The general framework proposed in this study is based on the observation that the occurrence of rockfall and small landslides increases over time prior to a larger failure (Huggel et al. 2005; Rosser et al. 2007; Szwedzicki 2003). Suwa (1991) and Suwa et al. (1991) proposed that the magnitude of the ultimate failure is proportional to the level of precursory behavior, and this suggests a dependence on the scale of the precursory events (Rosser et al. 2007). This phenomenon has been validated also for volcanic eruptions (that obey to the same precollapse behaviors as landslides, as demonstrated by Voight 1988) by the long-period observation of Hibert et al. (2017b). These authors continuously monitored, from 2007 to 2011, the Piton de la Fournaise volcano using seismically triggered video cameras, and analyzed the temporal evolution of the daily number of rockfalls. They found that the most active period, in terms of both the number of rockfalls occurrences and the volumes of material displaced, was the one immediately preceding the collapse of the Dolomieu crater. In this paper, instead of using the occurrence frequency of rockfall as a proxy for predicting a larger failure, the accumulated energy (Ae) recorded by a seismic network when the rock hits the ground (which is a function of the volume) was employed.
The whole framework is illustrated in Fig. 2 and is described in detail in the next section. To perform this framework, the first step is transforming seismic data recorded by the instruments into a data format that can be processed, in order to detect and classify the different seismic events by a rockfall detection and classification model. The second step is extracting all the seismic signals and features of the rockfalls, such as onset time and duration; then, Ae and the accumulated energy increment (ΔAe) in a time window are calculated, and, if ΔAe exceeds a fixed threshold, 1/Ae is calculated, a time of failure is automatically extrapolated, and a warning time declared. In parallel, as soon as the time of failure is computed, all the rockfalls detected in a fixed time window before the triggering of the first alarm time are localized in a topographical map to find the susceptible area from where the events probably originated.
Temporal forecasting
Let us assume that a series of rockfalls (R1, R2, ⋯, RM) is detected in a time period (t0 − t). Instead of the real event source energy, a parameter represented by the sum of the relative seismic energy (Es), measured in m2/s2 and recorded through the seismic network, is adopted in this study, in order to reduce the chance of introducing calculation errors of the environmental influence and the effects caused by the energy loss through propagation attenuation, rock fragmentation, and heat generation (Amitrano et al. 2005; Dammeier et al. 2011). Therefore, the relative seismic energy of the rockfall series is E1, E2, ⋯, EM, and the seismic signal time series of one rockfall recorded in component j and station k, is presented as xkj1, xkj2, ⋯, xkjN.
There are four three-component geophones employed in this case, so the sum of the relative seismic energy (Es) of one rockfall and the accumulated energy (Ae(t)) of the rockfall series from time t0 to t, respectively, are:
$$ {E}_s=\overset{k=4}{\sum \limits_{k=1}}\ \overset{j=3}{\sum \limits_{j=1}}\ \overset{i=N}{\sum \limits_{i=1}}{x}_{kji}^2 $$
(1)
$$ {A}_{e(t)}=\overset{s=M(t)}{\sum \limits_{s=M\left({t}_0\right)}}{E}_s\kern0.5em $$
(2)
where i, j, k, and s are the number of seismic samples, components, seismic stations, and rockfalls, respectively. M(t) is the accumulated number of rockfalls at the time t in the rockfall series.
In order to trigger the calculation of the time of failure, a sliding time window is created, and the accumulated energy increment (ΔAe(t)) in that sliding window at time t is continuously calculated according to Eq. 3. It is important to note that this parameter represents an empirical threshold (δ) below which the forecasting methods are not implemented because they would probably trigger false alarms. In the study area, the length of the sliding window was set equal to 1 h (the yellow area in Fig. 3) and stepped by 1 min, while the empirical threshold (δ) was set equal to 0.5 m2/s2. Such value can be calibrated once that more data and experience on the specific site are gathered.
$$ \Delta {A}_{e(t)}=\overset{s=M(t)}{\sum \limits_{s=M\left(t-1h\right)}}{E}_s $$
(3)
The times when ΔAe(t) exceeds a certain threshold (δ) (here called “alarm time”, ta) are shown as red circles in Fig. 3. If ΔAe is still over the threshold, the warning is maintained, indicating that the energy in the system calculated in the reference time window is still high.
Consistently with Voight (1988), who extended the application of the Fukuzono (1985) method to a set of different variables, including seismic quantities, in this framework a modified version of the classic Fukuzono-Voight method is proposed. The material failure law described by the Fukuzono-Voight method is presented in Eq. 4. As demonstrated by Voight, the seismic quantity (Ω) (such as the square root of cumulative energy released) is an observable quantity suitable for early warning, and A is a constant. Similarly, this method was also applied in the studies of Amitrano et al. (2005). Therefore, in this case, the accumulated energy (Ae) of rockfalls is adopted as \( \dot{\Omega} \), and the forecasting is made through the linear fit of the inverse of Ae. The forecasting method used in this study is shown in Eq. 5. Generally, this observable quantity should represent Ω (and not \( \dot{\Omega} \) as in our case), and the forecasting method of Fukuzono should use the inverse of \( \dot{\Omega} \) against time (for example, if Ω is the displacement, the forecasting is made through the inverse velocity). In this case, since Ae is already characterized by very abrupt accelerations, extrapolating the 1/Ae line until it intercepts the time axis gives approximately the same results as using the inverse of the rate of Ae, but the rate of Ae generates a much noisier time series (Fig. 3).
$$ {\dot{\Omega}}^{-\alpha}\cdotp \ddot{\Omega}-A=0 $$
(4)
$$ {A_{e(t)}}^{-1}-A=0 $$
(5)
In practice, once the first ta is declared, a linear fitting processing is initialized using Ae(t)−1 (as in Eq. 6) from time ta − 1 h to ta in a sliding window to obtain the slope (d) of the linear fit. The function of the linear fit is shown in Eq. 7, and Eq. 8 shows how to calculate the supposed collapse time (tc). To show the results more clearly, a normalized accumulated energy (Ne) was adopted by adding 0.99 to the denominator, since Ae of rockfalls measured in m2/s2 is typically a very small number (in the order of 10−2–10−5). An example of this procedure employing real data gathered at Torgiovannetto is shown in Fig. 3.
$$ (d)=\mathrm{linear}\ \mathrm{fit}\ \left(\frac{1}{A_{e\left(t-1h\right)}},\frac{1}{A_{e\left(t-1h+1\right)}},\cdots, \frac{1}{A_{e(t)}}\kern0.5em \right) $$
(6)
$$ d\left({t}_c-{t}_a\right)+\frac{1}{A_{e\left({t}_a\right)}+0.99}=0 $$
(7)
$$ {t}_c={t}_a-\frac{1}{d\left({A}_{e\left({t}_a\right)}+0.99\right)} $$
(8)
where tc − ta is the lead time and the warning period is tc2 − ta, where we define tc2 as the last forecasted time of collapse calculated at the last alarm time ta2. In our case, the linear fit is performed by the ployfit function in MATLAB.
Spatial estimation
As soon as ta and tc are calculated, all the rockfall events occurred before, within a fixed time window, can be localized on a topographic map and the susceptible area can be found consequently.
In this case, the method of seismic polarization (polarization bearing, P-B method) was adopted for rockfall localization. The method uses the polarization from a three-component sensor to calculate the source back azimuth through finding the correct P wave from event signal, which is commonly used in earthquake localization (Flinn 1965; Jurkevics 1988). Vilajosana et al. (2008) and Guinau et al. (2019) extended the technique to rockfall localization.
Starting from this point, the P-B method was optimized by means of an overdetermined matrix based on the geophone network. A confidence weight for each sensor was proposed according to the received signal quality and the reliability of the calculated back azimuths. Moreover, three marker parameters were compared with properly select frequency bands in seismic polarization: energy, rectilinearity, and special permanent frequency band. Finally, 30 of 96 frequency bands with the strongest energy are suggested to perform the P-B localization (P-B-30E). One example of P-B-30E localization from an in situ test consisting in manually released rockfall is shown in Fig. 4. For each rockfall, the coordinates of the starting and ending points were measured, and the trajectory was recorded using two video cameras. In this case, four impacts were picked and automatically localized in a topographic map (Fig. 4b). All the estimated positions are closely distributed along the real rockfall trajectory; the maximum error is impact #4 with 48.2 m, and the minimum error is impact #1 with 10.2 m. The details about this method are described in Feng et al. (2020b).
Moreover, the early warning framework can also be implemented with alternative or complementary localization methods, such as those based on arrival times (Gracchi et al. 2017), beam-forming (Lacroix and Helmstetter 2011), and amplitude source location (Pérez-Guillén et al. 2019).
Rockfall detection and classification
All the data are examined with an ad hoc program, DEtection and STorage of ROckfall (DESTRO, Feng et al. 2020a), to detect and classify all the seismic events that occurred in the monitoring period. The events have been grouped in the following classes: earthquake (EQ), rockfall (RF), and tectonic tremor (TR); other microevents and ambient noise are also detected and classified but not considered in our analysis. The examples of signals and frequencies of these events are shown in Fig. 5. Moreover, the event spatial scales are also classified as: point event (P), very local event (vL), local event (L), slope scale event (S), and regional event (R), which means that a certain event has been detected and classified correctly by respectively one component, one seismic station (at least two components), two seismic stations, and more than two seismic stations. Given their very local nature, point events are generally considered as noise and therefore not considered in the elaborations.
The validation of DESTRO has been done through a 2-day-long test where 90 rocks taken from the site were manually released along the slopes of Torgiovannetto quarry and recorded with two cameras and the seismic network. A detailed description of the experiment can be found in Feng et al. (2019). To describe the classification capability of DESTRO, we defined three variables, true positive (TP), false positive (FP), and false negative (FN), respectively, equal to the number of rockfall correctly classified as rockfall, the number of events misclassified as rockfall, and the number of rockfalls misclassified as noise. The recall (TP/(TP + FN)) of DESTRO in the experiment is 98% (88 rockfalls (TP) correctly detected out of the 90 released; other two rockfalls (FN) misclassified that being characterized by a particularly low signal). This value represents the confidence against false negative (i.e., the probability of missing an event is only 2%). On the other hand, 21 events (FP) in excess were detected, for a total of 109 signals classified as rockfalls. Even a manual check on these 21 extra signals made it impossible to distinguish them from the verified rockfalls. Probably these represent actual involuntary rockfalls caused by the passage of the experiment operators or even small natural rockfalls that went unnoticed. This means that the precision (TP/(TP + FP)) of DESTRO (i.e., of detecting real rockfalls from raw seismic data) is ≥ 81% (where 81% represents the assumption that the 21 extra events detected were all errors). The details of this program are described by Feng et al. (2020a).