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# Detection of rock bridges by infrared thermal imaging and modeling

## Abstract

Characterization of rock discontinuities and rock bridges is required to define stability conditions of fractured rock masses in both natural and engineered environments. Although remote sensing methods for mapping discontinuities have improved in recent years, remote detection of intact rock bridges on cliff faces remains challenging, with their existence typically confirmed only after failure. In steep exfoliating cliffs, such as El Capitan in Yosemite Valley (California, USA), rockfalls mainly occur along cliff-parallel exfoliation joints, with rock bridges playing a key role in the stability of partially detached exfoliation sheets. We employed infrared thermal imaging (i.e., thermography) as a new means of detecting intact rock bridges prior to failure. An infrared thermal panorama of El Capitan revealed cold thermal signatures for the surfaces of two granitic exfoliation sheets, consistent with the expectation that air circulation cools the back of the partially detached sheets. However, we also noted small areas of warm thermal anomalies on these same sheets, even during periods of nocturnal rock cooling. Rock attachment via rock bridges is the likely cause for the warm anomalies in the thermal data. 2-D model simulations of the thermal behavior of one of  the monitored sheets reproduce the observed anomalies and explain the temperature differences detected in the rock bridge area. Based on combined thermal and ground-based lidar imaging, and using geometric and rock fracture mechanics analysis, we are able to quantify the stability of both sheets. Our analysis demonstrates that thermography can remotely detect intact rock bridges and thereby greatly improve rockfall hazard assessment.

## Introduction

Erosion processes in rock masses depend on properties of the intact rock and the characteristics of brittle structures1 such as joints and other discontinuities. In steep bedrock landscapes, rockfalls are a dominant process that modulate overall erosion rates2,3,4,5,6,7,8. In some terrains (particularly granitic), rockfalls are often manifest by detachments of rock flakes9,10,11,12,13,14 (exfoliation sheets) that form along surface-parallel fractures known as sheeting joints15,16,17,18. Exfoliation sheets are typically connected to the rock by so-called “rock bridges”, areas of intact (unfractured) rock where sheeting joints have yet to propagate. Sheeting joints develop close to the surface, usually at less than 30 m depth17 and are characterized by extensive fractures whose surface trace length (known also as persistence) can exceed 100 m18,19.

Sheeting joint persistence and rock bridge area both directly influence cliff stability20,21,22. For example, the presence of rock bridges over as little as just a few percent of the detachment surface can significantly increase the factor of safety by increasing apparent cohesion of a potentially unstable rock mass23,24,25,26. In addition, recent work27,28 has begun to explore the contribution of rock bridges to defining various failure modes for rockfalls and rock slides. These studies clearly indicate that the spatial distribution of rock bridges is critical to understanding the stability of jointed rock masses.

The importance of rock bridges and step-path geometries for slope stability analysis has been recognized for several decades29,30,31,32,33,34,35. The contribution of rock bridges has been implemented in numerical models of rock slope stability using apparent cohesion36,37,38 or areas of intact rock24,39,40,41,42,43. In addition, techniques for mapping rock bridges on images and/or textured 3-D point clouds have substantially improved in recent years24,34,35,42,44,45. Nevertheless, these post-rockfall mapping methods are subject to exposure biases46,47 which depend on the accessibility and/or visibility of the outcrop. For example, the spatial resolution of the image and/or 3-D point cloud, the completeness of the 3-D point cloud (presence of shadow areas), as well as the quality of color contrasts used to distinguish fresh and pre-existing weathered fractures limit the ability of observers to provide reliable rock bridge extent percentages. More fundamentally, these studies seek to identify former rock bridges, which are only exposed after rock detachment has occurred, whereas evaluating the contribution of rock bridges to rockfall hazard requires detection of rock bridges prior to detachment. Aware of these limitations, one recent study26 proposed to work directly before a rockfall event to map intact rock bridges by combining observations from remote sensing and field mapping investigations. Based on the Digital Rock Mass Rating48 (DRMR), these authors made recommendations to improve digital mapping of discontinuity traces and defined new factors to quantify the intact rock bridge trace intensity49,50 R21. However, their modified DRMR only represents rock bridges in the form of discontinuous traces that delimit “rock mass bridges” and, so far, remote detection of an actual intact rock bridge has not yet been performed.

To address this need, we employed InfraRed Thermography (IRT). IRT has been used successfully to detect loose rock sections (represented by cooler thermal anomalies) in the fields of civil engineering51,52 and mine safety53,54,55,56,57, with increasing applications in earth sciences58,59,60. In landslide science, IRT has been coupled with Terrestrial Laser Scanning (TLS) to obtain information on the degree of fracturing of a rock mass61. Compact rocks have a greater thermal inertia than fractured rocks, therefore fractured rocks should experience faster temperature variations. Consequently, by repeating IRT surveys during cooling and heating phases, it is possible to highlight thermal anomalies, which are strongly controlled by the degree of fracturing and weathering of the rock. This approach was applied to rock wall surfaces by several authors59,62,63,64,65 for mapping open fractures and specific features such as rock cavities, seepage, highly fractured zones and weathered zones. To optimize the thermal mapping of fractured areas for stability analysis, one study66 developed a MATLAB toolbox (THIMRAN). However, even though IRT is a fast, efficient and easily reproducible measurement technique, this study pointed out that IRT results provide only 2-D images, whereas a correct interpretation of thermal contrasts would require a 3-D model.

Due to the influence of direct sun radiation, daytime thermograms are affected by cold thermal anomalies in areas partially shaded from the sun. Cross-validation with 3-D data thus enables detection of anomalies generated solely by the surface geometry. Given these diurnal artifacts, the best IRT acquisitions are obtained during the night59. Other authors67 successfully classified natural elements along a rock slope (vegetation, debris, talus, intact rock, weathered and fractured rock) according to their thermal signature in early night. The thermal classification then helps to better delineate hazardous areas subject to rockfalls. By combining IRT with TLS and ground-based InSAR, one recent study68 characterized the stability of rock cliffs that are difficult or dangerous to access. The associated rockfall susceptibility map was based on extremely high surface temperature anomalies, structural analysis, and measurement of displacements supported by geomechanical analyses68. However, it should be noted that the influence of emissivity is rarely highlighted in most of these studies, even though it can be the main source of a difference in temperature between two rock types or between intact and oxidized surfaces69,70. An accurate estimate of the emissivity of the different studied surfaces is therefore needed to correctly interpret thermal contrasts.

Here we demonstrate the potential of IRT to remotely detect intact rock bridges. We report the results of infrared thermal monitoring, calibrated in temperatures and corrected for emissivity effect, performed over four hours on a 1000-m-tall cliff: the southeast face of El Capitan in Yosemite Valley (California, USA). We interpret the observation of warmer thermal anomalies at the surface of two well-delimited exfoliation sheets (Boot Flake and Texas Flake) as signs of the presence of rock bridges. This interpretation is supported by the results of 2-D thermal modeling that reproduces the Boot Flake’s thermal behavior. We derive an estimate of the size of the potential rock bridges by draping the IRT images on the TLS topographic data, which we then used to evaluate the stability conditions of both monitored exfoliation sheets.

## Study Site

Our study focused on the vertical-to-overhanging granitic rock wall of El Capitan, located in Yosemite Valley, California, USA, within the central Sierra Nevada (Fig. 1a,b). Over the past 150 years, rockfalls have been intermittently documented9 from the southeast face of El Capitan, with 45 rockfalls recorded6,71 between 1857 and 2017. Exfoliation sheets are characteristic of the Yosemite landscape72,73, and are ubiquitous on El Capitan. Boot Flake and Texas Flake are particularly impressive exfoliation sheets (Fig. 1d,e) and are famous features given their location along the popular rock climbing route “The Nose”.

The cliff face of El Capitan was initially exposed by glacial erosion (and later, by rockfalls72,74) into late-Cretaceous plutons of the Yosemite Valley Intrusive Suite that is part of the greater Sierra Nevada Batholith75,76. These plutons are mainly composed of granite and granodiorite, with lesser quantities of diorite, tonalite, aplite and pegmatite73,75,77,78,79,80. Two main geologic units form the majority of the southeast face of El Capitan (Fig. 1c): El Capitan Granite (Kec, 106.1 ± 0.1 Ma) and Taft Granite (Kt, 106.6 ± 0.7 Ma)76,77,79,80,81. These rocks were intruded by the diorite of North America (Kd, 104.0 ± 0.4 Ma) and Leaning Tower Granite (Klt, 104.1 ± 0.1 Ma)76,77,79,80,81. Other geologic units forming the southeast face of El Capitan include the tonalite of the Gray Bands (Kgb) on the upper part of the southeast face (Fig. 1c), as well as two types of dikes (aplite dikes (Kap) and dikes of the Oceans (Kdo) along with some pegmatite76,77,79,80,81. The exfoliation sheets forming Boot Flake and Texas Flake, which are the focus of our study, consist of Kec with some thin intrusions of Kap and Kdo (Fig. 1d). These two sub-vertical flakes are relatively flat and have almost the same spatial orientation (mean dip direction: 153°; mean dip angle: 81°) on the cliff. The respective dimensions of Boot Flake and Texas Flake (measured on TLS data) are: maximum length: 17.4 m and 26.4 m; maximum width: 5.1 m and 11.3 m; mean width: 3.9 m and 7.8 m; mean thickness: 25 cm and 72 cm; and mean aperture: 8 cm and 33 cm.

## Materials and Methods

Our research focused on using thermal imaging technology to identify warm (attached) and cool (detached) parts of exfoliation sheets to quantify rock bridge area for quantitative rockfall stability analyses. Here, we outline our methods of thermal panorama image data acquisition. We subsequently used numerical modeling to calibrate and confirm our image-based findings.

### Thermal imaging

#### Data acquisition and technical specifications of the thermal camera

We acquired thermal images (thermograms) of the southeast face of El Capitan using a FLIR T-660 thermal imager (IRT camera) mounted on a GigaPan82 EPIC Pro robotic device (Fig. 2a). We collected data during the evening of 8 October 2015 beginning at 17:45 Pacific Standard Time (PST); October is one period in which maximum thermal changes to rock faces have been documented to occur in previous studies of rock exfoliation in Yosemite12. To increase the resolution of the data in specific areas of interest (Boot Flake and Texas Flake), we monitored these areas (one thermogram every 20 minutes) between 18:20 and 21:40 PST using a fixed tripod for the IRT camera. The FLIR camera has an infrared resolution of 640 × 480 pixels and is equipped with a 5-million-pixel digital RGB camera. The camera possesses a field of view of 25° × 19°, a focal length of 25 mm, an infrared spectral range of 7.5–14 µm, and a temperature measurement range of −40 °C to +2,000 °C with an accuracy83 of ±1 °C. The thermal sensitivity (or thermal resolution) is quantitatively defined by the Noise Equivalent Temperature Difference (NETD)70,84, which represents “the difference between the temperature of an observed object and the ambient air temperature that generates a signal level equal to the noise level85. The lower the NETD, the more the camera is able to distinguish small thermal contrasts. For the FLIR T-660 imager, the NETD is less83 than 0.02 °C at 30 °C.

#### Thermographic calibration protocol

The temperature of a target object measured by an IRT camera depends on its emissivity and is affected by direct radiation reflected by the object (Supplementary Fig. 1). Emissivity (which varies between 0 and 1) is defined by the ratio between the radiant energy emitted by the surface of the measured object and that emitted by a perfect blackbody under the same conditions (temperature, direction of observation, and wavelength)86,87,88. For low-emissivity materials, it is essential to accurately measure this value because small variations in emissivity can lead to large variations in the corrected temperatures83,89. The temperature of an object is also influenced by radiation from both the camera and object environment57,83,90 (Supplementary Fig. 1). To correct for these artifacts and obtain temperature values close to the real surface temperatures of the object, it is mandatory to calibrate the apparent temperatures measured by the camera. This is performed by specifying several parameters including the emissivity of the object, the reflected apparent temperature, the ambient air temperature and relative humidity, and the distance between the target and the camera83,89.

#### Determination of calibration parameters

We calibrated the emissivity of El Capitan Granite at the foot of the southeast cliff face from reference temperatures given by four platinum resistance temperature detectors (Pt100 type sensors with accuracy91 of ±0.1 °C at 20 °C) fixed to the rock (Fig. 2b). With the data from these sensors, along with the reflected apparent temperature, ambient air temperature, and relative humidity (presented subsequently), we calibrated the thermal camera to find the reference temperatures on the thermogram, which provided a mean emissivity value of 0.81.

To measure the reflected apparent temperatures from El Capitan Meadow, we fixed reflective paper at the base of the cliff (Fig. 2d) to obtain a reflective signal (i.e., diffuse reflector method83,88,89,92). Although we were positioned more than 1 km from the rock wall (Fig. 2a), the reflective paper was accurately detected with the IRT camera (Fig. 2c). The temperature reading directly provides the value of the reflected apparent temperature (Table 1).

Ambient air temperature and relative humidity affect the transmission of infrared radiation (i.e., the atmospheric transmission factor70,87,88). We measured these parameters using a pocketsize digital thermohydrometer (measurement resolution93 of 0.1 °C and 1% for temperature and relative humidity, with accuracy93 of ±1 °C and ±3% between 30% and 80% relative humidity (±5% outside this range), respectively). To calibrate temperatures at different acquisition times, we measured the reflected apparent temperature, relative humidity and ambient air temperature before each acquisition.

The final parameter needed for calibration of the thermographic data is the distance between the camera and the object of interest. We obtained this data using an Optech ILRIS-LR terrestrial lidar system (TLS) located at 1,037 m distance (as measured between the thermal camera in El Capitan Meadow (Fig. 1a) and the aluminum foil reflector). Our TLS device (which we also used to collect a 3-D point cloud of the southeast face of El Capitan on 7 October 2015) has a manufacturer-specified accuracy94 of 7 mm at a range of 100 m. Our resultant point cloud is composed of about 38 million points with a mean point spacing, equivalent to the spatial resolution, of 15 cm. We georeferenced the point cloud data onto an existing 1 m aerial Digital Elevation Model (DEM) of Yosemite Valley using Iterative Closest Point (ICP) algorithms95,96 implemented in CloudCompare97 software. The georeferenced point cloud allowed us to extract the geometry, spatial orientation, and dimensions of Boot Flake and Texas Flake, located at a distance of 963 m and 938 m, respectively from the IRT camera.

#### Post-processing and registration of thermal images

We generated the final El Capitan thermal panorama by stitching 30 individual thermal images using Kolor98 Autopano Giga software. To correct the shifts of some pixels due to the handling of the camera when specifying updated calibration parameters (i.e., when monitoring Boot Flake and Texas Flake), we exported all IRT images with the same temperature scale and co-registered these using roto-translations. We used the function imregister (part of the Image Processing Toolbox99) to automatically align images of the same size from the Intensity-based automatic image registration algorithm in MATLAB. Thus, each image at ti+1 was aligned on the image at ti so as to avoid abrupt intensity (temperature) changes between two acquisitions. Ultimately, our registration process achieved sub-pixel (<0.75 m at 950 m distance) accuracy.

### Thermal modeling

To demonstrate the link between rock bridge existence and warm thermal anomalies, we performed 2-D thermal modeling using COMSOL100 Multiphysics® software. Here, we present only the modeling of the thermal behavior of the lower part of Boot Flake for brevity.

#### Geometry and physical properties of the material

We determined the geometry of Boot Flake based on the TLS model and a profile photograph101 of the east side of the flake taken by rock climbers. The flake is approximately 17 m length with a total surface area of 75.5 m2. Given the overall symmetry of the flake (the lower west “toe section” notwithstanding), we modeled only the lower half of the flake (with area of 37.8 m2 and length of 8.5 m; Fig. 3a). In accordance with measurements performed on temperature profiles and 3-D “thermo-photorealistic” models (see the third section of the results), a length of 1.4 m was attributed to the rock bridge. We estimated the thickness of the flake to be 25 cm based on relative measurements from the profile picture. From the TLS model, we measured the average distance between the rock wall and the flake surface to be 33 cm (as measured along the vertical profile in Fig. 3b). Thus, we assigned an aperture of 8 cm for the crack. In addition, we assigned a 1.5 m thickness for rock wall in the model in order to generate a sufficiently large thickness contrast with the flake.

The intrinsic properties of the granite (Table 2, and assumed homogenous and isotropic) chosen for 2-D thermal modeling come from the COMSOL Multiphysics® software’s integrated material library. Only the emissivity was added to the basic properties and only the thermal conductivity was adjusted from 2.9 to 2.2 W/(m∙K) in order to obtain lower surface temperatures, and thus closer to the in situ measured values.

#### Heating-cooling cycles applied to the 2-D model

To obtain a realistic temperature field for the Boot Flake model, we applied several daily heating and cooling cycles (twelve in total) to the model, coincident with standard modeling practices e.g.,102,103. The flake surface temperature and temperature depth propagation were calculated at 5-minute intervals. We estimated solar radiation using the methodology of Duffie and Beckman103 and with a declination based on Cooper’s equation104. In mid-October, and for a south-southeast exposure at 1,740 m, the solar radiation is zero between 18:30 and 05:30 PST and reaches its maximum value (725 W/m2) at 12:00 PST (Fig. 3c). We modeled ambient air temperature fluctuations with a pseudo-sinusoidal curve function varying between 4.8 °C (06:30 PST) and 20.8 °C (13:30 PST). These values are based on temperature data from a nearby weather station105 at Yosemite National Park Headquarters, located at Yosemite Village (Fig. 1a). The initial temperature of the entire geometry was set at 10.0 °C at 06:30 PST (t = 0; beginning of the warming cycle) such that it is equal to the constant temperature fixed at the inner limit of the rock wall (Fig. 3a,c). At 06:30 PST, the solar radiation affecting the outer flake surface is 80 W/m2 (Fig. 3c) and for this simulation, we considered the atmospheric pressure as constant and equal106 to 830 hPa at an elevation of 1,700 m.

The geometry defined in Fig. 3a requires imposing thermal insulation conditions on its upper and lower edges to take into account the symmetry of Boot Flake and the extension of the rock wall, respectively. The insulation condition was not applied to the lower end of the flake where it is in contact with air (Fig. 3a). The influence of air is modeled by convective heat fluxes that take into account the ambient air temperature, the inclination angle, and the length of the surface on which the condition applies. In our model, we use two types of convection: (1) external natural convection (in yellow in Fig. 3a); and (2) external forced convection (in orange in Fig. 3a). The first, induced by a gradient of temperature, is applied at the crack level along the outer edge of the rock wall and the lateral edge of the rock bridge. The second, produced by a wind circulation (taken as constant at 3.9 m/s to obtain lower, and more realistic surface temperatures) affects the entire flake and allows more efficient cooling.

## Results

### Thermal signatures of El Capitan geologic units

The IRT panorama of El Capitan (Fig. 4a) acquired on 8 October 2015 at 17:45 PST shows calibrated temperatures ranging between 33.1 °C at the base of the cliff and 15.8 °C at the top. Thus, we observe an increase of 17.3 °C for a 950 m increase in elevation (i.e., +1.8 °C/100 m). Given the average adiabatic lapse rate for a dry atmosphere is around 2.0–3.0 °C for every 300 m increase in elevation107, this temperature range is two to three times higher than expected for neutral surroundings. This variance is due to the different incidence angles of infrared thermal radiation (Supplementary Fig. 1); the top of El Capitan reflects a portion of the colder sky (typical temperature of −60 °C on a thermal camera108) and the base of the cliff reflects a portion of the warmer ground (talus slope, vegetation). Thus, the minimum and maximum temperatures measured on El Capitan are overestimated (i.e., too low and too high, respectively). The coldest temperatures (<19.0 °C) are located in the upper part of cliff and correspond to southeast-facing rock faces that are in shadow beginning in the mid-afternoon in October. The warmest thermal signatures (>30.0 °C) occur at the base of the cliff, directly above southwest-facing talus slope areas that are heated until the late afternoon. For the remainder of the rock wall, the majority of the coldest zones correspond either to overhangs and southeast-facing facets or to lighter-colored rock areas. The data (Fig. 4d through 4g) show that temperatures also vary depending on lithology. For example, Kd (a dark-colored rock; Fig. 4f) displays higher temperature values (by about 3.6 °C) compared to Kec (a light-colored rock; Fig. 4f). However, this temperature deviation should be interpreted with caution because their different color implies different emissivities. By selecting a more appropriate emissivity value to the darker color of the Kd lithology (i.e., 0.95; measured value for the Glacier Point Granodiorite, a rock type outcropping to the east of Yosemite Village which shows similarities (color, structure) with Kd), the average temperature difference between these two rock types is 2.0 °C (Fig. 4e). Thus, the difference in emissivity alone does not explain the thermal contrast highlighted between Kec and Kd; the texture and mineralogical composition of the rock must therefore play a significant role. In summary, the three main factors that explain the observed thermal signatures are: (1) the incidence angle of infrared thermal radiation (related to the viewing angle of the camera); (2) the geometric features of the cliff (overhangs, dihedrals, buttresses, etc.); and (3) the emissivity and color of the different rock types.

The thermographic calibration protocols allow global corrections that mitigate the effect of these artifacts on the thermograms. However, further geometric corrections to the whole panorama are required to obtain a more realistic vertical temperature gradient. Thus, to learn more about the thermal signature of rockfall-prone exfoliation sheets (i.e., at a much smaller scale compared to the entire face of El Capitan), we focused our monitoring on the Boot Flake and Texas Flake area (Fig. 4b,c) where the incidence angle can be considered as constant, and the geometry and rock types show only minor variations. Thus, if thermal anomalies are visible at the scale of several hundred meters, the most interesting ones, in the aim of detecting potential rock bridges, should be coincident with the local anomalies.

### Detection of potential rock bridges for Boot Flake and Texas Flake

The most notable observation from the Boot Flake and Texas Flake thermal data are that the flakes always remain colder than the surrounding cliff areas (Figs 5 and 6) between 17:45 and 21:40 PST (i.e., during a cooling sequence). In addition, they display non-homogeneous temperature patterns: a warm thermal anomaly is visible in the center of Boot Flake (Figs 5d and 6b) and in the lower part of Texas Flake (Figs 5h and 6c). Although the bell-shaped thermal anomaly in Boot Flake becomes less pronounced over time, it remains visible on every temperature profile. The best thermal contrast occurs at 17:45 PST and is better visualized using a temperature scale ranging from 21.0 °C to 24.5 °C. At that time, the coldest part of the lower edge of Boot Flake (blue triangle in Fig. 5b) and the warmest adjacent area outside the flake (yellow area) present a difference of temperature of 3.6 °C. For Texas Flake, the coldest portion is located on its upper edge (Fig. 5f); the temperature difference here is 2.6 °C. At the flake surface, both warmer thermal anomalies indicate a maximum temperature of 23.3 °C (red triangles), which correspond to respective deviations of 2.1 °C and 1.8 °C with the blue triangles of Boot Flake and Texas Flake (Fig. 5d,h).

We favor rock bridges connecting the flakes to the cliff as the source of the warm thermal anomalies because other explanations (e.g., rock type variations, chockstones wedged behind the flakes) do not account for the observations. In terms of location, Boot Flake’s central anomaly coincides approximately with the position of a dark circular dike (Dike of the Oceans) situated at the same height (Fig. 5a). However, this dike only partially crosses the west side of the warm anomaly. The absence of a structure or color change on the east side of the flake favors the presence of a rock bridge in this area (at least on the east side). In the bottom part of Texas Flake, the lack of a dike that might be invoked to account for the thermal anomaly strengthens the rock bridge hypothesis suggested for Boot Flake. Additionally, Fig. 1e shows that Texas Flake must be held to the rock wall at its base by an area that we called “rock attachment” (unlike the rock bridge, this area is not delimited on all sides). Nevertheless, the data suggests that the warm thermal anomaly of Texas Flake defines two parts of potential attachment given the decrease and then increase in warm signal moving up and to the west from the rock attachment area. Thus, to the west of this zone, we assume the existence of a small rock bridge where the temperature profile has been drawn (Fig. 7c −2.0 m2). This assumption is reinforced by the knowledge that the top of this potential rock bridge is 13.9 m below the top of Texas Flake at 1,699.2 m (Supplementary Fig. 2), i.e. well below the position of the uppermost chockstones (8.5 m below the top of Texas Flake) highlighted in Fig. 1f. The chockstones do not appear as thermal anomalies due likely to the cooling that can occur around the chockstones behind the flake - the ventilation is sufficient to prevent a detectable thermal bridge.

The non-homogeneous temperature pattern is explained more rigorously by consideration of the cooling that is possible behind the flakes due to air circulation. In the area of a potential rock bridge, the rock would conduct heat into the flake from the cliff behind. Meanwhile, air circulation in the fracture between the flake and the cliff would result in lower temperatures in the flake away from the rock bridge.

### Dimensions of potential rock bridges

Given our hypothesis that the warm thermal anomalies are where the flakes are still attached to the cliff, we used a combination of thermal imaging and TLS to determine the surface areas of potential rock bridges. By measuring the distance between the greatest slope breaks on the temperature profiles (Fig. 5d,h), we obtained preliminary maximum rock bridge lengths of 2.8 m and 1.9 m for Boot and Texas flakes, respectively. We subsequently used the temperatures (22.1 °C for Boot Flake and 22.6 °C for Texas Flake) associated with the plot position of these slope breaks to determine the minimum temperatures (or temperature thresholds) to be applied to the thermograms to display only a color contrast within warm thermal anomalies. Thus, all portions of the flake with temperatures below the temperature thresholds (i.e., where the rear fracture is open) appear in black on the thermograms (Fig. 5c,g). This procedure allows better delimitation of the external contours (individualization) of the warm thermal anomalies. By draping these “thresholded thermograms” on the TLS data, we estimated the maximum surface area (6.9 m2 and 2.0 m2) of the inferred rock bridges for Boot and Texas flakes, respectively (Fig. 7a to 7c). Remotely identifying the area of potential attachment of rock bridges holding the flake onto the wall while still attached is an achievement never before quantified in this detail or with this methodology.

### Modeled temperature profiles versus measured profiles

At the end of the twelve simulated daily heating and cooling cycles, the results of the 2-D thermal modeling of the lower part of Boot Flake are similar to those of the IRT monitoring, namely that the rock bridge is always warmer than the detached part of the flake during nocturnal rock cooling (Fig. 8a). Although the modeled surface temperatures are, on average, 4 °C higher than those measured (resulting from several potential sources, including the use of generalized, rather site-specific, modeling parameters – Table 2; see also the Discussion section), all the profiles show a bell-shaped anomaly at the rock bridge position (Fig. 8b). This result therefore confirms our hypothesis on the presence of a rock bridge in the center of Boot Flake. However, the thermal modeling also shows that the length of the warm thermal anomaly (2.0 m) is greater than that of the rock bridge (1.4 m) (Fig. 8b). Consequently, the previously estimated dimensions (length, surface area) of the potential rock bridges are overestimated.

Based on this finding, we then tested two simple slope-break profile methods to try to find the “true” size of the rock bridges from the temperature profiles: (1) the Full Width at Half Maximum (FWHM) method; and (2) the Inflection Point (IP) method. The FWHM is given by the distance at which the profile reaches half its maximum value whereas the IP method gives the distance at which the curve reaches its inflection point in the thermal anomaly area. To improve the representativeness of this approach, additional numerical simulations were performed by varying the rock bridge length by a 10 cm increment between 0.3 m and 2.7 m. The results (Fig. 9a) clearly demonstrate that surface temperatures increase with increasing rock bridge length. However, it should be noted that beyond a rock bridge length of 1.7 m, increasing surface temperatures are no longer regular and tend to reach a limit value (Fig. 9a). We interpret this “saturation” as a geometric artifact since beyond 1.7 m, the rock bridge length becomes equal to or larger than the maximum thickness of the modeled geometry (1.8 m). The IP method provides the best results above a 0.7-m-long rock bridge; below this value, the FWHM method provides values closer to the real values (Fig. 9b). Nevertheless, the estimated lengths always remain smaller than the real lengths: on average, 0.11 m less with the IP and 0.17 m less with the FWHM (Fig. 9b).

By applying both slope-break profile methods to the full temperature profile of Boot Flake at 17:45 PST, we find a rock bridge length between 2.1 m (IP) and 2.16 m (FWHM) (Fig. 10). Given the calculated average errors for the IP and FWHM methods, these two estimates become, respectively, equal to 2.21 m and 2.33 m. Therefore, in accordance with the results of Fig. 8b, the new values are well below the 2.8 m initially measured (Fig. 5d) for the warm thermal anomaly detected at Boot Flake. They are, however, very close to those obtained (2.25 m and 2.21 m, respectively) with modeling a 2.4-m-long rock bridge (i.e., orange profile in Fig. 10). Thus, 2.4 m appears to be a very good approximation of the true length of the potential rock bridge of Boot Flake. Convincingly, the profile simulated with this rock bridge length reproduces almost exactly (2.0 °C instead of 2.1 °C) the maximum temperature difference measured along Boot Flake (Fig. 10). However, it should be noted that the shape of the modeled profiles does not reproduce the gradual decay of the surface temperatures observed between the rock bridge location and the flake edge (Fig. 10). The reasons for this difference and the consequences of this decrease in temperature on the stresses acting within the rear fracture are addressed in the Discussion section.

We take from these analyses that an overestimation of length of 0.4 m (2.8 m against 2.4 m) can be attributed to the IRT measurements. This suggests that for El Capitan Granite (the main rock type forming Boot and Texas flakes) a reduction factor of 1/7 should be applied to the thermal anomaly length to determine the true rock bridge length. Additionally, although we did not vary the rock bridge width in our thermal modeling, we assume that this reduction factor must similarly be applied to their width (i.e., 2.46 m against 2.11 m for Boot Flake). Thus, a reduction in surface area of 26.5% is necessary for determining more realistic rock bridge surface areas.

### Stability analysis

Knowing the dimensions of the rock bridges of Boot and Texas flakes, as revealed by the IRT/TLS analyses, offers an unprecedented opportunity to quantitatively analyze their stability. Here, we perform two types of analyses. The first uses fracture mechanics theory to compare the self-weight-induced stresses of the flakes to their ability to resist fracture at their respective rock bridge interfaces. The second analysis uses an empirical index of rock bridge attachment area to estimate the overall stability.

#### Fracture mechanics analysis of partially detached flakes

Brittle rock masses are subject to failure through propagation of fractures at their points of attachment (i.e., crack tips at rock bridges). As such, the use of fracture mechanics theory109 is well suited to analyzing flake stability. Here, we use linear elastic fracture mechanics to investigate the stability of Boot and Texas flakes. For simplicity, we assume that the only mechanism acting to cause instability is the self-weight of each flake’s rock mass, which drives gravitational forcing and subsequent outward rotation of the top of the flakes away from the cliff face (Fig. 7d,e). Thus, we envision the flake to be rotating outwards (inducing a tension at the top of the rock bridges). We idealize the somewhat complicated geometry of each of the flakes to coincide with geometrical fracture mechanics solutions available in the literature, and then compare the results (namely the stress intensity factor, KI – a measure of the stress state at crack tips) to the measured fracture toughness values (KIc) of the actual El Capitan rock mass (from laboratory testing). When KI equals or exceeds KIc, instability and rapid fracture (i.e., leading to rockfall) is expected. Exact solutions for Boot and Texas flake’s geometry are not available, however, a 3-D solution based on stress concentration factor methods110 (Eq. 25.7) captures the flakes fracture geometry fairly well (i.e., a disk-shaped attachment point of radius b, separating two parallel planes).

The fracture mechanics solution assumes that an outward moment (M = mgΔ) caused by the self-weight (mg) of the flake (0.50 MN for Boot; 3.37 MN for Texas) acts at a distance of Δ (0.21 m for Boot; 0.69 m for Texas) away from the point of attachment of the flake due to the separation of the base of the flake from the wall (Fig. 7d,e). The moment thus works to pry the partially-detached flake away from the parallel rock wall. Whereas the solution correctly mimics the geometry of Boot Flake, the analogy with Texas Flake is not quite correct since we know that Texas Flake is not cantilevered like Boot Flake (i.e., Texas Flake is partially supported at its base – Figs 1e and 7e). Still, this exercise is helpful for examining the role of rock bridges on the fracture geometry.

For Mode I (tensile) fracture, the stress intensity factor is given110 (Eq. 25.7) by:

$${K}_{I}=\frac{3}{2}\frac{M}{{b}^{2}\sqrt{\pi b}}\,\cos \,\theta$$
(1)

where θ is the orientation angle at which the stress is being evaluated along the edge of the rock bridge (maximum at θ = 0). For Boot Flake, the moment and radius length (b, equal to half the IRT-derived and corrected rock bridge length; i.e., 1/2 × 6/7 × 2.8 m) are, respectively, 0.11 MN·m and 1.2 m. For Texas Flake, these values are 2.33 MN·m and 0.8 m (b equal to 1/2 × 6/7 × 1.9 m). The resulting Mode I stress intensity factors are 0.06 MPa√m and 3.44 MPa√m, for Boot and Texas flakes, respectively.

The fracture toughness (KIc - measured as a critical stress intensity factor) of El Capitan Granite has recently been tested111 to be approximately 1.32 MPa√m. Thus, whereas KI for Boot Flake is well below KIc (indicating stability), the stress intensity factor for Texas Flake exceeds KIc (indicating failure should have already occurred). We therefore believe the small rock bridge identified by the IRT analyses for Texas Flake is not the only point of attachment and explore this, as well as proximal features (the chockstones and the rock attachment area highlighted in Figs 1f and 7a), in more detail in the next section.

Fracture mechanics analysis also allows us to explore how failure might be reached at Boot Flake. If the rock bridge were to deteriorate and became smaller (decreasing b in Eq. 1), KI would equal KIc when the rock bridge length (i.e., 2b) reached approximately 0.7 m (b = 0.34 m). Alternatively, critical fracture conditions could be reached if the loading condition changed (e.g., an increase in rotation moment by continued deflection (Δ) away from the wall or an increase in stresses beyond that by self-weight). This could take place, for example, by either thermal stress effects12,13,112 or the influence of water (or ice)113,114,115 pressure that could seasonally build up behind the flakes. Further, the fracture toughness that the stress intensity factors are evaluated against could be somewhat lower, given that the value used in our analyses is based on laboratory testing on relatively intact rock samples; the rock bridge interface is likely more highly weathered and would have a lower fracture toughness. Still, these analyses offer at least some insight into understanding the stability of the current geometry of these exfoliation sheets.

#### Evaluation of rock bridge attachment areas

Several post-rockfall event studies involving detailed mapping of rockfall scars28,36,45, characterization of discontinuity persistence25,26,34,41,42,44,116, and back analysis modeling23,24,38,40,117,118,119 have shown that rock bridge area (i.e., the presumed attachment area that holds a rock protrusion to the main cliff face) can vary between 0.2% and 45% of total scar surface, with an average value28 of about 22 ± 12%. To see where on this spectrum Boot Flake and Texas Flake may lie, we calculated the ratio between the maximum area of potential rock bridges and the total area of the exfoliation sheets.

The warm IRT anomaly observed on Boot Flake has a maximum length of 2.8 m (Fig. 5d) and a maximum surface area of 6.9 m2 (Fig. 7b). By applying the 26.5% IRT-based reduction factor (described previously) over the area of this anomaly, the effective maximum attachment area of the potential rock bridge is on the order of 5.1 m2. Given the total area of Boot Flake is equal to 75.5 m2 (Fig. 7b), the surface area of the rock bridge represents 6.8% of the total area. Thus, these results indicate that the rock bridge attachment area is about three times lower than the previously referenced28 average value, which ranks this value in the most “critical” part of the spectrum highlighted by the empirical data. However, the fracture mechanics analyses tell us otherwise, and for critical fracture conditions to be reached, the rock bridge area would need to reduce to 3.0 m² (for a 0.7 m long rock bridge and assuming a 4.3 m attachment width across the flake), representing only 4% of the total surface area of the flake.

We can perform similar analyses for Texas Flake. For the smaller IRT-identified rock bridge near the center of the flake, the effective (corrected) attachment area ([1–26.5%] × 2.0 m2 = 1.5 m2) is only 0.8% of the total area of Texas Flake (178.1 m2). This value is at the lower end of the 0.2% to 45% area of attachment spectrum identified in the literature and suggests (similar to the fracture mechanics results) that Texas Flake should be close to failure (or already detached completely). However, we also identified an additional rock attachment area from our IRT measurements along the base of the flake (Fig. 7a) with corrected attachment surface area of 13.1 m2 ([1–26.5%] × 17.8 m2). Aided by the IRT measurements, we deduce that the flake is attached here by approximately 7.3% of the total surface area – a result very similar to that calculated for Boot Flake (6.8%). Texas Flake further offers a unique opportunity to verify the model results, as the crack aperture behind the flake is sufficiently large to allow for direct inspection (indeed, the “Nose” climbing route travels between Texas Flake and the adjacent rock wall). Such inspection (Supplementary Fig. 2) reveals that the primary area of attachment is, in fact, at the base of the flake, represented by the warmest temperatures in Fig. 7c. Above this area, discrete, fractured rock sheets are in direct contact with the flake and the adjacent rock wall, providing a partial thermal link that appears as the purple and pink colors in Fig. 7c. By contrast, the uppermost chockstones highlighted in Fig. 1f, which are located 5.4 m above the 2.0 m² thermal anomaly (Supplementary Fig. 2), do not generate detectable thermal anomalies. A potential rock bridge or a compact wedged rock sheet that sits at the same location is therefore the likely source of the 2.0 m2 thermal anomaly in Fig. 7c. Thus, we assume that the dashed portion of Fig. 7e is actually intact rock across at least part of the width of the flake. The rock strength (at least in its unweathered form) and geometry of the two flakes therefore appear to be sufficient to resist outward gravitational forcing, even with only a ~7% rock bridge area connection to the cliff.

## Discussion

### IRT and thermal modeling comparison

Given the complexity of thermal imaging, and of course, from data collected in a natural setting, it is not surprising that discrepancies exist when comparing the IRT data to idealized thermal models. One significant difference between the results of the IRT monitoring and the 2-D thermal modeling is that the modeled temperatures are all higher than those measured (Figs 6b and 8b). This difference can come from the basic parameters used in the model to characterize the physical properties of the El Capitan Granite but also from the sources of external infrared radiation that affect the measurements of the calibrations parameters (more details are provided in Supplementary Appendix A). However, since the surface temperatures measured in the Boot Flake and Texas Flake areas have not been verified using thermoresistances, the IRT images are subject to a calibration bias that precludes identifying the source of the discrepancy between the observed and modeled temperatures. An additional substantial difference between the measured and modeled results is related to the shape of the modeled temperature profile which does not reproduce the “gradual” increase of the surface temperatures between the flake edge and the rock bridge location seen in the measured data (Fig. 10). On this flake’s portion, we observe a level of constant temperatures that mainly results from the constant thicknesses of the crack and flake applied to the model (Supplementary Appendix A). However, this is not the case in reality (Fig. 3b). In addition, it should be noted that this difference in behavior arises because the thermal conduction model is in 2-D, whereas 3-D effects obviously govern the actual thermal response of the flake and rock bridges. Finally, it is important to specify that the method used to measure the warm thermal anomaly size depends on several choices and manual procedures, including the determination of temperature thresholds, the texturing of IRT images on TLS data, and the drawing of 3-D polylines within the TLS data. In terms of uncertainty, a cumulative error of ±5% (Supplementary Appendix A) can be attributed to the estimate of the rock bridge surface at the end of this procedure.

### Potential of IRT monitoring for hazard assessment

Our monitoring of Boot and Texas flakes with IRT methods show that both flakes are always colder than the surrounding rock walls in the evening (Fig. 6). At the end of the day, exfoliation sheets with open cracks are characterized by colder thermal signatures (detection criterion) that result from air circulation that envelops and cools the detached parts (analogous to a heat transfer fin in engineering applications). Since the instruments used in our study can acquire IRT panoramas of 1,000-m-tall cliffs (Fig. 4), they could conceivably detect flakes over an entire cliff, thereby opening up a powerful new rock instability monitoring technique (assuming that partially detached flakes are more susceptible to rockfalls).

However, our study also revealed that cold thermal anomalies can exist for reasons other than partial detachment; these include areas with overhangs or with light-colored rocks. Consequently, classification of thermal anomalies must be conducted by superimposing67,68 the thermograms with digital photographs or by generating 3-D “thermo-photorealistic” models (Fig. 7). In addition, image rectification120,121 and topographic corrections must be applied to thermograms in order to account for the influence of lens distortion, incidence angle, spatial orientation, and emissivity for each pixel. Thus, utilizing our methodology for the purpose of examining an entire rock wall is not without substantial complications. That said, we have shown that coupled TLS-IRT data are very helpful for classifying topographic artifacts and for drawing 3-D maps of exfoliation sheets. Thus, we find the methodology worth exploring to determine the potential for estimating cliff-wide hazards. Even if this were to be found intractable (due to the investment needed in instrument and study area calibration), it might be used for long-term hazard assessment on a smaller scale. For example, capturing repeat surveys of the same areas of a rock cliff (e.g., repeating the data capture for Boot and Texas flakes) could possibly provide a path forward for IRT-based monitoring, in that the additional work required for calibration of the thermograms would not necessarily need to be performed in as rigorous detail each time data were collected. The repeat data might show trends in rock bridge area reduction (i.e., fracture growth). Coupled with fracture mechanics analyses like those we have presented, it may be possible to forecast when rock bridge area decreases such that the attachment length indicates fracture instability (i.e., when the stress intensity factor approaches the critical fracture toughness).

Recent studies12,13 have clearly shown the link between diurnal and seasonal temperature cycles and exfoliation flake movement. This is particularly true of thin, partially detached rock sheets that have the ability to deform. Boot and Texas flakes are such features, and being located on a south facing wall, may be particularly prone to show a thermal contrast between diurnal cycles. This, coupled with the fact that both flakes are significantly detached from the main rock cliff might suggest that they are ideal for identification and utilization with IRT methods. As we were able to verify that warmer areas did, in fact, coincide with rock attachment points, the corollary would hold that the method should be able to identify detached areas, even if the flakes are not as “open” as Boot and Texas flakes. Thus, it is plausible that less detached flakes might still be identifiable with IRT, even if their thermographic signal was not as strong. Obviously, additional calibration of the reflected signals would be needed. This might be particularly true were studies to be conducted on north-facing slopes where a smaller thermal differential (and stress) might be expected. Although our study did not incorporate thermal stresses into the analysis of stability, it is likely that a thermal signal could be contributing to the stresses acting on both Boot and Texas flakes. This, along with exploration of other possible stressors (e.g., the chockstone wedging mechanism122) could lead to higher stress intensity factors and possibly increased rock bridge deterioration. Thus, there are many avenues in need of further explanation; we expect the present study will assist with at least some of these.

## Conclusion

Quantifying rockfall hazards from steep cliffs continues to pose significant challenges, particularly due to our inability to measure the areal extent of rock bridges that fundamentally control the stability of partially detached rock masses. We explored the use of IRT technologies, in conjunction with TLS surveying, to remotely characterize the stability of two well-delimited granitic exfoliation sheets that form partially detached flakes on a nearly 1,000-m-tall vertical cliff. By examining the nocturnal cooling of the flakes, we found that both monitored flakes always remain colder than the surrounding rock walls. We interpret this colder thermal signature as a result of the air circulation that envelops and cools the detached parts from the cliff. This thermal signature could be used to improve the remote detection of partially detached exfoliation sheets in cliffs many hundreds of meters high, and from distances exceeding 1 km. Detailed analysis of our resulting thermograms for the flakes also revealed that the temperature patterns are not homogeneous – warmer thermal anomalies are visible on the surface of both monitored flakes. We interpret these local warmer areas as a sign of the presence of rock bridges that conduct heat from the cliff behind the exfoliation sheets. The detection of rock bridges from our IRT analysis is supported by the results of 2-D thermal modeling of an exfoliation sheet connected to the rock wall by a central rock bridge. By draping the IRT images on the TLS topographic data, we were able to measure the potential rock bridge surfaces and other areas of contact between the flakes and the adjacent wall, and thus evaluate flake-stability conditions. Based on the same geometric data, we supplemented this estimation with further analysis of fracture mechanics of both monitored exfoliation sheets. This study demonstrates that thermal imaging, combined with high-resolution 3-D topography, has significant potential for rockfall susceptibility and hazard analysis.

## Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

## References

1. Selby, M. J. A rock mass strength classification for geomorphic purposes: with tests from Antarctica and New Zealand. Zeit. fur Geomorph. NF 24, 31–51 (1980).

2. Wieczorek, G. F. & Jäger, S. Triggering mechanisms and depositional rates of postglacial slope-movement processes in the Yosemite Valley, California. Geomorphology 15, 17–31 (1996).

3. Hungr, O., Evans, S. G. & Hazzard, J. Magnitude and frequency of rock falls and rock slides along the main transportation corridors of southwestern British Columbia. Canadian Geotechnical Journal 36.2, 224–238 (1999).

4. Rosser, N. J., Lim, M., Petley, D. N., Dunning, S. A. & Allison, R. J. Patterns of precursory rockfall prior to slope failure. Journal of Geophysical Research 112, F04014 (2007).

5. Abellán, A., Calvet, J., Vilaplana, J. M. & Blanchard, J. Detection and spatial prediction of rockfalls by means of terrestrial laser scanner monitoring. Geomorphology 119.3-4, 162–171 (2010).

6. Stock, G. M. et al. Historical rock falls in Yosemite National Park, California (18572011) (U.S. Geological Survey Data Series 746, 2013).

7. Strunden, J., Ehlers, T. A., Brehm, D. & Nettesheim, M. Spatial and temporal variations in rockfall determined from TLS measurements in a deglaciated valley, Switzerland. Journal of Geophysical Research 120.7, 1251–1273 (2015).

8. Williams, J. G., Rosser, N. J., Hardy, R. J., Brain, M. J. & Afana, A. A. Optimising 4D approaches to surface change detection: improving understanding of rockfall magnitude-frequency. Earth Surface Dynamics 6.1, 101–119 (2018).

9. Wieczorek, G. F. & Snyder, J. B. Rock falls from Glacier Point above Camp Curry, Yosemite National Park, California (U.S. Geological Survey Open-File Report, 99–385, 1999).

10. Stock, G. M., Martel, S. J., Collins, B. D. & Harp, E. L. Progressive failure of sheeted rock slopes: the 2009–2010 Rhombus Wall rock falls in Yosemite Valley, California, USA. Earth Surface Processes and Landform 37.5, 546–561 (2012).

11. do Amaral Vargas, E., Velloso, R. Q., Chávez, L. E., Gusmão, L. & Do Amaral, C. P. On the effect of thermally induced stresses in failures of some rock slopes in Rio de Janeiro, Brazil. Rock mechanics and rock engineering 46.1, 123–134 (2013).

12. Collins, B. D. & Stock, G. M. Rockfall triggering by cyclic thermal stressing of exfoliation fractures. Nature Geoscience 9.5, 395–400 (2016).

13. Collins, B. D. et al. Thermal influences on spontaneous rock dome exfoliation. Nature Communications 9.1, 762, 12p. (2018).

14. Matasci, B. et al. Assessing rockfall susceptibility in steep and overhanging slopes using three-dimensional analysis of failure mechanisms. Landslides 15.5, 859–878 (2018).

15. Bradley, W. C. Large-scale exfoliation in massive sandstones of the Colorado Plateau. Geological Society of America Bulletin 74.5, 519–528 (1963).

16. Bahat, D., Grossenbacher, K. & Karasaki, K. Mechanism of exfoliation joint formation in granitic rocks, Yosemite National Park. Journal of Structural Geology 21.1, 85–96 (1999).

17. Hencher, S. R., Lee, S. G., Carter, T. G. & Richards, L. R. Sheeting joints: characterisation, shear strength and engineering. Rock Mechanics and Rock Engineering 44.1, 1–22 (2011).

18. Martel, S. J. Progress in understanding sheeting joints over the past two centuries. Journal of Structural Geology 94, 68–86 (2017).

19. Jahns, R. H. Sheet structure in granites: its origin and use as a measure of glacial erosion in New England. The Journal of Geology 51.2, 71–98 (1943).

20. ISRM (International Society for Rock Mechanics) Commission on standardization of laboratory and field tests: suggested methods for the quantitative description of discontinuities in rock masses. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 15, 319–368 (1978).

21. Dershowitz, W. S. & Einstein, H. H. Characterizing rock joint geometry with joint system models. Rock mechanics and rock engineering 21.1, 21–51 (1988).

22. Dershowitz, W. S. & Herda, H. H. Interpretation of fracture spacing and intensity (The 33th US Symposium on Rock Mechanics (USRMS), American Rock Mechanics Association, 1992).

23. Elmo, D., Clayton, C., Rogers, S., Beddoes, R. & Greer, S. Numerical simulations of potential rock bridge failure within a naturally fractured rock mass (Proceedings International Symposium on Rock Slope Stability in Open Pit Mining and Civil Engineering, Slope Stability 2011, 18–21, 2011).

24. Matasci, B., Jaboyedoff, M., Ravanel, L. & Deline, P. Stability assessment, potential collapses and future evolution of the west face of the Drus (3,754 m asl, Mont Blanc Massif) (Engineering Geology for Society and Territory-Volume 2, Springer, Cham, 791–795, 2015).

25. Elmo, D. & Stead D. Applications of fracture mechanics to rock slopes (Rock Mechanics and Engineering Volume 3: Analysis, Modeling & Design, Chapter 23, CRC Press, 2016).

26. Tuckey, Z. & Stead, D. Improvements to field and remote sensing methods for mapping discontinuity persistence and intact rock bridges in rock slopes. Engineering Geology 208, 136–153 (2016).

27. Bonilla–Sierra, V., Scholtès, L., Donzé, F. V. & Elmouttie, M. DEM analysis of rock bridges and the contribution to rock slope stability in the case of translational sliding failures. International Journal of Rock Mechanics and Mining Sciences 80, 67–78 (2015).

28. de Vilder, S. J., Rosser, N. J. & Brain, M. J. Forensic analysis of rockfall scars. Geomorphology 295, 202–214 (2017).

29. Jennings, J. E. A mathematical theory for the calculation of the stability of slopes in open cast mines. In Planning Open Pit Mines, Proceedings 21.2, 87–102, Johannesburg (1970).

30. Einstein, H. H., Veneziano, D., Baecher, G. B. & O’reilly, K. J. The effect of discontinuity persistence on rock slope stability. In International journal of rock mechanics and mining sciences & geomechanics abstracts, 20.5, 227–236, Pergamon (1983).

31. Goodman, R. E. & Kieffer, D. S. Behavior of rock in slopes. Journal of Geotechnical and Geoenvironmental Engineering 126.8, 675–684 (2000).

32. Diederichs, M. S. Rock fracture and collapse under low confinement conditions. Rock Mechanics and Rock Engineering 36.5, 339–381 (2003).

33. Kemeny, J. Time-dependent drift degradation due to the progressive failure of rock bridges along discontinuities. International Journal of Rock Mechanics and Mining Sciences 42.1, 35–46 (2005).

34. Frayssines, M. & Hantz, D. Failure mechanisms and triggering factors in calcareous cliffs of the Subalpine Ranges (French Alps). Engineering Geology 86.4, 256–270 (2006).

35. Stock, G. M. et al. High-resolution three-dimensional imaging and analysis of rock falls in Yosemite Valley, California. Geosphere 7.2, 573–581 (2011).

36. Eberhardt, E., Stead, D. & Coggan, J. S. Numerical analysis of initiation and progressive failure in natural rock slopes—the 1991 Randa rockslide. International Journal of Rock Mechanics and Mining Sciences 41.1, 69–87 (2004).

37. Fischer, L., Amann, F., Moore, J. R. & Huggel, C. Assessment of periglacial slope stability for the 1988 Tschierva rock avalanche (Piz Morteratsch, Switzerland). Engineering Geology 116.1, 32–43 (2010).

38. Gischig, V. et al. Composite rock slope kinematics at the current Randa instability, Switzerland, based on remote sensing and numerical modeling. Engineering Geology 118.1, 37–53 (2011).

39. Stead, D., Eberhardt, E. & Coggan, J. S. Developments in the characterization of complex rock slope deformation and failure using numerical modelling techniques. Engineering Geology 83.1, 217–235 (2006).

40. Sturzenegger, M. & Stead, D. The Palliser Rockslide, Canadian Rocky Mountains: characterization and modeling of a stepped failure surface. Geomorphology 138.1, 145–161 (2012).

41. Agliardi, F., Crosta, G. B., Meloni, F., Valle, C. & Rivolta, C. Structurally-controlled instability, damage and slope failure in a porphyry rock mass. Tectonophysics 605, 34–47 (2013).

42. Paronuzzi, P., Bolla, A. & Rigo, E. 3D stress–strain analysis of a failed limestone wedge influenced by an intact rock bridge. Rock Mechanics and Rock Engineering 49.8, 3223–3242 (2016).

43. Sampaleanu, C. I. A. The role of intact rock fracture in rockfall initiation (M.Sc. thesis, Simon Fraser University, Vancouver, Canada, 2017).

44. Paronuzzi, P. & Serafini, W. Stress state analysis of a collapsed overhanging rock slab: A case study. Engineering Geology 108.1, 65–75 (2009).

45. Spreafico, M. C., Franci, F., Bitelli, G., Borgatti, L. & Ghirotti, M. Quantification of intact rock bridges and rock mass fragmentation after failure by means of remote sensing techniques (Proceedings of the 2nd Virtual Geoscience Conference, Bergen, Norway, 96–97, 2016).

46. Matasci, B. Rockfall susceptibility assessment and remote geological mapping with LiDAR point clouds (Ph.D. thesis, University of Lausanne, Lausanne, Switzerland, 2015).

47. Elmo, D., Donati, D. & Stead, D. Challenges in the characterisation of intact rock bridges in rock slopes. Engineering Geology 245, 81–96 (2018).

48. Monte, J. M. Rock mass characterization using laser scanning and digital imaging data collection techniques (M.Sc. thesis, University of Arizona, Tuscon, United States, 2004).

49. Tuckey, Z. An integrated field mapping-numerical modelling approach to characterising discontinuity persistence and intact rock bridges in large open pit slopes (M.Sc. thesis, Simon Fraser University. Burnaby, Canada, 2012).

50. Tuckey, Z., Stead, D. & Eberhardt, E. An integrated approach for understanding uncertainty of discontinuity persistence and intact rock bridges in large open pit slopes (Proceedings of Slope Stability 2013, Brisbane, Australia, 189–204, 2013).

51. Hui, T. H. H. & Sun, H. W. Trial application of thermal infrared image and seismic sympathetic vibration techniques for slope investigation (GEO Technical Note No. TN1/2005, 2005).

52. Wu, J. H., Lin, H. M., Lee, D. H. & Fang, S. C. Integrity assessment of rock mass behind the shotcreted slope using thermography. Engineering Geology 80.1, 164–173 (2005).

53. Kononov, V. A. Pre-feasibility investigation of infrared thermography for the identification of loose hanging wall and impending falls of ground. Safety in Mines Research Advisory Committee GAP 706, 1–26 (2000).

54. Kononov, V. A. Infrared thermography of loose hanging walls. Safety in Mines Research Advisory Committee GAP 820, 1–102 (2002).

55. Vogt, D., Brink, V. Z., Brink, S., Price, M. & Kagezi, B. New technology for improving entry examination, thereby managing the rockfall risk in south african gold and platinum mines (CSIR, Center for Mining Innovation, Auckland Park, Johannesburg, South Africa, 2010).

56. Liu, S., Xu, Z., Wu, L., Ma, B. & Liu, X. Infrared Imaging Detection of Hidden Danger in Mine Engineering (In Progress in Electromagnetics Research Symposium Proceedings, Suzhou, China, 125–129, 2011).

57. Prendes-Gero, M. B., Suárez-Domínguez, F. J., González-Nicieza, C. & Álvarez-Fernández, M. I. Infrared thermography methodology applied to detect localized rockfalls in self-supporting underground mines (In ISRM International Symposium-EUROCK 2013, International Society for Rock Mechanics, 2013).

58. Frodella, W., Morelli, S., Gigli, G. & Casagli, N. Contribution of infrared thermography to the slope instability characterization. In Proceedings of world landslide forum 3, 144–147 (2014).

59. Pappalardo, G., Mineo, S., Zampelli, S. P., Cubito, A. & Calcaterra, D. InfraRed thermography proposed for the estimation of the Cooling Rate Index in the remote survey of rock masses. International Journal of Rock Mechanics and Mining Sciences 83, 182–196 (2016).

60. Thiele, S. T., Varley, N. & James, M. R. Thermal photogrammetric imaging: a new technique for monitoring dome eruptions. Journal of Volcanology and Geothermal Research 337, 140–145 (2017).

61. Squarzoni, C. et al. Terrestrial laser scanner and infrared thermography in rock fall prone slope analysis (Geophysical Research Abstracts 10, 2008).

62. Adorno, V., Barnobi, L., La Rosa, F., Leotta, A. & Paratore, M. Contributo della tecnologia laser scanner e termografia IR nella caratterizzazione geomeccanica di un costone roccioso (Atti 13a conferenza Asita, Bari 14, 2009).

63. Teza, G., Marcato, G., Castelli, E. & Galgaro, A. IRTROCK: A MATLAB toolbox for contactless recognition of surface and shallow weakness of a rock cliff by infrared thermography. Computers & Geosciences 45, 109–118 (2012).

64. Teza, G., Marcato, G., Pasuto, A. & Galgaro, A. Integration of laser scanning and thermal imaging in monitoring optimization and assessment of rockfall hazard: a case history in the Carnic Alps (Northeastern Italy). Natural Hazards 76.3, 1535–1549 (2015).

65. Baroň, I., Bečkovský, D. & Míča, L. Application of infrared thermography for mapping open fractures in deep-seated rockslides and unstable cliffs. Landslides 11.1, 15–27 (2014).

66. Teza, G. THIMRAN: MATLAB Toolbox for Thermal Image Processing Aimed at Damage Recognition in Large Bodies. Journal of Computing in Civil Engineering 28.4, 04014017 (2013).

67. Mineo, S., Pappalardo, G., Rapisarda, F., Cubito, A. & Di Maria, G. Integrated geostructural, seismic and infrared thermography surveys for the study of an unstable rock slope in the Peloritani Chain (NE Sicily). Engineering Geology 195, 225–235 (2015).

68. Mazzanti, P., Brunetti, A. & Bretschneider, A. A new approach based on terrestrial remote-sensing techniques for rock fall hazard assessment (Modern Technologies for Landslide Monitoring and Prediction, Springer Berlin Heidelberg, 69–87, 2015).

69. Bramson, M. A. Infrared Radiation: A Handbook for Applications (Wolfe, W.L. (Series Editor), Optical Physics and Engineering, Honeywell Inc. Radiation Center, Plenum Press, New York, 1968).

70. Pajani, D. La thermographie du bâtiment: Principes et applications du diagnostic thermographique (Editions Eyrolles, 2012).

71. Stock, G. M., Guerin, A., Avdievitch, N., Collins, B. D. & Jaboyedoff, M. Rapid 3-D analysis of rockfalls. GSA Today 28.8, 28–29 (2018).

72. Matthes, F. E. Geologic history of the Yosemite Valley (U.S. Geological Survey Professional Paper 160, 1930).

73. Huber, N. K. The geologic story of Yosemite National Park (U.S. Geological Survey Bulletin 1595, 1987).

74. Stock, G. M. & Uhrhammer, R. A. Catastrophic rock avalanche 3600 years BP from El Capitan, Yosemite Valley, California. Earth Surface Processes and Landforms 35.8, 941–951 (2010).

75. Bateman, P. C. Plutonism in the central part of the Sierra Nevada batholith, California (U.S. Geological Survey Professional Paper 1483, 1992).

76. Ratajeski, K., Glazner, A. F. & Miller, B. V. Geology and geochemistry of mafic to felsic plutonic rocks in the Cretaceous intrusive suite of Yosemite Valley, California. Geological Society of America Bulletin 113.11, 1486–1502 (2001).

77. Calkins, F. C., Huber, N. K. & Roller, J. A. Geologic bedrock map of Yosemite Valley, Yosemite National Park, California (U.S. Geological Survey Map I-1639, 1985).

78. Peck, D. L. Geologic map of the Yosemite quadrangle, central Sierra Nevada, California (U.S. Geological Survey Map I-2751, 2002).

79. Putnam, R. L., Glazner, A. F., Law, B. S. & Stock, G. M. Geologic map of the southeast face of El Capitan, Yosemite Valley, California. Geological Society of America Map & Chart Series MCH 106 (2014).

80. Putnam, R. L. et al. Plutonism in three dimensions: Field and geochemical relations on the southeast face of El Capitan, Yosemite National Park, California. Geosphere GES01133-1 (2015).

81. Reid, J. B. Jr., Evans, O. C. & Fates, D. G. Magma mixing in granitic rocks of the central Sierra Nevada, California. Earth and Planetary Science Letters 66, 243–261 (1983).

82. Gigapan Systems Available from, http://www.gigapan.com (2013).

83. FLIR, User’s manual FLIR T6xx series. Publ. n. T559600 (2014).

84. Meola, C. & Carlomagno, G. M. Recent advances in the use of infrared thermography. Measurement Science and Technology 15.9, R27 (2004).

85. Minkina, W. & Dudzik, S. Infrared thermography: errors and uncertainties (John Wiley & Sons, 2009).

86. ISO 9288:1989: Thermal insulation - Heat transfer by radiation - Physical quantities and definitions (ISO catalogue 1989. Retrieved 2015-03-15).

87. Holst, G. C. Common sense approach to thermal imaging (Washington, DC, USA: SPIE Optical Engineering Press, 2000).

88. Monchau, J. P. Mesure d'émissivité pour la thermographie infrarouge appliquée au diagnostic quantitatif des structures (Doctoral dissertation, Paris Est, 2013).

89. Usamentiaga et al. Infrared thermography for temperature measurement and non-destructive testing. Sensors 14.7, 12305–12348 (2014).

90. Shannon, H. R., Sigda, J. M., Van Dam, R. L., Hendrickx, J. M. & McLemore, V. T. Thermal camera imaging of rock piles at the Questa Molybdenum Mine, Questa, New Mexico (Proceedings America Society of Mining and Reclamation, 1015–1028, 2005).

91. Heraeus FK1020 PT100 1/3 DIN – Resistance Thermometer – Heraeus (2008). Available from, https://www.distrelec.ch/en/resistance-thermometer-heraeus-fk1020-pt100-din/p/17680937.

92. Sacadura, J. O. Influence de la rugosité sur le rayonnement thermique émis par les surfaces opaques: Essai de modèle. International Journal of Heat and Mass Transfer 15.8, 1451–1465 (1972).

93. Littoclime, Thermo-hygromètres portatifs DIGIHUM2 - Thermo-hygromètre digital de poche Available from, http://www.littoclime.net/thermometre-hygrometre-digital-de-poche-format-stylo,fr,4,DIGIHUM2.cfm (2014).

94. Teledyne Optech, ILRIS-LR Terrestrial Laser Scanner, Summary Specifications Sheet Available from, http://www.teledyneoptech.com (2018).

95. Besl, P. J. & McKay, N. D. A method for registration of 3-D shapes. IEEE Transactions on pattern analysis and machine intelligence 14.2, 239–256 (1992).

96. Chen, Y. & Medioni, G. Object modelling by registration of multiple range images. Image and vision computing 10.3, 145–155 (1992).

97. Girardeau-Montaut, D. CloudCompare (version 2.7) (GPL software, EDF RandD, Telecom ParisTech). Available from, http://www.cloudcompare.org (2015).

98. Kolor, Autopano Giga (version 3.0.8) (Computer software). Available from, http://www.kolor.com (2014).

99. MATLAB, Intensity-based image registration (imregister) Available from, http://ch.mathworks.com/help/images/ref/imregister.html (2018).

100. COMSOL, COMSOL Multiphysics® (version 5.3) (Comsol AB, Computer software,). Available from, https://www.comsol.com (2017).

101. Alpinist LLC, Speed Series Part II: Sean Leary and Dean Potter Available from: http://www.alpinist.com/doc/web11w/wfeature-speed-leary-potter/2 (2001).

102. Carslaw, H. S., & Jaeger, J. C. Conduction of heat in solids 2nd edition (Oxford: Clarendon Press, 1959).

103. Duffie, J. A. & Beckman, W. A. Solar engineering of thermal processes 4th edition (John Wiley & Sons, 2013).

104. Cooper, P. I. The absorption of radiation in solar stills. Solar energy 12.3, 333–346 (1969).

105. Western Region Climate Center Yosemite Park HQ, California (Station #049855). Available from, https://wrcc.dri.edu/cgi-bin/cliMAIN.pl?ca9855 (2018)

106. Sable Systems International, Altitude & Barometric Pressure Available from, https://www.sablesys.com/support/technical-library/barometric-pressure-vs-altitude-table (2018).

107. Hemond, H. F. & Fechner, E. J. Chapter 4 - The Atmosphere (Chemical Fate and Transport in the Environment 3rd edition, Academic Press, 311–454, ISBN 9780123982568, 2015).

108. Vollmer, M. & Möllmann, K. P. Infrared thermal imaging: fundamentals, research and applications 2nd edition (John Wiley & Sons, 2017).

109. Lawn, B. Fracture of Brittle Solids 2nd edition (Cambridge Univ. Press, 1993).

110. Tada, H., Paris, P. C. & Irwin, G. R. The Stress Analysis of Cracks Handbook 3rd edition (ASME Press, New York, 2000).

111. Sandrone F. Personal Communication (École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2018).

112. Eppes, M. C. et al. Deciphering the role of solar-induced thermal stresses in rock weathering. GSA Bulletin 128.9-10, 1315–1338 (2016).

113. Bottelin, P., Levy, C., Baillet, L., Jongmans, D. & Gueguen, P. Modal and thermal analysis of Les Arches unstable rock column (Vercors massif, French Alps). Geophysical Journal International 194.2, 849–858 (2013).

114. D’Amato, J. et al. Influence of meteorological factors on rockfall occurrence in a middle mountain limestone cliff. Natural Hazards and Earth System Sciences 16.3, 719–735 (2016).

115. Draebing, D., Krautblatter, M. & Hoffmann, T. Thermo‐cryogenic controls of fracture kinematics in permafrost rockwalls. Geophysical Research Letters 44.8, 3535–3544 (2017).

116. LeBaron, A. M. The effect of rock bridges on blasting fragmentation in sedimentary rock (M.Sc. thesis, University of Utah, Salt Lake City, United States, 2011).

117. Karami, A., Greer, S. & Beddoes, R. Numerical assessment of step-path failure of northwest wall of A154 Pit, Diavik Diamond Mines (Potvin, Y. (Ed.), Proceedings of the 2007 International Symposium on Rock Slope Stability in Open Pit Mining and Civil Engineering, CSIRO, Perth, Australia, 293–305, 2007).

118. Grøneng, G., Nilsen, B. & Sandven, R. Shear strength estimation for Åknes sliding area in western Norway. International Journal of Rock Mechanics and Mining Sciences 46.3, 479–488 (2009).

119. Lévy, C., Baillet, L., Jongmans, D., Mourot, P. & Hantz, D. Dynamic response of the Chamousset rock column (Western Alps, France). Journal of Geophysical Research 115, F04043 (2010).

120. Rodríguez-Martín, M., Lagüela, S., González-Aguilera, D. & Martínez, J. Thermographic test for the geometric characterization of cracks in welding using IR image rectification. Automation in Construction 61, 58–65 (2016).

121. Usamentiaga, R. & García, D. F. Infrared Thermography Sensor for Temperature and Speed Measurement of Moving Material. Sensors 17.5, 1157 (2017). (pp. 1-21).

122. Bakun-Mazor, D., Hatzor, Y. H., Glaser, S. D. & Santamarina, J. C. Thermally vs. seismically induced block displacements in Masada rock slopes. International Journal of Rock Mechanics and Mining Sciences 61, 196–211 (2013).

123. MATLAB, Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) Available from, https://ch.mathworks.com/help/matlab/ref/pchip.html (2018).

124. Guerin, A. et al. Common problems encountered in 3D mapping of geological contacts using high-resolution terrain and image data. European Journal of Remote Sensing 48.1, 661–672 (2015).

## Acknowledgements

The authors acknowledge the Swiss National Science Foundation (SNSF, grants#: 200020_146426 and 200020_159221), the U.S. National Park Service (USNPS, Scientific Research and Collecting Permit#: YOSE-2015-SCI-0124), and the U.S. Geological Survey (USGS) for supporting this research. The authors would also like to warmly thank Andrea Billi, Editorial Board Member of Nature Scientific Reports, as well as both anonymous reviewers who provided us with very constructive advice and corrections. We also thank Mickael Barsive and Joanna Kaiser from COMSOL France, as well as Thierry Luthy from COMSOL Multiphysics GmbH, for advice on using the Heat Transfer Module. Finally, special thanks to Stephen J. Martel of the University of Hawaii for his advice on fracture mechanics analysis and Olivier Dubas for his active participation in the October 2015 data acquisition. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.

## Author information

Authors

### Contributions

A.G., M.J. and M.-H.D. conceived the project and designed the experiment. A.G., M.J., B.D.C. and G.M.S. collected the data and interpreted the results. A.G. processed the data. A.G., M.B. and C.L. jointly conducted the modeling with input from M.J. and Y.Y.P., B.D.C. and B.M. developed the stability analyses. A.G. wrote the paper with input from M.J., B.D.C., M.-H.D., G.M.S. and B.M.

### Corresponding author

Correspondence to Antoine Guerin.

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### Competing Interests

The authors declare no competing interests.

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Guerin, A., Jaboyedoff, M., Collins, B.D. et al. Detection of rock bridges by infrared thermal imaging and modeling. Sci Rep 9, 13138 (2019). https://doi.org/10.1038/s41598-019-49336-1

• Accepted:

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• DOI: https://doi.org/10.1038/s41598-019-49336-1

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