An Update on Techniques to Assess Normal Mode Behaviour of Rock Arches by Ambient Vibrations Short Communication: Ambient Vibration Modal Analysis of Rock Arches using Enhanced Frequency Domain Decomposition and Covariance-driven Stochastic Subspace Identification

10 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 15 their ability to resolve characteristics of closely spaced or complex higher-order modes. Therefore, we investigate two algorithms 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 conventional seismological techniques. Results from enhanced Enhanced frequency Frequency domain Domain decomposition Decomposition and parametric covarianceCovariance-driven stochastic Stochastic subspace Subspace identification Identification modal analyses showed 20 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 and provide stableincluding their corresponding modal damping estimatesratios. In addition, due to preservation of phase information, enhanced frequency domain decomposition allows for direct and convenient three-dimensional visualization of mode shapes. These advanced techniques provide more detailed characterization of dynamic parameters, which can be monitored to detect structural changes indicating 25 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 and Identification (SSI-COV) modal analyses on four natural rock arches that were previously analysed using frequency-dependent polarization analysis (PA). Our results show that EFDD and SSI-COV are well suited to determine the natural frequencies, damping ratios, and mode shapes of these geological structures. For clear resonant modes, these techniques reproduce the results by Geimer et al. (2020). In 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 410 traces in a single plot, allowing for a user-friendly analysis of the dynamic response. The singular vectors resulting from EFDD can be directly interpreted as the three-dimensional modal deflection vector at each providing rapid and convenient visualization of normal mode shapes.


Introduction
Natural rock arches are rock landforms formedform by erosion (Bruthans et al., 2014;Ostanin et al., 2017) and are 30 major tourist attractions worldwide. However, ongoing erosion weathering will eventuallycan lead to partial failure or complete collapse of these landforms, posing a hazard to visitors; prominent recent 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). Currently, sBecauseAs arches occur in a variety of forms and settings, simple tools for rock arch stability assessment do not exist., and 35 current practices often include site-specific However, it might be achieved by geo-mechanical characterization of the rock mass including the analysis of rock samples and yingication ofnumerical geotechnical modelsing (Budetta et al., 2019). In the pastrecent decades, the stability of engineered structures, such as buildings and bridges, havehas been increasingly analyzed using and monitored by measuringmeasurements of their vibrational properties associated with resonanceusing measurements of their their vibrational properties. Understanding this dynamic response to ambient loading , summarized underformings the 40 basis for the field of structural health monitoring (SHM, Doebling et al., 1996). More recently, Ambient The idea of SHM wasconcepts have been transferredapplied at vibration surveys ofto natural rock arches and other geologicalrock features formations have recently been employed to improve site characterization and hazard assessment associated with failure of these structuresfeatures (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). Therefore, pMeasurements of pPassive seismic measurements are thus increasingly usedthen 45 provide as a provide non-invasive means to monitor dynamic behavior and evaluate stability relating in the presence of to natural or anthropogenic stimuli, which is especially valuable at culturally important sites, where more taking rock samples or drillingsinvasive or destructive monitoring techniques (e.g., taking rock samples) may should be avoided or isare not be permitted..
Passive One possibility to perform a relative sstability assessment by ambient vibrations ioften nvolvesinvolves 50 repeated or continuous measurements or continuous surveillance of thea structure to, and comparing themonitoring for deviations current vibrational properties toin long-termbaseline structural observationsdynamic behavior. This dynamic behavior, which is to detect deviations, possibly indicating upcoming failure. For more quantitative assessments, however, individual features must be numerically modelled. Moore et al. (2020) computed the three-dimensional static stress field for a set of natural rock arches to detect features with high internal tensile stresses, which may be more prone to tensile crack growth 55 and thus failure. However, these models required estimates of rock density and Young's modulus as input parameters. While density can be retrieved from rock samples, the Young's modulus can be derived from vibrational properties by minimizing the error between observed and modelled resonance attributes (Moore et al., 2018;Moore et al., 2020;Geimer et al., 2020).
The dynamic properties of a structure can be characterized by its natural frequencies, corresponding mode shapes 60 (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 precipitation or mass wasting events), variations in resonant frequencies arise primarily due to changes in rock mass stiffness, which is in turn correlatedcan be associated with rock damage and environmental influences, such as temperature and moisture (Colombero et al., 2021;Bottelin et al., 2013). 65 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 % less than two weeks prior to collapse of a 21,000 m 3 rock column, which they attributed to progressive failure. For Mmore quantitative assessments, of stress conditions prior to failure howeverrequire , individual features mustto be numerically modelled with realistic values for rock density and Young's modulus. With density constrained by rock samples, Young's modulus can be derived from dynamic properties by 70 minimizing the error between observed and modelled resonance attributes (Moore et al., 2018;Moore et al., 2020;Geimer et al., 2020). Such model validation facilitates the estimation of the three-dimensional stress field, used by Moore et al. (2020) to identify Moore et al. (2020) computed the three-dimensional static stress field for a set of natural rock arches to detect featuresarches that may be more prone to tensile crack growth and structural failure.with high internal tensile stresses, which may be more prone to tensile crack growth and thus failure. However, these models required estimates of rock density and 75 Young's modulus as input parameters. While density can be retrieved from rock samples, the Young's modulus can be derived from vibrational properties by minimizing the error between observed and modelled resonance attributes (Moore et al., 2018;Moore et al., 2020;Geimer et al., 2020). Therefore, measurements of frequencies over time are potentially valuable for structural health monitoring and earlywarning of failure, especially at culturally sensitive sites like rock arches where more invasive monitoring techniques may not 80 be permitted.
Currently, simple tools for rock arch stability assessment do not exist. One monitoring approachSHM involves repeated measurements or continuous surveillance of the structure by ambient vibration monitoring, and first-order analysis often and comparingcompares the current resonance attributes (i.e., resonant frequency, mode shapes, damping ratios) with an empirical models to detect any deviations from the long-term behavior, which that could be interpreted as precursors of 85 failure (e.g., Häusler et al., 2021). For more quantitative assessment, however, individual features need tomust be numerically modelled. Moore et al. (2020) computed the three-dimensional static stress field for a set of natural rock arches to detect features with high internal tensile stresses, which may be more prone to tensile crack growth and thus failure. However, these models required estimates of rock density and Young's modulus as input parameters, as well as calibration of boundary conditions. While density can be retrieved from rock samples, few observational tools are available to compute the Young's 90 modulus of rock masses and even fewer to calibrate structural boundary conditions. However, these can be estimated using numerical models by minimizing the error between observed and modelled resonance attributes, with the resonant frequency being the dominant contributor (Moore et al., 2018). Therefore, Geimer et al. (2020) performed ambient vibration modal analysis on a large set of natural rock arches, calibrating the material properties and boundary conditions of numerical supporting stability assessment.
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 the 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 and in order tocan 100 help to identify and track resonant frequencies (Weber et al., 2018). Frequency-dependent polarization analysis (PA) provides a tool to estimate the modal deflection at resonance (Burjánek et al., 2012;Geimer et al., 2020). However, these basic spectral analysis techniques fall short when applied to more complex systems, such as cases with closely spaced and overlapping modes, which that 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 105 techniques are necessary to providefor refined modal analysis suitable forsupporting structural health monitoring of rock landforms and provide providing accurate input parameters for stability assessment using numerical models. Among these, Enhanced Frequency Domain Decomposition (EFDD, Brincker et al., 2001a;Brincker et al., 2001b) is a promising approach to identify resonant frequencies, damping, and polarization attributes, and is well-suited to distinguish closely spaced modes.
The Covariance-driven Stochastic Subspace Identification (SSI-COV) is an alternative time-domain technique that is 110 especially beneficial for accurate estimates of modal damping ratios (van Overschee, 1996). Since their introduction, both techniques have becaome standard methods have found wide application on for structural analysis of engineered structures (e.g., Brincker and Ventura, 2015), and have been. The two techniques were compared, on various engineered structures providingyielding similar results (Cheynet et al., 2017;Brincker et al., 2000). Using these same complementary techniques, Bayraktar et al. (2015) found a good agreement between EFDD and SSI-COV in their study on historical masonry arch bridges 115 with resonant frequencies and damping ratios comparable to the natural rock arches and bridges studied here. RecentlyIn additionFurthermore, frequency domain decomposition was has also been applied on natural structuresfeatures, such as sedimentary valleys, glaciers, and rock slope instabilities (Poggi et al., 2015;Preiswerk et al., 2019;Häusler et al., 2021Häusler et al., , 2019Ermert et al., 2014), while the application of SSI-COV has remained remains restricted to artificial engineered structures.
In this study, we focus on analyze the modal characteristics of four natural rock arches in Utah (USA), previously 120 investigated by Geimer et al. (2020) and Moore et al. (2020). As these arches exhibit various spectral complexities that which complicate dynamic analyses interpretation, we apply two operational modal analysis techniques --EFDD and SSI-COV -to improve identification and characterization of normal modes. The strengths of these techniques lie in the analysis of closely spaced and hidden modes, and in the preservation of phase information between different components, allowing for the direct retrieval of mode shape information at each sensor location. Our results highlight the value and versatility of EFDD and SSI-125 COV techniques for structural health characterization and monitoring and in geologic hazard applications, which we propose is useful in across a broad range of natural geomorphologic features beyond our studied landforms, for example,such as rock slope instabilities and freestanding rock towers (Bottelin et al., 2013(Bottelin et al., ,2021Häusler et al., 2021;Moore et al., 2019). Ambient vibration data processed in this study were collected at four natural rock arches in Utah by Geimer et al. 130 (2020). These consist of three single-station measurements conducted using a Nanometrics Trillium Compact 20-s seismometer (TC 20-s, sites: Rainbow Bridge, Corona Arch, Squint Arch) and two array measurements using three-component Fairfield Zland 5-Hz nodal geophones with synchronous recording (sites: Squint Arch and Musselman Arch). Table 1 summarizes the arch measurements, including data acquisition length, site coordinates, and number of sensors deployed. Prior to processing, all data were corrected using the respective instrument response (to velocity units of m/s), and the mean and 135 linear trend were removed.
In the study by Geimer et al. (2020), Rainbow Bridge showed clear normal modes, although the higher-order modes are partly overlapping (Figure 1a, b). We include this arch in our study as an example of having for well-defined modes. For Corona Arch (Figure 1c), the numerical models by Geimer et al. (2020) predicted two modes between 5 and 6 Hz, but only one single peak could clearly be observed in the experimental data ( Figure 1d). Therefore, we include this arch as an example 140 of having a possibly hidden mode. At Squint Arch, the opposite phenomenon was observed: while two peaks were observed in the power spectrum between 10 and 15 Hz, only one resonant frequency was predicted by the numerical model (Figure 1e, f). Finally, the large array data set acquired at Musselman Arch provides the possibility to test the techniques presented here presented techniques withusing a dense sensor array, highlighting the value of retained phase information.

Figure 2: a) Frequency response of three example single-degree-of-freedom (SDOF) systems with different damping ratios (%) and
165 their superposition (a multi-degree-of-freedom-system, MDOF). The mMarkers indicate the -3 dB or half-power points of each response curve, which are used to compute damping by the half-power bandwidth technique (see Equation 1). Note that mode 2 and 3 merge to one single mode bell, which causes an overestimation of the modeal damping at 10.5 Hz (4.1% instead of 3.0%). The third mode at 10.7 Hz cannot be observed (i.e., is hidden) in the MDOF response responsepower spectra. , b) Impulse response function of a structure resulting, for example, from active excitation. Damping can beis determined from , for example, by the logarithmic

decrement technique (Equation 2) by determiningmeasuring the amplitudes separated by one period .
Modal shapes informration can be retrieved by polarization analysis (PA), for example, using the approach by Koper and Hawley (2010) and applied to rock arches by Moore et al. (2016) and Geimer et al. (2020). These single-station techniques are easy to use and provide reliable modal parameters in the case of well-separated resonant modes. However, they fall short in case of closely spaced or overlapping modes, as the mode bells are not visible or are not corresponding to the underlying 175 resonant mode (e.g., Papagiannopoulos and Hatzigeorgiou, 2011;Wang et al., 2012). This is illustrated atfor the example of three single-degree-of-freedom (SDOF, see Appendix A) systems illustrated in Figure 2a: one well-defined mode at 9 Hz is damped withat 2%, whereas two closely-spaced modes at 10.5 and 10.7 Hz are damped with 3 and 4%, respectively. The superposition of the three SDOF provides the resulting response of the multi-degree-of-freedom (MDOF) system (black line in Figure 2a), which is observed in the power spectrum. Analysis of the well-separated fundamental mode is straight forward, 180 as the peak corresponds to the resonant frequency and applying the half-power bandwidth technique provides the correct damping ratio of 2%. In contrast, the peaks of both higher modes merge to one single mode bell at 10.5 Hz with an apparent damping estimate of 4.5%. Therefore, the superposition of the two modes results in a broadening of the mode bell and consequently to an overestimation of damping. Furthermore, the third mode cannot be detected at allin the power spectra. In addition to thedamping overestimation by close and hidden modes, the half-power bandwidth technique tends to overestimate 185 damping due to spectral leakage (Seybert, 1981)(REF?) and related broadening of the mode bell.
The most direct estimate of modal damping ratios is obtained by active source experiments. T where the structure studied is excited artificially and energy dissipation is measured, for example, in the time domain by the logarithmic decrement (Cole, 1973, see Figure 2b): Here, represents the amplitude at time and refers to the damped natural period (i.e., the inverse of the resonancet frequency). For small damping, this can be simplified and solved for : Geimer et al. (2020) applied this technique to a set of small-sized natural rock arches by stomping on the ground next to the structure and applying a band-pass filter around the resonant frequency. While damping ratios originating from active source 195 experiments can be considered good estimates, values measured by passive techniques are subject to larger uncertainties (up to 20 % is no exceptionpossible, e.g., Au et al., 2021;Döhler et al., 2013;Gersch, 1974;Griffith and Carne, 2007;Au et al., 2021;Döhler et al., 2013;Gersch, 1974).

Enhanced Frequency Domain Decomposition
We processed three-component ambient vibration data using Enhanced Frequency Domain Decomposition (EFDD) 200 , which is a standardn output-only modal analysis technique used in civil engineering (e.g., Brincker et al., 2001a;Brincker et al., 2001b;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 the detection of close and hidden modes that are not visible in the power spectrum. The first singular 205 value shows peaks at the dominant natural frequencies of the system., If present, with peaks on higher singular values indicating overlapping secondary (i.e., non-dominant) modes result in the elevation ofelevated 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., shape) of the chosen mode. These processing steps up to this point are formingrepresent the Frequency Domain Decomposition method as described by Brincker et al. (2001b). The half-power 210 bandwidth technique could now be applied on the singular values to estimate damping, as the bias by modeal superposition is now tackledaddressed. However, spectral leakage mightmay still broaden the mode bell.
A more accurate technique to estimate modal damping is the Enhanced FDD (EFDD) technique, introduced byIn order to estimate modal damping, Brincker et al. (2001a). tTHere the mode bell of around each resonant frequency is picked manually and transformed to the time domain, providing the impulse response function (see Figure 2b). Energy decay in the 215 linear part of the impulse response function is expressed by the damping ratio ζ, which can be determined using the logarithmic decrement technique (Cole, 1973). Linear regression of the zero-crossing times within the linear part of the decay curve additionally provides an updated estimate of the resonant frequency. The advantage of EFDD over the half-power bandwidth method is that the damping estimate is not based on only three picks but on a curve fitting approach, which reduceds the errors introduced by noise. However, EFDD still tends to overestimate damping due to spectral leakage (e.g., Bajric et al., 2015). 220 Detailed description of the EFDD processing workflow applied in this study can be foundis in Häusler et al. (2019) andHäusler et al. (2021), whoich applied the algorithm technique to on unstable rock slopes.

Covariance-driven Stochastic Subspace Identification
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 an 225 output-only 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 mannerleast-square sense, i.e., minimizing the sum of squares of the residualsmisfit between modelmodelled and observed data. Because it is a time-domain approach, overestimation of damping due tofrom spectral leakage is avoided. The most important processing parameter is the maximum 230 lag time between two time samples used for computing the covariance matrices, which should be two to six times larger than the longest eigenperiod of the structure. Other user-controlled parameters include the number of possible modes (i.e., poles), the accuracy threshold for modal frequency and damping, variations, the minimum maximum spectral distance between two adjacent modesinside 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 neighbouringneighboring frequencies. We applied the SSI-COV 235 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 240 for "resonancet mode" and is used hereafter to be in line with the terminology in the field. Parameter combinations for every arch can be found in Table A1 in the appendix. 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. We applied the SSI-COV algorithm 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). Using these same complementary techniques, 245 Bayraktar et al. (2015) found a 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.
Results from SSI-COV (and other SSI variants) are illustrated using stability diagrams (e.g., Figure 1c3c). Initially, the structure's response is modelled 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 250 expected number of modes in order to established an overdetermined mathematical model. The resulting resonant frequencies for each mode at every model run are plotted in the stability diagram (blue crosses in Figure 1c3c). 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 Figure 1c3c). 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 255 clustered using hierarchical clustering, grouping poles with similar characteristics to the final resonant modes of the structure.

4 Results
We observe the first three resonant modes of Rainbow Bridge determined by the single-station measurement at 1.1, 2.2, and 2.5 Hz (Figure 1a3a-c). While the fundamental mode (f1 at 1.1 Hz) is distinctly separated from other spectral peaks, the second and third modes (f2, f3) occur at closely spaced frequencies but are clearly identified by the elevated second singular 260 value. Damping is estimated at between 0.6 and 1.3 % for all three modes (Table 1). For the fundamental mode, we estimate damping at 0.9 and 0.6 % using EFDD and SSI-COV, respectively, which is significantly lower than estimated by Geimer et al. (2020) using the half-power bandwidth method (2.4 %). Modal vectors (i.e., azimuth and incidence angle with a lower hemisphere projection) derived by EFDD are very similar to the polarization analysis (PA) results of Geimer et al. (2020) with some minor differences for f3 and a 180° ambiguity in the azimuth of the nearly horizontally polarized mode f2. Note that 265 Geimer et al. (2020) allowed a polarity flip for mode shapes with sub-horizontal incidence angles equal to or larger than 85° in order to compare to numerical models. As SSI-COV and EFDD provide similar results, we only compare values from EFDD to PA in Table 2 and provide SSI-COV results in Table A2 in the appendix.

superimposed with increasing number of poles. Each pole is marked with a blue cross, stable poles (in terms of resonant frequency, mode shape, and damping ratio) are marked with a red circle. Unstable poles (i.e. blue crosses at distance from stable poles) arise from noise fitting. Subplots d) to f) and g) to i) are the same as a) to c) for Corona Arch and Squint Arch, respectively. Photographs in panels a) and d) from Moore et al. (2020).
The singular value plot of Corona Arch reveals two distinct spectral peaks at 2.7 and 5.3 Hz (Figure 1d3e-f). However, 280 the second singular value also peaks at ~5.3 Hz, indicating the presence of a closely spaced mode at that frequency. Therefore, we confirm the interpretation of two close modes proposed by Geimer et al. (2020). However, EFDD and SSI-COV suggest nearly identical frequencies for f2 and f3 (5.3 Hz) while Geimer et al. (2020) selected more separated frequencies (5.0 and 5.4 Hz) based on PA and numerical modelling and PA. Modal vectors for f1 and f2 resolved by PA and EFDD are in good agreement, however azimuth and incidence differ for f3. While EFDD and SSI-COV gave similar values for modal incidence 285 of f3 (44° and 54°, respectively), PA estimated incidence at 73°. Damping is estimated between 0.9 and 2.0 % for all three modes, with 0.9 and 1.4 % for the fundamental mode (via SSI-COV and EFDD, respectively). These values are again slightly lower than the half-power bandwidth estimates of Geimer et al. (2020, 1.9 %). Damping ratios for f2 and f3 are between 1.5 and 2.0 % with good agreement between EFDD and SSI-COV providing similar damping ratios within the expected uncertainty range. 290 For Squint Arch, we observe two closely spaced modes at 11.5 and 12.5 Hz, and a third mode at 19.9 Hz ( Figure 1g3h, -i). The second mode was not analysedanalyzed by Geimer et al. (2020) as it could not be confirmed as a separate mode by numerical models. A mode splitting phenomenon, for example, caused by anisotropy, might be a potential explanation for the inability to replicate this mode in homogeneous numerical models. Our analysis of the second mode suggests the modal vector has a steeper incidence angle of 49°, and is therefore oriented 60° from f1. If the two spectral peaks were 295 analysedanalyzed separately by PA, the match between PA and EFDD is very good (see values in brackets in Table 2). Geimer et al. (2020) determined modal damping by applying the logarithmic decrement technique to a series of decaying time-series responses created by an active-source impulsive impact. The resulting damping ratio of 1.6 % is in perfect agreement with the estimation by SSI-COV but differs slightly from the EFDD output result (2.4 %). We demonstrate the ability of EFDD to retrieve the full normal mode shapes at the example ofThe first two modes of Squint Arch, where can also be resolved from EFDD analysis of data acquired by a nodal geophone array during a separate experiment isare available (Figure 3b4b, raw power spectra are shown in Figure A1 a-c in the appendix) with sufficient station density to extract the full mode shapes. We note that modal frequencies for f1 and f2 increased by about 1 Hz compared to the 310 single-station measurement., which we attribute to seasonal variations in the dynamic response due to temperature effects (11.5°C for the single-station and 16°C for the nodal measurement, respectively, see also Starr et al., 2015). No other higher modes are visible on the singular value plot, during this measurement. either because the noise level of the nodal geophones is too high or because the modes were not excited during the survey. Modal vectors for the first two modes at all stations resulting from EFDD analysis are shown in Figure 3c4c and d. We were not able to define a set of SSI-COV parameters that 315 could successfully reproduce the observed modes., possibly owing to the low excitation level at the small arch combined with the higher instrument noise level of the nodal geophones compared to the broadband instruments used in the single-station measurements.  Geimer et al. (2020). Visualization of the three-dimensional modal vectors for each station determined 330 by EFDD are shown in Figure 4d5d-g. The first two modes are full-span, first-order bending modes in the horizontal and vertical directions, respectively. The third mode is a nearly symmetric second-order vertical bending mode with its node point at the centrecenter of the arch. Mode four is a slightly asymmetric second-order horizontal bending mode with the node point shifted towards the eastern abutment. Modal damping ratios for the first three modes for Musselman Arch are estimated at 1.3, 1.0, and 1.9% with EFDD and 1.3, 1.1, and 1.6% with SSI-COV.to be between 1.3 % and 1.9 %, with EFDD and SSI-COV 335 providing similar results (Table A2 in the appendix).

Figure 45: a) Photograph of Musselman Arch, b) first nine singular values of the EFDD analysis. Solid markers indicate the resonant peaks, while vertical colored linesopen circles indicate the extent of the corresponding mode bell used for damping estimation, c)
340 singular value plot with SSI-COV poles superimposed: stable frequency (blue cross) and stable mode shape (red circle). d) to g): 3D normal mode shapes at the first four resonant frequencies (3.38 Hz, 4.15 Hz, 5.62 Hz, and 6.58 Hz) with projections onto the X-Z and X-Y planes. The mode shapes are normalized to the reference indicated by the cyan marker. For better visibility, only one of the two parallel geophone lines is displayed, while the mode shapes of both lines are comparable (see appendix Figure A12). Photograph panel a) by Kathryn Vollinger.

Modal identification
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. The largest discrepancies between EFDD and/ SSI-COV on one side and PA is 6 and 4° on the 350 other side for the azimuth and incidence is 6 and 4°, respectively. These discrepancies are in the range of the misfit between field obesrvationsobservations and the numerical models presented by Geimer et al. (2020). Therefore, we stateconclude that, for well-separated modes, all techniques provide identical results with in the range of the uncertainties.
At Corona Arch, only two resonant peaks can beare observed in the power spectra between 1 and 7 Hz. The first includes both horizontal components, while the second also includes the vertical component. However, tThe numerical model 355 by Geimer et al. (2020) predicted two close modes at the location of the second peak, which supported their interpretation of a close hidden mode. However, the same model also showed a significant misfit between model prediction and observed data.
With EFDD and SSI-COV, the presence of two close modes can be verified, (note the elevated second singular value in Figure 4e). At the example of Corona Arch we demonstrate that EFDD and SSI-COV are strong techniques to detect close and hidden modes. 360 At Squint Arch, the power spectra shows two resonant peaks between 10 and 14 Hz, both including all three components. However, the numerical models by Geimer et al. (2020) only predicted one single mode. Consequently, they interpreted this doublet-peak as the signature of one mode alone. In contrast, EFDD and SSI-COV independently indicate two closely-spaced modes. Therefore, these two techniques help to identify the resonant modes at the example of Squint Arch.
At Musselman Arch, all four resonant modes between 1 and 10 Hz can beare observed in the power spectra, as the 365 resonant frequencies are well separated. However, the large array dataset results in 3 timesx 32 power spectra to be analyzed, which causesis an extensive analysis effort ( Figure A1). In addition, the power spectra do not provide a direct statementevidence if the peaks corresponding to one single mode or if there are additional close modes. Here, EFDD combines all input traces in one single plot, providing a direct illustration of the resonant modes and statesindicates that no close modes are present. In addition, the mode shapes can be directly plotted by evaluating the singular vectors. Therefore, the EFDD 370 analysis of the large Musselman Arch dataset provides a demonstration case forof the user-friendliness and simplicity of the EFDD technique.

Damping estimates
For most resonant modes, EFDD and SSI-COV provide comparable damping ratiosresults within the expectedanticipated range of uncertainty range of up to 30%. However, we observe that EFDD results in 30 to 35% higher 375 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 observations as an effect of damping overestimation through broadening of the resonance peak, caused by spectral leakage (e.g., Bajric et al., 2015). For Squint Arch, Geimer et al. (2020) performed an active source experiment, resulting in a modal damping ratio of 1.6% for the fundamental mode, which is in perfect agreement with the values obtained by SSI-COV (1.6%). This supports our interpretation that damping ratios obtained by EFDD (2.4% 380 for the fundamental mode at Squint Arch) might be overestimated.
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 63 and 26% higher than estimated by EFDD. This is illustrated in Figure 6, where wWe compare damping ratios estimatedobtained by the various techniques for each arch in Figure 2. The resonant frequency and damping ratio derived by each technique are used to model a single-degree-of-freedom (SDOF) system, which 385 is superimposed on the singular value plots. The amplitude of the modelled 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 (Figure 6a). 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 a SDOF system. SSI-COV and EFDD perform equally well reproducing the mode bells of f2 and f3. At Corona 390 Arch, only SSI-COV is capable of reliably reproducing the mode bell of the fundamental mode, and the half-power bandwidth again overestimates damping (Figure 6b). The second and third mode are well reproduced by both SSI-COV and EFDD. SSI-COV and the active-source impact test yielded identical damping ratios for the fundamental mode of Squint Arch (Figure 6c).
For higher order modes, the discrepancy between EFDD and SSI-COV is smaller on all arches. However, as for Rainbow Bridge, a similar match could not be obtained for f1 and f2 by any of the techniques. For f3, EFDD produces a slightly better 395 match with the singular values, while SSI-COV appears to marginally overestimate damping. We applied Enhanced Frequency Domain Decomposition (EFDD) and Covariance-driven Stochastic Subspace Identification (SSI-COV) modal analyses on four natural rock arches that were previously analysed by Geimer et al. (2020) 405 using frequency-dependent polarization analysis (PA). Our results show that EFDD and SSI-COV are well suited to determine the natural frequencies, damping ratios, and mode shapes of these geological structures. For clear resonant modes, these techniques reproduce the results by Geimer et al. (2020). In 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 410 traces in a single plot, allowing for a user-friendly analysis of the dynamic response. 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 normal mode shapes.
Modal damping estimation from both SSI-COV and EFDD are not based on individual picks on the power spectra, as it is the case for the half-power bandwidth technique. EFDD damping evaluation is based on the shape of the entire mode 415 bell in the frequency domain, while SSI-COV is a parametric time-domain technique. Therefore, modal damping determined by these techniques is expected to be more robust than the half-power bandwidth picking technique, which is highly sensitive to spectral smoothing, resulting in over-estimates of damping at Rainbow Bridge and Corona Arch. This is supported by good agreement between EFDD, SSI-COV results and damping determined for the active impulse measurement at Squint Arch.
However, SSI-COV results are likely to be closer to the physical damping ratio than EFDD, as limitations in spectral resolution 420 can lead to a broadening of the normal mode bell, and thus overestimation of modal damping determined in the frequency domain. While EFDD performed well in all cases studied here, SSI-COV failed in one instance using nodal geophones, likely as a result of low signal to noise ratios and suggesting a possible limitation of the technique or instrumentation requirement for dynamic analysis of geological features with low ambient excitation. which we attribute to seasonal variations in the dynamic response due to temperature effects (11.5°C for the single-station and 16°C for the nodal measurement, respectively, 425 see also Starr et al., 2015).
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 increase the robustness of the results, especially for more complex dynamic systems, providing refined modal characterization. Improving the accuracy and our understanding of resonance properties helps generate more refined numerical 430 models, which can in turn lead to more accurate rock arch stability assessment. Future efforts in modelling the dynamic response of rock arches (and other geological structures) should additionally involve calibration of the modal damping ratio, as we have shown this parameter can be reliably measured. In addition, we recommend integrating material anisotropy in numerical models for rock arches exhibiting a complex dynamic response with closely spaced or split modes that could not be replicated using homogeneous models. 435

Differences between surveys at Squint Arch
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°C for the single-station and 16°C for the nodal measurement, 440 respectively, see also Starr et al., 2015). Seasonal effects are also expected to influence modal damping ratios and mode shapes (Häusler et al., 2021).
Another difference between the two surveys is that only two modes can be detected by the geophone array. This is likely an effect of the higher self-noise of the Zland geophones compared to the broadband TC -20-s seismometer, which might be higher than the excitation level of the higher modes (e.g., Brincker and Larsen, 2007). We were also not able to find a set 445 of SSI-COV parameters that could reliably reproduce the resonant frequencies, which is again attributed to the lower signalto-noise ratio of the geophone array datsetdata (e.g., Brincker, 2014;Liu et al., 2019).
Our results encourage adaptation and widespread application of EFDD and SSI-COV modal analysis techniques, which are commonly used in civil engineering, and complement existing seismological techniques for dynamic analysis of geological features. Both techniques might be well-suited for future near real-time monitoring of the structural integrity 450 of geological features beyond rock arches, for example, rock slope instabilities, unstable glaciers, and freestanding rock towers.

Conclusions
We applied Enhanced Frequency Domain Decomposition (EFDD) and Covariance-driven Stochastic Subspace Identification (SSI-COV) modal analyses on a set of four natural rock arches that were previously analyzed by Geimer et al. 455 (2020) 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 geological structureslandforms, including close, hidden, and higher resonant modes. For well-separated resonant modes, these techniques reproduce the results by Geimer et al. (2020). 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 460 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.to the information forof 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 normal mode shapes. 465 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 arch stability assessment. Future efforts in modelling the dynamic response of rock arches (and other geological 470 structuresfeatures) should additionally involve calibration of the modal damping ratio, as we have shown this parameter can be measured on complex structures.
Our results encourage adaptation and widespread application of EFDD and SSI-COV modal analysis techniques, which are commonly used in civil engineering, andto complement existing seismological techniques for dynamic analysis of geological 475 features. Both techniques might beare well-suited for future near real-time monitoring of the structural integrity of geological features , however 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's surface structureslandforms. 480

Appendix A
The frequency response ( ) of a single-degree-of-freedom (SDOF) system is given by being the angular frequency and being the angular resonant frequency. refers to the modal damping ratio and is the imaginary unit (see, for example, Chopra, 2015). Table A1: SSI-COV input parameters as defined by Cheynet (2020). Ts: time lag for covariance calculation (two to six times the natural period), Nmin: minimal number of model order, Nmax: maximum number of model order, ε cluster: maximal distance 490 inside each cluster. Frequency accuracy (ε frequency), MAC accuracy (ε MAC) and damping accuracy (ε zeta) are set to 0.01, 0.05 and 0.04 for all analyses, respectively. The band-pass filter was chosen such that the resonant peaks observed in the spectra are included. Nmax and ε cluster were testes in a trial and error approach to obtain stable poles that match the first three observed resonant modes.  Mode f1 3.4 1.3 n/a n/a Mode f2 4.2 1.1 n/a n/a Mode f3 5.6 1.6 n/a n/a 500 Figure