This study investigates earthquake-induced wave dynamics at mountain summits, particularly at the Matterhorn (Switzerland) and Tre Cime di Lavaredo (Italy). Full wavefield modeling is utilized to simulate the induced resonant oscillations and amplification of seismic signals at the summits compared to adjacent valleys. The simulated amplification (up to 10 times) in the summit depends on the characteristics of motion direction, topography, and presence of permafrost. Major resonance modes are identified at Matterhorn at frequencies of 0.4 Hz and 1.4 Hz. Higher resonance frequencies above 2 Hz are obtained at the smaller rock formation Tre Cime di Lavaredo, indicating mountain-specific resonances. We demonstrate that the presence of a permafrost body inside the mountain tends to mitigate seismic amplification by up to 30%. However, this effect is dependent on the amount of permafrost and the wavelength of the seismic waves. Locations of potential slope instabilities on the mountain’s surface are identified based on the dynamic stress changes during the simulated earthquake. We find that locations of stress amplification are mainly at the mountain flanks and are influenced by azimuthal characteristics of the incoming wave. The approach and findings presented in our study have the potential to improve hazard assessments for earthquake-induced slope instabilities at mountains.
Article
Predicting earthquake-induced wavefield and stress dynamics in high-alpine mountains using full waveform modeling
https://doi.org/10.21203/rs.3.rs-5156490/v1
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Landslides and rockfalls are significant hazards for residents and infrastructures, especially in mountainous regions. Their occurrence is increasing due to climate warming and the subsequent degradation of permafrost [1, 2, 3], as well as heavy rainfall [4, 5, 6, 7, 8]. Moreover, seismic activity and waves from earthquakes can induce landslides and rockfalls, or destabilize mountain slopes progressively. This is a phenomenon that is recorded globally [9, 10, 11] and observed on regional scales, e.g., Italy [12], Japan [13], New Zealand [14], China [15], Guatemala [16], Alaska [17], or Papua New Guinea [18]. Furthermore, earthquakes have the potential to induce snow avalanches [19].
Seismic wave amplification in mountainous regions, particularly during earthquakes, is a known phenomenon [20, 21, 22]. Previous experimental and numerical studies have investigated site effects and seismic amplification due to topography. Weber et al. [23] conducted ambient noise measurements at the summit of Matterhorn and observed significant signal amplification of up to 9 times on average at the mountain summit compared to nearby valleys. Furthermore, they suggest the high potential for earthquake-induced rockfalls or landslides due to the measured and simulated resonances at distinct frequencies. Massa et al. [24] and Weber et al. [23] also showed the existence of preferential motion directions within the topographic formation, indicating complex wavefield dynamics resulting in resonances and signal modulations in the mountains. The importance of considering full wave form modeling to analyze and assess earthquake-induced landslides is shown by Dahal et al. [25].
Previous numerical studies have mainly utilized generic two-dimensional models to estimate site amplifications considering factors such as wave frequencies, material properties, and topographic features [26, 27, 28, 29, 30]. These studies have shown that topographic features can significantly enhance seismic signal amplification, particularly with steeper slopes and strong material contrasts. Additionally, the incidence angle of seismic waves influences amplifications [31]. Contrarily, the presence of glaciers in the valley or frozen layers in the subsurface have the potential to damp and reflect waves and hence to reduce site amplifications [32, 29]. To model resonance frequencies, Weber et al. [23] performed numerical modal analyses, while wavefield dynamics were neglected. The development of seismic resonances in complex structural environments, such as volcanoes, was studied, e.g., by Jousset et al. [33], Sturton & Neuberg [34], and Limberger & Rümpker [35]. However, the complex seismic response of specific mountains is poorly modeled.
Potential damping effects of subsurface permafrost layers on wave amplifications and elastic properties of the rock were studied using experimental and numerical methods [32, 36, 37]. Lindner et al. [38] analyzed seismic long-term measurements from the mountain Zugspitze (Germany) and found that seasonal permafrost melts affect seismic wave travel times and hence velocities. Nevertheless, the potential influence of permafrost degradation inside mountains on seismic amplification at summits has not been adequately explored or modeled in existing research, despite its relevance in the context of climate change promoting mountain instabilities.
The emission of seismic waves through rock masses causes a dynamic stress perturbation that adds up to the background static stress fields [39]. The effect of dynamic stress perturbations on seismic triggering has been studied based on both theory and observations [40, 41]. While their effect on aftershock sequences is debated [42, 43, 44, 45], they are often involved to explain seismic sequences occurring in volcanic and fluid-rich environments [46, 47, 48, 45]. In the context of landslides and rockfalls, the role of dynamic stress has been addressed in models relying on modal analysis [49] and mechanical analysis [39, 50]. The effects of fractured structures inside rocks on slope instabilities were studied by Burjánek et al. [51] using observational data linked with numerical simulation of ambient vibrations. However, these studies do not specifically consider temporal earthquake-induced wavefield and stress dynamics in realistic high-altitude alpine environments including permafrost effects and full-waveform modeling.
In our study, we address these gaps by utilizing highly resolved digital elevation models (DEM) and numerical models to estimate the characteristics and spatial distribution of frequency-dependent signal amplifications at prominent topographic structures. Specifically, we model the Matterhorn in Switzerland and Tre Cime di Lavaredo (Dolomites) in Italy as case examples in two regions that are known for frequent major rockfalls. The two mountains are characterized by considerably different sizes. Through full-wavefield simulations, we investigate how these mountains respond to incoming seismic waves. Additionally, we examine the potential impact of permafrost degradation on seismic signal amplification at the Matterhorn's summit. Finally, we establish a linkage between the simulated peak ground velocities at the mountain and stress field dynamics, offering insights into predictions of location-dependent slope instabilities during earthquakes.
Our research aims to provide an improved approach for estimating wavefield dynamics and stress field changes in mountainous regions, which is crucial for the local hazard assessment of earthquake-induced landslides, rock falls, and avalanches.
Model configuration
We utilize a high-resolution DEM obtained from the Federal Office of Topography of Switzerland (Dataset: swissALTIRegio, www.swisstopo.admin.ch) to accurately represent the topographic shape of the Matterhorn. The DEM has a resolution of 10 meters and covers an area of approximately 10 km x 10 km (Fig. 1) around the Matterhorn. The modeled elevation ranges from -2000 m (below sea level) to the peak elevation of 4478m. The wavefield and geometrical model are resolved using a minimum of two elements per wavelength and a polynomial order of 4 (~order of nodes per element). This ensures accurate representation of the wavefield and the geometry of the models. Absorbent layers (not depicted in Fig. 1) are attached to the actual model domain to ensure sufficient wave damping at the model boundaries. In order to analyze a wide range of frequencies, we use an Ormsby wavelet as the source, which has a trapezoidal frequency spectrum that is flat between 0.3 and 3.5 Hz (Fig. S1), to account for the typical frequency range observed in terms of mountain resonances [23, 22]. To simulate the incoming seismic wave from a local or regional earthquake, we model a vertical plane wave excited by three force vectors (in the X, Y, and Z directions) at each cell of the mesh at elevation z=0. This represents a plane wave excitation with both P- and S-waves. Synthetic receivers across the summit are located along the NS-axis and EW-axis across a length of 5 km at intervals of 100 m.
The subsurface properties of our reference model include a linear velocity gradient, with a velocity of VP=3000 m/s at the summit and VP=6000 m/s at z=-2000m (Fig. 1). This represents lower velocities near the surface and higher velocities in the crust. The velocity values at the surface are based on the geological composition of the highly weathered and fractured Gneiss in the mountain top region [52], while higher velocities are typically expected in deeper regions with more compressed bedrock. The shear wave velocity VS is always equal to VP/1.7. We assume lateral homogeneity in the velocity distribution. Additionally, the Q-values (QP=200, QS=100), which represent the energy dissipation associated with seismic waves, are assumed to be constant throughout the model.
Frozen rock, such as permafrost, is typically characterized by higher seismic velocities and denser material due to the presence of ice in fractures and voids [36, 37]. To model scenarios with permafrost inside the mountain (used later in Fig. 6), we systematically decrease the amount of permafrost and increase the seismic velocities by up to 50% compared to the reference model depicted in Fig. 1.
For the Tre Cime di Lavaredo (peak elevation of 2999 m) in Italy, we utilize a DEM derived from INGV (Italian National Institute of Geophysics and Volcanology, https://tinitaly.pi.ingv.it/) with a 10-meter resolution. The subsurface properties of the Tre Cime di Lavaredo model are based on the same velocity model as the Matterhorn. The mountains in the Dolomites typically show relatively thin stratigraphic layers, which are neglected in our idealized models. Hence, the main distinction between the modeled mountains is their topography, which sets the focus on topographic effects on wave dynamics in our study. The corresponding 3D model of the Tre Cime di Lavaredo is given in the supplementary material (Fig. S2).
All simulations in this study are performed using the commercial software package Salvus [53], which utilizes the spectral element method to compute the propagation of elastic waves. In order to facilitate wider accessibility to similar calculations, alternative software options such as OpenSWPC [54], Seissol [55], or SpecFEM [56] offer equivalent capabilities and are open source.
Seismic responses of Matterhorn (Switzerland) and Tre Cime di Lavaredo (Italy)
Synthetic seismograms are extracted from receivers positioned along the summit of the mountain (Fig. 2A and B). The wavefield (Fig. 2C-E) exhibit higher signal amplitudes at the central locations of the profiles compared to stations situated in adjacent valleys. The central receivers are located at the summit. These prolonged and amplified signals are predominantly observed on the horizontal components of ground motion, particularly in the north-south (NS) direction (Fig. 2D).
The variation between the receivers along the north-south axis and the east-west axis is negligible, suggesting it may be attributed to the azimuthal symmetry of Matterhorn. Resonances with prolonged durations of 10 seconds are primarily observed on the NS component at the summit. Such resonances and signal amplification can be explained by reflections off the mountain flanks resulting in constructive interferences and reverberations of the waves trapped in the summit of the mountain. Aside the resonances at the central stations, a scattered wavefield can be observed. This wavefield is likely arising from multiple reflections off the sides of the mountain, as well. However, compared to the prolonged resonances, such reflections are not trapped inside the upper part of the mountain.
An analogue simulation is performed for the Tre Cime di Lavaredo. Receivers located on the summits exhibit amplified signal amplitudes compared to stations in nearby valleys, both in the north-south (NS) and east-west (EW) directions (Fig. 3C and D). Along the EW-axis, the resonances extend eastwards from the center station due to the extended rock formation of the Tre Cime di Lavaredo (Fig. 3D) in east direction. However, this extension is absent along the NS-profile, indicating significant dependency on the ridge-like geometry of the Tre Cime di Lavaredo. In general, the resonances have a duration of approximately 6 seconds. Similar to the results for Matterhorn, a weaker scattered wavefield is observed apart from the center stations.
Regarding the Matterhorn, notable spectral amplification is observed on the horizontal components (Fig. 4A and B), peaking at magnitudes up to 10 times larger than the signal amplitude at a valley station. Distinct resonant frequencies, notably between 0.4 Hz and 2.5 Hz, are discerned, with the most prominent peaks occurring at 0.4 Hz and 1.4 Hz on the NS-component (Fig. 4B). A minor peak is at 1.8 Hz on the EW-component. Conversely, vertical amplitudes are comparatively subdued, underscoring a horizontal seismic energy amplification.
Concerning Tre Cime di Lavaredo, horizontal components similarly display substantial amplification, with signal magnitudes reaching approximately seven times larger amplitudes compared to valley station signals of the same frequency (Fig. 4C and D). Again, vertical amplitudes are reduced relative to horizontal motions. Noteworthy resonant frequencies of 0.8 Hz and 1.0 Hz (NS-component and EW-component), 2.0 Hz (EW-component), and 3.0 Hz (Z-component) are identified (Fig. 4D). However, the most prominent peak is at approx. 2.0 Hz, along with the highest amplification. Distinct frequency peaks persist up to of 3.0 Hz. This excitation of higher frequencies at Tre Cime di Lavaredo can be attributed to the comparatively small size (H=600 m, B=2000 m) in contrast to the Matterhorn’s dimensions (H=2000 m, B=5000 m). Interestingly, the lower frequency peak of 0.8 Hz is dominant on the NS-component (perpendicular to the EW oriented rock formation), while the higher frequency peak of 2.0 Hz is dominant on the EW-component. This indicates combined dependency of the resonance modes on motion direction, wavelength as well as summit geometry and extension.
The calculation of PGV (Peak Ground Velocity) maps offer insights into the distribution of maximal ground motion amplitudes during earthquakes. Through computed PGV (Fig. 5) maps, we elucidate the maximal ground motion amplitude at Matterhorn, as well as Tre Cime di Lavaredo. The analysis of the response to a 0.4 Hz wave frequency reveals widespread peak values across the Matterhorn summit (Fig. 5A), attributable to the elongated wavelength of the vertically incoming P- and S-wave. Adjacent summits and ridges experience significantly lower excitation from the seismic event. However, the amplitude of higher-frequency waves (e.g. 1.0 Hz) may be amplified by smaller mountains, as shown for the Tre Cime di Lavaredo (Fig. 4C and D, Fig. 5B). At the Tre Cime di Lavaredo, notably elevated PGV values manifest at the three summits of Tre Cime di Lavaredo and Monte Paterno, situated eastward. The plateau of the mountains exhibits a modest PGV increase as well, indicating larger scale amplification. In addition to the three-dimensional PGV maps, we study further details using highly resolved 2D models in the next sections.
The effect of permafrost on signal amplification
We assess the potential impact of permafrost on seismic amplifications by postulating heightened density and wave velocity within a certain subsurface layer inside the mountain, modeled by an ice shield below the surface (Fig. 6A). This model serves as an idealized approximation based on the thermal model conducted by Noetzli & Gruber [57] and permafrost distribution inside mountains described in Arenson et al. [58]. In our model, the permafrost body's dimensions systematically decrease in thickness and extension towards the valley, assuming that the melt starts at lower altitudes. The bedrock below the ice shield is assumed to be ice-free. Five models are utilized for our study. Model 1 presupposes an idealized large permafrost body inside the mountain (thickness below the summit is 600 m). Models 2-4 represent diminishing ice bodies, while model 5 represents the mountain without any frozen material in the rock, resulting in reduced density and velocity due to air or water occupying fractures, voids, and cracks.
Spectral analyses are performed for the synthetic receiver situated atop the summit, revealing two prominent peaks at approx. 0.4 Hz and 1.4 Hz (Fig. 6B), consistent with simulated peaks derived from a three-dimensional model (Fig. 4B). As permafrost diminishes, spectral amplitudes at 0.4 Hz and 1.4 Hz decrease. This decrease is significantly more pronounced at 0.4 Hz than at 1.4 Hz. For frequencies > 2 Hz, the effect on the amplification is negligible. However, depending on the permafrost distribution and mountain geometry, higher-frequencies could be affected as well in other scenarios, as shown in Fig. 4. Similar to previous results, the effect is notably enhanced for horizontal motions compared to vertical motions (Fig. 6B). We performed a sensitivity study, which involves 30 models with different combinations of permafrost thickness (varied between 100 m and 600 m) and seismic velocity increase, caused by frozen water in the permeated rock (varied between 10% and 50% increase). It reveals that the maximum mitigation (~30%) in PGV occurs in case of a thick and dense permafrost layer (see Fig. 6C). This suggests that both a decrease in the thickness of the ice shield and a decrease in the amount of ice within the shield itself lead to higher PGV values, while the sensitivity is slightly higher for the ice thickness than the velocity increase. However, it is unlikely that very small ice bodies have the potential to significantly impact the wavefield with frequencies less than 3 Hz.
An explanation for this mitigation is the contrast in material properties between frozen layers and unfrozen layers within the mountain. When seismic waves encounter the rock-ice discontinuity, they are partly reflected. In addition, the transmitted waves are less likely to be trapped in the summit due to their elongation of the wavelength when entering a material with higher seismic velocities. This results in a more efficient transportation out of the permafrost region after reflections at the free surface, thereby preventing the generation of resonances.
Dependencies on earthquake azimuth and incidence angle
In this section, we analyze the seismic waves approaching the Matterhorn from the north and south directions using a two-dimensional cross section. We consider non-vertical incidence angles and use an angle of 45° in both cases (north and south). These models serve as the foundation for the simulation results presented in Fig. 7, which is why we focus on two dimensions in this section. We find that, consistent with the three-dimensional model (Fig. 5), the maximum amplitude occurs at the summit in the horizontal component (NS-direction) in all three cases (0° vertical incidence, 45° south, and 45° north). In the case of a vertical incoming plane wave, the amplitudes at the northern and southern mountain flanks are similar due to the symmetry of the mountain and the incident wave. However, when waves approach from the north and impact the southern parts of the formation, the horizontal ground motions in the valley to the south of the Matterhorn slightly increase (Fig. 7B). As a result, the contrast between maximum amplitudes at the summit and amplitudes in the southern valley slightly decreases. Conversely, when the wave arrives from the south, the horizontal PGV distribution is comparable to the results obtained with a vertical polarization of the incoming wave (Fig. 7C). Interestingly, the vertical ground motion is slightly increased at the very steep part of the southern flank (>4200 m). Although the directional characteristics of the wave influence the distribution of PGV, the mountain experiences the maximum oscillation at the summit in horizontal directions under all circumstances, which supports the findings depicted in Fig. 4.
Dynamic stress changes during an earthquake
In order to identify locations of potential slope instabilities during an earthquake, we explore the linkage from seismic ground motion (in Fig. 7) to the consequential stress distribution. Thus, the distribution of the dynamic stress field is calculated. The underlying deviatoric stress, denoted as dQ, refers to material deformation (e.g., by seismic waves) without a change in volume of the material. Hence, the initial hydrostatic stress is subtracted from the deviatoric stress. Using the second invariant J2of the two-dimensional total stress tensor 
derived from the numerical simulation, the scalar of the deviatoric stress dQ can be expressed by
using the hydrostatic stress (or pressure)
Although the stress calculation is performed in two-dimensions, the out-of-plane stress 
must be considered. Note that the z-axis is directing out of the x-y-plane and not in vertical direction. The detailed derivation of the formulation is provided in the supplementary material. By examining the peak dynamic stress per mesh element across all simulation time steps, we can ascertain the stress distribution throughout the two-dimensional model of Matterhorn.
While the summit of the Matterhorn experiences the highest magnitude of motion (Fig. 7), the peak stress at the surface manifests on the mountain's flanks (Fig. 7). For vertically incident waves, a slight elevation in stress is observed on the southern flank, particularly within elevations spanning approximately 3800 m to 4300 m, while the northern flank exhibits comparatively weaker stress variations (Fig. 7G). Moreover, an escalation in stress is simulated at the topographic salient point situated on the northern side of the mountain, at an altitude of approximately 3400 m.
In case the wave arrives from north, the stress is significantly increased compared to a vertical oncoming wave. The location of increased stress on the southern flank is shifting to lower altitudes and covers a larger area. Conversely, the stress on the northern flanks intensifies strongly in magnitude and encompasses a significant broader area, if the waves are approaching from the north, (Fig. 7H). An explanation could be surface waves from north, that are generated by the incident incoming wave. A wave coming from the south leads to a strong increase in stress on the southern flank at an elevation higher than 3700 m (Fig. 7I). Interestingly, there is no increase in stress observed on the northern flank in this scenario. Overall, a non-vertical incoming wave generally leads to broadened areas of increased stress on mountain flanks, although minor effects of the PGV at the summit. The variation in results for waves approaching from the south and north suggests that the distribution of stress on the mountain flanks depends on the azimuth of the earthquake source and on the specific topographic effects of the mountain. Nevertheless, an increase of deviatoric stress at the southern flank of Matterhorn is derived in every scenario, which is interestingly supported by reports about huge rockfalls mainly at the south flank of Matterhorn on September 10th in 2023 (https://explorersweb.com/rockfalls-eiger-matterhorn/).
To obtain absolute stress estimates, we consider two potential scenarios involving PGV values at the Summit. According to the research conducted by Cauzzi et al. in 2017 [59], during a magnitude 4.4 earthquake near Matterhorn in Vallorcine in 2005, the PGV values at the surface ranged from approximately 0.1 cm/s for light shaking to a maximum of 3 cm/s for strong shaking, close to the epicenter. Considering these findings, we assume Scenario A with a peak amplitude of 1 cm/s at the summit (ten times higher than 0.1 cm/s in the valley), and Scenario B with values of approximately 30 cm/s at the summit (and 3 cm/s in the valley). Incorporating these assumptions, we can conclude that the maximum stress levels at the mountain’s flank can reach approximately 40 kPa during light shaking and can exceed 1 MPa during strong shaking (Fig. 7).
We simulated the resonant seismic response of Matterhorn and Tre Cime di Lavaredo using full waveform modeling and demonstrated seismic amplifications at the summits of the mountains. These amplifications depend on the frequency of the incoming wave, the potential presence of permafrost, as well as the geometrical characteristics of the rock formation. Although the reasons for rockfalls might be due to a combination of multiple factors (e.g., heavy rainfall, melting permafrost, preexisting fractures), short-term dynamic stress changes could also play a significant role in triggering slope instabilities [60, 61]. Mainly earthquakes with a large magnitude trigger landslides immediately; however, frequent seismicity with small-magnitude earthquakes can destabilize the rock progressively [61], resulting in delayed slope failures. Hence, we furthermore analyzed the dynamic stress variations during an earthquake to identify locations of elevated hazard in terms of slope instabilities (Fig. 7). Our results underscore the significance of incorporating realistic topographic models in conjunction with wave characteristics and stress computations.
Although the general database of seismological recordings at mountains is limited, the findings of our study mostly confirm the observations from existing previous studies that have investigated the effects of steep topography on seismic amplification. In the following, we point out the most important comparisons.
Weber et al. [23] conducted an experimental study at the Matterhorn and performed numerical modal analysis to identify distinct resonance. Their observations highly support our model results of mountain-specific resonances at the Matterhorn at frequencies around 0.4 Hz (Fig. 4B and 6A), with dominant motion in horizontal directions, especially in the NS-direction. However, we additionally can identify resonances at 1.4 Hz and a minor peak at 1.8 Hz, which can be explained by signal modulations due to reflections inside the mountain summit. Weber et al. [23] analyzed ambient noise, which might not reflect the full spectrum of potential resonance modes and frequency-dependent amplifications that could be additionally excited by earthquakes. Nevertheless, they detected noise amplifications of 9 times on average and 14 times as a maximum at the summit of Matterhorn, comparable to simulated amplifications (~10 times) in our study. In addition to Matterhorn, they measured resonances of 1.8 Hz and 2.3 Hz at the mountain Großer Mythen, which is considerably smaller in size. These findings validate the presence of higher frequency resonances that depend on the mountain's volume and dimensions, as we demonstrated through the comparison of synthetics derived from models of Matterhorn and the smaller Tre Cime di Lavaredo. Moreover, Leinauer et al. [22] performed seismic measurements at a mountain flank of the Hochvogel (Austria) and suggest an amplification of ground motion due to topographic effects of a factor of 2-11, which further supports the estimations based on our models. A comparison of resonance frequency and amplification factors found in existing studies and our study are depicted in Fig. 8, indicating the tendency of higher resonance frequencies and lower amplification factors for smaller mountains. The relatively broad frequency range of resonances between 1.5 and 3.0 Hz observed at Hochvogel in Austria [22] could be explained by the less distinctive mountain shape compared to Matterhorn, Tre Cime di Lavaredo, or Großer Mythen. This likely extenuates the generation of distinct resonances due to less freedom of oscillation.
The findings of Massa et al. [24], who conducted a comprehensive review of topographic effects based on experimental data collected over four decades, confirm predominant horizontal motions. Furthermore, they observed that wavefields tend to be polarized perpendicular to the main orientation of the topographic formation. This supports our observation of major lower-frequency motion in the NS-direction at the Tre Cime di Lavaredo, which has a main rock formation orientation in the EW direction (Fig. 3).
Regarding the influence of permafrost on seismic site response, our study is consistent with the findings of Yang et al. [32], who used 2D models to investigate the effect of horizontal permafrost layers. They concluded that permafrost can alter site effects, specifically attenuating wave fields. We studied permafrost effects in a steep slope mountain region and demonstrate the importance of considering secondary effects of degrading ice in the mountain that mitigates seismic amplifications (Fig. 6). In summary, we find that the presence of permafrost can mitigate signal amplifications by up to 30%, based on the studied scenarios. In addition to the increased likelihood of slope instabilities caused by melting ice in the mountains, there is also a heightened amplification of earthquakes. This means that alpine regions face an even greater risk if permafrost degradation continues due to ongoing climate warming, posing a significant hazard in the coming decades [62, 63].
Poursartip et al. [30] and Shen et al. [31] used generic 2D models of valleys and hills to examine the effects of topographic irregularities and wave incidence angles on ground motions. They found that the incidence angle influences the amplification; however, the magnitude of amplification depends on the topographic feature dimensions and wave frequency. If the wave is efficiently trapped within the feature, amplifications tend to be significantly heightened [30], which is shown by our models as well. Furthermore, our findings show that the incident angle does not necessarily lead to significantly increased amplification of signals with relatively low frequencies (< 2 Hz). However, we provide evidence that the locations characterized by strong stress changes during an earthquake are influenced by the azimuth and incidence angle (Fig. 7), a factor that was not addressed in the aforementioned studies.
As a further possible application, we evaluated a larger three-dimensional model of Matterhorn to incorporate additional stress dynamics at the summit, which may be missing in two dimensions. This model, shown in the supplements in Fig. S3, demonstrates the dynamic stress linked to the previously discussed PGV results in Fig. 5A. We observe heightened stress on both the southern and northern flanks of Matterhorn, consistent with the two-dimensional stress patterns presented in Fig. 7. However, the three-dimensional stress is largely distributed across the flanks, e.g., as seen in the shifted location of increased stress on the southern flank to the lower southeastern ridge of the mountain, which is not captured using a 2D model.
The seismic response and associated dynamic stress changes during an earthquake are influenced by earthquake source characteristics, such as the angle at which the earthquake strikes, the azimuth of the epicenter, the primary frequency of the seismic waves, and the overall magnitude and mechanism of the event. As a result, different mountains may experience different impacts depending on their sensitivity to higher-frequency local earthquakes, or to regional and teleseismic events. Thus, the findings in our study suggest mountain-specific case studies and experiments to obtain reliable estimates.
For future applications, more complex velocity structures including stratigraphic layers along with smaller-scale fractures and faults in different geological settings should be studied to account for further geology-dependent effects. Nonetheless, these considerations go beyond the scope of this study. Most importantly, simulation results should be verified and calibrated by observational data collected at various mountains [23, 22], which improves the estimation of absolute PGV and stress values. Due to extreme terrain and inaccessibility, seismological data recorded at steep and high mountains is rare. This challenges reliable estimations of the seismic response of a mountain as a function of earthquake characteristics, e.g., magnitude, focal mechanism, and azimuth. Hence, a broader database with mountain-specific seismic characteristics would be valuable for future studies.
In our study, we used numerical models along with detailed DEM to simulate wavefield propagation and dynamic stress changes in specific mountains induced by seismic waves from earthquakes.
We modeled mountain-specific responses with distinct resonance frequencies, such as the major peaks at 0.4 Hz and 1.4 Hz at the Matterhorn. The dimensions and extensions of rock formations influence the frequencies and locations of amplifications. This is evident in the case of Tre Cime di Lavaredo, where amplification occurs along the formation extension in the east-west direction, with higher frequencies due to smaller rock formations. We also found that dominant amplification occurs on horizontal components, which verifies the findings of previous studies. For the first time, the effect of permafrost inside the mountain on wavefield propagation is modelled. We found that a permafrost body reduces the amplifications in the summit by up to 30%, indicating a higher hazard in case of permafrost degradation in the coming decades. However, this effect is frequency-dependent and varies with the thickness and amount of frozen material within the mountain. The increased dynamic stress induced by earthquakes at the mountain suggests locations on the mountain flanks that indicate a higher probability of slope instabilities during an earthquake. Furthermore, the changes in stress distribution depends on topography, wave incidence and azimuth, resulting in significantly increased stress at the flank oriented towards the seismic source.
We are hopeful that our study will help to enhance experimental and numerical hazard assessments of earthquake-induced rockfalls and landslides, as the likelihood of these events is expected to increase worldwide in the future.
Acknowledgements
Will be added after peer-review.
Author contribution
F.L. initiated the study, performed the numerical simulations, created the figures, and wrote the first version of the manuscript. G.R. and F.L. optimized the general approach and models. J.P.K and F.L. discussed geological aspects and interpretations. T.D. advised the calculation and interpretation of the deviatoric stress fields. All authors edited and finalized the manuscript.
Conflict of interest
The authors declare that they have no conflict of interest.
Data availability
The simulation scripts will be provided by the corresponding author upon request.
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No competing interests reported.