Natural rock arches are rare and beautiful geologic landforms with important cultural value. As such, their management requires periodic assessment of structural integrity to understand environmental and anthropogenic influences on arch stability. Measurements of passive seismic vibrations represent a rapid and non-invasive technique to describe the dynamic properties of natural arches, including resonant frequencies, modal damping ratios, and mode shapes, which can be monitored over time for structural health assessment. However, commonly applied spectral analysis tools are often limited in their ability to resolve characteristics of closely spaced or complex higher-order modes. Therefore, we investigate two techniques well-established in the field of civil engineering through application to a set of natural arches previously characterized using polarization analysis and spectral peak-picking techniques. Results from enhanced frequency domain decomposition and parametric covariance-driven stochastic subspace identification modal analyses showed generally good agreement with spectral peak-picking and frequency-dependent polarization analyses. However, we show that these advanced techniques offer the capability to resolve closely spaced modes including their corresponding modal damping ratios. In addition, due to preservation of phase information, enhanced frequency domain decomposition allows for direct and convenient three-dimensional visualization of mode shapes. These techniques provide detailed characterization of dynamic parameters, which can be monitored to detect structural changes indicating damage and failure, and in addition have the potential to improve numerical models used for arch stability assessment. Results of our study encourage broad adoption and application of these advanced modal analysis techniques for dynamic analysis of a wide range of geological features.

Natural rock arches form by erosion (Bruthans et al., 2014; Ostanin et al., 2017) and are major tourist attractions worldwide. However, ongoing weathering can lead to partial or complete collapse, posing a hazard to visitors; prominent examples include collapse of London Bridge (Australia) in 1990 (Woodroffe, 2002), rockfall from Landscape Arch (USA) in 1991 and 1995 above a hiking trail (Deseret News, 1991), and collapse of the Azure Window in Malta in 2017 (Satariano and Gauci, 2019). As arches occur in a variety of forms and settings, simple tools for stability assessment do not exist, and current practices often include site-specific geomechanical characterization and numerical modeling (Budetta et al., 2019). In recent decades, the stability of engineered structures, such as buildings and bridges, has been increasingly analyzed using measurements of their vibrational properties associated with resonance. Understanding this dynamic response to ambient loading forms the basis for the field of structural health monitoring (SHM; Doebling et al., 1996). More recently, SHM concepts have been applied at natural rock arches and other geological formations to improve site characterization and hazard assessment associated with failure of these features (e.g., Bottelin et al., 2013; Burjánek et al., 2018; Iannucci et al., 2020; Kleinbrod et al., 2019; Mercerat et al., 2021; Moore et al., 2018). Passive seismic measurements then provide a non-invasive means to monitor dynamic behavior and evaluate stability in the presence of natural or anthropogenic stimuli, which is especially valuable at culturally important sites where more invasive or destructive monitoring techniques (e.g., taking rock samples) may not be permitted.

Passive stability assessment often involves repeated or continuous measurements of a structure to monitor deviations in baseline structural dynamic
behavior. This dynamic behavior is characterized by natural frequencies, corresponding mode shapes (i.e., structural deflection at those frequencies),
and damping ratios (e.g., Chopra, 2015). While damping describes internal energy dissipation and radiation out of the system, resonant frequencies
are primarily a function of stiffness and mass. As the mass of a rock landform is approximately constant over time (in the absence of mass-wasting
events), variations in resonant frequencies arise primarily due to changes in rock mass stiffness, which can be associated with rock damage and
environmental influences, such as temperature and moisture (Colombero et al., 2021; Bottelin et al., 2013). As internal crack growth accumulates
during progressive failure, stiffness and thus frequencies are anticipated to decrease; for example, Lévy et al. (2010) reported a drop in
resonant frequency of about 20 % in less than 2 weeks prior to collapse of a 21 000

Modal analysis techniques used in structural health monitoring of geological features rely primarily on identification of spectral attributes from
in situ ambient vibration data. Power spectra visualizations provide a means for first interpretation, often leading to identification of resonant
frequencies that can be confirmed through numerical modeling (Moore et al., 2018), while site-to-reference spectral ratios may be used to eliminate
source and path effects in order to identify and track resonant frequencies (e.g., Weber et al., 2018). Selecting the maximum peak directly on the
power spectra to determine the resonant frequency is usually referred to as “peak picking”. By additionally selecting the frequencies left and right
of the resonant peak, for which the power drops by 3

Frequency-dependent polarization analysis (PA) provides a tool to estimate the modal deflection at resonance (Burjánek et al., 2012; Geimer et al., 2020a). However, these spectral analysis techniques fall short when applied to more complex systems, such as cases with closely spaced and overlapping modes, which have identical or similar frequencies but different mode shapes. In addition, phase information is not preserved across separate recording stations, impeding precise determination of mode shapes for higher modes. Thus, new techniques are necessary for refined modal analysis supporting structural health monitoring of rock landforms and providing accurate input parameters for stability assessment using numerical models. Among these, enhanced frequency domain decomposition (EFDD; Brincker et al., 2001a, b) is a promising approach to identify resonant frequencies, damping, and polarization attributes, and it is well-suited to distinguishing closely spaced modes. Covariance-driven stochastic subspace identification (SSI-COV) is an alternative time domain technique that is especially beneficial for accurate estimates of modal damping ratios (van Overschee, 1996). Since their introduction, both techniques have become standard methods for analysis of engineered structures (e.g., Brincker and Ventura, 2015) and have been compared, yielding similar results (Cheynet et al., 2017; Brincker et al., 2000). Using these complementary techniques, Bayraktar et al. (2015) found good agreement between EFDD and SSI-COV in their study on historical masonry arch bridges with resonant frequencies and damping ratios comparable to the natural rock arches and bridges studied here. Furthermore, frequency domain decomposition has been applied to natural features, such as sedimentary valleys, glaciers, and rock slope instabilities (Poggi et al., 2015; Preiswerk et al., 2019; Häusler et al., 2021, 2019; Ermert et al., 2014), while application of SSI-COV has remained restricted to engineered structures.

In this study, we analyze the modal characteristics of four natural rock arches in Utah (USA) previously investigated by Geimer et al. (2020a). As these arches exhibit various spectral complexities which complicate dynamic analyses, we apply two operational modal analysis techniques – EFDD and SSI-COV – to improve identification and characterization of normal modes. Our results highlight the value and versatility of EFDD and SSI-COV for structural characterization and monitoring in geologic hazard applications, which we propose is useful across a broad range of geomorphologic features beyond our studied landforms, such as rock slope instabilities and rock towers (Bottelin et al., 2013; Häusler et al., 2021; Moore et al., 2019).

Location, span, and data acquisition characteristics for each arch investigated (coordinates in WGS84).

Ambient vibration data processed in this study were collected at four natural rock arches in Utah by Geimer et al. (2020a). These consist of three
single-station measurements conducted using a Nanometrics Trillium Compact 20

In the study by Geimer et al. (2020a), Rainbow Bridge showed clear normal modes, although the higher-order modes are partly overlapping (Fig. 1a
and b). We include this arch in our study as an example of having well-defined modes. For Corona Arch (Fig. 1c), the numerical models by Geimer
et al. (2020a) predicted two modes between 5 and 6

In previous studies of the dynamic response of natural rock arches, the resonant frequencies of the landform were determined by selecting the local
maxima of the power spectra of the recordings, which is so-called peak picking (Starr et al., 2015; Moore et al., 2018, e.g.,

Mode shape information can be retrieved by polarization analysis (PA), for example, using the approach by Koper and Hawley (2010) as applied to rock
arches by Moore et al. (2016) and Geimer et al. (2020a). These single-station techniques are easy to use and provide reliable modal parameters in the
case of well-separated modes. However, they fall short in the case of closely spaced or overlapping modes, as the mode bells are not visible or do not
correspond to the underlying resonant mode (e.g., Papagiannopoulos and Hatzigeorgiou, 2011; Wang et al., 2012). This is illustrated for the example
of three single-degree-of-freedom (SDOF; see Appendix A) systems in Fig. 2a: one well-defined mode at 9

The most direct estimate of modal damping ratios is obtained by active source experiments in which the structure studied is excited artificially and
energy dissipation is measured, for example, in the time domain by the logarithmic decrement

Here,

Active source experiments can be considered to provide good estimates of damping ratios, but their application is restricted to structures that can be excited artificially (without inducing damage, Magalhães et al., 2010). In contrast, passive (i.e., ambient vibration) experiments can be applied to a broad range of structures but are subject to larger uncertainties (up to 20 % is possible, e.g., Au et al., 2021; Döhler et al., 2013; Gersch, 1974; Griffith and Carne, 2007).

We processed three-component ambient vibration data using enhanced frequency domain decomposition (EFDD) (Brincker et al., 2001a, b; Brincker and Ventura, 2015; Michel et al., 2010). The method first computes the cross-power spectral density between all input traces and for every discrete frequency. Next, singular value decomposition for each frequency provides the singular values and singular vectors. The singular values can be understood as the collection of virtual SDOF systems of the structure, which enables detection of close and hidden modes that are not visible in the power spectrum. The first singular value shows peaks at the dominant natural frequencies of the system. If present, overlapping secondary (i.e., non-dominant) modes result in elevated higher singular values. Resonant frequencies are then determined from analysis of the singular value plot and the singular vector at the identified frequencies gives the three-dimensional modal vector (i.e., mode shape) of the chosen mode, with higher singular vectors representing the mode shape of non-dominating modes. These processing steps represent the frequency domain decomposition method described by Brincker et al. (2001b). The half-power bandwidth technique could now be applied to the singular values to estimate damping, as the bias by modal superposition is now addressed. However, spectral leakage may still broaden the mode bell.

A more accurate technique to estimate modal damping is the enhanced FDD (EFDD) technique introduced by Brincker et al. (2001a). Here the mode bell
around each resonant frequency is picked manually and transformed to the time domain, providing the impulse response function (see Fig. 2b). Energy
decay in the linear part of the impulse response function is expressed by the damping ratio

The second technique used in this study is the covariance-driven stochastic subspace identification (SSI-COV) method (Peeters and De Roeck, 1999; Van Overschee and De Moor, 1993; van Overschee, 1996). Like EFDD, SSI-COV is a modal analysis technique frequently used in civil and mechanical engineering. Contrary to EFDD, SSI-COV is a time domain parametric technique, which searches for the best set of modal parameters (resonant frequencies and modal damping) representing the observed structural response in a mathematical manner, i.e., minimizing misfit between modeled and observed data. Because it is a time domain approach, overestimation of damping from spectral leakage is avoided. The most important processing parameter is the maximum lag time between two time samples used for computing the covariance matrices, which should be 2 to 6 times larger than the longest eigenperiod of the structure. Other user-controlled parameters include the number of possible modes, the accuracy threshold for modal frequency and damping, the maximum spectral distance inside a cluster, and the variation of the minimum modal assurance criterion (e.g., Allemang and Brown, 1982), which is a measure of the similarity of the mode shape at neighboring frequencies. We applied the SSI-COV algorithm software by Cheynet (2020), which is based on the implementation by Magalhães et al. (2009) and was used for comparison to EFDD on long suspension bridges (Cheynet et al., 2017). We followed the parameter suggestions by Cheynet (2020) and chose the parameters in a trial-and-error approach such that the first three resonant modes were reproduced (see Table A1). As SSI-COV establishes a mathematical model of the structure studied, the dynamic response can be defined by poles and zeros (in the sense of mathematics of complex numbers). Therefore, the term “pole” can be used as representative for “resonant mode” and is used hereafter to be in line with the terminology in the field. Since SSI-COV is a parametric method, its resulting resonant frequencies should be verified by a frequency domain technique to prevent misinterpretation by model overfitting.

Results from SSI-COV (and other SSI variants) are illustrated using stability diagrams (e.g., Fig. 3c). Initially, the structure's response is modeled with a low number of modes (poles), which is continuously increased to the maximum number of poles defined by the user. The maximum number of poles should be chosen to be significantly larger than the expected number of modes in order to establish an overdetermined mathematical model. The resulting resonant frequencies for each mode at every model run are plotted in the stability diagram (blue crosses in Fig. 3c). Repeated poles, i.e., identical or very similar values for resonant frequencies, damping, and mode shape, represent stable poles and can be identified as vertical stacks of poles in the stability diagram (red circles in Fig. 3c). Poles not fulfilling the user-defined accuracy criteria are not interpreted as stable poles and are scattered at arbitrary values as a result of noise fitting. Stable poles are clustered using hierarchical clustering, thereby grouping poles with similar characteristics to the final resonant modes of the structure.

Overview of resonant frequencies, modal damping ratios derived by EFDD and SSI-COV, and modal vectors (azimuth and incidence angle) estimated by EFDD and polarization analysis (PA) for Rainbow Bridge, Corona Arch, and Squint Arch. The values in brackets for Squint Arch are derived by PA if

n/a: not applicable.

We observe the first three resonant modes of Rainbow Bridge determined by the single-station measurement at 1.1, 2.2, and 2.5

The singular value plot of Corona Arch reveals two distinct spectral peaks at 2.7 and 5.3

Modal analysis of Squint Arch.

For Squint Arch, we observe two closely spaced modes at 11.5 and 12.5

We demonstrate the ability of EFDD to retrieve the full-length normal-mode shapes at Squint Arch, where data acquired by a nodal geophone array during
a separate experiment are available (Fig. 4b; raw power spectra are shown in Fig. A1 a–c). We note that modal frequencies for

We performed EFDD and SSI-COV modal analyses on geophone array data acquired at Musselman Arch, revealing the first four resonant modes at 3.4, 4.2,
5.6, and 6.6

At Rainbow Bridge, the first three spectral peaks are well-separated. Therefore, simple peak picking on the power spectra provides the same resonant
frequencies as obtained by EFDD and SSI-COV. All techniques provide comparable values for azimuth and incidence of the modal vector. The largest
discrepancies between EFDD/SSI-COV and PA are 6 and 4

At Corona Arch, only two resonant peaks are observed in the power spectra between 1 and 7

At Squint Arch, the power spectra show two resonant peaks between 10 and 14

At Musselman Arch, all four resonant modes between 1 and 10

First nine singular values of the EFDD analysis of Musselman Arch.

We observed that the second singular value at Musselman Arch is elevated to about 85

For most resonant modes, EFDD and SSI-COV provide comparable damping results within the anticipated range of uncertainty. However, we observe that EFDD results in 30 % to 35 % higher damping ratios for the fundamental modes of Rainbow Bridge, Corona Arch, and Squint Arch, as well as for the first higher mode of Squint Arch. We interpret this observation as an effect of damping overestimation through broadening of the resonant peak caused by spectral leakage (e.g., Bajric et al., 2015).

Singular value plots with spectra of single-degree-of-freedom systems modeled by using input data from EFDD (red), SSI-COV (blue), and the half-power bandwidth technique (cyan). At Rainbow Bridge

Damping ratios obtained by the half-power bandwidth technique at Rainbow Bridge and Corona Arch are 75 % and 53 % higher than those estimated by SSI-COV, and they are 63 % and 26 % higher than estimated by EFDD. This is illustrated in Fig. 7, where we compare damping ratios obtained by the various techniques for each arch. The resonant frequency and damping ratio derived are used to model an SDOF system, which is superimposed on the singular value plots. The amplitude of the modeled SDOF is normalized to the maximum amplitude of the first singular value. The mode bell of the fundamental mode of Rainbow Bridge is well-reproduced by EFDD and SSI-COV, but damping is overestimated by the half-power bandwidth technique (Fig. 7a).

Rainbow Bridge had the lowest fundamental frequency in the study by Geimer et al. (2020a), who used the same settings to compute the power spectra for all arches, including arches with higher resonant frequencies such as Squint Arch. Therefore, it is likely that these parameter settings were not ideal to resolve the low resonant frequencies of Rainbow Bridge with sufficient resolution. Therefore, we interpret the discrepancy between the half-power bandwidth and EFDD and SSI-COV as a result of strong spectral leakage for the half-power bandwidth technique. However, neither SSI-COV nor EFDD is able to perfectly reproduce the mode bell due to its slightly asymmetric shape, likely reflecting the oversimplified assumption of an SDOF system. At Corona Arch, SSI-COV is capable of reproducing the mode bell of the fundamental mode, and the half-power bandwidth again overestimates damping (Fig. 7b). SSI-COV and the half-power bandwidth technique yielded identical damping ratios for the fundamental mode of Squint Arch (Fig. 7c). For higher-order modes, the discrepancy between EFDD and SSI-COV is smaller on all arches.

The normal-mode analysis of Squint Arch resulted in different resonant frequencies for the single-station broadband measurement in February 2018 and
the geophone array measurement in April 2018. We attribute this shift in frequency to seasonal variations and mainly temperature differences
(11.5

We applied enhanced frequency domain decomposition (EFDD) and covariance-driven stochastic subspace identification (SSI-COV) modal analyses to a set of four natural rock arches previously analyzed by Geimer et al. (2020a) using frequency-dependent polarization analysis (PA). Our results show that EFDD and SSI-COV are able to determine the natural frequencies, damping ratios, and mode shapes of these landforms, including close, hidden, and higher resonant modes. For well-separated resonant modes, these techniques reproduce the results by Geimer et al. (2020a). In the case of more complex spectra, EFDD and SSI-COV are able to extract additional modal details not resolved with PA. EFDD facilitated identification and interpretation of closely spaced (i.e., spectrally overlapping) and hidden modes at Corona and Squint arches. EFDD additionally combines information for all input traces in a single plot, allowing rapid analysis of the dynamic response, especially when compared to picking the resonant peaks and determining polarization information on every station spectrum individually. The singular vectors resulting from EFDD can be directly interpreted as the three-dimensional modal deflection vector at each station, providing rapid and convenient visualization of mode shapes.

While modal analysis via peak picking and subsequent PA has been shown to be satisfactory for adequately spaced spectral peaks and strongly amplified resonant frequencies, here we demonstrate that more sophisticated modal analysis techniques can provide refined modal characterization for more complex dynamic systems. Improving the accuracy and our understanding of resonance properties could in turn help generate more refined numerical models, facilitating more accurate arch stability assessment. Future efforts in modeling the dynamic response of rock arches (and other geological features) should additionally involve calibration of the modal damping ratio, as we have shown that this parameter can be measured on complex structures.

Our results encourage widespread application of EFDD and SSI-COV modal analysis techniques, which are commonly used in civil engineering, to complement existing seismological techniques for dynamic analysis of geological features. Both techniques are well-suited for future near-real-time monitoring of the structural integrity of geological features beyond rock arches: for example, rock slope instabilities, unstable glaciers, and freestanding rock towers. EFDD and SSI-COV are only two methods out of many other available algorithms for modal analysis, including other SSI variants and curve fit FDD (Peeters and De Roeck, 2001; Jacobsen, 2008). Therefore, future research could explore the potential of these techniques for applications involving modal analyses and monitoring of Earth surface landforms.

Data used in this study and originating from the study by Geimer et al. (2020b) are available at

The paper was written by MH with significant contributions from all co-authors. PG, RF, and JM acquired seismic data. MH carried out data processing and software development of EFDD modal analysis. PG performed data curation and validation of results. All authors reviewed and approved the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the Eidgenössische Technische Hochschule Zürich (grant no. 0-20361-17) and the US National Science Foundation under grant no. EAR-1424896.

This paper was edited by Jens Turowski and reviewed by two anonymous referees.