Deformation monitoring between 2004 and 2011 at the rock slope instability above Randa (Switzerland) has revealed an intriguing seasonal trend. Relative dislocation rates across active fractures increase when near-surface rock temperatures drop in the fall and decrease after snowmelt as temperatures rise. This temporal pattern was observed with different monitoring systems at the ground surface and at depths up to 68 m, and represents the behavior of the entire instability. In this paper, the second of two companion pieces, we interpret this seasonal deformation trend as being controlled by thermomechanical (TM) effects driven by near-surface temperature cycles. While Part 1 of this work demonstrated in a conceptual manner how TM effects can drive deep rock slope deformation and progressive failure, we present here in Part 2 a case study where temperature-controlled deformation trends were observed in a natural setting. A 2D discrete-element numerical model is employed, which allows failure along discontinuities and successfully reproduces the observed kinematics of the Randa instability. By implementing simplified ground surface temperature forcing, model results were able to reproduce the observed deformation pattern, and TM-induced displacement rates and seasonal amplitudes in the model are of the same order of magnitude as measured values. Model results, however, exhibit spatial variation in displacement onset times while field measurements show more synchronous change. Additional heat transfer mechanisms, such as fracture ventilation, likely create deviations from the purely transient-conductive temperature field modeled. We suggest that TM effects are especially important at Randa due to the absence of significant groundwater within the unstable rock mass.
 The role of ambient temperature cycles as a driving mechanism of large rock slope instabilities is often considered negligible. While thermo-elastic stresses in the near-surface region subject to annual temperature changes (here termed the thermal active layer) have been shown to cause seasonal movement patterns (e.g., Mufundirwa et al. ; Gunzburger et al. ; Krähenbühl ), only a few cases are reported where thermal effects were conclusively proven to drive deeper instability deformation. One notable example is the Checkerboard Creek landslide in Canada, where Watson et al.  demonstrated that thermo-elastic induced stress changes at shallow depth can force deformation at greater depth. Borehole extensometer data from 26 m showed a dislocation rate increase during cold periods, which was related to thermo-elastic contraction in the thermal active layer. The discontinuous nature of the slope instability was suggested to play a key role in allowing these movements.
 In Part 1 of this study [Gischig et al., 2011a], we investigated thermomechanical (TM) forcing of rock slope deformation induced by surface temperature cycles with the help of simplified numerical models. The models implemented both purely elastic behavior (no failure) and elastic behavior combined with failure along prescribed discontinuities. This conceptual study aimed to support interpretation of seasonal deformation trends observed in monitoring data from the Randa instability in the southern Swiss Alps. Therefore, the two kinematic failure modes controlling the Randa instability, translational sliding and toppling, were investigated. Our models demonstrated that the thermo-elastic reaction of near-surface bedrock subject to temperature cycling induces stress changes at depths below the thermal active layer as a result of steep topography. If discontinuities at depth are critically stressed, these stress changes can induce irreversible slip. Subsequent stress redistribution due to slip along discontinuities leads to increasing stresses at slip fronts and can eventually result in slip front propagation. Thus, TM-induced stress changes can drive progressive failure of an unstable rock slope. The magnitude of the effect depends on the elastic properties of the medium, the amount of critically stressed discontinuities within the system, and on the post-failure constitutive behavior of discontinuities (e.g., slip-weakening).
 In this paper, we present the case study of the Randa rock slope instability, where up to eight years of monitoring data from both the surface and boreholes suggest that thermal effects may control ongoing slope deformation. We first introduce the current Randa slope instability and summarize results from previous investigations at the site. We then describe relevant details of the monitoring system and present data from 2004 to 2011. Deformation time series are interpreted in combination with insights gained from the conceptual study presented in Part 1 and with the help of site-specific numerical models. In the final section, we discuss the role of TM effects as a driving mechanism of progressive rock slope failure.
2. Randa Rock Slope Instability
 The current rock slope instability above the village of Randa in the southern Swiss Alps (Figure 1) is the legacy of two catastrophic rockslides in 1991, which released in total ∼30 million m3 of crystalline rock [Schindler et al., 1993]. About 22.5 million m3 of orthogneiss that forms the lower part of the present scarp failed in April 1991, followed by retrogressive failure of ∼7 million m3 of the overlying paragneiss and schists in May 1991 [Sartori et al., 2003]. The resulting failure surface forms an 800 m high cliff reaching to 2300 m a.s.l., which is composed of a sub-vertical orthogneiss face below 1900 m a.s.l. overlain by paragneiss and schists (Figure 1). Geodetic monitoring initiated after these failures revealed that a sizable rock mass remains unstable and currently moves at rates of greater than 20 mm/yr [Jaboyedoff et al., 2004]. The volume of this remaining unstable rock mass was estimated to be ∼6 million m3 [Sartori et al., 2003; Gischig et al., 2009].
 In 2000, a collaborative research program was initiated with the goal of understanding the internal structure and kinematics of the unstable rock mass, as well as investigating temporal deformation trends. A comprehensive, multicomponent monitoring system was installed, creating a so-called in situ laboratory (Figure 2a). Monitoring components and data will be described in detail in Section 3. Geological, geotechnical, and geophysical investigations were performed and interpreted to constrain a 3D geological, structural and kinematic model of the unstable slope [Willenberg et al., 2008a, 2008b; Heincke et al., 2005, 2006; Spillmann et al., 2007]. Various borehole logs, fracture imaging, and borehole georadar were performed in three boreholes of 50 and 120 m depth. The results were combined with periodic inclinometer and extensometer surveys in casing cemented into the boreholes. Deformation was found to be localized along persistent, large-scale discontinuities (>15 m long) that cut the borehole, and dislocation vectors across these discontinuities could be resolved. Observed deformation patterns were best interpreted as toppling, with dislocation along discontinuities dipping into the slope and block rotation in between. Thus, the internal structure and deformation kinematics in the upper part of the instability to a depth of ∼120 m were constrained (Figure 1b) [Willenberg et al., 2008a, 2008b]. It should be noted that no evidence of persistent, downslope dipping discontinuities was discovered from investigations in the uppermost 120 m of the unstable rock mass.
 Research activities conducted between 2000 and 2005 were mostly limited to the accessible area at the top of the instability, while information remained sparse for the inaccessible failure surface. This limitation was later alleviated through application of remote-sensing techniques, including ground-based radar interferometry (GB-DInSAR), airborne laser scanning (LiDAR) and photogrammetry, and additional geodetic measurements [Gischig et al., 2009, 2011a]. GB-DInSAR displacement maps confirmed the toppling behavior in the uppermost portion of the slope. Additionally, the lower boundary of the instability could be identified, formed by a persistent basal sliding surface, as well as a lateral release plane bounding the unstable rock mass to the south. Structural analysis of LiDAR and orthophoto data helped complete the structural models, and confirmed that two kinematic failure modes control instability movement: toppling in the upper portion above 2150 m and translational sliding below. Translational sliding occurs on a set of discontinuities dipping downslope, which could be observed on the scarp but were seemingly absent in the upper portion of the rock mass investigated in previous research phases. Although some of these sliding discontinuities have persistence of >200 m, it was assumed that intact rock bridges separate individual discontinuities of this set. Two-dimensional numerical models of the discontinuous rock mass presented in Section 6 confirmed the feasibility of this kinematic model.
3. Monitoring System and Components
 A comprehensive monitoring program was initiated at the Randa rock slope to record temporal trends in instability behavior [Willenberg et al., 2002]. An overview of the current monitoring system is shown in Figure 2a. Two crack extensometers (labeled Z9 and Z10) measure normal dislocation across tension fractures at the ground surface, which have trace lengths of ∼50 m and apertures of ∼0.2 −0.5 m. Three sub-vertical boreholes were drilled to depths of 120 and 50 m (labeled sb120, sb50n, and sb50s). The 50 m holes were equipped with inclinometer casing and the 120 m hole with inclinometer/extensometer casing. Between 2001 and 2008, bi-annual borehole inclinometer and extensometer surveys were performed to detect and measure dislocation along discontinuities at depth (Figure 2b). Two in-place inclinometers were installed in borehole sb120 in 2003, spanning active discontinuities that dip into the slope at ∼45° and show normal faulting with a minor opening component. These vertical inclinometers measure tilt along two orthogonal axes. Multiplying tilt in radians by the instrument base-length of 1.87 m gives the horizontal component of the dislocation vector of the lower block with respect to the upper block. Pore pressure is measured at the bottom of each borehole with (non-vented) piezometers isolated in slotted increments, sensing absolute pressure. Crack extensometers, in-place inclinometers, and piezometers are vibrating-wire (VW) type sensors, and have accuracies reported by the manufacturer of 150 μm, 90 μm, and 3.5 kPa, respectively. These values refer to the accuracy of absolute readings. For our purposes, however, temporal changes relative to the initial value at installation are of interest (i.e., the time series), which have better accuracies than absolute recordings. These can be deduced from the noise level of the recorded time series, and are better than 90 μm (crack extensometers), 50 μm (in-place inclinometer), and 0.2 kPa (piezometers). This includes random error as well as systematic daily errors likely arising from unbalanced voltage changes of the solar-power system. The VW sensors also include embedded temperature sensors for thermal correction. Thermal drifts for the instruments are 40 μm/°C, 32 μm/°C, and 0.35 kPa/°C, for crack extensometers, in-place inclinometers, and piezometers, respectively. We applied temperature correction only to surface crack extensometer data, as temperatures recorded by all VW borehole sensors varied by less than 0.5°C. Temperatures recorded by these embedded sensors provide an approximate record of air temperature around the instruments both at the ground surface and in the borehole. However, air temperature in the borehole may not always be identical to that of the surrounding rock, since thermal equilibrium between the two is not always ensured.
 A vertical thermocouple array was installed in shallow bedrock in 2008 to obtain accurate information about near-surface thermal conditions. The array consists of nine sensors distributed in a 4 m deep borehole (sensor accuracy = 0.5°C). The spacing between sensors increases from 20 mm at the surface to 2 m at the bottom of the borehole, providing optimal data coverage where temperature changes are strongest. The sensors were embedded in grout that has similar thermal conductivity to the surrounding rock. Relevant meteorological parameters at the site, namely air temperature, barometric pressure, relative humidity, and rainfall, have been recorded since 2008. The rain gauge is not heated, so only provides reliable rainfall data in summer. Note that no snow height sensor was deployed at Randa. As the approximate date of first significant snowmelt, we use the time when piezometric pressure in borehole sb50s increases sharply, acknowledging that snowmelt may begin earlier, lasts for several weeks, and is spatially variable. Examination of data from snow monitoring stations in the region (provided by SLF, 2011) revealed that the pressure increase in sb50s usually occurs within less than ten days of the snowmelt onset observed from snow height data [Alpiger, 2010]. Thus, our measure is a sufficiently accurate indicator for the onset of snowmelt, and also corresponds to the time when snowmelt may affect the instability due to groundwater pressure changes. The date of first lasting snowfall is known either from the temperature measured at crack extensometer Z9 (2004–2008) or from rock temperature data and a webcam installed at the site (2008–2011). Meteorological sensors, thermocouples, and VW sensors are recorded by a Campbell Scientific CR10X data logger; the sampling rate for all sensors was 60 min before 2008, and 30 min afterwards.
 A second monitoring system based on fiber optic (FO) strain sensors was installed at the site in 2008 [Moore et al., 2010]. This system is able to record measurements with an accuracy of a few microstrain (dislocations of micrometers over a 1 m base-length) at a sampling rate of 100 Hz. Two 0.8 m base-length FO extensometers were installed at the ground surface across the opening tension fractures, co-located with VW extensometers. In borehole sb120, six 1.5 m base-length FO axial extensometers were installed across steeply dipping fractures at three depths to measure the vertical component of dislocation (Figure 2b); the instruments were installed in pairs for redundancy. The FO monitoring system thus provides independent and complementary deformation data to that from the VW sensors.
 The monitoring systems described above provide information only about relative dislocations across active discontinuities. To increase spatial data coverage, a 3D local geodetic network was installed in 2008 and surveyed monthly over the course of one year. This network was tied into the larger, valley-scale geodetic network surveyed since 1995 [Jaboyedoff et al., 2004]. The absolute displacement of reflectors in the local network could thus be determined. Distances and angles within the local network of 34 retro-reflectors were measured with a Leica total-station (type TPS1201, accuracies: ±1″ in angle; ±1 mm + 1.5 ppm in distance). Twenty points were monitored relative to 14 stable points within the network such that redundancy for the monitored points was assured. The maximum base-length within the network was 250 m. In total, 11 surveys were performed at intervals of 30 to 45 days. The network geometry, processing procedure and measurement results are fully described by Gischig et al. [2011a]. In addition to the magnitude and direction of displacement vectors, time series of absolute displacement data were obtained for all reflector points with resolvable movement.
4. Monitoring Data
 Figure 3 presents four years of monitoring data measured prior to 2008. All time series except piezometric pressure were low-pass filtered to remove signals with periods less than 3 days. Doing so, we can avoid high-amplitude daily signals, making it easier to interpret the long-term signals of interest. Figures 3a and 3b show data from the VW extensometers spanning tension fractures Z9 and Z10, together with temperature from their built-in sensors. Measurements show that these two fractures open at an average long-term rate of ∼2 mm/yr. A considerable annual signal can be seen superimposed on the linear trends, with opening rate strongly enhanced in winter and reduced after snowmelt so that even some fracture closure occurs in summer. The amplitude of this signal is stronger at fracture Z10, as the sensor is installed on a small rock face that remains snow-free throughout most of the winter. The temperature record at this sensor shows a nearly sinusoidal annual trend, as opposed to the temperature at fracture Z9, which is steady at 0°C when covered by snow. The sensor at Z9, in contrast to Z10, lies in a small depression that is covered with long-lasting drifted snow (maximum snow thickness: ∼1.5 m). Temperatures drop below zero in autumn, when there is at most only a thin snowpack, and then rebound toward zero when the snowpack is thick enough to insulate the ground from air temperature variations.
 Figure 3c shows horizontal relative dislocation between the ends of the in-place inclinometer at 68 m depth. This is equivalent to the horizontal component of relative dislocation across the monitored fracture. Dislocation increases nearly linearly with time at an average rate of about 1.8 mm/yr. However, a slight annual signal is also superimposed, with rate increases up to ∼2.7 mm/yr in winter and rate decreases to ∼0.8 mm/yr in summer. This variation becomes clearer in the de-trended time series shown in Figure 3d. Periods when the ground temperature is above or below zero, as inferred from the temperature sensor in extensometer Z10, are shown colored red or blue, respectively. The dislocation rate increases sharply as soon as the temperature falls below 0°C and decreases gradually after snowmelt. It should be noted that temperature sensors in the borehole inclinometer show constant temperature throughout the entire measurement period.
 Piezometric pressure at the bottom of boreholes sb50s and sb120 is shown in Figure 3e. Temperatures measured by these sensors are constant at 3.5°C (sb50s) and 4.1°C (sb120). Pressure data from sb50n are not available as the sensor malfunctioned after installation. Pressure in borehole sb120 is nearly constant at a value corresponding to the air pressure at the altitude of the sensor, indicating that the water table is below 120 m depth. However, pressure measurements in borehole sb50s indicate the existence of a groundwater table at ∼3 m above the sensor (i.e., at 47 m depth). The height of the water column increases to 4 m after the onset of snowmelt, then decays slowly back to 3 m over the course of the year. Borehole sb50s is located only ∼50 m from the dry borehole sb120. Therefore, it is likely that water in sb50s is part of a localized, perched groundwater body. The onset of snowmelt, indicated by the piezometer record from sb50s, does not correlate with the onset of increased deformation rates in inclinometer data, nor does it coincide with the onset of decreased rates (Figure 3d). Snowmelt usually begins around April, whereas increased dislocation rates start around November.
 With the addition of new FO sensors in 2008 in borehole sb120, the inclinometer previously at 68 m was moved to an active discontinuity at 12 m depth. Data from this instrument since relocation are shown in Figure 4a, and with the linear trend removed in Figure 4b. For the detrended time series, unfiltered data are also shown in the background indicating the sensor noise level. The vertical temperature profile from the shallow bedrock array during the same time period is presented in Figure 4c. Rock surface temperature, measured at the top of the sensor array, and ambient air temperature are shown in Figure 4d. Temperatures at 4 m depth range between 2.5°C in April and 10°C in September. The ground surface temperature decreases rapidly to 0°C around first snowfall, and then continues decreasing slowly to a minimum value of about −0.5°C in January.
 Contemporary inclinometer and temperature data show a similar relationship as observed for the previous inclinometer record from 68 m depth (compare to Figure 3). Dislocation rates increase when the rock surface rapidly cools at the time of first snowfall and decrease after snowmelt as the rock warms. In falls of 2008 to 2010, the first cooling events correlate with a sudden increase of dislocation rate (see blue arrows in Figure 4b). These are followed by a short-lived return to summer dislocation rates and then gradually increase to higher rates over several weeks. Dislocation rates decrease again as soon as the rock surface begins to warm after snowmelt. A sharp decrease to the dislocation rates prevailing in summer occurs as soon as the snow has disappeared and rock can efficiently warm (see red arrows in Figure 4b). Thus, comparison between rock surface temperature and fracture dislocation measured at 12 m depth shows that deformation rates react to temperature changes not only on an annual basis, but also within shorter time intervals. Piezometric pressure data from sb50s (Figure 4e) reveal the onset of snowmelt as a sharp increase of water pressure. The record of daily precipitation since June 2009 is also shown in Figure 4e. Times of maximum water pressure or heavy rainfall do not correlate with significant acceleration phases in inclinometer data.
 Note that in contrast to the sensor position at 68 m, the built-in temperature sensor at 12 m depth measures a slight annual signal of about 0.5°C amplitude (average temperature: 4.3°C). Calculations attempting to fit temperature data from the 4 m deep rock array using a thermal diffusivity of 1.9 × 10−6 m2/s showed that this amplitude is consistent with thermal wave penetration to 12 m depth. The direct temperature influence on the instrument (32 μm/°C) is too small to create the observed inclinometer signal; this may only account for an annual signal of 16 μm peak-to-peak, while the actual annual variation has a peak-to-peak amplitude of >300 μm.
 Data from FO extensometers at 38, 40, and 68 m depth in sb120 are shown in Figure 5a, and with the linear trend removed in Figure 5b. Note that these sensors measure the vertical component of dislocation across discontinuities (i.e., shortening) as opposed to the inclinometer, which measures the horizontal component. For comparison, the detrended times series from the inclinometer at 12 m depth is also shown (Figure 5c). Although FO and inclinometer sensors measure different components of fracture dislocation at different depths, the signals mimic each other well: changes in dislocation rate occur in a nearly synchronous manner with the maximum dislocation rate during winter and minimum rate in summer. Data from the FO sensor shows several abrupt steps, which do not correlate with signals from the inclinometer (Figures 5b and 5c). The largest step occurs at 40 m depth in fall 2009 and indicates rapid axial extension of about 50 μm. Such transient events interrupting the long-term shortening rate are commonly observed on all FO sensors, however, more frequently in winter when dislocation rates are greater. The origin of these events is still unclear, though they have been interpreted to result from intermittent activation of sliding on nearby discontinuities [Moore et al., 2010]. Such signals may be absent in inclinometer measurements, because the noise level of the inclinometer (<50 μm) does not allow observation of such small transient dislocations.
 Absolute displacement time histories for several reflectors in the local geodetic network between 2008 and 2009 are shown in Figure 6a, and the network geometry is shown in Figure 6b. Only points having a complete set of 11 measurements are presented. Displacement vectors have average trend and plunge angles of 135° and 30°, respectively. Plunge angles range from 10 to 40°, as expected for toppling along discontinuities dipping about 50 to 80° into the slope. The measurement standard error is estimated to be ±2.5 mm. The total displacement of these points accumulated during the measurement period is 8 to 12 mm. Displacements between subsequent measurements are too small to be considered significant at the given error margin. The trends, however, are consistent for all points: most of the observed displacement occurs in winter (7 to 10 mm), while displacements in summer cannot be considered significant. Geodetic measurements thus support observations of an annual deformation trend from inclinometer and FO extensometer monitoring data.
5. Interpretation of Monitoring Data
 In situ monitoring at the Randa rock slope instability has revealed a clear seasonal pattern of changing deformation rates both across active discontinuities at depth and at the surface. An abrupt change to higher rates occurs around the time of first snowfall as the rock rapidly cools, while a decrease in deformation rate occurs after the onset of snowmelt when the rock warms. Significant variations in deformation rate do not correlate with heavy rainfall events or with increased water pressure after snowmelt. Volumetric expansion due to ice formation in fractures is also considered secondary; the water supply to fractures, which is required for sustaining ice formation throughout the winter [Matsuoka and Murton, 2008], is limited due to the low groundwater table and the large fracture aperture. Instead, near-surface rock temperatures appear to control the temporal deformation pattern, as measured by independent monitoring systems. We conclude that the observed temporal deformation trend is not a local effect but represents the behavior of the entire unstable rock mass.
 Deformation signals related to surface temperature changes have been previously reported and related to thermo-elastic [Wyatt et al., 1988; Bonaccorso et al., 1999] or freezing effects [Wegmann and Gudmundsson, 1999; Matsuoka, 2001]. Different authors also discuss the role of temperature changes in driving movements of single blocks [Gunzburger et al., 2005] or deformation of shallow slope instabilities in brittle rock [Krähenbühl, 2004]. Vargas et al.  further suggest that strong temperature changes can induce fracture propagation and lead to rockfall, and that thermal cycling can contribute to fatigue at fracture tips. Few studies report similar effects at large slope instabilities with active discontinuities reaching below the thermal active layer. One notable case is the Checkerboard Creek landslide in Canada, described by Watson et al. , where a seasonal trend comparable to that observed in our monitoring data has been measured. A rather exotic case is also described by Bonaccorso et al.  for a landslide at a volcanic crater rim subject to heating and cooling through changing fumarole activity. Measured displacement patterns were explained by thermo-elastic contraction and expansion of the entire landslide body, while a portion of the downhill displacement was shown to be irreversible.
 In our case at Randa, seasonal opening and closing of surface tension fractures is readily explained by direct thermo-elastic expansion and contraction of near-surface rock blocks. However, seasonal deformation trends at depths where temperatures are essentially constant can only be interpreted in association with induced thermomechanical effects, as investigated with conceptual models in Part 1 of this study [Gischig et al., 2011b]. These simplified models revealed how surface temperature cycles induce strain and stress signals at depths below the thermal active layer due to thermomechanical stress transfer and topography. If a sufficient number of critically stressed discontinuities are present within the rock mass, TM-induced stresses promote slip along discontinuities and lead to failure propagation at slip fronts. We hypothesize that the temporal behavior of the Randa instability can be interpreted within the framework of TM forcing, and here attempt to apply the findings from the conceptual study to the Randa slope instability. Two hypotheses explaining the observed signals are presented:
 1. The annual deformation signal represents strictly the thermo-elastic reaction of the rock mass to near-surface temperature changes in the presence of steep topography. Thermo-elastic induced strains at depth are superimposed on an ambient, underlying displacement trend driven by some other mechanism, such as stress corrosion. Thermo-elastic induced stress changes at depth are too small to cause failure along discontinuities, but create reversible strain signals modifying the deformation time series.
 2. Thermomechanical induced stress changes at depth, although small in amplitude, are sufficient to promote slip along discontinuities and result in a seasonally variable deformation rate. The effect is irreversible and can be regarded as a mechanism driving time-dependent, progressive rock slope failure.
 In the following section we explore both hypotheses through 2D numerical simulations.
6. Numerical Models
 Similar to the conceptual study of TM effects [Gischig et al., 2011b], 2D numerical modeling of the Randa rock slope instability was performed using the discrete-element software UDEC (Universal Distinct Element Code, Version 4.0, Itasca International, Inc., 2008). Both a purely elastic model and one allowing failure along prescribed discontinuities were explored. The models were based on results of a previous study at the Randa instability [Gischig et al., 2011a], which used analysis of structural data and displacement patterns to create a 2D kinematic model implemented in UDEC. Two kinematic failure modes were identified: toppling at higher altitudes and sliding along a basal rupture surface below. Here we explore the reaction of the Randa instability to seasonal thermal forcing by adopting this same kinematic model (including geometry, discontinuity orientations, material and discontinuity properties, and boundary conditions) in a coupled TM mode. The model geometry is presented in Figure 7a. As with the conceptual models, zero-displacement boundary conditions were applied at the base. For the side boundaries, zero-displacement conditions were used for horizontal movements, while vertical displacements were allowed (i.e., roller boundaries). Thus, shear stresses parallel to the side boundaries remain zero throughout. The mesh size was set to 3 m for the uppermost 20 m of the model, 8 m in deeper portions, and 50 m farther from the region of interest. Implemented rock mass material properties are presented in Table 1 and Table 2, and are identical to those used in the previous kinematic study [Gischig et al., 2011a]. Intact rock properties were derived from laboratory tests [Willenberg, 2004], while rock mass properties were estimated using the Geological Strength Index (GSI) approach [Hoek et al., 2002]. A Mohr-Coloumb failure criterion including slip-weakening was assigned to discontinuities. Peak strength values were derived by Gischig et al. [2011a] in an attempt to reproduce the observed kinematic behavior at Randa (Table 2). All discontinuities sets were represented in the model as fully persistent, although limited persistence due to intact rock bridges was assumed from structural analyses. Intact rock bridges or varying in situ discontinuity roughness are implicitly modeled by assigning higher apparent peak cohesion and friction angles to the fully persistent discontinuities. Thus, different peak friction angles, cohesion and tensile strength values were chosen for each of the four sets. Note that these are not based on direct field measurements but rather on modeling back-calculation fitting the observed deformation patterns. Thermal diffusivity and the coefficient of thermal expansion were kept the same as in previous conceptual models, derived from analysis of in situ rock temperatures and literature values [Clauser and Huegens, 1995; Gueguen and Palciauskas, 1994]. A sinusoidal temperature time history with one year period and 20°C peak-to-peak amplitude was applied at the ground surface, as estimated from measured temperature data.
|Young's modulus [GPa]||32||21|
|Young's modulus [GPa]||26||14|
|Thermal diffusivity [m2/s]||1.9e-6||1.9e-6|
|Thermal expansion [1/K]||8e-6||8e-6|
|Peak friction angle||46°||31.0°||31.7°||33.6°||35.3°|
|Peak cohesion (MPa)||8.3||1.2||2.1||4.4||6.5|
|Peak tensile strength (MPa)||5.2||0.6||1.0||2.1||3.1|
|Residual friction angle||27°||27°||27°||27°||27°|
|Residual cohesion (MPa)||0.05||0.02||0.02||0.03||0.03|
|Residual tensile strength (MPa)||0||0||0||0||0|
|Joint normal stiffness (GPa/m)||10||10||10||10||10|
|Joint shear stiffness (GPa/m)||5||5||5||5||5|
6.1. Purely Elastic Model
 Figure 7b shows the peak-to-peak amplitude of vertical tilt (∂uy/∂z) induced by shallow thermo-elastic stress changes. Tilt amplitudes are greatest in the thermal active layer (i.e., the depth where temperature changes are less than about 0.1°, here ∼20 m). However, annual tilt variations also occur in regions of constant temperature behind the steep slope face and behind breaks in topography (up to 10 μrad). Associated with these strain variations, stress changes are induced at depths of constant temperature. Figure 7c presents profiles of the peak-to-peak amplitude of thermo-elastic induced vertical tilt (∂uy/∂z) and vertical strain (ezz) predicted by the elastic model at the location of borehole sb120. Also shown are the peak-to-peak amplitudes of horizontal tilt and vertical strain signals measured with in-place inclinometers and FO strain sensors in the same borehole. We find that the predicted thermo-elastic strains and tilts are significantly smaller than the measured signals. Various effects may be included in the model to enhance the magnitude of thermo-elastic strains and tilts at depth, such as lateral temperature variations, material heterogeneities, and 3D topography. However, it is unlikely that including these effects will significantly reduce the observed discrepancy, since the model predictions differ from measured values by several orders of magnitude.
6.2. Discontinuum Model
 The implemented discontinuity distribution is shown in Figure 8a, which was based on geological observations depicted in Figure 1b. Absolute block displacements accumulated over 10 years of thermal cycling are also shown in Figure 8a. The net absolute displacement at the top of the unstable area is ∼100 mm, while it is only ∼5 mm at the toe of the instability. Discontinuities that have reached their strength limit solely due to gravitational stress are shown as black lines in Figure 8b, whereas those that failed during the subsequent 10 years of TM cycling are indicated as orange lines. Several additional discontinuities have failed. A number of monitoring locations were chosen in the model to visualize the temporal behavior. These include a tension fracture at the back of the instability (C1), discontinuities at depth that show either toppling or sliding dislocation (D1–3, B1–2, respectively), as well as two points on the ground surface (S1–2).
 The annual temperature cycle applied at the ground surface is shown in Figure 9a, while Figure 9b presents the modeled horizontal relative dislocation (analogous to the measured dislocation in Figure 4a) induced across three neighboring, but different, toppling discontinuities: D1, D2, and D3 at 14, 48, and 68 m depth, respectively (Figure 8b) (left-lateral shear sense is positive). At all depths, a clear annual trend is visible that is controlled by slip along discontinuities (i.e., not only due to discontinuity compliance). Slip (i.e., irreversible shear dislocation) is induced when the stress state along the discontinuity is modified such that it meets the failure criterion. Stress changes that do not result in slip only create compliant (reversible) dislocations. Details on the shear and normal stress paths along discontinuities are not explicitly shown here, since this topic is discussed in detail in the companion paper. There, shear dislocation time series along with corresponding stress paths are shown and interpreted for discontinuities in simplified scenarios. In the present study, at a depth of 14 m, the dislocation rate changes from negative to positive after the surface temperature has reached a minimum and begins to warm (Figure 9b). At 68 m depth, the opposite is observed with an abrupt change to positive dislocation rate when the surface temperature is at a maximum and begins to cool. At 48 m depth, the greatest dislocation rates occur predominantly during the negative cycle of surface temperature. TM-induced relative dislocations also show consistent long-term rates ranging from 0.2 to 0.8 mm/yr. Although these rates are lower than the 1 to 2 mm/yr values measured across active discontinuities (Figure 4a), they are of a similar order. Shear dislocation at points B1 and B2 on the basal sliding surface are shown in Figure 9c (right-lateral shear sense is negative). An annual variation in dislocation is present at both locations, superimposed on long-term rates of −0.1 and −0.25 mm/yr at B2 and B1, respectively. For the annual signal, displacement increases and decreases at relatively constant rates with rapid reversals at times when temperatures change from T > 0°C to T < 0°C, and vice versa. Thus, the greatest dislocation rates (i.e., most negative) along the basal sliding surface occur predominantly in summer. This is contradictory to the measured and modeled dislocation trend for discontinuities at the top of the instability. Note that there are no field measurements available from the basal sliding region that could verify this model result.
 Simulated fracture opening at the ground surface across the discontinuity that delineates the back of the unstable area (C1) is shown in Figure 9d. The fracture opens at an average rate of ∼1.1 mm/yr and exhibits a strong annual signal with amplitude of 1.4 mm, both of which are similar to the measured values at tension fractures Z9 and Z10 (see Figure 3a). The phase of the predicted annual signal also matches observations with closure during summer and opening in winter. Time histories of horizontal and vertical absolute displacements predicted by the model at two points on the ground surface (S1, S2) are presented in Figure 9e. Long-term displacement rates are consistent at ∼6 mm/yr, which is lower than the 14 mm/yr measured at the top of the instability (the absolute maximum rate of 30 mm/yr is measured at the headscarp). However, these values are still within the same order of magnitude, and also well within the error expected for such simulation. The surface monitoring points all show a similar annual signal: vertical displacement rates increase at the beginning of winter, while horizontal displacement rates gradually increase toward the middle of winter. The ratio of vertical to horizontal displacements reveals a plunge angle of about 27°, which matches geodetic measurements.
 Figure 10 shows comparison of measured and modeled annual signals both at the ground surface and at depth. All model results were shifted in time by a constant value to provide the best fit between observed and modeled surface temperatures (Figure 10a). This shift was performed to transfer model time, which is relative to an arbitrary zero reference (i.e., when the time-dependent surface boundary conditions were initiated), to a meaningful time scale corresponding to the observation period. Deformation time series were detrended to highlight annual variations. The measured and modeled amplitudes of surface fracture opening match one another well (Figure 10b). The measurements, however, lag behind the modeled signal by about 40 days. At depth, the predicted amplitudes of TM-induced signals are of the same order as measured values (Figure 10c). Recall that this was not the case in the purely elastic model, where amplitudes were 1 to 2 orders of magnitude less than the measured values (Figure 7b). Times when displacement rates change in the model vary with depth, and differ by –120 to +40 days from the observed changes. The modeled displacement trend at a surface point matches reasonably well with geodetic measurements (Figure 10d), although with a time lag of the rate change of +90 days. Modeled absolute displacement rates are predominantly higher in winter as observed in all geodetic monitoring data.
6.3. Discussion of Numerical Models
 Results of discontinuum numerical modeling demonstrate the feasibility of TM effects as a driving mechanism for slope deformation at the Randa instability. Simulated relative and absolute displacement rates are smaller than the measured values but within the same order of magnitude. Annual variations in displacement rates were induced both at depth and at the ground surface, with modeled amplitudes similar to those measured in the field. Along toppling discontinuities, increased deformation predominantly occurs in early winter and decreases in summer, which is in accordance with monitoring data. However, predicted times of displacement rate changes did not fully match field measurements, which showed a more synchronous change at various depths within the toppling rock mass. Model results, on the other hand, exhibited spatial variability in displacement onset times. Furthermore, modeled times of increased dislocation along the basal sliding surface were found to be opposite of those at the top of the instability. The onset of slip is determined by the time when the stress path touches the failure envelope, which depends on the orientation of discontinuities with respect to the ambient stress field. At the basal sliding surface, this occurs in summer, while for the toppling discontinuities slip initiates in early winter. Thus, if TM cycling drives rock slope failure, times of increased displacement rate are not strictly limited to winter, but can also occur in summer. The model shows that times of displacement rates changes vary spatially and as a function of complex local kinematics.
 Assumptions made in the 2D model of the Randa instability include purely conductive heat transfer, isotropic and homogenous mechanical and thermal properties, spatially uniform surface temperature, as well as a simplified applied surface temperature forcing. Deviations from these assumptions may alter the presented results. The surface temperature distribution at Randa is certainly heterogeneous, controlled by varying insolation due to the changes in aspect and altitude, as well as non-uniform snow cover in winter. While the top of the instability is covered by snow throughout the winter, the steep failure surface is largely snow free and experiences annual temperature variations different from snow covered areas. As shown by Berger , lateral temperature variations can also create thermo-elastic strains at depth. Thus, the spatially homogeneous temperature forcing used in our models may underestimate true thermo-elastic strains. Furthermore, the real temperature field at Randa appears to be influenced by advective disturbances, such as temporary water infiltration and air ventilation in deep fractures [Moore et al., 2011; Weeks, 2001]. Given the large void space within the fractured, unstable rock mass (up to 17% predicted by Heincke et al. ), such processes are likely to be abundant at the instability. Advective thermal disturbances generally have a cooling effect at depth. The resulting contraction along discontinuity surfaces can result in reduction of normal stress and enhancement of TM-induced slip. Furthermore, the Randa slope has a steep 3D topography. Since TM effects depend on the presence of topography, some error due to our 2D approximation is expected. Anisotropy and heterogeneity of rock properties, in addition to varying surface temperature, are also expected to contribute additional heterogeneity to the TM-induced stress field [Harrison, 1976]. Hence, we believe that simplifications made in our model and described above generally tend to underestimate the effect of TM forcing. One additional factor not considered in our models, and which may reduce TM-induced strains at depth, is a contrast in compliance between the uppermost meters of the rock mass (where temperatures fluctuate) and the underlying medium. Fracture compliance decreases with normal stress [Zangerl et al., 2008; Goodman et al., 1968], and higher compliance can be expected at shallow depths where normal stresses are low and the rock is more weathered. A highly compliant layer of rock in the near-surface would tend to attenuate strains transmitted to depth.
 As shown with conceptual models [Gischig et al., 2011b], net TM effects depend strongly on rock mass mechanical properties, such as discontinuity strength and material stiffness. Strength influences the amount of critically stressed discontinuities within the model ready to slip under small stress changes. Elastic properties determine the efficiency of stress transfer to depth. However, changes in these parameters influence not only TM effects, but also alter the modeled kinematic behavior of the instability [Sjöberg, 2000]. In our models of the Randa instability, kinematics involves both sliding and toppling, but the detailed kinematics is complex and also depends on a number of model parameters. Thus, the sensitivity of TM effects to stiffness and strength is strongly nonlinear and difficult to quantify. However, results from conceptual models also apply qualitatively to the Randa-specific models. Increasing the elastic modulus (i.e., increasing model stiffness) enhances TM effects, while reducing discontinuity strength produces more critically stressed discontinuities within the model able to react on TM-induced stress changes.
 Through both analysis of monitoring data from the Randa rock slope instability, as well as conceptual and site-specific numerical models, we have demonstrated that deep-seated, irreversible rock slope deformation can be driven by TM effects. TM forcing of progressive slope failure is promoted by steep topography, stiff rock mass elastic properties, and the presence of critically stressed discontinuities. Reversible thermo-elastic strains extending below the thermal active layer occur wherever competent rock extends to the surface of a slope [Harrison and Herbst, 1977]. However, irreversible strains can be induced when the slope is already in a near-critical state, i.e., a large number of discontinuities are critically stressed. At the Randa instability this is the case, since the rock mass is currently unstable and is therefore expected to contain a large number of critically stressed discontinuities susceptible to small TM-induced stress changes.
 Coupled TM modeling of the Randa study site, with slip allowed along discontinuities, was able to reproduce the order of magnitude of measured displacement rates and the amplitude of seasonal variations. However, some aspects of the observed temporal displacement trend could not be reproduced. The measured displacement time series at the ground surface and different depths revealed spatially synchronous changes in the onset of increased displacement rates at all monitoring points (Figures 5b and 5c). The modeled displacement time series, however, showed substantial phase shift between different locations (Figure 10). Such spatial variation in onset times may reflect model complexity introduced by the composite kinematic behavior. Since monitoring data show no similar phase shift (Figure 5), we assume that additional processes (disregarded in our models) may occur at Randa, which help synchronize the temporal instability behavior. One candidate may be air ventilation in open tension fractures [Moore et al., 2011]. In contrast to purely conductive heat transfer assumed in our model, air convection through open fractures is more efficient at communicating surface temperature changes to depth. As soon as the ambient air temperature decreases below the mean rock temperature at depth and crack convection begins, relatively rapid cooling can decrease normal stress at interlocking asperities along fractures and initiate TM effects. Field observations of warm air vents in the snowpack confirm winter-time convective air circulation, and are described in detail by [Moore et al., 2011]. Enhanced TM effects through air ventilation may also partly explain why modeled displacement rates are about half of the observed values. However, neglecting lateral temperature variations through heterogeneous insolation and snow cover, together with natural material heterogeneity, may also lead to discrepancies between the modeled and measured displacement rates and trends.
 Incremental damage accumulation induced by cyclic loading (in the present case through temperature changes) resembles the process of cyclic fatigue observed in laboratory experiments and described by many authors [e.g., Attewell and Farmer, 1973; Brown and Hudson, 1973; Scholz and Koczynski, 1979]. Similar to the laboratory scale, this meso-scale fatigue process involves irreversible slip on pre-existing discontinuities, as well as propagation of fractures by cycling loading. In laboratory stress cycling experiments, the maximum applied stress determines whether a sample reaches quasi-static equilibrium, in which it can bear a theoretically infinite number of cycles, or ruptures after a certain number of cycles [Attewell and Farmer, 1973]. Micro-cracks in the sample propagate slowly or suddenly depending on whether the peak stress intensity factor at crack tips (a function of the maximum applied stress) approaches or exceeds a critical value. Analogous in a rock mass subject to cyclic loading, the degree of criticality dictates whether the rock mass becomes stable in a quasi-static equilibrium, or if it approaches a limit as damage accumulates after which it will fail catastrophically.
 Similar to fatigue acting on pre-existing discontinuities, intact rock between fracture tips and at interlocking asperities can be affected by micro-scale damage through stress cycling. The simplified constitutive laws for discontinuities used in our models dictate that slip occurs only when stress exceeds the specified strength limit. At the micro-scale, however, brittle rock does not only accumulate irreversible damage at rupture, but also at stresses far below the ultimate strength [Bieniawski, 1967]. Micro-cracking is commonly thought to initiate at stress levels between 30 and 50% of the unconfined compressive strength [Brace et al., 1966], while coalescence of micro-cracks occurs at about 70 to 90% of the rupture strength [Bieniawski, 1967]. Cyclic loading of intact rock at stress levels above the limit required for crack coalescence (e.g., through TM forcing) leads to accumulation of microscopic damage, and to progressive strength degradation [Attewell and Farmer, 1973]. Thus, micro-scale cyclic fatigue can contribute to progressive failure of intact rock bridges and interlocking asperities, as also suggested by Vargas et al. . However, at the stress amplitudes and annual period typical for TM effects investigated here (<100 kPa below the thermal active layer), stress corrosion (sometimes referred to as static fatigue) may be equally important and also contribute to progressive failure [Scholz and Koczynski, 1979].
 In most case studies of deep-seated rock slope instabilities reported in literature, water pressure changes due to heavy rainfall or snowmelt are suggested to influence deformation rates to a greater degree than TM effects, and water pressure variations are now routinely included in slope stability analyses [Terzaghi, 1943; Wyllie and Mah, 2004]. TM-induced stress changes at depths below the thermal active layer at Randa are estimated to be less than 100 kPa for both shear and normal stress, which is similar to water pressure changes of ∼10 m head. Given the evidence that TM effects play a predominant role at the Randa instability, the question arises as to why water pressure changes do not affect deformation rates as reported at other sites. Pressure data in Figure 3 show that the continuous groundwater table is below 120 m depth. Pockets of perched groundwater exist (borehole sb50s), and show maximum pressure changes of about 15 kPa, which are in the range of stress changes expected by TM effects. Water seepage at the 1991 failure surface observed from field visits and time-lapse photography only occurs within the orthogneiss, while no springs exist in the overlying paragneiss and schists [Alpiger, 2010]. Further, it is improbable that the water table is elevated in winter due to ice formation preventing free drainage at the cliff face, since substantial ice formation on the failure surface has not been observed. It is likely that the entire unstable rock mass lies above the continuous groundwater table throughout the year, and is thus outside its range of influence. Such groundwater conditions are a consequence of the highly fractured rock mass and ridge topography. Water from rainfall and snowmelt drains either as surface run-off or infiltrates rapidly to depths below the unstable volume. We suggest that effects of water pressure are secondary due to the absence of a continuous groundwater table within the unstable rock mass. In contrast to many other slope instabilities, TM effects at Randa are not masked by a more dominant water pressure influence. Similar hydrogeological conditions may prevail at similar sites in strongly fractured slopes, or arid climates. Groundwater conditions within instabilities may also change as damage within the rock mass accumulates. Progressive opening of fractures increases hydraulic conductivity of the rock mass and results in lowering of the water table. In consequence, water pressure effects decrease until the rock mass is fully drained [e.g., Amann et al., 2006]. Thus, for instabilities initially controlled by water pressure, TM effects may become increasingly important as they approach an advanced stage of progressive failure.
8. Summary and Conclusion
 Through combined analysis of simplified numerical models [Gischig et al., 2011b], deformation monitoring data from the current Randa rock slope instability, as well as complex site-specific numerical models, we demonstrate the role of TM effects as an important driving mechanism of deep rock slope deformation. The key outcomes of our study are summarized as follows:
 1. Cyclic thermal expansion and contraction in the shallow subsurface creates stress changes at greater depths (>100 m), which can induce slip along discontinuities if these are already close to failure. Subsequent stress redistribution increases stresses at discontinuity slip fronts and can lead to slip front propagation. Thus, TM-induced damage is not limited to shallow bedrock subject to seasonal temperature variations (and the attendant thermo-elastic response), but can penetrate to depths well below the thermal active layer.
 2. Stress propagation to depth is dependent on the elastic properties of the rock mass and TM effects become greater in more competent rock. Furthermore, the net TM effect depends on the amount of critically stressed discontinuities within the rock mass, which are sensitive to small stress changes. The more discontinuities close to failure (at constant rock mass stiffness), the stronger TM effects become.
 3. TM effects depend strongly on rock slope kinematics. Discontinuity orientation and relative location within the rock slope control both the efficiency and onset time of slip induced by TM stress changes.
 4. TM effects are enhanced when discontinuity constitutive behavior includes slip-weakening. Stress redistribution after peak-strength failure alters the stress state of surrounding critically stressed discontinuities, making them more sensitive to TM effects. Including slip-weakening also enhances propagation of discontinuity slip fronts.
 5. At the current Randa slope instability, deformation time series from depths of 12, 38, 40, and 68 m reveal a seasonal deformation trend of increased displacement rates in winter and slower rates in summer. Contrary to most rock slope instabilities controlled by changing pore water pressures, phases of increased displacement rates do not correlate with snowmelt or heavy rainfall, but rather with bedrock cooling at the onset of winter.
 6. Numerical modeling of the Randa instability showed that TM effects are a feasible driving mechanism of slope deformation. Modeled deformation rates match measured values within an order of magnitude, and the amplitudes of annual signals at depth were reproduced well. However, synchronous displacement rate changes observed in field data could only be partially reproduced.
 7. Due to steep ridge topography and the highly fractured nature of the rock mass, the groundwater table is low at the Randa instability. We suggest that TM effects are observable in part because the role of water is minor; TM effects at Randa are not masked by potentially stronger groundwater effects.
 8. The net TM effect can be interpreted as a meso-scale fatigue process within the rock mass, which involves incremental slip along critically stressed discontinuities, as well as micro-scale fatigue in intact rock driven by periodic TM loading.
 Geotechnical and borehole monitoring equipment were installed by Stump Foratec AG, who also performed inclinometer and extensometer surveys. Local geodetic measurements were conducted by the group for Geodetic Metrology and Engineering Geodesy at ETH Zurich (H. Ingensand). Thanks to Freddy Xavier Yugsi Molina for preparing orthoimages for map views in ArcGIS and to Kerry Leith and Reto Seifert for developing and installing the time-lapse camera at Randa. Special thanks also to Andrea Alpiger for her contributions on groundwater during her Bachelor's thesis. We also thank two anonymous reviewers and Erik Eberhardt for careful reviews that led to an improvement of the manuscript. This project was funded by the Swiss National Science Foundation (project 200020-112073).