E3S Web of Conferences 415, 07003 (2023)
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https://doi.org/10.1051/e3sconf/202341507003

A new statistical method to assess potential debris flow erosion
Gabriele Bertoldi1, Tommaso Baggio2*, Francesco Bettella2 and Vincenzo D’Agostino2
1Service for torrent control, Autonomous Province of Trento, via Giovanni Battista Trener 3, 38121 Trento, TN, Italy
2Department of Land, Environment, Agriculture and Forestry, University of Padova, viale dell'Università 16, 35020 Legnaro, PD,Italy

Abstract. Debris-flow erosion patterns were investigated for two adjacent catchments, Molinara and Val
del Lago creeks (Eastern Alps, Trento Province, Italy), where two debris flows were triggered by an intensestorm in the summer of 2010. Both basins have been inactive over the last two centuries. The debris flows
were activated by channel and bank erosion under stable bed conditions before the event. The erosive
process was analysed by combining a field campaign (two hundred cross sections were surveyed along thecreeks) and pre- and post-event LiDAR surveys. Data were analysed by selecting morphologicallyhomogenous channel reaches and deriving for each reach: erosion depth, creek width, eroded volume andpeak discharge. Investigating the frequency distribution of the erosion depth we found out that it follows an
EV1 probability distribution. On this basis, a new approach has been proposed to predict event volumes
when the expected maximum potential depth erosion is known. The procedure would be of high interest inpredicting debris flow volume in mountain channels characterized by long silent periods.

1 Introduction
Villages and infrastructure in Alpine regions areexposed to rapid mass movements, including debrisflows. Several studies aimed to understand theseprocesses, but fundamental questions concerning hazard
assessment, mass growth along the flow path, and thevariability of the processes remain open [1–3]. Adopting
a qualitative approach, the first studies on erosion
showed a high variability of the process and a strong
influence of the local channel slope.
The methods generally used to quantify the expected
debris-flow volume initially focused on the field
estimation of entrainable sediment. Such criteria arebased on a geomorphic reach-by-reach estimation ofsediment availability along the stream network [4,5].
More recently, the quantification of debris flow erosion
is increasingly supported by high-resolution topography
surveys and digital surface analysis [6,7] combined into
the geomorphic estimation procedures.
A database collected by [8] shows a statisticalfingerprint of erosion depth. Other studies proposed
empirical equations able to predict the depth of erosion
based on channel and basin geomorphic variables suchas channel slope [9–11]. In contrast, [12,13] did not finda major slope influence on erosion. Combining fieldsurvey data with high-resolution digital terrain models
DTM, [2] emphasized the flow front height to be the key
variable in erosion processes. Laboratory experiments
and numerical models have took into consideration
bulking and de-bulking processes along with sedimentconcentration, yield and shear stresses [14–16].
However, studies that require the back-calculation ofcomplex field conditions suffer from difficulties indescribing the channel bed geology, the boundary
conditions and the basin/channel morphometry/morphology [8].
The two debris flows that occurred in the Molinara andVal del Lago catchments, investigated in this study,
enlarge the knowledge of in-channel erosion processes
through the analysis of data and conditions scarcely
investigated in the literature. The research objective is
to test the hypothesis that at the scale of aformative/highly erosive debris-flow a statisticalfingerprint of the erosion depth exists and it can be used
as a tool to predict the expected debris-flow volume.

2 Study area and the 2010 event
The study area is composed of two adjacent catchments
located in Eastern Trentino (Alps, Northern Italy,
11.288091°, 46.140849°) on the NW oriented side of theCostalta peak (1955 m a.s.l., Fig. 1). Both basins aremainly covered by forest (Norway spruce and larch).
Their channels are incised on a thin alluvial Quaternary
cover over a massive porphyritic platform (Ring andRichter, 1994). The drainage network is well developed
and rainstorms normally produce flood events without
significant bed load transport. Effectively, there is no
documentation noting the occurrence of significant
debris flows in the 200 years preceding the severe eventthat occurred in summer of 2010.
The catchment area of the Molinara torrent is 0.88 km2,extending from 1113 to 1955 m a.s.l. The mean slope ofthe catchment and main channel is high, equal to 70%
and 37%, respectively. The total stream network is 4 km

* Corresponding author: Tommaso Baggio, tommaso.baggio@unipd.it© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0(https://creativecommons.org/licenses/by/4.0/).

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long, 1.7 km of which constitutes the main stream. The
Molinara torrent network flows in a deeply incisedvalley and is characterised by low sinuosity. The
catchment of Val del Lago torrent has an extension of
0.42 km2 and the elevation ranges from 1024 to 1722 ma.s.l. The mean slope of the main channel (1.5 km long)is 28%.

village of Campolongo di Pinè damaging roads andhouses (Fig. 1). After-event field inspections did notindicate particular sediment source areas other than thechannel heads and the subsequent channel reaches
eroded by the debris-flow passage (Fig. 2).

3 Material and methods
The Val Molinara and Val del Lago debris flows were
investigated by means of a field survey of the torrent
reaches affected by the flow passage, available LiDAR
surveys acquired before and after the event, correctionof field survey through LiDAR data and statistical data
analysis. Accounting for advances in debris-flow
erosion [17] and event evidence, three basic conditions
can be assumed. i) the channel bed was quite stable after
the first storm, ii) the erosion depth Z (Fig. 2 and 3) was
generated during the second highest debris-flow front(eyewitness of two surges), and iii) the average heightof the debris flow front (Zd) resulted from the difference
between the average maximum flow depth (observed in
the field in accordance with the top flow-width line
joining opposite banks) and pre-event channel bed
elevation (Fig. 3).
Accurate field investigations were conducted in thesummer of 2011 estimating the sediment yield following
the methodologies described by [5]. The torrent network
was divided into homogeneous reaches in terms ofslope, bed/banks morphology and sediment transport
type. For each reach, representative cross-sections weremeasured (maximum of 3 sections) using a rangefinder(Disto Trupulse® 360B, precision 0.1 m and 0.1°). The
shape of the section was approximated to a trapezium,
and the following measurements were taken (Fig. 3):- Top (B) and bottom (b) widths of the channel
section where erosion and bank collapse occurred
regardless of the traces of the debris-flow passage;- Section heights of the right and left bank (Hr, Hl)corresponding to the maximum width of erosion B;- Maximum cross-section flow widths (b) andheights (hr, hl) with respect to the post-event thalweg
elevation, whose shape is clearly the result of the debrisflow passage. The channel bed average erosion (Z) was
estimated in the surrounding reach of each cross-section
by different types of field evidence (Fig. 2-D and Fig. 3)[18]. Afterwards, the following variables werecalculated: the erosive yield rate Y (m3/m), the eroded
sediment volume (V) and the top-flow planimetric areaAc, determined through the B values.
Multi-temporal DTMs were provided by the Geological
Services of the Trento Province and the difference
between pre- and post-event digital surface models(DoD) allowed for the correction and validation of thefield survey data. The pre-event DTM is the standard
model of the Province acquired in 2006-2007 (meandensity: 1.5 pts/m2). The derived DTM has a 1-meter
grid resolution with a vertical and horizontal accuracyof ± 0.30 m and ± 1.00 m respectively. The post-event
DTM was obtained during an ad hoc flight (meandensity: 10 pts/m2) performed in the autumn of 2010.
The DTM of 1-m grid resolution has a vertical andhorizontal accuracy of ± 0.10 and ± 0.30 m respectively.

Fig. 1. Molinara and Val del Lago catchments (contour lines
interval of 50 m); the crossed polygons represent the debrisflow deposits. A,B,C and D indicate the location of the photos
in the Figure 2.

Fig. 2. Photos of the debris-flow torrents in the upper reaches(A), middle reaches (B), and lower reaches (C and D); field
traces are highlighted in B and D.

On the 14th of August 2010, a storm hit the study areaat 2:45 p.m. (CEST time) and continued until 5:00 a.m.
of the following day, discharging a cumulated rainfall
amount of 169.1 mm (rain gauge located 5 km from thetwo catchments). The bulk of the storm affected the areafrom 11:45 p.m. to 4:45 p.m. and was characterised by
two main bursts of rainfall lasting two hours: maximum
1-hour rainfall intensity of 39.3 mm and 38.2 mm,
separated by 45 minutes of low-intensity rainfall. The
return period of the event was estimated equal to 100years for the 3-hour maximum rainfall (73.1 mm) andgreater than 200 years for the 6-hour maximum rainfall(156.3 mm). During the first storm, the basin hillslopes
were partially saturated, due to previous rainfalls, andthe debris-flood discharge increased considerably.
During the second rainfall burst, massive destabilisation
originated from the channel heads and triggered debrisflow surges in both basins. The surges progressively
entrained sediment from the channel bed, enlarging inthis way the debris-flow volumes. The Val del Lagodebris flow filled the retention check dam (Fig. 1),
whereas the Molinara torrent debris flow flooded the2E3S Web of Conferences 415, 07003 (2023)
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The resulting DoD could provide an error estimation of
the ground differences between the pre- and post-event
surfaces (Weathon et al., 2010). Nevertheless, the low
resolution of the pre-event DTM and the significant
irregularity of the banks made difficult a precise
assessment of the local variation of the channel
characteristics (B and heights of each bank Hr, Hl). Theintegration of remote sensing data with field survey wasthen necessary to generate the erosion-related variables(channel yield rate Y and volume V). In this context, weused the cross-section post-event surveys (Z, Zd, Hr, Hl,hr, hl, b, B; assumed to be the more correct) as primary
information and then we accounted for them to obtainan adjusted DoD based on matching the cross-section
field surveys.

DoD values (ZDoD). Underestimation was significant andcan be expressed by the following equation:
𝑍𝐷𝑜𝐷 =𝑍𝐹𝑖𝑒𝑙𝑑(1)𝑞where the coefficient q showed a value of 0.39 for
Molinara (p value < 0.01, R2 = 0.46) and 0.44 in Val delLago (p value < 0.01, R2 = 0.49). Thanks to equation 1
the final correct depths of erosion reached maximum
values (Zmax) of 4.86 and 3.74 m in the Molinara and Val
del Lago torrents respectively.
The analysis of the datasets of erosion depth (Z)indicated that are significantly skewed with a long tailfor the largest values. The Z sample characteristics interms of the cumulative distribution function (CDF) are
shown in Fig. 4a, where the variable has beennormalised to the dimensionless depth: Zr = Z/Zmax,
being Zmax the maximum Z measured in each stream. TheZr sample has proven not to be adaptable to theexponential and logistic probability distributions(Kolmogorov-Smirnov test, confidence level of 95%),
while it resulted adaptable to all three right-skewed
distributions. Comparing the measured Z values withthose expected from the CDF of the log-normal, Fuller
and EV1 distributions, the latter was the most accurate(RMSD equal to 0.22 and 0.17 for the Molinara and Val
del Lago basins, respectively). Afterwards, thehypothesis that the datasets of the two basins belonged
to the same population was tested focusing on thedimensionless erosion depth. The statistical analysis
used the non-parametric Mann-Whitney test (95%confidence level) assuming the null hypothesis
corresponding to samples with equal distribution(Gumbel, Fuller, log-normal), and the alternative
hypothesis to samples with different distributions. The
analysis proved that the Z/Zmax values of the adjacentcatchments belong to the same population with a p-value<< 5%. The joint Molinara-Val del Lago Zr sample (176values) has an average of 0.308 and a standard deviationof 0.228. This sample has been shown to follow an
extreme value EV1 (Extreme Value Type I) probability
distribution at best (Fig. 4b). The function of the
cumulative probability (P) of non-exceedance can thenbe written as:

Fig. 3. Sketch of cross-section measurements with the main
field evidences. Tiny letters indicate the variables of the flow
traces, capital letters indicate the variables of the erosion
evidences.

The analysis of the erosion depths (Z) has considered
their statistical distribution with the aim to test theexistence of a characteristic erosion pattern produced bya severe debris flow after a long period of inactivity.
Data were tested both through continuous asymmetricprobability distributions and a number of symmetricaldistributions. In particular, following the suggestions of[19], the adaptation considered the exponential,logistics, Gumbel, Fuller and log-normal distribution
and was completed applying the Kolmogorov-Smirnov
test with a confidence level of 95%. The mostappropriate distributions were selected by comparing
the values of root mean square deviation (RMSD).

4 Results

𝑃=𝑒

In summer 2011, the field survey consisted of 190 crosssection measures in the Molinara catchment,
corresponding to 155 homogenous reaches. Regarding
the Val del Lago catchment, a total of 73 cross-sections,grouped into 45 homogeneous reaches were surveyed.
The DoD calculation of debris-flow volume and depthsof erosion for the Molinara torrent provided a volume of68400 m3 (with an error of the estimate: Errv,high = 13
100 m3, [7]) and 8700 m3 (Errv,high = 2700 m3) in the Val
del Lago catchment.
Both DoD volumes weresubstantially confirmed by the post-event surveys of theTorrent Control Service (Trento Province authority).
Afterwards the average erosion depths from the fieldsurvey and LiDAR analysis were compared. The
analysis highlighted that field observations havesystematically underestimated the erosive depths,
evidencing a linear correlation between field (ZField) and−𝑒 −𝑦=𝑒−𝑒−𝛼(𝑍𝑍−µ)𝑚𝑎𝑥(2)

where y is the reduced variable of the distribution andthe parameter estimation (method of moments) yields:
α= 5.628 and µ=0.206.
The corrected adaptation of the sample to the EV1
probability distribution was positively verified even bymeans of [20] test assuming a confidence level equal to95%. Equation (2) was then tested to recalculate theeroded volume Ve through the assumption of a number(n) of equal-spaced Zr,i (=Zi/Zmax) intervals as follows:
𝑉𝑒 = ∑𝑖=0,𝑛(𝑝𝑟,𝑖 𝑍𝑟,𝑖 𝑍𝑚𝑎𝑥 𝐿𝑏𝑚 )(3)

where pr,i is the relative probability density (Equation 2)
associated with the normalised erosion rate Zr,i of the ith interval, L is the total length of the erodible channels
and bm the mean torrent width. The computation of Ve
assuming 10 equal intervals of Z/Zmax (class width of3

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0.1) provided values of eroded material equal to 63000
m3 in the Molinara channel and 8000 m3 in the Val delLago reach. These volumes do not vary significantlywith an increase in the number n of the intervals (i.e.,
13% reduction in volume for n=100) and substantiallyagree with those calculated through DoD analysis.

monitored cross-sections within a network ofexperimental debris-flow catchments) and spatial
continuity of the information provided by intensive
post-event surveys.
Research funds BIRD-DOR 2021 and 2022, TeSAF Dep.
University of Padova, Prof. V. D'Agostino

References
1.

Iverson, R.M.; Reid, M.E.; Logan, M.; LaHusen,R.G.; Godt, J.W.; Griswold, J.P., Nat. Geosci., 4,116–121, 2011
2. Schürch, P.; Densmore, A.L.; Rosser, N.J.;
McArdell, B.W., Geology, 39, 827–830, 2011
3. McCoy, S.W.; Kean, J.W.; Coe, J.A.; Tucker,G.E.; Staley, D.M.; Wasklewicz, T.A., J. Geophys.
Res. Earth Surf., 117, 2012
4. Hungr, O.; Morgan, G.C.; Kellerhals, R., Can.
Geotech. J., 21, 663–677, 1984
5. D’Agostino, V.; Marchi, L., DFHM, 1096–1106,
2003
6. Cavalli, M.; Tarolli, P.; Marchi, L.; Dalla Fontana,G., Catena, 73, 249–260, 2008
7. Favalli, M.; Fornaciai, A.; Pareschi, M.T.
Geomorphology, 111, 123–135, 2009
8. Hungr, O.; McDougall, S.; Bovis, M., DFHM,135–158, 2005
9. Rickenmann, D.; Zimmermann, M.,
Geomorphology, 8, 175–189, 1993
10. Conway, S.J.; Decaulne, A.; Balme, M.R.;
Murray, J.B.; Towner, M.C.,Geomorphology, 114,556–572, 2010
11. Baggio, T.; Mergili, M.; D’Agostino, V.,
Geomorphology, 381, 2021
12. Breien, H.; De Blasio, F. V.; Elverhøi, A.; Høeg,K., Landslides, 5, 271–280, 2008
13. Marchi, L.; Cavalli, M., Environ. Manage., 40,493–503, 2007
14. Egashira, S.; Honda, N.; Itoh, T. Phys. Chem.
Earth, Part C Solar, Terr. Planet. Sci., 26, 645–650, 2001
15. Hussin, H.Y.; Quan Luna, B.; Van Westen, C.J.;
Christen, M.; Malet, J.-P.; Van Asch, T.W.J.
Hazards Earth Syst. Sci, 12, 3075–3090, 2012
16. Rickenmann, D.; Weber, D.; Stepanov, B.,
DFHM, 2, 883–894, 2003.
17. Berger, C.; McArdell, B.W.; Schlunegger, F., J.
Geophys. Res. Earth Surf., 116, 2011
18. Aulitzky, H., Interpraevent, 139–174, 1992.
19. Vidal, I. A Stoch. Environ. Res. Risk Assess., 28,571–582, 2014
20. Klemes, V. Water Resour. Res., 12, 1325–1326,1976
21. Bovis, M.J.; Jakob M., Earth Surface Processes andLandforms, 24, 1039-1054, 1999

Fig. 4. a) Cumulative frequency (F%) distribution of thenormalised depth of erosion (Z/Zmax); b) probabilistic plot andadaptation line (Equation 2) to the EV1 distribution: reduced
variable y versus Z/Zmax.

5 Discussion and conclusions
The mobilised sediment volumes in the triggering areas
were negligible compared to the final magnitudes. The
investigated two adjacent debris flows were dominatedby in-channel sediment entrainment, equal to 68400 and
8700 m3 in the Molinara and Val del Lago basinsrespectively. The maximum erosion depths (4.9 m and3.7 respectively) are comparable to those found by[8,10].
According to [21], long silent periods have been
confirmed as preparation for important sediment
recharge, particularly for the geological settings ofvolcanic and compact metamorphic rocks. When thehazard has to be assessed in these lowfrequency/apparently stable mountain streams, thedifficulty of the geomorphic estimates of the expected
sediment volumes makes the identification of astatistical erosion pattern a practical and valuable tool.
As already shown by [8], the distribution of the erosion
depth Z has been confirmed to be markedly
asymmetrical towards the right tail. Starting from thescale of the expected maximum deepening of the bed(Zmax), recalculating the volume - accomplished bymeans of the fitting probability distribution EV1(Equation 2) and a simplified sediment budget(Equation 3) - has proven to be very precise, requiring
only a few data. The computation for the Molinara/Val
del Lago basins has obviously benefitted from the backanalysis and thus from the Zmax field measurement, andparameter (α and μ) adaptation. Nevertheless, thecomputational method could be quite promising andinnovative when used in a purely predictive task.
Indeed, assuming the channel length to be invariant
under erosion and the average channel width quasiinvariant, Zmax could be measured by carrying out
geophysical surveys to estimate the sediment thickness.
A certain degree of subjectivity would remain in thedistribution parameters but the proposed model for realcases and their comparison with the measurable valuesof bank heights before the event would help to make areliable choice. Additional verifications are thenrecommended, in terms of the number of events (e.g.,4