E3S Web of Conferences 415, 01002 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341501002

Analysis of superelevation and debris flow velocities atIllgraben, Switzerland
Amanda Åberg1,2*, Jordan Aaron2,1, Tjalling de Haas3, Brian McArdell1, Jacob Hirschberg2,1
1Swiss Federal Institute for Forest, Snow and Landscape Research WSL, CH8903 Birmensdorf, Switzerland
2Department of Earth Sciences, ETH Zürich, 8092 Zürich, Switzerland
3Department of Physical Geography, Utrecht University, 3584 CS Utrecht, the Netherlands

Abstract. The forced vortex equation, based on the cross-stream inclination of a flow surface as it passes
through a bend, is a common approach to estimating debris flow velocities. Here, we present the preliminary
results of a study of superelevation and the correction factor k, used to adapt the forced vortex equation todebris flows, based on data from the Illgraben torrent in Switzerland. The definition of the radius ofcurvature, a factor in the calculation of superelevation velocities, is not found to exercise a large influenceon the calculated velocities when using high resolution aerial images, with the choice of cross-section
location and k-factor exercising a more significant influence. The k-factors found here fall within the range
previously reported in the literature, ranging from approximately 1 to 7, and a previously suggested nonlinear relationship with Froude numbers is evident in the dataset. Following the debris flow season of 2022,
the study will be continued with additional debris flow events and the investigative methods will be extended
to include high-resolution LiDAR sensors installed along the Illgraben torrent.

1 Introduction
Hazard evaluations of debris flows are criticallydependent on good estimates of debris flow velocity anddischarge [1]. One commonly used method for post-hoc
estimation of debris flow velocity is the superelevation
approach, which is based on the cross-stream inclinationof the flow surface as it experiences centripetal
acceleration passing through a bend [2, 3]. Thoughinitially used for pure water flows, the forced vortex
equation has been adapted to non-Newtonian fluids suchas debris flows by incorporating a correction factor k [4,5]:𝑅𝑐 𝑔∗(1)
𝑣𝑑𝑓 = (∆ℎ)0.5
𝐵𝑘
where g* represents the bed-normal component ofacceleration due to gravity, Rc the radius of curvature,Δh the difference in elevation of the flow surface
between the inner and outer bend, and B the width of theflow. Note that eq. 1 is valid for rectangular crosssections. The correction factor k has been suggested tobe connected to the viscosity and vertical sorting of adebris flow [4, 6]. Literature values of k are typicallybetween 1 to 5 [4, 6] but may be as high as 10 [7].
Further, k-values have been found to depend on theFroude number, converging to 1 for supercritical flow
conditions and increasing exponentially for subcriticalflow conditions [5, 8]. This was found both forexperiments as well as k-factors derived from events at
the Illgraben torrent. In addition to the uncertainties

related to k, the definition of the radius of curvature fornatural bends has been contentious because it isgenerally based on a subjective assessment andinfluenced by the scale of the media used [9].
Though the superelevation method is commonlyused for post-hoc assessment of debris flow velocity,evaluation of the accuracy of the results is often difficult
due to the lack of independent velocity reference data.
In this study, we use high-resolution orthophotos anddigital elevation models (DEM’s), as well as
independent front velocity estimates from the 20192021 seasons in the well-instrumented Illgraben torrent
in Switzerland, to investigate the challenges related todefining the radius of curvature. Furthermore, wepresent the preliminary results of an analysis ofappropriate k-values to fit the superelevation velocities
to the measurement station velocities. During the debris
flow season of 2022, additional data has been collectedusing newly installed, high-resolution LiDAR scanners
along the torrent in addition to the continued collectionof high-resolution drone-based images, both of which
are expected to contribute to further insights.

2 Site description
The well-known Illgraben torrent in canton Valais,
Switzerland, has been instrumented and monitored since
the year 2000 [10, 11]. Along the channel a large
number of check dams have been constructed to stabilise* Corresponding author: amanda.aaberg@wsl.ch© 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/).

E3S Web of Conferences 415, 01002 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341501002

Fig 1. Overview figure of the study location: a) orthophoto from the lower section of the Illgraben torrent with locations of checkdams, the measurement station at CD29 and the investigated bends (flow direction from left to right), b) bend 1 with the twotested radii of curvature and cross-sections used to collect Δh for radius 1a, c) bend 2 with the three tested radii of curvature andcross-sections used to collect Δh for radius 2c.

this active torrent. The measurement station where dataon bulk flow properties are collected is located at thedistal end of the fan at check dam (CD) 29, shortly
before the channel’s confluence with the river Rhône(Fig. 1). The superelevation analysis presented here
focuses on two bends; one located in immediateproximity to CD27 approximately 480 m from CD29,
and the other 340 m further upstream, shortly
downstream of CD25 (Fig. 1).

superelevation, Δh, for a total of 10 debris flow events,
though the number of cross-sections with reliable data
range from four to 10 for the different events.
3.2 Reference velocities
The superelevation velocity has been compared tovelocity estimates obtained using geophone detection of
the front arrival at various check dams. Resultspresented here are based on the measurements obtained
from travel times between CD27 and CD28, which is thenearest available segment to the investigated bends. The
instrumentation for velocity estimates was re-installed
in 2019 and is sometimes affected by problems related
to interpretation of the geophone signals. Consequently.
three events have been removed from the analysis.

3 Methods
3.1 Superelevation extraction
Extraction of superelevation values was conducted in
ESRI ArcMap 10.8.1 using orthophotos and DEM’swith a resolution of 5 to 10 cm from the 2019-2021
debris flow seasons [12, 13]. This topographic data has
a typical error <10 cm in x, y and z directions. For eachseason a channel centreline was defined along a
Euclidian midpoint between the channel banks using theorthophotos. Two bends were then selected for thisinitial analysis based on their relative proximity to themeasurement station and their lack of overly steep banks
to help facilitate mudline identification. A range of radii
were manually fitted to these bends, such that thesmallest and largest plausible radii of curvature wereestimated (Fig. 1). Each bend radius was then fitted with
10 cross-sections evenly spaced along its arc length. For
bend 2, which is interrupted by a check dam, all crosssections were placed upstream of the check dam and,hence, do not cover the full arc length. The mudline
elevations were then manually collected in the inner andouter bend for each cross-section using the orthophotos
and corresponding DEM’s. This resulted in data for

3.3 Superelevation velocities and k
Superelevation velocities for each event were calculatedusing eq. 1 and k = 1. These velocities have then beenused to examine the influence of differing radius ofcurvature. Additionally, for each event and each bend,
k-factors were back-calculated for all available crosssections, and a mean k and its standard deviation werethus derived for each bend and event, averaging thevalues obtained using different bend radii. This standard
deviation represents some of the uncertainty in the Δhand B estimates, but does not incorporate the uncertainty
related to the geophone velocities. The back-calculated
k-factors were then compared to various flow bulk
properties including Froude numbers, which werecalculated using the same front velocity and acorresponding frontal flow depth obtained at CD29
using a radar sensor.

2E3S Web of Conferences 415, 01002 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341501002

before the measurement station, they also produce fewer
k-factors < 1, which are therefore more in line with those
proposed in the literature.
Event 2, which displays the largest k-factor, shows asignificant discrepancy between the velocities obtained
between CD27-28 and CD28-29 and based on video
observations, the CD28-29 velocity is overestimated.
The event did not have a clear front, but 1.7 m/s wasdeemed to be a reasonable velocity for the early stagesof the flow, whilst surges producing the maximum
mudline elevation during following stages more likely
had velocities of approximately 4 m/s at CD29. With theextracted superelevation such a velocity would haveresulted in k = 1.26 ± 0.61 for bend 1 and 0.89 ± 0.71 forbend 2, significantly lower than those based on thegeophone velocities (Table 1).
Surprisingly, no clear relationships were identified
between the back-calculated k-factors and othervariables related to flow depth or bulk density, but kdoes appear to be correlated with the Froude number(Fig. 3). This nonlinear relationship supports the resultsof [5, 13], which were based on experimental work andfield observations at Illgraben.
Superelevation velocities derived from theindividual cross-sections at bend 2 show an accelerating
trend toward the check dam. We attribute this to the

4 Results and discussion
In this study, the variability in the estimatedsuperelevation velocities resulting from using different
radii of curvature was limited, with the maximum
difference between the medians for the same bend beingapproximately 1 m/s (Fig. 2). Meanwhile, the velocities
calculated for individual cross-sections can varyconsiderably for the same bend and event. Local
topographical features have a large influence on
individual superelevation measurements, indicating that
careful consideration of local topography is necessary
when choosing locations for cross-sections and that
ideally, many cross-sections should be used to achieve areasonable mean value for Δh in a bend.
The back-calculated k-factors ranged between 0.83
and 7.00 (Table 1), with the exception of one extreme
outlier that was likely the results of a faulty front arrivalmeasurement. Though k should be equal to or exceed 1[8], the standard deviation indicates that a value of 1 iswithin the range of the uncertainties for the one eventwith k < 1. The front velocities measured between CD27
and 28 were used as reference velocities due to theproximity to the investigated bends. Because they areoften slightly lower than those measured immediatelyFig. 2. Boxplots illustrating the differences in median values
for superelevation velocities calculated using a constant kfactor of 1 and variable radii of curvature for: a) bend 1 with
two radii of curvature and b) bend 2 with three radii ofcurvature. The number of data points for each radius is: 1a –79 points, 1b – 73 points, 2a – 68 points, 2b – 58 points, 2c –46 points.

Fig. 3. Back-calculated k-factors for each event and bend using thefront velocity estimate from CD27 to CD28 plotted against
corresponding Froude number for a) bend 1 and b) bend 2. Error
bars show standard deviations and k = 1 is marked with a solidline. Numbers reference events as listed in table 1.

Table 1. Velocities based on travel times between geophones, Froude numbers, back-calculated k-factors and superelevation
velocities calculated using eq. 1 and k = 1 for the seven debris flow events for which front velocities between CD27 and CD28
were available. Numbered subscripts for k and superelevation velocities denote bend location.
Geophone velocities
Superelevation velocities
Date
vCD28-29 (m/s)
vCD27-28 (m/s)
Fr nr
k1 ± st. dev
k2 ± st. dev
v1v212019-06-21
6.62
5.32
1.04
0.83** ± 0.53 1.32 ± 0.72
4.59
5.89
2
2019-07-26
8.69
1.7
0.52
7.00 ± 3.49
4.90 ± 3.91
4.34
3.45
3
2019-08-11
6.95
5.16
1.20
2.19 ± 0.71
1.32 ± 0.60
7.55
5.77
4
2020-06-29
1.23
0.6
0.18
*2.58
5
2021-06-24
8.18
5.41
1.09
1.54 ± 0.70
6.44
62021-07-16
2.78
3.08
0.77
2.39 ± 1.20
2.91 ± 1.50
4.62
5.04
7
2021-08-07
2.32
2.41
0.64
4.75 ± 0.49
4.95 ± 2.16
5.24
5.23
* Unreliable velocity data for CD27-28 producing unreasonably large k-factor.
** [8] indicates that k values should be >= 1, the value here likely results from uncertainties in the estimation procedure.
- Superelevation data missing.

3

E3S Web of Conferences 415, 01002 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341501002

presence of a hydraulic draw-down effect as the flow
approaches the check dam, and this will be investigatedfurther using LiDAR data collected during the 2022
debris flow season.
The distance between the locations of theinvestigated bends and the channel segment along which
the front velocity is estimated may introduce someerrors. Velocities obtained between CD27 and 28 areoften slightly lower than those between CD28 and 29(Table 1), and one of the bends is located at an additional
distance upstream. Additionally, the maximum velocity
and flow depth may not necessarily occur at the flow
front for Illgraben debris flows, where events often lacka well-defined bouldery front. In ongoing work, theLiDAR sensors will enable a more accurate
determination of the local debris flow velocity,
specifically at bend 2.
Additionally, disturbances to the flow due to local
irregularities may produce crossing wave fronts leaving
multiple mud traces on the channel banks [14]. Whenaffected by such phenomena, the post-event mudlines do
not represent the maximum cross-channel inclination of
the flow surface. The influence of such effects alongwith splashing caused by turbulent flow or roll-waves
will likewise be examined further.

velocity estimates obtained through the superelevation
method.

References
1.

L. Bugnion, B.W. McArdell, P. Bartelt and C.
Wendeler, Landslides 9, 2, pp.179-187 (2012)
2. J.E. Costa, Physical geomorphology of debris
flows, in Developments and applications ofgeomorphology, pp. 268-317, Springer, Berlin,
Heidelberg (1984)
3. S.P. Pudasaini, M. Jaboyedoff, Landslides 17, 6,pp.1377-1392 (2020)
4. O. Hungr, G.C. Morgan, R. Kellerhals, Can.
Geotech. J. 21, 4, pp. 663-677 (1984)
5. C. Scheidl, B. McArdell, G. Nagl, D.Rickenmann,
Debris flow behavior in super-and subcriticalconditions, in 7th International Conference onDebris-Flow Hazards Mitigation (2019)
6. D.F. VanDine, Res. Br., BC Min. For., Victoria,BC, Work. Pap. 8 (1996)
7. M.H. Bulmer, O.S. Barnouin‐Jha, M.N. Peitersen,
M. Bourke, J. Geophys. Res. Planets 107, E5,pp.9-1. (2002)
8. C. Scheidl, B.W. McArdell, D. Rickenmann, Can.
Geotech. J. 52, 3, pp. 305-317. (2015)
9. A.B. Prochaska, P.M. Santi, J.D. Higgins, S.H.
Cannon, Landslides 5, 4, pp. 431-444 (2008)
10. M. Hürlimann, D. Rickenmann, C. Graf. Can.
Geotech. J. 40, 1, pp. 161-175 (2003)
11. B.W. McArdell, Int. J. Eros. Control Eng. 9, 4,pp.194-198 (2016)
12. T. de Haas, W. Nijland, B.W. McArdell, M.W.
Kalthof, Front. Remote Sens. 2, p.5 (2021)
13. T. de Haas, B.W. McArdell, W. Nijland, A.S.
Åberg, J. Hirschberg, P. Huguenin, Geophys. Res.
Lett. 49 (2022)
14. T.C. Pierson, Geol. Soc. Am. Bull. 96, 8, pp.10561069 (1985)
15. J. Aaron, R. Spielmann, B. McArdell, C. Graf
DFHM8, submitted (this proceedings volume)
16. R. Spielman, J. Aaron, B. McArdell, DFHM8,submitted (this proceedings volume)

5 Conclusions and outlook
The accuracy of the forced vortex equation forestimating debris flow velocity was investigated bycomparing existing velocity estimates from themeasurement station with those calculated using theforced vortex equation, based on an analysis of mud line
elevations extracted from bends using high-resolution
UAV imagery. The definition of the bend radii has beenhighlighted as a source of uncertainty [9] and, though
relevant, the preliminary results presented here indicatethat the choice of cross-section locations and the valueof k are far greater sources of uncertainty in the
superelevation approach to velocity estimation. Similar
to the work of [5, 8], k-factors back-calculated here
approached 1 for supercritical flow and became
increasingly large for subcritical flow and decreasing
Froude numbers.
Despite the rare level of monitoring and data
available from the Illgraben torrent, the use of thesuperelevation method to assess debris flow velocities ischallenging. During the debris flow season of 2022, datawill continue to be collected using improved
measurement methods at CD29, along with UAV-based
topographical data and orthophotos. Additionally, anumber of newly installed LiDAR sensors will facilitate
observations of front flow velocities and continuous
observations of surface velocity vectors, flow depths,
and flow surface geometry [15, 16]. Detailed pre- andpost-event topography will be available, along with thetiming of maximum flow depth and surface inclination
as well as the presence and timing of surges and crosswave maxima. This data will thus provide a newwindow into the physical meaning of the k-factor andhopefully enable improvements to the accuracy of4