After the first 9 years of observations (1990–1998), Deganutti et al. (2000) identified a rainfall intensity threshold for debris-flow occurrence in the Moscardo catchment. A time interval of at least 6 h with null or negligible precipitation (≤0.2 mm) was chosen for separating the rainfall events (Deganutti et al., 2000). Duration and mean intensity of triggering rainstorms were computed from the onset of precipitation to the passage of the debris flows at the stage measurement stations. This choice implies an approximation in rainfall duration because debris flows are recorded on the alluvial fan (Fig. 1) some minutes after their initiation in the upper sector of the catchment, but this discrepancy is small if compared to the uncertainties that commonly affect the time of occurrence of landslides and debris flows in the Alps (Palladino et al., 2018).

The threshold defined by Deganutti et al. (2000) is confirmed by the more recent data (Fig. 3a). Moreover, it is possible to identify an upper limit, with the same exponent, for the rainstorms that triggered debris flows in the Moscardo Torrent. Both the lower critical threshold and the upper limit were fitted empirically, a procedure that we consider acceptable due to the relatively small sample size. The two rainfall thresholds have the form 1I=a⋅D-0.7, where a is 15 for the lower threshold and 50 for the upper limit.

The different rain gauges location during the monitoring period (Fig. 1 and Table 4), and the coarse time resolution in 1990–1998 (60′ data), which can lead to underestimating the triggering rainfall (Marra, 2019), had a limited impact on the critical rainfall thresholds, which lie in a rather narrow intensity belt.

The above-mentioned study by Deganutti et al. (2000) had also shown that debris flows in the Moscardo basin were triggered by rainstorms which had a minimum of 21 mm of total rainfall and at least 60′ rainfall intensity of 12.6 mm h-1. These thresholds were applied to the months between June and September of the entire precipitation dataset for detecting the rainstorms that have the potential of causing debris flows in the Moscardo (potentially triggering rainstorms), regardless if they caused debris flows. The rain gauges considered were Pramosio (1990–1998), Rio Lares (1999–2006 and 2011–2016), and La Musa (2016–2019). After removing the rainstorms that have caused debris flows, a sample of non-triggering events was obtained and is plotted in the duration-intensity graph of Fig. 3b.

The filter on total rainfall amount and 60′ intensity led to exclude several low-intensity precipitation events: 85 non-triggering rainstorms were extracted from the database. It is worth noting that the automatic extraction of rainstorm events leads to the identification of duration and rainfall quantities that can hardly be compared to the expert-driven event identification. While the expert-based event definition can leverage the availability of debris-flow timing information and unravel the role and importance of rain and hiatuses, the automatic procedure relies only on thresholds of rainfall amount and intensity, and the separation of showers. As such, on average the automatically extracted rainfall events tend to be longer than the expert-identified ones as they include rainstorm tails.

Most of the non-triggering rainstorms identified in this way plot above the empirical threshold for debris-flow occurrence, and a few ones (all of them for a duration longer than 10 h) also above the upper threshold of triggering rainstorms. Non-triggering rainstorms lying above the critical rainfall threshold confirms that, even in a catchment with abundant sediment supply, like the Moscardo, intense rainfall is not the only factor required for debris-flow occurrence. A time interval – even though short – for sediment recharge is probably needed between a debris flow and the next one. The time series of Moscardo, however, shows that some debris flows took place a few days after the previous one (11, 19, and 20 July 1993; 24 and 27 September 2012; 11, 13, and 22 July 2016). Although triggered by different rainstorms and resulting in different debris flows, from the point of view of sediment transfer from the source areas to the alluvial fan, these events can be ascribed to the same debris-evacuation episode. A more comprehensive analysis of rainfall-related variables and their interaction with sediment recharge periods between consecutive debris flows (Pastorello et al., 2018), which is outside the presentation of experimental data proposed in this paper, could shed more light on the processes that control debris-flow occurrence in the Moscardo catchment.

The debris flows in the Moscardo Torrent occurred in a time interval of 119 d, from the beginning of summer (4 June 2019) to early autumn (30 September 1991), with 18 out of 30 events occurring in the first 50 d (Fig. 4).

Precipitation data were analyzed to explore a possible control of intense rainfall on the seasonal distribution of the debris flows. In the first years of observation, the rain gauge of Pramosio shows that most of the potentially triggering rainstorms, as defined in the previous section, occurred in June and July (24 rainstorms versus 15 in August and September). The most recent period, however, shows a much more balanced distribution of intense rainstorms, both at Rio Lares (24 events in June and July versus 27 in August and September) and at La Musa (10 events in June and July versus 9 in August and September). If we consider the entire observation period (1990–2019), the distribution of potentially triggering rainstorms does not show a disproportionate number of events in the first 50 d, and the occurrence of such storms across the debris-flow season is almost constant (Fig. 4). The absence of a rain gauge that covers the entire monitoring period prevents more definite conclusions, but the analysis of available rainfall data indicates that the differences in the occurrence of intense rainstorms between June–July and August–September are not closely related to the higher frequency of the debris flows in the first part of summer.

A factor that could explain the more frequent occurrence of debris flows in the first part of summer is the larger availability of debris stored in the channel bed in the debris-flow initiation areas after winter and the snowmelt period. Visual field observations support this interpretation, but the absence of systematic measurements of variations in sediment availability does not permit us to confirm it.

Figure 5a shows the days of debris-flow occurrence in the 30 years of observations: the tendency toward the earlier occurrence of debris flows is weak (linear regression coefficient r=-0.17) and not significant (p value of 0.363).

Most debris flows (69 %) occurred in the second half of the day, with nine events from 12:00 to 18:00 LT and nine from 18:00 to 00:00 LT: this is consistent with the triggering commonly caused by summer convective rainstorms, which mostly take place in the afternoon or the evening (Fig. 5b). This result is especially visible in the warmest months, from June to August, whereas four out of the six debris flows observed in September occurred from 00:00 to 06:00 LT. For 4 out of the 30 events, the hour of occurrence is not known: these debris flows were not recorded by the monitoring system and are only known through post-event observations (Table 2).

Figure 6 plots the number of debris flows in each year and the cumulative number of debris flows in the observation period. The absence of debris flows in 2007 and 2008 could be referred to the construction of check dams along the main channel in the middle and lower parts of the basin (Cucchiaro et al., 2019b) that trapped most of the material and prevented the debris flows to reach the alluvial fan, but this does not apply to the years 2013–2015. We could argue that the large evacuation of sediment from the catchment in two large debris flows of September 2012 (Table 2) required a few years of sediment recharge before new debris flows were produced. However, the lack of data on the variations of sediment storage in the source areas does not permit us to verify this interpretation. Similar mechanisms could be advocated to explain the absence of debris flows between 1999 and 2001. Artificial channel widening and the construction of bed sills (Fig. 2) could have favored the deposition of some minor flow events, but these factors did not prevent the occurrence of debris flows in the following years. The number of potentially triggering rainstorms in the years 1999–2001, 2005, and 2013–2015, when no debris flows were recorded, was compared with that of the years in which debris flows did occur. The years 2007–2010 were not included in the analysis because no rainfall data are available (Table 4). The mean number of potentially triggering rainstorms in the years without debris flows (2.61), weighted by the number of years of observations at each rain gauge, is lower than in years with debris flows (4.86). However, large variability exists in the number of potentially triggering rainstorms, both in the years with (standard deviation of 2.29) and without (standard deviation of 2.33) debris flows. The absence of debris flows in some years in an active catchment like the Moscardo is probably caused by a combination of extrinsic, weather-related factors, i.e., the occurrence of a small number of intense rainstorms, and factors intrinsic to the catchment, i.e., the effect of control works and temporary scarcity of mobilizable debris.

We underline that our record consists of debris flows that reached the alluvial fan and were recorded there by the installed instrumentation or documented through post-event surveys. Field observations, although not systematic, have provided evidence of other flow events that did not reach the alluvial fan (for instance, on 20 July and 20 August 2001), not even as debris floods, but the record of these small events is not complete.

The mean frequency between 1990 and 2019 is one event per year including three debris floods (Table 2). The frequency is similar or lower than other instrumented basins in the Alps if the monitoring stations at the basin outlet or on the alluvial fan are considered: Réal (France): one event per year (Hürlimann et al., 2019), Illgraben (Switzerland): three to five debris flows every year plus some debris floods (Hürlimann et al., 2019), Gadria (Italy): 1.3 events per year between 2011 and 2017 (Comiti et al., 2014; Coviello et al., 2020).

The analysis of the debris-flow hydrographs permitted us to define their basic characteristics, such as maximum flow depth, time to peak, and surge duration. These characteristics have been used to derive triangular hydrographs and dimensionless hydrographs that could be employed to define realistic inputs for debris-flow mathematical models. Such hydrographs could also allow comparisons with hydrographs recorded in other instrumented catchments.

The analysis was performed examining individual debris-flow surges. In the Moscardo, the water level before the occurrence of a debris flow is negligible if compared to the maximum depth of debris-flow surges: the start of a surge is easily identified at the first rise of the hydrograph. Some uncertainties may arise regarding the end of the recession phase, which occurs when the flow level becomes almost stable or a new surge begins. Data from 62 surges were collected, i.e., on average approximately two surges per debris-flow event. The maximum flow depth was computed as the difference between the surge's peak and the level before the start of the flow rise. A minimum rise of 0.5 m was adopted for the analysis: this value is a trade-off between the need to consider all relevant flow surges and the requirement of minimizing the risk of including small fluctuations in the analysis that are not representative of the debris-flow behavior in the studied channel. Table 5 reports basic statistics on the time to peak, the duration of the recession, and the maximum flow depth. In six cases, while the rising limb and the debris-flow peak are clearly defined and allowed the calculation of time to peak and maximum flow depth, the duration of the recession phase could not be properly recognized (for instance, because of substantial channel aggradation).

We have devised, based on the data in Table 5, the triangular hydrographs presented in Fig. 7 to provide a simplified representation of the debris-flow hydrographs in the downstream reach of the Moscardo Torrent. We used the summary data related to duration and the maximum depth of the debris-flow surges to define three triangular hydrographs related to different levels of event severity: a.

a hydrograph characterized by the median value of the time to peak, flow depth, and recession duration;

Hydrograph “b” features fast arrival of the front, high flow depth, and long duration (hence high peak discharge and volume too): all these characteristics define a challenging debris-flow event. The opposite conditions characterize the low-severity hydrograph (“c”). The synthetic representation provided in Fig. 8 enables us to define, based on a simple statistical analysis of the data collected, the worst scenario expected in the studied channel, as far as the maximum flow depth and the shortest time to peak of debris flows are concerned. In the absence of mathematical modeling that could provide more in-depth information, this can be of some use for the planning of mitigation measures.

A more detailed representation of hydrographs' shape was achieved by averaging the recorded hydrographs of the debris-flow surges. This analysis was performed on the debris flows recorded between 2002 and 2019: data for 12 surges for both the upstream (E, Fig. 1) and downstream (D) measuring stations were available. The debris flows recorded between 1990 and 1998 were excluded because of coarser time resolution from 1990 to 1994 and variable cross-sectional geometry. Dimensionless hydrographs were generated normalizing the flow depth by its maximum value and the time by the total surge duration. The flow peaks were aligned to preserve the sharp shape that is a distinctive feature of debris-flow hydrographs. Finally, the ordinates were averaged, and mean debris-flow hydrographs were obtained, one for each measuring station (Fig. 8). Since the duration varies from surge to surge, average debris-flow hydrographs were computed only for the interval covered by at least 50 % of the hydrographs, which led us to consider six hydrographs for both the considered measuring cross-sections. The choice of averaging the hydrographs on the interval covered by at least 50 % of the events is intended to limit the impact on the results of the peculiar characteristics of single surges.

The dimensionless hydrographs at the two measuring stations, which are quite close to each other (Fig. 1) and have similar cross-sectional geometry, display similar shapes (Fig. 9). It can be noted that the precursory surge that precedes the peak is somewhat larger in the upstream monitoring station, whereas the ordinates of the intermediate part of the recession phase are higher in the downstream cross-sections and lower in the last part of the recession limb of the hydrograph. These differences could indicate that flow propagation, even in a short (76 m) channel reach, causes non-negligible changes in hydrographs' shape. In our particular case, the hydrograph deformation, due to the downstream propagation of the debris-flow wave along the channel reach between the stations, does not lead to a spreading of the hydrograph but its shrinkage. It would be interesting to verify if this behavior, which is in contrast to the hypothesis that debris flows behave as kinematic waves, will be confirmed by further observations in this torrent and will be also observed in other catchments.