Time forecast of a break-off event from a hanging glacier

A cold hanging glacier located on the south face of the Grandes Jorasses (Mont Blanc, Italy) broke off on the 23 and 29 September 2014 with a total estimated ice volume of 105.000 m. Thanks to accurate surface displacement measurements taken up to the final break-off, this event was successfully predicted 10 days in advance, enabling local authorities to take the necessary safety 5 measures. The break-off event also confirmed that surface displacements experience a power law acceleration along with superimposed log-periodic oscillations prior to the final rupture. This paper describes the methods used to achieve a satisfactory time forecast in real time and demonstrates, using a retrospective analysis, their potential for the development of early-warning systems in real time. 10


Study site
The Whymper glacier is located on the south face of the Grandes Jorasses (Mont Blanc massif, Italy) between 3900 and 4200 m asl (Fig. 1). The front of the glacier is about 90 m wide and its surface area amounts 25,000 m 2 . This very steep cold hanging glacier (about 40 • ) lies above the village of Planpincieux and the Italian Val Ferret, a famous and highly frequented tourist destination both 70 in winter and summer. In 1997, six boreholes were drilled down to the bed and temperature profiles were measured, indicating basal temperatures below the freezing point (below −1.6 ± 0.4 o C) at all locations (Pralong and Funk, 2006). Historical data and morphological evidence indicate that the glacier experienced recurrent break-off events that can be dangerous, particularly in winter, when the initial ice avalanche can drag snow in its path. This hanging glacier periodically broke off in the 75 past leading to large avalanches that flowed down into ::::::

Break-off event history
The glacier broke off several times during last 100 years. Some of these events have been observed and reported: -On 21 December 1952, after an intensive snowfall period, a huge avalanche was released 80 below the Grandes Jorasses which destroyed a 200-year old forest and blocked the bottom of the Val Ferret over a distance of more than 1 km. The avalanche volume was estimated at more than 1,000,000 m 3 . It is not clear whether the snow avalanche was triggered by an ice avalanche from the Whymper glacier.
-In August 1993 and July 1996, the glacier released ice avalanches of 80,000 and 24,000 m 3 , 85 respectively. These ice avalanches did not reach the bottom of the valley. According to Pralong and Funk (2006) the formation of the upper crevasse was observed 2.5 90 years before failure .

Present monitoring: 2009-2014
The survey primarily consisted of surface displacement measurements with an automatic total station and GPS as well as close-range photogrammetry (Margreth et al., 2011). Two reflectors set on the rock on both sides of the glacier were used as reference, and several reflectors mounted on stakes 95 were directly drilled into the ice, so that their exact positions could be monitored (Fig. 2). Because of instrument problems, the seismic activity unfortunately could not be monitored as initially planned.
Starting in 2010, surface displacements were continuously recorded at several stakes at 2-hour intervals (when the prisms were visible, i.e., good weather conditions) with the aim to timely detect an impending ice avalanche (Margreth et al., 2011). Using the same correction technique as de-100 scribed by Faillettaz et al. (2008) :::::: (section :::: 4.1), the surface displacements could be determined with an accuracy better than 1 cm, allowing surface velocities to be inferred.
In parallel to the monitoring program, a safety concept for the valley floor was developed considering several scenarios of falling ice volumes. The different ice avalanche scenarios were simulated using the 2-dimensional calculation model RAMMS :::::::::::::::::: (Christen et al., 2010) . The necessary safety 105 measures were defined according to the local avalanche danger level and the potential volume of a break-off event (Margreth et al., 2011).

The 2014 break-off event
From 2010  of September, the monitoring was operational up to the final break-off. By chance, there was one reflector on each of the two unstable parts and one on the stable part, (Fig. 2).

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Striking qualitative analogies with those of the 2005 Weisshorn event (Faillettaz et al., 2008) can be highlighted.
1. This steep cold hanging glacier experiences periodic break-off events.
2. The geometrical configuration of the glacier is similar before each break-off, with an upper crevasse spanning the whole glacier ::::: width and a clear thickening of the glacier towards its 6. The whole break-off occurred in two steps; a minor section at the left side of the glacier was released first.
3 Previous findings on cold glacier break-off

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Based on a retrospective analysis, the main conclusion drawn by Flotron (1977) and Röthlisberger (1981) was that the forecast of a break-off event from a hanging glacier was possible using surface displacements alone. The principle is to fit the characteristic acceleration of the surface motion with a power law behavior of the form: 140 where s(t) is the displacement (in meters) at time t (in days), s 0 a constant in meters, u s the constant velocity of the upstream stable part (in md −1 ), t c the critical time (in days), θ < 0 (without units) and a (in md −θ ) the parameters characterizing the acceleration. In this way, the critical time t c , i.e., time at which the theoretical displacement becomes infinite, could be evaluated using such empirical law. Although the break-off event would necessarily occur earlier, this critical time rep-145 resents the upper limit of the break-off timing. Moreover, an oscillating pattern superimposed on the power law acceleration of the surface displacements was evidenced prior to the 2005 Weisshorn event (Pralong et al., 2005;Faillettaz et al., 2008). This peculiar glacier dynamics was shown to be a log-periodic oscillating process superimposed on this acceleration (for appearance and interpretation see ::::::::::::::: Pralong (2006) and : Faillettaz et al. (2015)). The time evolution of the surface displacement mea-150 surements can be described with the following equation (after Sornette and Sammis, 1995;Pralong et al., 2005): where C : is : the relative amplitude (without units), λ the logarithmic frequency (in days) and D the phase shift of the log-periodic oscillation (without units).

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Thanks to a combined analysis of surface displacement and seismic measurement, Faillettaz et al.
(2011a) were able to obtain a coherent quantitative picture of the damage evolution process developing before the 2005 Weisshorn break-off. They have suggested three regimes in the evolution of the failure process leading to the break-off event: (i) A first stable phase related to a self-organizing regime, where diffuse damage accumulates 160 within the glacier, with a proliferation of dislocation-like defects.
(ii) A transitional phase where the damage process goes on, micro-cracks grow and start merging in a homogeneous way. Log-periodic oscillations appear and reveal the hierarchical structure of the fracture process under development.
Note that this constitutes a unique dataset not only because of the great accuracy and long measurement period but also due to available surface displacement data up to a few hours prior to the 185 break-off event. Whereas surface velocities at Stake 4 are approximately constant (Fig. 4), the three other stakes show a clear acceleration which is typical for an unstable situation. According to this observation we can expect that the glacier section around Stake 2, 13 and 14 will break-off, while the section around Stake 2 will remain stable (section 2.4).

Application to forecasting
190 Previous findings (section 3) were applied in order to forecast the breaking-off event in real time.
As soon as a significant increase in velocity was detected, the same procedure was followed as in Faillettaz et al. (2008). We periodically fitted surface displacements of all stakes to a power law (Eq. 1) and a log periodic oscillating behavior (Eq. 2). The nonlinear least-squares curve-fitting was performed using the Levenberg-Marquardt algorithm. Because the results depend on the initial    Table 1 contains the values of the parameters in Eq. 2, taking λ = 2 d. Note that measurements are available up to the final break-off for 3 prisms (i.e., Stake 13, Stake 2 and Stake 4) and stopped on 16 September for Stake 14, i.e., 8 days before the first break-off.

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It appears that the power law behavior describes well the surface displacements with a maximum discrepancy of about 5 cm for Stake 14 (8 days before break-off), about the same order of magnitude as the one observed during the 2005 Weisshorn event (Fig. 5). However, residuals indicate an oscillating pattern. When using the log-periodic function (Eq. 2), the agreement between measured and fitted values (dashed gray line) becomes significantly better, with an accuracy of the order of mag-210 nitude of the measurement accuracy (less than a centimeter). Results show that the critical time can be expected around the 3rd October for both stakes, which is fairly close to the observed break-off.
Note that such an approach can be used to investigate how far in advance a reliable time forecast is possible (see section 5.4).
However, even if Stake 14 is located on a section that broke off earlier, no significant differences 215 could be detected. Our approach is not able to detect whether the break-off will occur all at once or as successive small events. Now when considering the entire dataset for Stake 13 (where measurements could be recorded up to the break-off) using the same method, it appears that the amplitudes of the oscillations superimposed on the power law acceleration become even larger close to the break-off -they reach values 220 up to 30-40 cm (Fig. 6). Such a broad oscillating pattern has never been observed before, confirming that the jerky motion of the glacier (with oscillating nature) might have a physical origin (see Section 5.2).
Stake Initial data ± 1cm noise ±5cm noise fests itself in data by log-periodic corrections to scaling. Several mechanisms may lead to this partial 245 breaking of the continuous symmetry. Thanks to a combined analysis of surface displacements and seismic measurements, Faillettaz et al. (2011a) suggest that it results from the dynamic interactions between newly developed micro-cracks, as shown by Huang et al. (1997) and Sahimi and Arbabi (1996).
To identify the log-frequency, we analyzed the data in the same way as Faillettaz et al. (2008) 250 with a Lomb periodogram analysis (Press, 1996;Zhou and Sornette, 2002b), which is designed to analyze non-uniformly sampled time series. This method enables us to determine f Lomb as a function of cos(2πf Lomb t). The parameter λ in Equation 2 can then be evaluated easily as λ = e 1/f Lomb .

Accurate determination of break-off occurrence
As critical time t c given by power law or log-periodic fit indicates when surface displacements become theoretically infinite. However, the break-off event is expected before t c . When fitting in real time the surface displacements with both power law and log-periodic behavior, it is not only 295 possible to assess the critical time but also the time at which the derived velocities are expected to reach a given threshold (for example 50 cm d −1 or 1 m d −1 ). Fitting and estimating the time at which the velocity reaches a given threshold provides a more accurate way to predict the break-off event. We developed a software based on this idea by fitting in real time the measurements with both power law and log-periodic behavior and thus provide an estimate of the break-off time.

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According to our knowledge, it is not possible to know in advance the velocity at which breakoff will occur. However, from previous events (Weisshorn 1973and 2005event, Flotron (1977; Röthlisberger (1981); Faillettaz et al. (2008)), it seems that break-off occurs between 50 cm d −1 to 1.2 m d −1 , but this is based on a restricted number of events.
Taking threshold surface velocities of 50 cm d −1 and 1 m d −1 , our analysis (using Eq. 2) per-305 formed every days from the 12 September to 16 September suggested that break-off could occur between the 23 September (v th = 50 cm d −1 ) and the 29 September (v th = 1 m d −1 ). Note that the two breaking-off events occurred exactly at these two days, which were forecasted 10 days in advance. Following this analysis, alert was immediately sent to the authorities leading them to close the endangered area one week before the event. Note that the definition of the velocity threshold has 310 an influence on the prediction itself, as we saw nearly one week is needed for the glacier to accelerate from 50 cm d −1 to 1 m d −1 . The precise prediction would also not only be based on a correct fit of the surface displacement data but also on a guess of this parameter. We suggest to choose 40 cm d −1 as a conservative threshold to define a safe break-off danger time interval. :: It :: is ::: not ::: yet :::: clear :::::: which :::: value ::: has ::: to :: be ::::::::: considered :::::::: according :: to ::: the :::::: results ::::: from :: the :::::: events :::::::: analyzed :: so ::: far. : panels) account for the fitting procedure.

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First, the prediction is better when using log-periodic fit than power law fit. This retrospective analysis shows that the prediction is correct after 12 September, i.e., 11 and 17 days before the break-off with a confidence interval becoming less than than 10 days with a log-periodic fit.

Overall recommendations
According to the knowledge gained from the different studies on Weisshorn, Mönch and Grandes Jorasses glaciers, accurate data are required to forecast an impeding break-off event. As the amplitudes of the log-periodic oscillations are increasing towards the break-off (from 5 cm one week 335 before the break-off to 40 cm at the break-off), the confidence of the time forecast strongly depends on the precision of the surveying data. To ensure a satisfactory forecast about one week in advance, a surveying accuracy better than half of the expected log-periodic amplitudes, i.e., 2.5 cm, is required.
In this study an accuracy of 1 cm was achieved with an automatic total station (Leica theodolite TM1800 combined with the DI3000S Distometer). The sampling rate needs to be adapted to the  The break-off ( :::::::::: corresponding :: to : 0 :: on :::::: x-axis) ::::::: occurred :: on :: the : 23 rd ::::::: September : ( : a : ) and 29 th September : (b). Bottom :::::: Vertical :::: grey ::: line :::::::: represents :: the ::::::: observed :::::::: break-off. :::: Right: Error in days on critical time fitted with power law (blue) and log-periodic (red) estimated from the 95% confidence interval : as :: a :::::: function :: of ::: the :::: time ::: prior :: to ::: the ::::::: break-off :::: event ::::::::: (t analysis ). Errors on v50 and v100 are similar to the errors on critical time, as they are directly derived from these fits. each days. A sampling rate of 2 hours was chosen in this study, ensuring thus several opportunities to obtain data every day. This technique can be performed in near real time and several measurements can be performed every day with a sufficient accuracy. Note that GPS measurements would 345 be a valuable alternative but this technique requires a long acquisition time and additional processing to achieve to required accuracy. Although independent of weather conditions, the power supply and data transmission are problems to be solved. This procedure based on power law/log-periodic oscillations regression requires at least two measurements points on the potentially unstable part of the glacier, so that the time evolution of surface motion at different points could be compared. It also 350 ensures that the results are not affected by stake/prism stability issues.
An alternative surveying technique is terrestrial Insar. The advantage of this technique is that no installation on the glacier (potentially dangerous) is required. However the data accuracy which can be expected with this monitoring system is not completely clear yet (Preiswerk et al., 2016) 6 Conclusions 355 Grandes Jorasses glacier broke off twice, on 23 rd and 29 th September 2014. In 2008, as it was suspected that a large part of this glacier is becoming unstable, a monitoring program was initiated.
At the time of the break-off, 4 stakes covering a large part of the glacier enabled surface displacement measurements up to the time of the break-off. By regularly analyzing the dataset, it was possible to forecast the event ten days in advance. In the following the local authorities closes the endangered area up to the final rupture.
It was possible to confirm for an impeding ice fall that a time series of surface displacements exhibits strong log-periodic oscillations superimposed on a global power law acceleration, as first discovered for the Weisshorn event (Faillettaz et al., 2015). In the immediate vicinity of the breakoff, such oscillations reached an amplitude of more than 40 cm, almost one order of magnitude larger 365 than revealed in previous findings. By fitting our recorded surface displacements, the critical time, i.e. time at which surface displacement become infinite, can be determined. Using this critical time value as an upper bound, a good time forecast could be achieved.
The inferred surface velocities immediately prior the two events were 0.5 m d −1 and 1.2 m d −1 , in the same range as for the Weisshorn event, suggesting that break-off of a cold hanging glacier 370 could occur as soon as surface velocities reached 0.5 m d −1 . We showed that evaluating the time at which extrapolated velocities (based on the log-periodic fit) reach a prescribed threshold (0.5 m d −1 and 1 m d −1 ) provides a significantly better forecast. However, in the present case, surface velocity increased from 50 to 100 cm/d in the order of one week. In practice, we suggest to use a critical velocity of v=0.4 ::: 0.5 m d −1 to determine the period of highly likely break-off occurrence. A 375 retrospective analysis based on this method showed that an accurate prediction of the phenomenon can be achieved two weeks before its occurrence using the last month of surface displacement data and 0.5 m d −1 and 1 m d −1 as velocity thresholds. Although enabling a crude estimation of the total unstable ice volume, this point based surveying procedure is not appropriate to determine whether the unstable ice mass will fall down in one event or disaggregate and give rise to several smaller events, 380 as no differences in the evolution of surface displacements were detected. This has consequences for the risk evaluation, as the resulting ice avalanche (and also the chain of processes resulting from its release) depends on the falling ice volume. To conclude, our results suggest that the presented monitoring and data processing techniques exploiting the log-periodic oscillating behavior can be applied in real time to forecast a break-off event on any cold unstable hanging glacier.