Displacement time series of active slopes from time-lapse cameras

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Displacement time series of active slopes from time-lapse cameras

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Andrea Manconi*, Franziska Glueer and Simon Loew

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Swiss Federal Institute of Technology, Department of Earth Sciences, Zurich, Switzerland

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*Corresponding author: [email protected]

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Abstract

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We propose a new algorithm based on Digital Image Correlation (DIC) to systematically process

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images acquired from time-lapse cameras (TLC) and retrieve displacement time series over

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active slopes. The aim of this work is to maximize the information obtainable from large datasets

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of TLC digital images acquired with different light and meteorological conditions, and to retrieve

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accurate information on the spatial and temporal evolution of surface deformation. We present

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our approach along with an example of application to an unstable slope in the Swiss Alps, more

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specifically in the Great Aletsch area, where glacier retreat has activated and/or reactivated

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several slope processes. At this location, a webcam was used to monitor the evolution of rock

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deformation towards a partial slope failure. We validate or results against ground based

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measurements, and discuss advantages and limitations of our approach, as well as future

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improvements.

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1. Introduction

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Measuring the spatial and temporal development of ground deformation is essential for the

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analysis and interpretation of various geological and geophysical phenomena (Angeli et al.,

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2000; Dermanis and Kotsakis, 2006; Dzurisin, 2006). Surface displacements can be retrieved

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with different monitoring methods, ranging from field-based instruments to remote sensing, and

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the choice of the appropriate methods and sensors depends on several factors, including

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technical specifications and site-specific conditions (Manconi, 2016). In this context, optical

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imagery acquired from different platforms (terrestrial, airborne, and space-borne) is beneficial

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(Leprince et al., 2008). Digital images can be exploited to visually identify changes of the scene

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occurring over time, but also to quantify the evolution of surface displacements. Moreover,

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image processing can be used to pass from visual (qualitative) information to semi-quantitative

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and/or quantitative assessment of deformation. Digital Image Correlation (DIC, known also as

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pixel-offset or feature-tracking) is a suitable method to detect surface deformation at high spatial

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and temporal resolutions. The implementation of DIC is very flexible, thus it can be applied to

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data acquired form different platforms, as well as different sensors (such as optical,

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multispectral, thermal, laser and also radar) on the Earth’s surface and also on other planets

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(Bridges et al., 2012; Delbridge et al., 2016; Scambos et al., 1992; Singleton et al., 2014; Walter,

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2011). Starting from the work of Scambos et al., 1992 measuring glacier velocities form satellite

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imagery, DIC methods have been further developed and widely used to investigate other

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phenomena such as earthquakes, active volcanoes, as well as unstable slopes (e.g., Manconi et

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al., 2014; Manconi and Casu, 2012; Travelletti et al., 2012; Van Puymbroeck et al., 2000).

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Moreover, recent advances allow us to analyze multitemporal imagery and retrieve displacement

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time series from optical and radar satellite imagery (e.g., Casu et al., 2011; Stumpf et al., 2017).

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Focusing on DIC analyses to retrieve deformation at unstable slopes, data obtained from

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satellite and/or aerial imagery have spatial resolutions (i.e., usually meters to tens of meters)

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often unsuitable to achieve reliable results in the target area. In addition, satellite sampling 2

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frequencies (revisit times) conflict with the evolution of such processes, which may experience

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surface velocity changes (accelerations) even within hours. For this reason, ground based time-

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lapse cameras (TLC) are being extensively employed to monitor mountain environments and

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achieve high spatial and temporal resolutions (Danielson and Sharp, 2013; Farinotti et al., 2010;

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Giordan et al., 2016; Travelletti et al., 2012; Vernier et al., 2012). This proliferation in the use of

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TLC in alpine areas reflects the increasing availability of reasonably priced but robust high

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resolution photo and/or video cameras, as well as of storage volumes and data transmission

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capabilities. The frequency of data acquisition can be relatively high, in the order of several

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pictures every minute, or even several frames per second with webcams. However, a number of

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problems have to be considered when dealing with TLC acquisition in mountain areas, as for

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example the effects of light conditions, shadowing, and/or meteorological variables. Moreover,

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when TLC data has to be used for DIC analyses, additional concerns might be related to data

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selection, as well as to the definition of the appropriate algorithm to fully exploit the available

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information. For this reason, straightforward and efficient retrieval of accurate displacement time

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series in alpine areas using DIC on TLC data can be a difficult task.

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Here we propose a new algorithm to select and process images acquired from TLC and monitor

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active slope deformation. We present our methodology by applying the approach to a case study

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located in the Swiss Alps, more specifically in the Great Aletsch Glacier area. There, several

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sectors of the slopes adjacent to the recent tongue of the Great Aletsch Glacier have been

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reactivated due to its retreat since the Little Ice Age (Kos et al., 2016). The paper is organized as

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follows. We first introduce the area of investigation and the characteristics of the monitoring

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framework. Then, we describe the steps from the raw data acquisition to the displacement time

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series recovery. Finally, we discuss the results obtained, as well as the limitations of the current

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implementation of the algorithm and its potential future improvement.

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2. Area of study and monitoring system

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The Great Aletsch Glacier is located in the Swiss Alps, and is currently the largest and longest

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glacier in the European Alps. The Great Aletsch valley is characterized by steep slopes primarily

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composed of crystalline rocks (mainly gneisses and granites of the Aare massif), which

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underwent multiple cycles of glacier advancements and retreat (Grämiger et al., 2017). In the

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vicinity of the glacier tongue (Figure 1a), the current deglaciation phase is progressively

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exposing the rock mass to stress changes locally causing slope instabilities which might

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eventually lead to failure. Surface deformation associated to slope instabilities in the Great

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Aletsch valley has been recently recognized and measured with space borne and ground based

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remote sensing methods (Kos et al., 2016; Strozzi et al., 2010). Since 2013, a geodetic

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monitoring system has been deployed to study into detail reversible and irreversible deformation

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of the rock slopes located in the vicinity of the glacier terminus. A combination of two Robotized

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Total Stations (RTS) and GNSS systems provide high resolution information to investigate the

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role of glacier cycles on the progressive degradation of the rock mass, as well as on the

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evolution towards slope failure processes. For a detailed description of the geodetic monitoring

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system we refer the reader to (Loew et al., 2017). Early in July 2015, local authorities have

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recognized the progressive formation of tensile fractures in the area named Silbersand (see

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photos in the Supplementary Material, S1), located on the southern slope of the Aletsch valley

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and adjacent to the current (summer 2015) position of the glacier tongue. To support the local

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authorities, the Mobotix M25 TLC camera (Figure 1b) was set to acquire images every hour over

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the area showing instability signs (see Figure 2). In addition, to monitor the potential evolution of

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surface displacements towards a slope failure, we have installed 3 reflectors to be measured

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with the RTS (Figure 1c). The installation of more points was initially planned to gain more

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information on the spatial extent of the active deformation; however, field access was limited due

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to the slope steepness and surface instabilities. The acquisition of multitemporal optical imagery

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was initially intended to perform visual interpretations of the potential evolution towards 4

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catastrophic failure of the slope, which could partially or totally block the Massa river outflow and

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eventually cause secondary hazards downslope. Subsequently, TLC imagery has been

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considered to retrieve the evolution of surface deformation with DIC algorithms. In the following

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we describe the steps applied from data acquisition to the generation of displacement time

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series relevant to active portions of the Silbersand slope.

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3. From TLC acquisitions to displacement time series

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The selection of TLC sampling rate and picture resolution is usually a function of the available

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power supply, as well as on the storage capacity (when data are saved on site), or the available

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bandwidth for data transmission. In our case, data were acquired at the maximum picture

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resolution available for the Mobotix M25 camera (2048x1536, i.e. 3Mpixel) and directly

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transmitted hourly via GSM network to a remote server located in Zurich. We collected

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continuously the pictures from July 8 to September 23, 2015, with few exceptions due to data

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transmission problems in the periods July 20 – 23 (no data available), and August 03 – 15 (only

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10% of pictures available). Due to this data loss, the final number of TLC raw pictures collected

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was reduced of about 30% compared to the sampling specifications; however, this has not

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affected the possibility to obtain valuable information over the whole time period.

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3.1 Supervised selection of suitable TLC pictures

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As mentioned in the introduction, the information obtained from a TLC installed in an open

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environment is deeply affected by continuous changes in light conditions, which may result in

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shadowing effects (cast shadows and/or indirect shadows), direct sunlight exposure, excess of

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darkness early morning and/or late evening. If the monitoring period continues for several

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months, light/shadowing conditions may vary also due to seasonal changes of sun angles.

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Moreover, obstacles and/or variability of the atmosphere (clouds, fog, heavy rainfall, snowfall,

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etc.) further reduce the amount of usable imagery. The selection of images suitable for DIC

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analyses can be performed manually, in order to guarantee visibility of the target area, and verify 5

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visually that lightning/shadowing properties are similar (Giordan et al., 2016). However, when the

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dataset is very large, the manual selection becomes inconvenient. Moreover, manual selection is

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suitable for back analyses, but not applicable in a context where near-real time monitoring

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applications might be necessary.

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Several algorithms were proposed to perform filtering/denoising of digital imagery (Dravida et al.,

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1984), remove shadows (Fredembach and Finlayson, 2006), as well as to identify similarities

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multitemporal acquisitions (Wang et al., 2005 and references therein). Here we implemented a

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supervised approach to find images suitable for the subsequent DIC analyses. To this end, we

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selected as reference image (master) one picture of the target area with good light conditions

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(i.e., limited cast shadows) and free of atmospheric problems. For this image (hereafter called

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master), we defined a characteristic function (CFM) as follows:

𝐶𝐹𝑀 (𝑥) =

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Σ𝑦 𝐼(𝑥) 𝑁𝑦

(1)

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where I is the intensity obtained by converting original RGB to gray values, and Ny the number of

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pixels in the vertical axis of the image (y). In other words, with the equation (1) we calculate the

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average intensity along the pictures’ horizontal axis (x), which is in our scenario the most

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affected by changes due to the light/shadowing cycles. Secondly, we modeled CFM with a

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polynomial function and obtained MCFM. The latter is iteratively compared to the characteristic

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functions of the other pictures of the dataset (i.e., characteristic functions for slave images, CFS).

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The fitness between MCFM and CFS was evaluated by considering the norm-1 of their difference,

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as follows:

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‖𝑀𝐶𝐹𝑀 − 𝐶𝐹𝑠 ‖1 ≤ 𝑘 ∙ ‖𝑀𝐶𝐹𝑀 − 𝐶𝐹𝑀 ‖1

(2)

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where k is the threshold defined for the selection of pictures similar to the master. The norm-1

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was selected as estimator in equation (2) because of its robustness to outlier values (Ke and 6

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Kanade, 2005). A collection of the different situations affecting the quality of the images acquired

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in our area of study, and the comparison between the MCFM and CFS, is provided in the

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Supplementary Material, S2. With the supervised selection strategy, only 36% of the available

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TLC imagery was selected as suitable for the subsequent DIC analysis. The comparison

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between the supervised selections relying on different norm-1 thresholds and manual selection

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based on subjective evaluation of the pictures shows that, by assuming k values between 3 and

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5, we obtain very similar performances, i.e. Simple Matching Coefficients are larger than 0.8

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(Sokal and Sneath, 1963). Values of k, as well as the polynomial order to be considered for

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MCFM (in our case a 4th polynomial order has been considered), are site specific and have to be

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calibrated for the area under investigation.

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3.2 Image pairing

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After the selection of the dataset, a criterion to pair images to be processed and obtain

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displacement time series has to be identified. If the temporal baseline is relatively short (in our

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case the sampling rate is 1 picture/hour), the detection of potential displacements via DIC

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approaches is limited by the image resolution (Leprince et al., 2007). On the other hand, in

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environments characterized by large slope movements, image pairs with rather long temporal

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baseline (e.g., several weeks or even months) can be affected by very large surface changes,

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and thus unsuitable for reliable DIC results. The minimum and maximum temporal baseline for

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pairs are thus also site-specific parameters. Due to preliminary knowledge of the study area and

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initial DIC tests, we expected deformation in the order of 10-20 cm/day, and we defined one day

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as minimum and seven days as maximum temporal baseline. Then, images are paired only if

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acquired at the same hour of the day, in order to mitigate the effects of cast shadowing. Such

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criterion resulted in a total of 1,123 pairs to be further processed with DIC.

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3.3 DIC analysis

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Before applying DIC and obtain surface deformation, images acquired at different times have to

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be aligned by relying on control points and/or on areas assumed to be stable over the

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investigation period (Giordan et al., 2016; Travelletti et al., 2012; Wang et al., 2015). In our

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pictures, part of the northern slope of the Aletsch valley within the observation time is imaged

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(opposite to the area of investigation, location Driest, where the TLC was installed, see also

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Figure 1 and Figure 2); therefore, we selected a sub-portion of this area as reference for pictures

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alignment (also referred to as co-registration step). Subsequently, DIC was used to evaluate

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internal misalignment of sub-portions (i.e., within Correlation Windows, CW) of the images to find

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potential deformation of the investigated area along two directions (i.e., orthogonal to the camera

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sight, the so-called camera view plane).

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In recent years, several DIC algorithms have been proposed, each one with different benefits

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and drawbacks (Heid and Kääb, 2012). Here we consider the approach presented in (Guizar-

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Sicairos et al., 2008), which operates in the frequency domain and obtains an initial estimate of

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the cross-correlation peak between master and slave images within search windows using a 2D

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Fast Fourier Transform (FFT). Moreover, the normalized root mean square error (NRMSE) is

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used as metric for cross-correlation accuracy (Fienup, 1997). The implementation of DIC

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approaches based on FFT are known to perform faster than other algorithms, and this is

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convenient in our scenario due to the large amount of pairs to process (Heid and Kääb, 2012).

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After preliminary tests, we selected the dimensions of the Correlation Windows (CW) as 16x16

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pixels. This is a compromise between computational time, accuracy, and expected spatial

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resolution of the displacement field (Delacourt et al., 2007). Following Nyquist-Shannon theory,

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spatial sampling was thus 8 pixels (in both directions orthogonal to the camera sight), and an

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oversampling factor 4x was applied to further increase the DIC theoretical accuracy (Leprince et

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al., 2007). We applied DIC on the 1,123 pairs and computed offset velocities (in pixels/day) for

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both directions orthogonal to the camera sight, as well as the NRMSE. We remind that the offset 8

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values retrieved with DIC are representative of the offset occurring within the correlation window,

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and the results are for convenience assigned to the center of CW. Thus, the final DIC results are

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represented as the displacements occurred within “cells”, with ground sampling distance in

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agreement with the DIC sampling.

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To compute time series, we defined a time step of 1 day, suitable for our investigation and

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available dataset and in agreement with the minimum temporal baseline used for image pairing,

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see also section 2.2), and stacked (averaged) “cell-wise” the obtained DIC velocities (see Figure

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3). Similar stacking approaches are frequently used to increase the signal-to-noise ratio in large

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remote sensing datasets (Gorelick et al., 2017; Stumpf et al., 2017). To further reduce noise

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effects and outliers on the final displacement time series results, we consider for each time step

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only cells presenting small NRMSE. Finally, the velocity results can be integrated over time to

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compute displacement time series for each cell over the area of interest.

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4. Results

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Figure 4 shows a relevant subset of the results obtained applying the time series approach

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described above (complete DIC time series results are presented in the Supplementary Material,

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S3). Looking at the spatial and temporal evolution of the slope, we note that the initial phase is

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characterized by minor displacements until August 19th. Before this date, signs of deformation

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can be noticed only at the interface between the glacier and the slope. Instead, clustering of

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cells with deformation becomes remarkable in the Silbersand area within a relatively short period

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(between August 19th and August 26th). At this stage, the portion of the active slope located on

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the western side (right side in the figure) entails the largest deformation values. There, the

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surface deformation pattern is bounded on the western side by the progressive opening of a

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tensile fissure, and its lateral evolution can be observed over time. A closer look to velocities and

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displacement time series for a point at this location (see Figure 5b) reveals an exponential

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evolution of the displacements between August 19th and August 22nd, followed by a sharp 9

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decrease of the deformation trend and a progressive reduction of the deformation until the end

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of our investigation period (September 23rd). The displacement time series measured at different

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locations (see Figure 6 a-e) within the active slope area show also an increase in the same time

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period, despite less pronounced. Considering the average ground pixel dimensions in the area

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analyzed, we obtain total displacements in the period July 8 and September 23, 2015 ranged

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between 0.5 and 5 meters. The temporal evolution of the DIC time series is very similar to the

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measurement obtained from the RTS at the point p65 (see Figure 6f); however, the total

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amplitude over the whole time period is sensibly smaller (see section 5 for details). The

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displacement maps obtained with DIC allowed us defining the total area affected by the

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deformation (~0.01 km2), as well as to identify subdomains within the active slope characterized

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by different deformation velocities.

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5. Discussion

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Monitoring spatial and temporal evolution of displacements in alpine environments is important

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to evaluate and understand the effects of climatic changes, as well as to deal with potential

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hazards associated to catastrophic failures (Huggel et al., 2012). Slope areas can be very

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difficult to access and/or affected by rapid surface changes, thus unsafe for in situ installations.

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For this reason, remote sensing approaches are often a suitable alternative. Recently, several

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studies have shown how active ground-based remote sensing techniques (e.g., Radar and

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LiDAR) can be used to gain accurate information on slope processes (Abellán et al., 2014;

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Kieffer et al., 2016; Rouyet et al., 2017). However, despite their versatility and achievable

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accuracies, these approaches might be hindered because of unsuitable logistics, poor revisit

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times, and/or high costs of installation and management for monitoring purposes. On the other

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hand, passive remote sensing methods such as the acquisition of optical imagery with TLC

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offers also the possibility to quantitatively monitor the evolution of surface deformation in

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different scenarios. Indeed, DIC approaches applied to multitemporal acquisitions of pictures of

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the same scene can be used to map and track deformation over space and time, provided that

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the coherence between analyzed images is preserved in the target area (Travelletti et al., 2012).

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In this study, we have shown how to leverage on large datasets of optical images acquired from

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an off-the-shelf TLC installed in high alpine environment. Using the herein proposed approach,

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we mapped and monitored slope deformation in a rock slide in the Silbersand area at the tongue

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of the Great Aletsch Glacier. The results have provided details on the deformation occurred,

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where early in summer 2015 signs of a potential failure were recognized from field inspections.

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The supervised selection of TLC imagery allowed to work on a homogeneous dataset and to

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decrease the amount of data to process with DIC, although retaining the information about the

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deformation occurred over space and time. In addition, the stacking approach allowed to

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increase the signal-to-noise ratio and, despite the relatively low resolution of the TLC data

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available, to achieve accuracies in the order of fractions of pixels in agreement with previously

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presented DIC time series approaches (Casu et al., 2011; Stumpf et al., 2017).

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The time series show that in the period between July and September 2015 a portion of the

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Silbersand slope has experienced rapid surface displacement, in particular during the period

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August 19th-22nd. By taking a closer look to the TLC pictures available, it is possible to note that

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on August 22nd a portion of the Silbersand slope has failed, and that the rock blocks and debris

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have reached the Massa river levels; however, the failed material was not sufficient to interrupt

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the river flow. Morphological details of this failure are evidenced on the high-resolution pictures

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acquired from a high resolution TLC installed closer to the Silbersand active slope in the event’s

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aftermath (see location of high resolution TLC in Figure 1 and details in Supplementary Material,

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S4).

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The final stages of this failure event have developed very rapidly (within 3-4 hours, see

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Supplementary Material, S5). Such a fast evolution could not be fully tracked by our analysis,

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because of the temporal step selected for the generation of the time series (1 day). In addition, 11

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the constraints imposed to perform a full back analysis of the dataset over the area of interest

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(i.e., coherence preservation for the time period analyzed), cause underestimations of

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displacements in the specific portion affected by the failure. A careful analysis of the time series

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evidences that most of the DIC cells at this location are below the NRMSE thresholds for most of

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the time period, and only after August 22nd start to show some reliable deformation probably

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associated to the post failure phases (see .gif animation Supplementary Material). The

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exponential evolution towards failure in the vicinity of the failing zone, as well as a noticeable

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drop of displacements after the event, is well in agreement with the independent results obtained

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from the RTS (see also Figure 6). Several additional slope failures have occurred since that

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time, mainly in the upstream direction and correlate with the progressive retreat of the Great

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Aletsch glacier tongue position.

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A number of limitations have to be taken into account when using the DIC results for a correct

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interpretation of the surface deformation retrieved. Indeed, depending on the acquisition

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geometry only portions of the actual displacement vectors can be retrieved with DIC. In our

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specific case, the camera-viewing angle is approximately orthogonal to the slope area analyzed

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(see also Figure 1); thus, the along- and across-slope displacements occur in the camera plane

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and can be fully tracked. However, in cases where the viewing geometry is not orthogonal to the

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slope, specific corrections have to be applied considering the topographic relief of the area of

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study (Travelletti et al., 2012; Giordan et al., 2016). Furthermore, we remind again that out-of-

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camera view plane deformation cannot be measured with DIC. This is an intrinsic limitation of

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the approach, which could be mitigated only by combining measurements from multiple

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acquisitions of the same area at the same time from different view angles, as already presented

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for satellite based imagery (Casu and Manconi, 2016). In our results, this problem is evidenced

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by the comparison between DIC and the point p65 measured with RTS. With DIC, we measure

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only average deformation occurring in the camera plane, while with RTS we retrieve the 3D

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deformation at a single point. Values are compatible in amplitude until August 22nd, i.e. the day 12

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of the failure occurring on the western portion; however, after this date the p65 shows a further

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increase of the displacements (despite with reduced trend), while the DIC time series fade

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progressively. We interpret these differences as an expression of localized large displacement

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occurring only in the vicinity of the installation of p65, which ended with the final failure of the

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target (not measurable with RTS after September 23rd). Thus, displacements measured at p65

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are not fully representative of the behavior of the entire slope. Similar local problems at targets

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measured with RTS have been experienced in other unstable slopes (Manconi and Giordan,

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2015). Despite local inaccuracies, our results suggest that the DIC methodology proposed is

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suitable to identify spatial and temporal changes in the deformation pattern from TLC, and to

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provide appropriate and additional information helping in the evaluation of the status of activity of

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an active slope area.

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6. Conclusions

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We have developed a method to derive displacement time series from of active slope processes

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from large datasets of time-lapse cameras acquisitions. The application to a case study in the

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Swiss Alps provided important hints on the spatial and temporal evolution of a rock slope failure

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occurring in the vicinity of the Great Aletsch glacier tongue. The surface acceleration toward

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failure has been retrieved and the behavior of the slope deformation is in agreement with the

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results obtained from ground based monitoring. Moreover, the lateral evolution of the

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displacements from the west to the east end of the slope evidenced a potential relationship

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between the slope activity and the glacier retreat. This will be further investigated and modeled

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to derive a better understanding of the physical processes underlying the interaction between

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the glacier and the Silbersand slope. At this stage, we tested the performance of the method on

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a back-analysis of TLC acquisitions; however, our approach can be straightforwardly

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implemented to perform near-real-time monitoring of surface deformation. Our results confirm

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that the use of low-cost TLC may provide quantitative and fairly accurate alternatives to measure

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the evolution of surface deformation in areas characterized by active slope processes and

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elsewhere.

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7. Acknowledgements

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We thank P. Schwitter Gmbh and Ruppen Michael from Odilo Schmid und Partner AG for

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support in the field operations.

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Figure 1. The Great Aletsch area, Swiss Alps. (a) Overview of the study region with indications of the Silbersand landslide area and locations of monitoring sensors (vertical scale exaggerated for visualization). (b) Mobotix M25 webcam and (c) Robotized Total Station Leica TM50, respectively, both installed at the Driest location.

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Figure 2. Example of the pictures acquired from the Mobotix M25. X and Y are the directions in the camera view plane, i.e. orthogonal to the camera sight, respectively. The target area of Silbersand is bounded by the red rectangle, and the unstable slope (yellow polygon) is in the center of the scene. The contact between the southern slope and the southern boundary of the Aletsch glacier is also visible in the pictures. The bottom of the picture (white shading) images a portion of the northern slope, i.e. Driest, where the TLC is installed. This area is considered as stable reference for DIC analyses (orange rectangle). Location of the RTS point targets (p61, p63 and p65) has been selected to monitor the potential evolution of the tensile fractures (black lines) identified early in July 2015.

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Figure 3. Schematic representation of the stacking approach used to generate time series from multitemporal TLC acquisitions. For each pair (Pi) the surface deformation velocity (vi) is computed with DIC. Velocity (𝑉𝑛 ) for each time step (tn) is then calculated as the average of DIC velocities falling within the defined temporal window. For example, average velocity for the time t2 is the average of v2, v3 and v4, as well as any additional DIC result covering the considered time step (see gray shading). This approach allows increasing the signal-to-noise ratio, as the average of random noise will tend to zero while real offsets of DIC results will be highlighted. Displacement time series for each DIC cell can be calculated by integrating average the resulting 𝑉𝑛 velocities over time.

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Figure 4. Cumulative surface displacement in the y camera direction (downslope) as measured from the DIC time series analysis. Dates of the relevant time steps shown is reported on top of each snapshot.

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Figure 5. (left) Total displacement between July 8 and September 23, 2015 (y camera axis) in the Silbersand area. (right) Surface velocity (gray bars) and displacement time series (orange solid line) for the point (a) over the period of investigation.

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Figure 6. (a-e) Displacement time series as measured from DIC for different points located in the Silbersand active slope (see location in Figure 5). (f-g) Three-dimensional displacement time series measured from the Robotized Total Station at different point targets (see location of the station and of the target points in Figure 1 and Figure 2)

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