture-controlled modes, as the title suggests, or is that the same for you?
Line 228: “For this reason we selected…”
Line 229: you are introducing a new terminology here: “highest relative modal mass (RMM) values”, please introduce this.
Line 243: The wording “…can be better appreciated…” does not seem to fit here.
Figure 4: Please label in black, then you can omit the outlines, but the contrast is still higher. You are not using the whole inclination-color-wheel from 90º to -90º. Please adapt so you have a better distinction in the colored arrows. Why did you change the view for panel (f) and (g)? Could you use view (f) for all of them? It seems to become overly complicated.
Figure 5: is (a) necessary? (b) please use different symbols for each direction. The color scheme might also not be visible for some or printed in grey scale. If you use black outlines, the difference in color becomes even harder to see for the smaller symbols. (c) What is the min -max color bar? How do the grey arrows compare to Figure 4 and the inclination-colored arrows? Why does the view change or is the arch deforming?
Table 1: Maybe this can go in the Supplementary Information? It is also confusing why the resonance mode no. 6 becomes 4 in the model but stays 6 in Figure 6. What is the reasoning behind changing the frequency from the field and in the different models? Why did you pick those four modes for the modeling and not the others?
Figure 6: Can you please use different symbols for the field data and the models. Color comment see above.
5 Discussion
The discussion reads rather length and not to the point. Can you be more specific and quantitative? For example, lines 287f, what do you mean by this sentence and how does your measurement and the shown results support the trapping of seismic energy?
If you want to discuss the hypothesis of e.g., “…increasing energy transmissivity at the fracture scale”, it would be good to state this and introduce the concept and how it relates to your study, e.g., in the introduction. Please review the hypothesis you are stating in the introduction and those in the discussion and see which of them are addressed by your study, and which are not. Please be clear about, what was observed in the field and what you modeled.
The part on the fractures and how it was implemented in the model reads very confusing, which is partly because new hypothesis and data (fracture sets) are introduced but also the basis for the whole study is unclear, e.g., what is the anticipated outcome of the model, and how does the measured resonance modes fit in? One thing that might help is to make the temporal and spatial (expected) damage/weaknesses explicit, and how that would show in the resonance frequency. This would also help to link it better to Figure 7.
Figure 7: See comments to colors and labels above, as well as the observed and modelled modes 1,2,3,6/4. (c ) why relative modal displacement? Why does it change? And why is mode 4 plotted with flipped axis?
6 Conclusion
Please come back to the bigger picture and the research question here. The writing is lengthy and ambiguous and vague. Please be precise on which hypotheses did you test and what are their outcome? What are the take home messages?
Tell the reader why these frequencies, why these modes are indicative of fractures, bulk properties, local changes? How does it relate to spatial scales, and how much would the mode shape change due to ongoing fracturing (temporal), thus how sensitive is this approach? What can we take away from this specific arch and apply it to other structures or arches?
Citation: https://doi.org/10.5194/egusphere-2024-1894-RC2
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2024-1894', Anonymous Referee #1, 07 Oct 2024
This paper presents a seismic response analysis of the Hunter Canyon Arch (Utah, US). Both field observations and numerical modeling are adopted in the study. An array measurement of the arch’s ambient vibrations was performed with nine simultaneously recording seismic stations. A polarization analysis was performed on the recorded data, as well as a modal analysis using the cross-correlation technique. Modal frequencies, shapes, and corresponding damping ratios were estimated. Based on photogrammetry, a 3D geometrical arch model was developed, and present joint sets were characterized. This model was implemented into a finite element solver. The numerical model analysis was focused on the effects of fractures and their propagation on the modal frequencies and shapes. The numerical results were compared with the observations (modal frequencies, shapes). The authors demonstrate that the fundamental and the first higher modes are sensitive only to the overall shape and bulk modulus of the arch, while the third higher mode is sensitive to the presence of fracture and its extent. They conclude that the higher modes of the slender structures are more sensitive to the fractures (if present). Therefore, higher modes could be better suited for the monitoring of the localized damage compared to fundamental modes. The manuscript's topic is original, interesting, and suitable for the Earth Surface Dynamics journal. The presented analysis is in-depth, and the discussions are focused. The manuscript is well-written and comprehensible. In conclusion, I recommend only minor revisions.Specific comments:The addressing of the observed modes as presented in the text is not very convincing in some cases. This might be due to poor visibility of the weak modal motions in Fig. 4 (that is, the small vectors are too small):• Mode 2 (Fig. 4b) is addressed as a second-order transverse bending mode, but no node is visible in the observed shape, all points seem to be in phase. This, in my opinion, contradicts the simulation’s result (Fig 5c).• Mode 3 (Fig. 4c) looks very similar to Mode 1. The comparison with the simulation is also not very clear. The direct comparison of the motions at the observation points might be helpful.The cross-correlation technique is not very well introduced. A brief introduction would be helpful, outlining the suitability compared to other methods (frequency-domain decomposition, stochastic subspace identification)The relative modal mass (RMM) parameter is not well introduced in the text, although it is used as a criterion for the mode selection presented in Fig 5. A brief description should be included in the text, or RMM can be omitted in the text and the figure.The damping was not considered in the numerical simulations, although the observed values are discussed in the text. The possible implementation of damping in the simulations would give more insight. This could be stressed in the manuscript.The discussion of the damping ratios of different modes is not very clear. The damping ratio of the fundamental mode is found to be small, and it is stated in the manuscript: “These low damping values indicate that seismic energy is trapped within the structure and unable to propagate to the surrounding rock mass (Häusler et al., 2021b).” In contrast, in the explanation of the stronger damping of the higher modes, it is stated: “We hypothesize this change could arise from increasing participation of the abutment fracture area, and corresponding fracture shear and normal compliance, in modal deflection of the arch. The hypothesis of increasing energy transmissivity at the fracture scale is also supported by our numerical modeling results.” This is not clear. In general, seismic energy could leave the structure through the base and side of the arch which are assumed transparent. Indeed, the compliant fractures should result in lower transparency and, thus, even lower damping. However, the energy dissipation in the fracture might increase the damping. Please reformulate this part of the discussion.The angle difference presented in Fig. 6 is not defined in the text. Is the difference for the observed modes rather in horizontal or vertical direction?The name of the arch should be mentioned in the manuscript's title. Other arches are not studied and not much discussed in the text in the context of the study (higher modes and fractures).Different symbols should be used for the simulated modal frequencies (for example, in Fig. 5). You can use caret or tilde.Citation: https://doi.org/
10.5194/egusphere-2024-1894-RC1 -
AC1: 'Reply on RC1', Guglielmo Grechi, 30 Oct 2024
We thank the reviewer for the insightful comments and suggestions. We appreciate the time and effort invested in the review process and have carefully considered each point raised. Below, we provide responses to each specific comment:
SC1.1) The addressing of the observed modes as presented in the text is not very convincing in some cases. This might be due to poor visibility of the weak modal motions in Fig. 4 (that is, the small vectors are too small):
R1.1) We agree that the normalized modal vector amplitude at some monitoring stations is in some cases very small, especially for the first three modes in correspondence with the stations closest to the arch’s abutment. However, these reduced amplitudes indicate small participation of those points in the modal deformation pattern at those resonance modes, hence we believe those vectors may enhance rather than hide the visualization and interpretation of modal shapes.
SC1.2) Mode 2 (Fig. 4b) is addressed as a second-order transverse bending mode, but no node is visible in the observed shape, all points seem to be in phase. This, in my opinion, contradicts the simulation’s result (Fig 5c).
R1.2) It is indeed true that no nodal point can be observed from experimental results in Figure 4b, but this is because the transverse or out-of-plane bending takes place on the Radial-Vertical (YZ) plane, thus mostly involving the pier/pillar. Due to the impossibility to measure ambient vibrations anywhere else than on the arch lintel, we could only interpret this mode of vibration based on our experience and numerical modeling results. This modal deflection pattern is very well reproduced by our numerical modeling results. However, we understand that Figure 5c may not represent this second-order bending mode correctly, hence we have decided to modify that panel to enhance the visualization of mode 2. To better clarify this point I’ve attached a sketch created using our numerical modeling results for mode 1 and 2, that should highlight how these two modes are indeed a first- and second-order transverse bending modes (see the supplementary file).
SC1.3) Mode 3 (Fig. 4c) looks very similar to Mode 1. The comparison with the simulation is also not very clear. The direct comparison of the motions at the observation points might be helpful.
R1.3) This is an example of how reduced participations in modal deformation patterns at specific locations can highlight differences between the measured resonance modes. In particular, the main differences in the experimental results between Mode 1 and Mode 3 can be derived by observing normalized modal vectors at stations H06 and H07, both in terms of magnitude and direction. Considering Mode 1, station H06 features a small amplitude, transverse-oriented modal vector that is in-phase with all the other vectors measured on the arch (i.e., from H01 to H05). For the same resonance mode, station H07 shows no significant participation, and this is well-described by the absence of a visible modal vector at that specific location. Considering Mode 3, the modal vectors for stations H06 and H07 describe a different configuration from Mode 1. Here, despite their reduced amplitude, those small amplitude vectors align toward a radial-oriented direction with respect to the arch. The direct comparison between experimental and numerical modeling results for each resonance mode, both in terms of normalized modal displacement amplitudes and angle differences between modal vector directions, is provided in Figure 6.
SC2) The cross-correlation technique is not very well introduced. A brief introduction would be helpful, outlining the suitability compared to other methods (frequency-domain decomposition, stochastic subspace identification).
R2) We added a new paragraph to the manuscript:
“[…] Cross-correlation modal analysis, also referred to as Natural Excitation Technique (NExT) (Farrar and James III, 1997), is a time-domain, output-only method used to estimate modal properties of vibrating structures by analyzing the cross-correlation functions between output response measurements under ambient excitation. We selected this analytical method over Frequency Domain Decomposition (FDD) and Stochastic Subspace Identification (SSI) techniques (Brincker et al., 2001; Van Overschee and De Moor, 1996) on the basis of computational efficiency. The robustness of the cross-correlation modal technique, compared to FDD and SSI, has been previously demonstrated in comparative studies (Bessette-Kirton et al., 2022; Häusler et al., 2021a), establishing it as a reliable approach for accurate modal property estimation of natural structures under ambient excitation.”
SC3) The relative modal mass (RMM) parameter is not well introduced in the text, although it is used as a criterion for the mode selection presented in Fig 5. A brief description should be included in the text, or RMM can be omitted in the text and the figure.
R3) We have updated the corresponding paragraph, adding more detail and references to introduce the meaning of the relative modal mass parameter and its implications in modal analysis:
“[…] For this reason, we elected to analyze only those modes characterized by the highest relative modal mass (RMM) values (Fig. 5b). This direction-dependent parameter is frequently employed in engineering studies as it measures the extent of mass participation at each resonance mode (Table 1), thus allowing for assessment of the significance of specific eigenmodes in describing the dynamic behavior of a structure (Aenlle et al., 2021; Mayes et al., 2015).”
SC4) The damping was not considered in the numerical simulations, although the observed values are discussed in the text. The possible implementation of damping in the simulations would give more insight. This could be stressed in the manuscript.
R4) We appreciate the suggestion to include damping in our numerical simulations. This is indeed an interesting line of research that could further refine our understanding of the system’s dynamic behavior, particularly by providing a more detailed representation of energy dissipation during free vibrations. However, we believe that incorporating damping, while valuable, is beyond the scope of this study. The main objective of our research is to investigate the sensitivity of resonance frequencies and mode shapes to discrete rock mass fractures, and our numerical models were specifically calibrated to match these resonance modes, and the insights we derive are based on the comparison of modal frequencies and shapes. Furthermore, while we have experimentally measured damping ratios and discussed them in the context of the observed resonance modes, incorporating these into the numerical modeling would require careful validation steps. It would necessitate an additional layer of complexity in our modeling, including time-consuming efforts to develop, calibrate, and validate models that realistically capture energy dissipation mechanisms. Given the scope of our current work, we believe it is appropriate to leave this task for future research.
SC5) The discussion of the damping ratios of different modes is not very clear. The damping ratio of the fundamental mode is found to be small, and it is stated in the manuscript:
Line 287–288 “[…] These low damping values indicate that seismic energy is trapped within the structure and unable to propagate to the surrounding rock mass (Häusler et al., 2021b).”
In contrast, in the explanation of the stronger damping of the higher modes, it is stated:
Line 292–2945 “[…]. We hypothesize this change could arise from increasing participation of the abutment fracture area, and corresponding fracture shear and normal compliance, in modal deflection of the arch. The hypothesis of increasing energy transmissivity at the fracture scale is also supported by our numerical modeling results.”
This is not clear. In general, seismic energy could leave the structure through the base and side of the arch which are assumed transparent. Indeed, the compliant fractures should result in lower transparency and, thus, even lower damping. However, the energy dissipation in the fracture might increase the damping. Please reformulate this part of the discussion.
R5) Thank you for these insightful comments regarding the discussion of damping ratios across different modes. We have tried to improve the abovementioned paragraph by adding a more in depth discussion. We interpreted the very low damping values observed for Modes 1 and 2 (≤1%) as primarily resulting from an internal friction mechanism (i.e., material damping) and the minimal contact area of the abutment and base surfaces relative to the overall size of the arch. This limited contact area restricts pathways for seismic energy to dissipate into the surrounding rock mass, effectively trapping energy within the structure (radiation damping). The significant increase in the modal damping ratio observed at Mode 3 suggests the involvement of additional energy dissipation mechanisms.
Below is the revised paragraph incorporating the new discussion:
[…] “We hypothesize this change could arise from increasing participation of the abutment fracture area, and corresponding fracture shear and normal compliance, in modal deflection of the arch. In structural engineering, it is well-documented that frictional interfaces, such as joints and connections, contribute to energy dissipation due to micro-slip and frictional losses under dynamic loading (Cimellaro, 2023; Lazan, 1968). This phenomenon, often referred to as interface or slip damping, is critical in dynamic analysis and structural design. In engineering geology, the role of fractures in modifying energy dissipation mechanisms is less extensively studied, and there is a higher degree of uncertainty due to the complex nature of rock fractures (i.e., geometry, properties, contact areas). Despite this, it is reasonable that frictional interactions at fracture interfaces, as well as surface roughness, can contribute to increased energy dissipation within rock masses (Bandis et al., 1983; Goodman, 1980; Habaraduwa Peellage et al., 2024). The hypothesis of increasing energy dissipation at the fracture scale is also supported by our numerical modelling results.”
Our hypothesis is that, although we cannot precisely characterize the fracture behavior at the scale of our study, frictional mechanisms at the fracture scale—or even at the surface roughness scale—may enhance the damping of higher-order modes. This would lead to greater energy dissipation as seismic energy is partially transmitted and dissipated through these discontinuities.
SC6) The angle difference presented in Fig. 6 is not defined in the text. Is the difference for the observed modes rather in horizontal or vertical direction?
The angle difference presented in Figure 6 refers to the angular separation between the observed experimental and numerical modal vectors, regardless of orientation in horizontal or vertical directions. We retrieved the angle (θ) by evaluating the arccosine of the normalized dot product between vector pairs (i.e., experimental modal vectors were always used as reference). The equation was added to the text.
SC7) The name of the arch should be mentioned in the manuscript's title. Other arches are not studied and not much discussed in the text in the context of the study (higher modes and fractures).
R7) The title has been updated following the reviewer’s suggestion:
“Array-based ambient vibration modal analysis describes fracture-controlled mode shapes at the Hunter Canyon Arch (Utah, USA)”
SC8) Different symbols should be used for the simulated modal frequencies (for example, in Fig. 5). You can use caret or tilde.
R8) We have changed the labels for modeled and measured resonance modes in Figure 5 (see the supplementary file) as we agree that it could have been confusing. However, we believe it is not necessary to change the adopted symbols in other parts of the manuscript.
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AC1: 'Reply on RC1', Guglielmo Grechi, 30 Oct 2024
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RC2: 'Comment on egusphere-2024-1894', Anonymous Referee #2, 03 Nov 2024
The manuscript “Array-based ambient vibration modal analysis describes fracture-controlled mode shapes at a natural rock arch (Utah, USA)” by Grechi et al. shows how modal analysis of ambient vibrations can inform about the structural state of free-standing landforms. The approach is novel. The results exciting and promising. And this is also the flaw. The reader needs to be well versed in both fracture mechanics and stability, as well as in ambient seismic methods to be able to see the contribution this manuscript can make. The manuscript also lacks a bit in clarity of the structure and scope, and explanation of the major concepts and methods.
However, with some minor revision, this manuscript will be a great contribution to our understanding of the rate of fate of rock arches.
Detailed comments:
Title
The title reads quite complicated and not well descriptive of the main finding and scope.
- As far as I understand this is specific to the Hunter Canyon Arch and could maybe be also more general to natural rock arches in Utah. So, I would either make the title general to that “…at natural rock arches (Utah, USA)” or specific “at Hunter Canyon Arch (Utah, USA)”
- What is really the key finding here? – The title describes more of the method “Array-based ambient vibration modal analysis….” – isn’t the key finding rather that there is a mode shape that is indicative of fractures, which can be determined by ambient vibration analysis?
Abstract
The abstract, if read before the manuscript, reads quite confusing. This is partially due to unnecessary complicated sentence structure, as well as jargon and concepts that are not introduced.
- What is the motivation? What is the scope? What is the research question? How did you address this? If you could be a bit more straight forward/explicit on that, that would enhance the readability. E.g., the motivation is the degradation of free-standing rock arches by (progressive) damage evolution, the scope is to determine the structural state including the fractures, the research question/hypothesis is that the damage is affecting the mode shapes, your approach is to do an array-based ambient vibration modal analysis.
- Can you really show that it is the “Fracture generation and propagation” that leads to damage or is it disintegration and weakening of bonds, or buckling of force chains in the sandstone, too?
- These fractures lead to structural degradation but they are also the reason for the shape of the arch in the first place. What is the difference here (e.g., time-dependent, environmentally controlled material property changes)?
- Why do natural arches have a “high cultural value” and why does it matter here – you are not suggesting any way to preserve them? Or is it rather that not only the access (physically getting there) but also the accessibility (permission to get there) is limited. What you might want to point out is that only non-destructive methods are eventually allowed.
- Line 9ff: The sentence reads repetitive. Please simplify and here it would be great to get your hypothesis, so it is easier to follow why you are doing what.
- Some of the wording seems off: results “revealed”, and “assortment of 3D mode shapes”.
- Lines14-24: While this is exiting to read as a summary, it is hard to follow if you did not read the whole manuscript before. Please narrow down and ideally be more quantitative of the key finding and its implications.
1 Introduction
Imagine the reader skipped reading the abstract, they will not get what the motivation is. Starting with “Modal analysis” and in the following using jargon and very specific concepts without defining or introducing them makes it quite ambiguous, especially if the reader is not familiar with seismic methods including the deployment and analysis.
2 Study site…
The section reads quite unclear, and some of the methods could go into the supplements. For example, what do you mean by (line 82) “…appearance of precarious stability”? What is a (line 94) “fracture-bounded compartment”? How does the fracture sets link to your analysis or do you map them but then assume a continuum of the material and transmission of the arching stress? What is the difference between the mapped fracture sets and the fractures that degrade the stability (scale, openness, persistence…)? If the fractures (line 95f) have aperture how are your vibrations transmitted?
Figure 1 is rather confusing than providing a good overview. Only S1 and S2 fractures are mentioned later in the manuscript. The photos are very red and hard to see where the arch is and where the background rocks are. Can you provide one that just depicts the arch? Similar to figure 2a? Indicating fracture planes in red dashed lines is nearly invisible, even if not colorblind.
3 Modal analysis methods
3.1 This reads a bit confusing and could be linked to the fracture sets described before. For example, take Figure 2a and make it Figure 1. What do you mean by (line109) “but still encompassing the fracture-isolated rock volume”? What do you mean with (line 113) “Winds where calm”? -remember some readers might not understand why that piece of information is important and why you don’t report on other environmental conditions, e.g., T or RH. Please be clear on what this means, e.g., you mounted the samples with putty to provide good coupling, calm winds allowed for low environmental noise, and thus a high(er) signal to noise ratio of your measurements.
3.2 It is hard to follow what you did exactly. Especially, how you identified the “prominent peaks“ and why those are interpreted as the “resonance modes of the arch”. I see two options, one would be to share the code in the supplement, the other would be to detail the methods by which you picked the peaks.
In Figure 2b, please label the peaks you identified. They should match in all panels. To better compare it might be good to have the same y-axis for all directions. The x-axis differs between b panels and c and d, why? Can you combine c and d, and plot them under b, making the x axis not the mode but the frequency and just indicating the modes in all plots with a dashed/dotted line? – The dotted lines in b seem not to correspond to those in c and d.
How does the amplitude of cross-correlation (Figure 3a) enable “visualization of mode shapes along the linear array”? - which is not fully linear, as H08 is offset. Please be clear on what it shows. You have not defined what “mode shapes” are by now, and what do you mean by “relative modal displacement” (line 47). Please use the figure to better explain what the modal shape and changes thereof show, as well as what the composite vector and its inclination can tell. In this section you are describing the methods, not the results. Where can I find the cross correlograms for the other frequencies?
You are not referring to Figure 3b in the text at all. Is it necessary? Why is the direction of the ZZ component not indicated? There seem to be also some artefacts in the figure which are distracting. Maybe consider simplifying the 3D shaded model to outlines, e.g., sometimes less is more.
3.3 This section starts with a logical jump. 3.1 and 3.2 are about the physical arch and the measurement of the ambient vibrations. This section takes a step further by using the information from 3.1 on the structure, turning it into a numerical model arch. Then you use the measured vibrational modes to simulate the dynamics in the model arch. The aspect of how you set up the model geometry, what you include and what not reads very confusing. Please clarify between model setup (mesh, smoothing, spatial data, ...), material properties and structural features and with which assumption they have been implemented in the model, please also provide the boundary conditions. You seem to have different models you tried, please make their difference explicit and why you chose those. For example, what do you mean by (lines 182) “…generate these compliant mechanical zones.”, and “…a 1 m wide zone cutting…”, or (line 189) “…fractures were modelled as open zones”.
The dynamic modeling part is also not easy to understand as it is mixed in with structural aspects, without saying how the one affects the other. Please be clear on how you represent the dynamics in your models and how you get “data” from your model which is then compared with the field measurements. Please state how you compared the modeling and field data. It seems also that you are not showing the comparison of the first 4 resonance modes, but 1,2,3, and 6 (Figure 6).
4 Results
Here it would be good to link the figures better with the text – show and tell. For example, what does the Figure 2b really show? I can count and try to detect where the 9 peaks are, or you could label them and mark them throughout Figure 2 (see comment above).
Please present the results first and then interpret (and discuss, in the Discussion). For example, to highlight what the result of the spectral analysis is you need to make clear how this was done. Then state the results, you found 9 peaks in the arch which were not present in the reference station. The interpretation of those peaks is part of the discussion (strictly speaking). This is a pattern that is found throughout the results and makes it hard to discern what the result really are and what your interpretation is. This also leads to a repetition in the Discussion.
It would also be good if you could link your results to your research question and hypothesis, e.g., what was tested/looked for. For example, starting in line 225, it becomes unclear what you are aiming at in this manuscript: Are you interested in structural controls of the arch or fracture-controlled modes, as the title suggests, or is that the same for you?
Line 228: “For this reason we selected…”
Line 229: you are introducing a new terminology here: “highest relative modal mass (RMM) values”, please introduce this.
Line 243: The wording “…can be better appreciated…” does not seem to fit here.
Figure 4: Please label in black, then you can omit the outlines, but the contrast is still higher. You are not using the whole inclination-color-wheel from 90º to -90º. Please adapt so you have a better distinction in the colored arrows. Why did you change the view for panel (f) and (g)? Could you use view (f) for all of them? It seems to become overly complicated.
Figure 5: is (a) necessary? (b) please use different symbols for each direction. The color scheme might also not be visible for some or printed in grey scale. If you use black outlines, the difference in color becomes even harder to see for the smaller symbols. (c) What is the min -max color bar? How do the grey arrows compare to Figure 4 and the inclination-colored arrows? Why does the view change or is the arch deforming?
Table 1: Maybe this can go in the Supplementary Information? It is also confusing why the resonance mode no. 6 becomes 4 in the model but stays 6 in Figure 6. What is the reasoning behind changing the frequency from the field and in the different models? Why did you pick those four modes for the modeling and not the others?
Figure 6: Can you please use different symbols for the field data and the models. Color comment see above.
5 Discussion
The discussion reads rather length and not to the point. Can you be more specific and quantitative? For example, lines 287f, what do you mean by this sentence and how does your measurement and the shown results support the trapping of seismic energy?
If you want to discuss the hypothesis of e.g., “…increasing energy transmissivity at the fracture scale”, it would be good to state this and introduce the concept and how it relates to your study, e.g., in the introduction. Please review the hypothesis you are stating in the introduction and those in the discussion and see which of them are addressed by your study, and which are not. Please be clear about, what was observed in the field and what you modeled.
The part on the fractures and how it was implemented in the model reads very confusing, which is partly because new hypothesis and data (fracture sets) are introduced but also the basis for the whole study is unclear, e.g., what is the anticipated outcome of the model, and how does the measured resonance modes fit in? One thing that might help is to make the temporal and spatial (expected) damage/weaknesses explicit, and how that would show in the resonance frequency. This would also help to link it better to Figure 7.
Figure 7: See comments to colors and labels above, as well as the observed and modelled modes 1,2,3,6/4. (c ) why relative modal displacement? Why does it change? And why is mode 4 plotted with flipped axis?
6 Conclusion
Please come back to the bigger picture and the research question here. The writing is lengthy and ambiguous and vague. Please be precise on which hypotheses did you test and what are their outcome? What are the take home messages?
Tell the reader why these frequencies, why these modes are indicative of fractures, bulk properties, local changes? How does it relate to spatial scales, and how much would the mode shape change due to ongoing fracturing (temporal), thus how sensitive is this approach? What can we take away from this specific arch and apply it to other structures or arches?
Citation: https://doi.org/10.5194/egusphere-2024-1894-RC2
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Guglielmo Grechi
Jeffrey R. Moore
Molly E. McCreary
Erin K. Jensen
Salvatore Martino
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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