Predicting earthquake-induced wavefield and stress dynamics in high-alpine mountains using full waveform modeling

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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. Earth and environmental sciences/Solid earth sciences/Geophysics Earth and environmental sciences/Solid earth sciences/Seismology Earth and environmental sciences/Natural hazards Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Introduction 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. Results 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 J 2 of 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 10 th 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). Discussion 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. Conclusion 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. Declarations 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. References Haeberli, W., Schaub, Y., and Huggel, C. Increasing risks related to landslides from degrading permafrost into new lakes in de-glaciating mountain ranges. Geomorphology , 293 , 405–417, doi: 10.1016/j.geomorph.2016.02.009. 2017. Deline, P. et al. Ice loss from glaciers and permafrost and related slope instability in high-mountain regions. 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Supplementary Files LimbergerRuempkeretal2024SRsupplements.docx Cite Share Download PDF Status: Published Journal Publication published 04 Jul, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 19 Feb, 2025 Reviews received at journal 19 Feb, 2025 Reviewers agreed at journal 07 Feb, 2025 Reviews received at journal 08 Nov, 2024 Reviewers agreed at journal 24 Oct, 2024 Reviewers agreed at journal 22 Oct, 2024 Reviewers invited by journal 17 Oct, 2024 Editor assigned by journal 17 Oct, 2024 Editor invited by journal 11 Oct, 2024 Submission checks completed at journal 10 Oct, 2024 First submitted to journal 26 Sep, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5156490","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":365063232,"identity":"d1d1da72-bd55-4f75-9b74-bf3cb5d35375","order_by":0,"name":"Fabian Limberger","email":"data:image/png;base64,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","orcid":"","institution":"Goethe University Frankfurt","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Fabian","middleName":"","lastName":"Limberger","suffix":""},{"id":365063233,"identity":"3d2407fd-7896-46a8-b66f-0a51d48183d5","order_by":1,"name":"Georg Rümpker","email":"","orcid":"","institution":"Goethe University Frankfurt","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Georg","middleName":"","lastName":"Rümpker","suffix":""},{"id":365063234,"identity":"9b2d888f-0cc4-49d3-9205-059cce94f205","order_by":2,"name":"Jan Philipp Kruse","email":"","orcid":"","institution":"Goethe University Frankfurt","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Jan","middleName":"Philipp","lastName":"Kruse","suffix":""},{"id":365063235,"identity":"9cbf9836-93ab-4dee-90b3-22c9d375fcfe","order_by":3,"name":"Thibault Duretz","email":"","orcid":"","institution":"Goethe University Frankfurt","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Thibault","middleName":"","lastName":"Duretz","suffix":""}],"badges":[],"createdAt":"2024-09-26 07:23:34","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5156490/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5156490/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-08218-5","type":"published","date":"2025-07-04T15:58:24+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":66568883,"identity":"e810365f-09dc-4fc8-9d4a-5949c6a9ead2","added_by":"auto","created_at":"2024-10-14 11:21:05","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":537440,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eModel setup of the Matterhorn includes a high-resolution DEM (10m) and velocity gradient. The initial velocity is Vp = 3000 m/s at the summit and increases to Vp = 6000 m/s at a depth of -2000 m. The Q-values (Qp=200, Qs=100) describe the attenuation of the seismic waves due to dissipation of energy while the waves travel through the rock. The values are assumed to be constant throughout the model. To simulate the seismic response, a plane wave with a broad frequency spectrum ranging from 0.3 Hz to 3.5 Hz is used. The source wavelet is an Ormsby-function (Fig. S1), and the source mechanism consists of three force vectors in the X, Y, and Z directions, generating S- and P-waves. The dimension of the mountain is described using \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eH\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e(valley-summit elevation difference) and \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eB\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e (extension of the mountain’s base). Image data: ©Google Earth, 2024.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/6bb6b2220f7c8a891a0dfeee.png"},{"id":66568604,"identity":"41662906-0bd7-4ed0-9002-4bc71d36e345","added_by":"auto","created_at":"2024-10-14 11:13:05","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":918596,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eA and B: Location of synthetic receivers positioned along the east-west (EW, red) and north-south (NS, blue) axes, spanning across the summit for Matterhorn. C-E: Synthetic seismograms extracted along the NS- and EW-axis (left and right), for all three components (from top to bottom). The amplitudes of the seismic traces in each subfigure are normalized to the maximum amplitude of their respective figure. The black circle and rectangle in B correspond to the summit and valley data presented in Fig. 4. Image data: ©Google Earth, 2024.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/4f7b585c6b2b82eb54d9a09e.png"},{"id":66568612,"identity":"5ad10c06-c6ec-41f5-a31b-d9ff4c88a3dc","added_by":"auto","created_at":"2024-10-14 11:13:05","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":688775,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eA and B: Location of synthetic receivers positioned along the east-west (EW, red) and north-south (NS, blue) axes, spanning across the summits of the Tre Cime di Lavaredo. C-E: Synthetic seismograms extracted along the NS- and EW-axis (left and right), for all three components (from top to bottom). The amplitudes of the seismic traces in each subfigure are normalized to the maximum amplitude of their respective figure. The black circle and rectangle in B correspond to the summit and valley data presented in Fig. 4. Image data: ©Google Earth, 2024.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/a1915a1271b46a7c958af879.png"},{"id":66568607,"identity":"f2759261-f94f-4c8f-b0b7-fffc32665618","added_by":"auto","created_at":"2024-10-14 11:13:05","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":273279,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTemporal and spectral seismic response of Matterhorn (A and B) and Tre Cime di Lavaredo (C and D) to a source wave with frequencies ranging from 0.3 Hz to 3.5 Hz. The black line is the synthetic seismogram extracted at the summit location marked by the circle in Fig. 2 (Matterhorn) and Fig. 3 (Tre Cime di Lavaredo), respectively. The rectangle in Fig. 2 and Fig. 3 marks the location of the extracted (blue colored) data. Dominant amplification at the summit occurs primarily on the horizontal components along with distinct frequency peaks, indicating resonance modes.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/9e94c003254948ca8f160c28.png"},{"id":66569740,"identity":"dc0a49f5-7824-4eca-9a88-a34d60b290fc","added_by":"auto","created_at":"2024-10-14 11:29:05","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":1144976,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003ePGV distribution for the region at Matterhorn (A) and Tre Cime di Lavaredo (B). The dominant frequency of the source Ricker-wavelet is 0.4 Hz for Matterhorn and 1.0 Hz for Tre Cime di Lavaredo, based on findings from the spectral analysis. The thin black lines represent the mesh. Here, the PGV is calculated from the ground motion’s (EW, NS, ZZ) magnitude. The values (color scale) are normalized to the respective maximum. Image data: ©Google Earth, 2024.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/369648589559644eccbf5b31.png"},{"id":66568608,"identity":"2144fd01-f85e-4246-ad7d-6139c96a9ecc","added_by":"auto","created_at":"2024-10-14 11:13:05","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":394024,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eA: Two-dimensional models of Matterhorn including different amount of permafrost inside the mountain. B: Spectra of the vertical and horizontal components, extracted from the synthetic receiver atop the summit. The dashed black line represents the spectrum observed at a receiver located in the valley. C: Sensitivity of the mitigation in terms of permafrost thickness and seismic velocity increase compared to a model without a frozen layer in the rock. The dashed contour lines are isolines (5% steps).\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image6.png","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/2c994f0dfbd2439e82d6040f.png"},{"id":66570109,"identity":"81cbe808-3fff-4171-a5a4-cc95dbd803af","added_by":"auto","created_at":"2024-10-14 11:37:05","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":402015,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eMaximum horizontal (A-C) and vertical (D-F) ground motion amplitude and maximum deviatoric stress dQ (G-I) across all time steps during the simulated earthquake. An earthquake in the north and south of Matterhorn is modeled using an incident plane wave with an angle of 45°. The black arrows denote the P-wave propagation direction. The polarization of the S-wave is perpendicular to the arrows. The values are commonly scaled to show the difference between the amplitudes of the vertical and horizontal motion. Scenario A and B (in the title of G-I) provide estimates of absolute PGV and stress values, based on PGV observations during an earthquake provided by Cauzzi et al. [59].\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image7.png","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/c37a6fc47600caa9edf3e6c5.png"},{"id":66568886,"identity":"4534a77e-bd52-4950-b7c5-ff560fdfb842","added_by":"auto","created_at":"2024-10-14 11:21:05","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":220246,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eComparison of dominant resonance frequencies (span of the bar) and average amplification factors (numbers in bars) from existing studies and our findings. The four studied mountains are sorted according their size, from small to large (approximated height H and extension B, see Fig. 1). Note that the results from Weber et al. (2022) [23] and Leinauer et al. (2024) [22] are based on continuous ambient noise measurements and not specifically on earthquake signals.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"image8.png","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/ac3e746d994bdf399b0c62fd.png"},{"id":86180014,"identity":"6e22e4bd-5a1e-4198-a8ba-624c56297eb3","added_by":"auto","created_at":"2025-07-07 16:20:47","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":7454888,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/79911e36-6f34-450c-8f41-e54b9a3b5815.pdf"},{"id":66568882,"identity":"7a26b3e3-e0e0-46ff-a94d-cfe05e494c81","added_by":"auto","created_at":"2024-10-14 11:21:05","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":2147021,"visible":true,"origin":"","legend":"","description":"","filename":"LimbergerRuempkeretal2024SRsupplements.docx","url":"https://assets-eu.researchsquare.com/files/rs-5156490/v1/07232dd774c328c879261832.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Predicting earthquake-induced wavefield and stress dynamics in high-alpine mountains using full waveform modeling","fulltext":[{"header":"Introduction","content":"\u003cp\u003eLandslides 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].\u003c/p\u003e\n\u003cp\u003eSeismic 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].\u003c/p\u003e\n\u003cp\u003ePrevious 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 \u0026amp; Neuberg [34], and Limberger \u0026amp; R\u0026uuml;mpker [35]. However, the complex seismic response of specific mountains is poorly modeled.\u003c/p\u003e\n\u003cp\u003ePotential 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.\u003c/p\u003e\n\u003cp\u003eThe 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 Burj\u0026aacute;nek 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.\u003c/p\u003e\n\u003cp\u003eIn 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\u0026apos;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.\u003c/p\u003e\n\u003cp\u003eOur 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.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eModel configuration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe 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.\u003c/p\u003e\n\u003cp\u003eThe 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.\u003c/p\u003e\n\u003cp\u003eFrozen 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.\u003c/p\u003e\n\u003cp\u003eFor 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).\u003c/p\u003e\n\u003cp\u003eAll 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.\u0026nbsp;\u003c/p\u003e"},{"header":"Results","content":"\u003ch2\u003eSeismic responses of Matterhorn (Switzerland) and Tre Cime di Lavaredo (Italy)\u003c/h2\u003e\n\u003cp\u003eSynthetic 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).\u003c/p\u003e\n\u003cp\u003eThe 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.\u003c/p\u003e\n\u003cp\u003eAn 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.\u003c/p\u003e\n\u003cp\u003eRegarding 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.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eConcerning 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. \u0026nbsp;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 (\u003cem\u003eH\u003c/em\u003e=600 m, \u003cem\u003eB\u003c/em\u003e=2000 m) in contrast to the Matterhorn\u0026rsquo;s dimensions (\u003cem\u003eH\u003c/em\u003e=2000 m, \u003cem\u003eB\u003c/em\u003e=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.\u003c/p\u003e\n\u003cp\u003eThe 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.\u0026nbsp;\u003c/p\u003e\n\u003ch2\u003eThe effect of permafrost on signal amplification\u003c/h2\u003e\n\u003cp\u003eWe 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 \u0026amp; Gruber [57] and permafrost distribution inside mountains described in Arenson et al. [58]. In our model, the permafrost body\u0026apos;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.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eSpectral 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 \u0026gt; 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.\u003c/p\u003e\n\u003cp\u003eAn 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.\u0026nbsp;\u003c/p\u003e\n\u003ch2\u003eDependencies on earthquake azimuth and incidence angle\u003c/h2\u003e\n\u003cp\u003eIn 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\u0026deg; 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\u0026deg; vertical incidence, 45\u0026deg; south, and 45\u0026deg; 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 (\u0026gt;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.\u003c/p\u003e\n\u003ch2\u003eDynamic stress changes during an earthquake\u003c/h2\u003e\n\u003cp\u003eIn 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 \u003cem\u003edQ\u003c/em\u003e, 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 \u003cem\u003eJ\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003eof the two-dimensional total stress tensor \u003cimg src=\"data:image/png;base64,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\" width=\"23\" height=\"23\"\u003e derived from the numerical simulation, the scalar of the deviatoric stress \u003cem\u003edQ\u003c/em\u003e can be expressed by\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"690\" height=\"44\"\u003e\u003c/p\u003e\n\u003cp\u003eusing the hydrostatic stress (or pressure)\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" height=\"28\" width=\"662\"\u003e\u003c/p\u003e\n\u003cp\u003eAlthough the stress calculation is performed in two-dimensions, the out-of-plane stress \u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAAMCAYAAACNzvbFAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAEESURBVDhPrZHPDUVAEMbHq4C4OdICFw3QgQ4klKADKtABRycOjhSgAR2Q6GCeb0LeW4f3L++XbGZ3svvN7Dca79CfuR3xryiidV1TGIakaZqy1nU9bnwIvg+SJGHXdXkYBl6WRfa/IqJVVbHjOCJ2gnoo8Avy/aZpKI5jMk0Tx++/e0FEt20j3/clAfq+l2hZlsRvEVFd16nrOkmM40hpmlKWZWTbtuQ8z1MGB9q2VXJFUUhegAfTNMlgcDQMg/M8F29OoigSvzFM+A9wB+/gexAE6jyO+BI8wEOIPIMCKHTlrSiE0Ok8z9LV2eluD5dlKfur8FtRdAhbzgVhrOcc7jxgvgN0ic6N8hYCSgAAAABJRU5ErkJggg==\" width=\"21\" height=\"12\"\u003e\u0026nbsp;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.\u003c/p\u003e\n\u003cp\u003eWhile the summit of the Matterhorn experiences the highest magnitude of motion (Fig. 7), the peak stress at the surface manifests on the mountain\u0026apos;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.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn 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 10\u003csup\u003eth\u003c/sup\u003e in 2023 (https://explorersweb.com/rockfalls-eiger-matterhorn/).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTo 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\u0026rsquo;s flank can reach approximately 40 kPa during light shaking and can exceed 1 MPa during strong shaking (Fig. 7).\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eWe 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.\u003c/p\u003e\n\u003cp\u003eAlthough 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.\u003c/p\u003e\n\u003cp\u003eWeber 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 Gro\u0026szlig;er Mythen, which is considerably smaller in size. These findings validate the presence of higher frequency resonances that depend on the mountain\u0026apos;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 Gro\u0026szlig;er Mythen. This likely extenuates the generation of distinct resonances due to less freedom of oscillation.\u003c/p\u003e\n\u003cp\u003eThe 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).\u003c/p\u003e\n\u003cp\u003eRegarding 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].\u003c/p\u003e\n\u003cp\u003ePoursartip 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 (\u0026lt; 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.\u003c/p\u003e\n\u003cp\u003eAs 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.\u003c/p\u003e\n\u003cp\u003eThe 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.\u003c/p\u003e\n\u003cp\u003eFor 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.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eIn 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.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eWe 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.\u003c/p\u003e\n\u003cp\u003eWe 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.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eWill be added after peer-review.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contribution\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eF.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.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no conflict of interest.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe simulation scripts will be provided by the corresponding author upon request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eHaeberli, W., Schaub, Y., and Huggel, C. Increasing risks related to landslides from degrading permafrost into new lakes in de-glaciating mountain ranges. \u003cem\u003eGeomorphology\u003c/em\u003e, \u003cstrong\u003e293\u003c/strong\u003e, 405\u0026ndash;417, doi: 10.1016/j.geomorph.2016.02.009. 2017.\u003c/li\u003e\n\u003cli\u003eDeline, P. et al. 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Elevation-dependent warming in mountain regions of the world. \u003cem\u003eNature Climate Change\u003c/em\u003e, \u003cstrong\u003e5\u003c/strong\u003e, 424\u0026ndash;430, doi: 10.1038/nclimate2563. 2015.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-5156490/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5156490/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"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.","manuscriptTitle":"Predicting earthquake-induced wavefield and stress dynamics in high-alpine mountains using full waveform modeling","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-10-14 11:13:00","doi":"10.21203/rs.3.rs-5156490/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-02-20T04:20:39+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-02-19T07:32:51+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"154929638459913854409730213234654192983","date":"2025-02-07T20:32:10+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-11-08T11:45:22+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"230810934479509128831010696531890722822","date":"2024-10-24T19:51:51+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"134164820235277512312344980723904912517","date":"2024-10-22T15:56:17+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-10-17T15:34:59+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-10-17T07:25:25+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2024-10-11T17:23:38+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-10-10T10:25:24+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2024-09-26T07:18:44+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"03da2742-a8b9-4630-8860-622c43c780b7","owner":[],"postedDate":"October 14th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":38847809,"name":"Earth and environmental sciences/Solid earth sciences/Geophysics"},{"id":38847810,"name":"Earth and environmental sciences/Solid earth sciences/Seismology"},{"id":38847811,"name":"Earth and environmental sciences/Natural hazards"}],"tags":[],"updatedAt":"2025-07-07T16:12:34+00:00","versionOfRecord":{"articleIdentity":"rs-5156490","link":"https://doi.org/10.1038/s41598-025-08218-5","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-07-04 15:58:24","publishedOnDateReadable":"July 4th, 2025"},"versionCreatedAt":"2024-10-14 11:13:00","video":"","vorDoi":"10.1038/s41598-025-08218-5","vorDoiUrl":"https://doi.org/10.1038/s41598-025-08218-5","workflowStages":[]},"version":"v1","identity":"rs-5156490","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5156490","identity":"rs-5156490","version":["v1"]},"buildId":"-HB7Z8yhvgn0wM9Nzuekk","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}