Debris Flow Runout Predictions Based on the Average Channel Slope

Debris Flow Runout Predictions Based on the Average Channel Slope
Available online at www.sciencedirect.com
Engineering Geology 98 (2008) 29 – 40
www.elsevier.com/locate/enggeo
Debris-flow runout predictions based on the average channel slope (ACS)
Adam B. Prochaska a,⁎, Paul M. Santi a , Jerry D. Higgins a , Susan H. Cannon b
a
Colorado School of Mines, Department of Geology and Geological Engineering, 1516 Illinois Street, Golden, CO 80401 USA
b
U.S. Geological Survey, P.O. Box 25046 Mail Stop 966, Denver, CO 80225-0046 USA
Received 27 July 2007; received in revised form 16 December 2007; accepted 20 January 2008
Available online 13 February 2008
Abstract
Prediction of the runout distance of a debris flow is an important element in the delineation of potentially hazardous areas on alluvial fans and
for the siting of mitigation structures. Existing runout estimation methods rely on input parameters that are often difficult to estimate, including
volume, velocity, and frictional factors. In order to provide a simple method for preliminary estimates of debris-flow runout distances, we
developed a model that provides runout predictions based on the average channel slope (ACS model) for non-volcanic debris flows that emanate
from confined channels and deposit on well-defined alluvial fans. This model was developed from 20 debris-flow events in the western United
States and British Columbia. Based on a runout estimation method developed for snow avalanches, this model predicts debris-flow runout as an
angle of reach from a fixed point in the drainage channel to the end of the runout zone. The best fixed point was found to be the mid-point
elevation of the drainage channel, measured from the apex of the alluvial fan to the top of the drainage basin. Predicted runout lengths were more
consistent than those obtained from existing angle-of-reach estimation methods. Results of the model compared well with those of laboratory
flume tests performed using the same range of channel slopes. The robustness of this model was tested by applying it to three debris-flow events
not used in its development: predicted runout ranged from 82 to 131% of the actual runout for these three events. Prediction interval multipliers
were also developed so that the user may calculate predicted runout within specified confidence limits.
© 2008 Elsevier B.V. All rights reserved.
Keywords: Debris flow; Runout; Hazard mapping
1. Introduction
Debris-flow runout estimations are important for the
delineation of potentially hazardous areas on alluvial fans and
for the siting of mitigation structures. Existing runout estimation
methods require input parameters that can be difficult to accurately estimate. The purpose of this work was to develop a
simple method for preliminary estimates of debris-flow runout
distances that would be applicable for both fire-related and nonfire-related debris flows that emanate from confined channels
and deposit on well-defined alluvial fans. The scope of work
⁎ Corresponding author. 11131 W 17th Ave #106, Lakewood, CO 80215 USA.
Tel.: +1 303 241 0571; fax: +1 303 273 3859.
E-mail addresses: adamprochaska@hotmail.com (A.B. Prochaska),
psanti@mines.edu (P.M. Santi), jhiggins@mines.edu (J.D. Higgins),
cannon@usgs.gov (S.H. Cannon).
0013-7952/$ - see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.enggeo.2008.01.011
was to: (1) develop a model to estimate debris-flow runout
distance from easily measured topographic parameters, (2) test
the model using information from debris flow events independent of those used in its development, (3) provide the model
user with multipliers to the model output in order to obtain a
desired level of nonexceedance, (4) compare the accuracy of the
model to that of existing runout estimation methods, and (5)
compare the consistency of the model results to those from
scaled flume experiments.
2. Background
Most runout estimation methods for debris flows and other
mass movements fall into one of three categories: volume-based
models, dynamic models, and models based on topographic
parameters. These runout estimation methods have the limitation of requiring uncertain input parameters. This limitation is
discussed in the following sections.
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A.B. Prochaska et al. / Engineering Geology 98 (2008) 29–40
2.1. Volume-based models
As a first approximation, it had been proposed that debris-flow
runout distance could be related to event volume and deposit
geometry (Hungr et al., 1984; Hungr et al., 1987; VanDine, 1996;
Lo, 2000). Several authors have related the angle of reach of a
mass movement to its volume (Heim, 1932; Scheidegger, 1973;
Corominas, 1996). A mass movement’s angle of reach is the
declination of a line that connects the head of the failed mass to the
distal end of the deposit (Corominas, 1996). Rickenmann (1999)
developed an empirical relationship to relate a debris flow’s total
horizontal travel distance (L) to its volume (V) and the elevation
loss along the travel path (H) (Fig. 1). Ikeya (1981, 1989)
developed empirical relationships to estimate debris-flow runout
length from event volume and channel slope. Schilling and
Iverson (1997), Iverson et al. (1998), Griswold (2004), and Berti
and Simoni (2007) developed relationships between massmovement volumes and the inundated cross-sectional and planimetric areas. Cannon (1989) and Fannin and Wise (2001)
proposed that the initial volume of a debris flow and the rate at
which material is entrained or deposited along its travel path could
be used to estimate the total travel distance.
Volume-based models allow the likelihood of different debrisflow runout lengths to be estimated through frequency–magnitude
relationships. While it would be beneficial to map hazard zones
as a function of event volume, difficulties can arise in the
estimation of a probable range of flow volumes for a given
channel. Hungr et al. (1984) suggested that volume may be
estimated through a unit yield rate per length of channel or per
area of the drainage basin. However, generalized values for
these yield rates can be quite variable. For five debris-flow
events in British Columbia, the channel yield rate varied by a
factor of 3 and the area yield varied by a factor of 5 (Hungr
et al., 1984). Even after channels were divided into different
gradient and geologic classifications, the estimated channel
yield that could be expected from each classification still varied
by a factor of 2 (Hungr et al., 1984). Mizuyama (1982) showed
that for debris-flow events in central Japan, the area yield may
vary over several orders of magnitude. While detailed field
investigations of basins may produce reasonable estimates of
potential debris-flow volumes, these assessments may not be
justified for preliminary hazard mapping.
Fig. 1. Definition of the angle of reach, α, and β used in avalanche runout
predictions.
In addition to uncertainty in event volume, angle of reach and
channel yield rate estimations have the additional difficulty that an
initiation point of the debris flow must be predicted. Not all debris
flows initiate from a single failed landslide mass. In basins
recently burned by wildfires, rills and gullies contribute sediment
to the channel and subsequent scour and bulking of material can
occur over much of the channel length (Cannon et al., 2003). This
debris-flow initiation mode would make choosing a single point
of initiation impossible.
2.2. Dynamic models
A commonly advocated dynamic model to calculate debrisflow runout is the leading-edge model (Takahashi, 1981, 1991;
Hungr et al., 1984; VanDine, 1996; Lo, 2000):
v02 cos2 ðh0 hÞ
gh0 cosh0 2
1þ
xL ¼
ð1Þ
2v02
g Sf cosh sinh
where:
xL
v0
θ0
θ
g
Sf
h0
runout distance,
debris velocity,
entrance slope angle (channel slope),
runout slope angle (fan slope),
acceleration of gravity,
friction slope, and
debris flow depth, all in consistent units.
Other dynamic models are also presented by Cannon and
Savage (1988), Kang (1997), Lo (2000), Rickenmann (2005),
and Van Gassen and Cruden (1989). Dynamic models require
two parameters that can be difficult to accurately estimate,
namely the flow velocity and the frictional parameter. Flow
velocity may be back-calculated from previous events (Johnson,
1984) or predicted using flow equations. Velocity backcalculations require an estimate of a channel's radius of curvature. This estimate can be subjective and dependent on the
method of analysis (Prochaska et al., in review), although a
reasonable estimate of the radius of curvature may be obtained.
Velocity predictions require the selection of an appropriate
rheological model and its input parameters. Rickenmann (2005)
suggested that the frictional parameter (Sf ) could be estimated to
be slightly larger than the tangent of the fan slope. However, as
Sf approaches the tangent of the fan slope in Eq. (1), the
denominator approaches zero and this equation becomes increasingly sensitive to Sf. Sassa (1988) described the calculation
of a frictional parameter through the measurement of the friction
angle and pore pressure during shearing in a ring shear apparatus. This method has the disadvantages of the scarceness of
ring shear devices and limitations on the maximum particle size
that may be tested.
Numerical models also exist for the analysis of debris flows.
These models treat the flowing debris as either a continuum (e.g.
O'Brien et al., 1993; Hungr, 1995; McDougall and Hungr, 2003;
McArdell et al., 2007) or as distinct elements (e.g. Asmar et al.,
2003; González et al., 2003; Miyazawa et al., 2003). González
A.B. Prochaska et al. / Engineering Geology 98 (2008) 29–40
et al. (2003) discuss the advantages and limitations of continuum
and distinct element modeling techniques. Continuum models
have faster computational times and are better suited to model
viscous flows and pore pressure effects, while distinct element
models can better model particle segregation and active and
passive pressures. Although generalized calibration factors have
been obtained for numerical models (e.g. Ayotte and Hungr,
2000), these factors can be highly variable, even for different
debris flows from a single channel (McArdell et al., 2007). This
would present difficulties when applying these models in a
predictive sense. Numerical runout simulations are also limited by
the quality of the topographic data used and difficulties associated
with modeling the entrainment and deposition of debris along the
travel path (McArdell et al., 2007). Non-uniqueness of a solution
is also a problem during the back-analysis of an event.
With the use of the correct input parameters, dynamic models
have the potential to provide accurate runout lengths. They also
provide additional information, such as the flow velocity along
the runout path, the area of flow, and the peak discharge. However, these models also require the most sophistication during data
collection and analysis in order to estimate appropriate input
parameters. The cost of these detailed analyses may not be
warranted for preliminary hazard assessments.
2.3. Topographic parameters
To avoid the use of uncertain and highly variable input
parameters required in the classical avalanche runout equation
(Voellmy, 1955), snow avalanche runout estimations have been
made through a method that used easily measured topographic
parameters (Lied and Bakkehøi, 1980; Bakkehøi et al., 1983;
Lied and Toppe, 1989). This method predicted the angle of
reach, α, of an avalanche as a linear function of the average
slope of the travel path, β. α and β were both referenced from
the head of the failed snow mass (Fig. 1). Bathurst et al. (1997)
obtained encouraging results when this method was used to
estimate the delivery of debris-flow sediment into streams.
Although this runout estimation method avoids the prediction
of event volume, velocity, and frictional parameters, it still has the
disadvantage discussed previously of requiring knowledge of the
event initiation point, since the α and β angles are referenced from
the head of the failed mass.
3. Data sets
3.1. Field data set
Mapped debris-flow deposits described in the technical
literature were used to develop, test, and analyze the runout
model. A summary of the data set is shown in Table 1. The
majority of the debris-flow events in the field data set would be
classified as Size Class 3 or 4 (Jakob, 2005), which have volumes
that range between 1000 and 100,000 m3. The deposits of the
debris-flow events included in the field data set had been mapped
by the original researchers (Table 1) during field investigations.
These mapped deposits were illustrated in the references at scales
typically larger than 1:10,000. All debris-flow events in the data
31
set were single events that emanated from confined channels and
terminated in natural lobate deposits on well-defined fans. Events
were excluded from the data set if they deposited within a steep
channel, or if they deposited in, along, or against an obstruction
such as a river, an opposing bank of a higher-order stream, or a
road embankment. Excluding events that deposit within channels
may bias the data set towards longer-runout debris flows, however
flows that reach alluvial fans are generally more hazardous to
developments than those that deposit within channels. A larger
data set would have been desirable, but published values of
accurately measured, unimpeded debris-flow runout were rare.
The heavy horizontal lines on Table 1 divide the field data set
into four subsets based on geology, burn condition, and location.
Basins where a debris-flow event occurred more than three years
after a forest fire were categorized as being unburned, since most
fire-related debris flows occur in the first few years following a
fire before the vegetation becomes reestablished (Cannon and
Gartner, 2005). Debris flows from the burned basins were runoffgenerated, and material was entrained over much of the channel
length. While debris flows from the unburned basins also may
have entrained material along their travel paths, these events were
generated from discrete landslides.
The first field data subset in Table 1 contains ten debris-flow
events from burned basins underlain by sedimentary materials
in Utah and Colorado. The second subset contains six debrisflow events from unburned basins underlain by sedimentary
materials in California and British Columbia. The third subset
contains four events from unburned basins underlain by metamorphic and plutonic materials in British Columbia, California,
and Colorado. The fourth subset contains two events from
burned basins underlain by metamorphic and plutonic materials
from Wyoming and Montana, and one unburned basin underlain
by metamorphic and plutonic materials in Idaho. The first three
data subsets were used to develop the runout estimation model.
The fourth subset was withheld from model development in
order to provide a geographically independent test of the model.
3.2. Flume data set
Flume data collected from several sources in the technical
literature were compared to the developed model. Chau et al.
(2000) performed 10 tests on sand in a 3-meter-long flume that
was inclined between 26 and 32°. Runout slopes were either 0
or 10°, gravimetric water contents ranged from 25.5 to 31.2%,
and runout distances ranged from 18 to 170 cm.
Shieh and Tsai (1997) performed 69 tests on sand and gravel
in an 8-meter-long flume that was inclined between 15 and 21°.
The flume bed was roughened by gluing sand to it. Runout
slopes were 2 or 5°, gravimetric water contents ranged from 35
to 88%, test volumes ranged from 3600 to 51,300 cm3, and
runout distances ranged from 29 to 125 cm.
Various sources report the results of experiments conducted by
the U.S. Geological Survey (USGS) in their large-scale (95meter-long) debris-flow flume (Iverson, 1997; Major and Iverson,
1999; Iverson et al., 2000; Iverson, 2003; Iverson, 2005). This
flume is inclined at 31° and can accommodate as much as 20 m3
of up to gravel-sized material. Runout occurs on a 3-degree pad;
32
A.B. Prochaska et al. / Engineering Geology 98 (2008) 29–40
Table 1
Summary of the field data set
Basin name a
Year Location
Geology b Burn condition c Volume (m3) β (deg) α (deg) Runout length d (m) Reference
T3
2002 Santaquin, UT
Sed.
2001
1700
12.1
11.2
435
T4
2002 Santaquin, UT
Sed.
2001
15,200
12.5
11.0
694
T5
2002 Santaquin, UT
Sed.
2001
9900
15.4
13.4
703
T6
2002 Santaquin, UT
Sed.
2001
7600
18.1
16.9
390
T7
2002 Santaquin, UT
Sed.
2001
2400
21.7
20.1
249
T9
2002 Santaquin, UT
Sed.
2001
530
22.1
19.8
354
T2
T3
Compton
Bench Middle
Storm King
Mountain – G
Gulley #1
2004 Santaquin, UT
2004 Santaquin, UT
2004 Farmington, UT
Sed.
Sed.
Sed.
2001
2001
2003
N/A
N/A
N/A
17.9
12.1
22.5
14.8
11.7
18.3
723
238
294
1994 Glenwood Springs, CO
Sed.
1994
1000
21.8
19.1
183
1973 Port Alice, B.C.
Sed.
u.b.
22,000
22.2
18.2
368
Gulley #2
1975 Port Alice, B.C.
Sed.
u.b.
4500
20.7
18.3
305
Newton Canyon
Cathedral Gulch
1965 Malibu, CA
1925 Field, B.C.
Sed.
Sed.
u.b.
u.b.
4600–6100
80,000
7.1
23.0
6.8
20.1
400
635
Cathedral Gulch
1946 Field, B.C.
Sed.
u.b.
90,000
23.0
19.3
785
Cathedral Gulch
1962 Field, B.C.
Sed.
u.b.
24,000
23.0
22.6
221
Hummingbird
Creek
Mayflower Gulch
Surprise Canyon
Pilot Ridge
Twelve Kilometer
1997 Sicamous, B.C.
Met./Plut. u.b.
92,000
13.2
11.9
420
1961
1917
1997
1989
Met.
Met.
Plut.
Met./Plut.
17,000
26.5
500,000
7.6
460
19.8
8500–14,800 10.7
22.9
6.5
16.7
9.4
181
3178
142
475
Slough R.S.
1989 SW Montana
Met./Plut. 1988
N/A
17.2
15.8
400
Jughead
1997 Lowman, ID
Plut.
14,600
14.1
11.7
180
a
b
c
d
e
Climax, CO
Ballarat, CA
Stanislaus National Forest, CA
NW Wyoming
u.b.
u.b.
u.b. (1987)
1988
u.b. (1989)
McDonald and
Giraud (2002)
McDonald and
Giraud (2002)
McDonald and
Giraud (2002)
McDonald and
Giraud (2002)
McDonald and
Giraud (2002)
McDonald and
Giraud (2002)
unpublished e
unpublished e
Giraud and
McDonald (2005)
Cannon
et al. (1998)
Nasmith and
Mercer (1979)
Nasmith and
Mercer (1979)
Campbell (1975)
Jackson
et al. (1989)
Jackson
et al. (1989)
Jackson
et al. (1989)
Jakob et al.
(2000)
Curry (1966)
Johnson (1984)
DeGraff (1997)
Meyer and
Wells (1997)
Meyer and
Wells (1997)
Meyer
et al. (2001)
As reported within the reference.
Sedimentary (Sed.), Metamorphic (Met.), or Plutonic (Plut.) rocks.
Year of most recent forest fire (u.b. = unburned).
Horizontal distance from the fanhead point to the distal end of the deposit (Fig. 2).
Obtained from Richard Giraud of the Utah Geological Survey through personal communications in 2005.
runout lengths of the eight tests used in this study ranged from
11.9 to 31.6 m.
objective, reproducible location that attempts to provide a
center-of-mass characterization of fire-related debris flows that
initiate through progressive sediment bulking along much of
4. Methods
4.1. Development of the ACS model
Similar to the model shown on Fig. 1, the ACS model
predicts the runout angle (α) from the fanhead angle (β ).
However, rather than attempting to identify a point of initiation
for each event, we selected the point from which to reference the
α and β angles as the point on the stream profile that is halfway
(vertically) between the onset of deposition (which corresponds
to the fanhead point at the apex of the alluvial fan) and the
drainage divide, as shown on Fig. 2. This reference point is an
Fig. 2. Location of the point from which α and β were referenced in the ACS
model.
A.B. Prochaska et al. / Engineering Geology 98 (2008) 29–40
their channel length. While the erosion along the path of a firerelated debris flow (not necessarily from a completely burned
basin) may not be homogeneous and its initiation is difficult to
predict, the basin’s midpoint elevation provides a simple approximation. This reference point may have no physical meaning in
other cases, and it was lower in the basin than the initiating
landslides for the unburned debris-flow events. Several different
reference points were investigated before the one shown on
Fig. 2 was decided upon. The use of these other points was
decided against because of subjectivity in identification of their
locations, sensitivity in their use, or less of a physical basis to
their placement.
For each debris-flow event in the field data set, topographic
maps were used to construct a profile of the basin along the length
of the main channel. 1:24,000 scale topographic maps were used
for debris-flow events within the United States and 1:50,000 scale
topographic maps were used for events in Canada. These are the
largest-scale maps that are commercially available, and we
consider them sufficiently accurate for drainage-basin-scale
analyses. This profile was extended down the fan beyond the
farthest extent of the mapped runout and up to the drainage divide
directly above the head of the basin's main channel. The extents
of the mapped deposit were also located along the profile. The β
angle was identified for each basin by measuring the declination
of a line that connected the reference point to the fanhead point on
the profile. The fanhead point was selected from topographic
maps as being the point where a decrease in channel confinement
accompanied a decrease in the channel slope. Fig. 3 shows
examples of the fanhead points chosen for basins T3 and T4. The
α angle was identified for each basin by measuring the declination
of a line that connected the reference point to the farthest extent of
runout on the profile.
The accuracy of the data used in the model development is
limited by the amount of mapping detail and scale used by the
authors of the original references. Runout distance can be a
subjective measurement that varies between investigators based
on the specific facies or thickness of deposit that is used to
define the extent of runout (Jakob, 2005). Hungr et al. (1987)
reported that the limit of runout is a gradational process. They
33
advocated that the extent of runout should be defined as the
farthest extent of debris surge travel, downstream of which no
large-scale debris movement will occur. For studies that mapped
different deposit facies (e.g. Meyer and Wells, 1997), we
interpreted runout as the farthest extent of deposits mapped as
being debris-flow deposits. For sources that mapped different
levels of impact within the deposit (e.g. Jakob et al., 2000),
runout was interpreted to be the farthest extent of the direct
impact zone. For sources that did not map different facies or
impact zones, runout was interpreted as the farthest extent of the
mapped deposit.
Regression analyses of α versus β were performed. Dummy
variables, DA and DB, (Wonnacott and Wonnacott, 1990) were
used to code for the first three field data subsets through binary
independent variables. If regression analyses show that DA and
DB are both statistically insignificant, then the regression
equations to the individual data subsets share a common intercept. For the burned sedimentary subset, DA and DB were both
set equal to zero. For the unburned sedimentary subset, DA was
set equal to one and DB was set equal to zero. For the plutonic
and metamorphic subset, DA was set equal to zero and DB was
set equal to one. Best-fit linear regression equations were
obtained for each of the first three field data subsets. ANOVA
and t-tests were performed using MINITAB (statistical analysis
software) to investigate the statistical significance of the regression equations as a whole and of the individual terms of
these equations. After a statistically significant regression equation was obtained for each of the first three field data subsets,
tests were performed to identify whether these regression equations were statistically similar (Crow et al., 1960). The data used
to develop statistically similar regression equations were combined and a new regression equation was calculated for the
unified field data sets.
4.2. Test of the ACS model
The fourth field data subset was withheld from the model
development in order to be a geographically independent test of
the developed model. For the debris-flow events in this subset, the
runout (α) angles that were predicted from the basins' fanhead (β)
angles (using Eq. (3)) were used to calculate predicted runout
lengths from the basin profiles. These predicted runout lengths
were compared to the actual mapped deposit extents and were
reported as percentages of the actual runout.
4.3. Flume tests
Fig. 3. Fanhead locations chosen for basins T3 and T4.
For the flume data collected from the literature, actual runout
(α) angles and fanhead ( β ) angles were calculated trigonometrically based on the reported runout length, runout slope, flume
slope, and flume length for each test. To remain consistent with
the methodology of the field data set, α and β angles were
referenced from half-way up the length of the flume. However,
flume tests differ from field events, in that debris is released
from the head of the flume and no entrainment of material
occurs along the flow path. In most cases, the β angle was equal
to the flume slope since the flume was linear. The curved profile
34
A.B. Prochaska et al. / Engineering Geology 98 (2008) 29–40
Fig. 4. Regressions of α versus β for the first three field data subsets.
Fig. 5. The final developed ACS model and bounds to the data set. Also shown
are the three data points used to test the model.
of the USGS flume was modeled as being three linear segments:
a 75-meter section inclined at 31°, a 7-meter section inclined at
17°, and a runout section inclined at 3°. The 17-degree inclined
section was chosen as the intermediate angle between the other
two sections in order to represent the curved bottom of the
flume (Iverson et al., 1992). The β angle was referenced to the
junction of the 17-degree section and the 3-degree runout pad.
For tests where an additional length of confinement had been
added along the runout section, the β angle was referenced to
the farthest extent of the confinement.
5. Results
5.1. Development of the ACS model
Table 1 shows the measured fanhead (β) and runout (α)
angles for each debris-flow event in the field data set. Using the
dummy variables, the best-fit regression equation to the first
three field data subsets was (with α and β expressed in degrees):
a ¼ 0:80 þ 0:84b þ 0:06DA 0:42DB
R2 ¼ 0:97
ð2Þ
Fig. 4 shows the best-fit regression lines and equations corresponding to the first three field data subsets. Table 2 summarizes
the results of the ANOVA and t-tests to test for the significance of
the regression equations shown on Fig. 4. The low P-values in
Table 2
Summary of tests for significance of the best-fit regression equation
Regression results
P-values
a
Standard Error.
Attribute
Value
Slope
Slope SE a
Intercept
Intercept SE
DA
DA SE
DB
DBSE
Slope
Intercept
DA
DB
Regression
0.84
0.04
0.80
0.77
0.06
0.50
− 0.42
0.57
0.00
0.31
0.91
0.47
0.00
Table 2 indicate that each best-fit regression equation is
statistically significant, and that the slopes of each of these
equations are also statistically significant. However, the high Pvalues on DA, DB, and the intercept indicate that there is not a
statistically significant difference between the intercepts of the
three field data subsets, and also that the intercepts of the best-fit
regression lines are not statistically significant. Therefore, the
regression equations for the first three field data subsets were
recalculated while forcing the intercepts to zero, and t-tests
showed no significant differences (at a significance level of 0.005)
between the slopes of the three regression lines. As a result, the
three field data subsets were then combined into a single
regression equation with an intercept of zero, which is plotted on
Fig. 5:
a ¼ 0:88b
R2 ¼ 0:96
ð3Þ
The standard error of the slope of Eq (3) is 0.01, and α and β
are again expressed in degrees.
Forcing the intercept through zero makes physical sense as
well, so that α does not exceed β for basins with small β angles.
Fig. 5 also shows bounding lines α = β and α = 0.80β. The line
α = β is a physical upper-bound to the data representing no
runout, because by definition the runout angle α must be less
than the fanhead angle β (Fig. 2). The line α = 0.80β has no
physical or statistical basis, but is a convenient lower-bound
encompassing the data set. By representing a lower-bound to the
declination data, the equation α = 0.80β actually represents an
upper bound to the corresponding runout distances.
5.2. Test of the ACS model
The three data points withheld from model development to
permit an independent test are shown on Fig. 5. The runouts
predicted by Eq (3) ranged from 82 to 131% of the actual
runouts for these three events, as summarized in Table 3.
5.3. Confidence in prediction
The ACS model regression equation shown as Eq. (3) was
bounded by two sets of limits: prediction limits and confidence
A.B. Prochaska et al. / Engineering Geology 98 (2008) 29–40
Table 3
Results of the test of the ACS model
Basin
β
Actual Actual
(deg) α (deg) runout
(m)
Twelve
10.7
kilometer
Slough R.S. 17.2
Jughead
14.1
Predicted Predicted Predicted/actual
α (=0.88β) runout
runout (%)
(deg)
(m)
9.4
475
9.4
480
101
15.8
11.7
400
180
15.1
12.4
525
148
131
82
limits. For the regression equation to be used as a predictive
tool, the prediction limits are of interest because prediction
limits define the expected range of the response (α) for a new
observation (β ) at a given confidence level. For confidence
levels of 65 to 95% in 5-percent increments, MINITAB was
used to identify the responses (α values) at the prediction limits
for the actual β values of the field data set. These α values were
drawn through the reference points of the appropriate basin
profiles and the runout lengths predicted by the prediction-limit
α values were obtained. Runout obtained from the predictionlimit α value as a percentage of the runout obtained from the
regression equation was plotted versus the confidence level for
each basin; this graph was then summarized by producing box
and whisker plots at each confidence level.
Fig. 6 shows multipliers that can be applied to predicted
runout lengths of the field data set to achieve desired confidence
levels. The graph is used by selecting a desired confidence level,
which is plotted on the x-axis, and a certain percentage of the
field data set not to be exceeded, which is represented by the
box and whisker plots. The multiplier to the predicted runout
length is then read along the y-axis. For example, in order to be
70% confident that the estimated runout length would not be
exceeded by one-half of the field data set (represented by the
median value), the runout length obtained from the α calculated
using Eq (3) would need to be multiplied by 1.5.
5.4. Comparison to existing runout models
Predicted runout lengths for six events within the field data
set were compared to runout lengths that would be predicted by
35
Table 4
Volumes and mobility characteristics for the debris-flow events used in the
comparative analysis
Basin name
Volume
(m3)
Actual H
(m)
Actual L
(m)
Actual runout a
(m)
Gulley #1
Gulley #2
Hummingbird Creek
Mayflower Gulch
Pilot Ridge
Jughead
22,000
4500
92,000
17,000
460
14,600
650
660
1370
370
240
230
1560
1420
5540
800
770
1130
368
305
420
181
142
180
a
Horizontal distance from the fanhead point to the distal end of the deposit.
other runout estimation methods. The two estimation methods
used for comparison were those presented by Corominas (1996)
and Rickenmann (1999, 2005) that relate a debris flow's angle
of reach to its volume:
log ð H=LÞ ¼ 0:105 log V 0:012
ð4Þ
(Corominas, 1996)
L ¼ 1:9 V 0:16 H 0:83
ð5Þ
(Rickenmann 1999, 2005)
where:
H
L
V
elevation loss along the travel path (Fig. 1) (m),
total horizontal travel distance (Fig. 1) (m), and
event volume (m3).
The six debris-flow events chosen for comparison had
initiating landslides from which to reference H and L whose
locations were accurately described in their respective reference:
Gulley #1, Gulley #2, Hummingbird Creek, Mayflower Gulch,
Pilot Ridge, and Jughead. Event volumes and mobility characteristics for these six debris flows are presented in Table 4.
For each of the debris flows listed in Table 4, the event
volume was used to calculate a predicted angle of reach using
both Eqs. (4) and (5). Lines with declinations equal to these
angles were drawn through the location of the initiating
landslide on the basin profile for each event; debris-flow travel
was predicted to cease where these angle-of-reach lines intersected the basin profile. Table 5 summarizes the mobilities for
these six events that were predicted using Eqs. (4) and (5). For
three of these events, the predicted angle of reach was steeper
than, and thus fell below, the basin profile. For comparison, the
runout lengths predicted by Eq. (3) are also included in Table 5.
5.5. Volume effects on debris mobility
Fig. 6. Summary of multipliers to the predicted runout in order to have a certain
confidence of nonexceedance for the field data set. Whiskers extend 1 interquartile
range beyond the quartiles.
Previous research has shown that debris-flow mobility and
inundation areas increase with an increase in event volume
(Heim, 1932; Corominas, 1996; Schilling and Iverson, 1997;
Iverson et al., 1998; Griswold, 2004; Rickenmann, 1999, 2005).
Thus, the possibility of including event volume to improve the
predictive capabilities of Eq. (3) was explored. Using the data
36
A.B. Prochaska et al. / Engineering Geology 98 (2008) 29–40
Table 5
Predicted mobilities for the debris-flow events used in the comparative analysis
Basin name
Predicted H/L Predicted H0.83/L
Predicted runout a Predicted runout a Predicted runout b Predicted runout b Predicted runout b
from Eq. ( 4) from Eq. (5) (m− 0.17) from Eq. (4) (m) from Eq. (5) (m) from Eq. (4) (%) from Eq. (5) (%) from Eq. (3) (%)
Gulley #1
Gulley #2
Hummingbird Creek
Mayflower Gulch
Pilot Ridge
Jughead
0.34
0.40
0.29
0.35
0.51
0.36
a
b
c
0.11
0.14
0.08
0.11
0.20
0.11
775
678
B.C.P. c
580
B.C.P. c
B.C.P. c
892
561
B.C.P. c
739
B.C.P. c
B.C.P. c
211
222
N/A
320
N/A
N/A
242
184
N/A
408
N/A
N/A
64
106
130
91
73
82
Horizontal distance from the fanhead point to the predicted end of travel.
Predicted runout reported as a percentage of the actual runout. See Table 4 for actual runout distances.
The predicted angle-of-reach line fell below the channel profile.
from the field data set (Table 1), regressions of different
mobility characteristics versus volume were investigated. For
debris-flow events where a range of volumes had been reported,
the average value was used. Power law regressions were examined, based on trends observed by others (Corominas, 1996;
Rickenmann, 1999, 2005). Regression results are summarized
in Table 6.
5.6. Flume tests
Fig. 7 shows how the flume test results compare to the field
data set results on a plot of α versus β. Also shown on Fig. 7 are
the linear relationships previously shown on Fig. 5. Details of
this comparison are provided in the discussion section below.
6. Discussion of results
6.1. Development of the ACS model
The fact that no statistical difference was found between the
regressions to the burned sedimentary field data subset and the
unburned sedimentary field data subset indicates that both firerelated and non-fire-related debris flows within the field data set
had similar mobility characteristics. It tentatively appears that a
distinct runout length can be identified for debris flows, regardless of whether those flows are initiated by progressive sediment
bulking or by a single failed landslide mass. In addition, since
no statistical difference was observed between sedimentary
basins and metamorphic and plutonic basins, this tentatively
indicates that debris flows within the field data set had similar
mobility characteristics regardless of geology. Unfortunately,
the analyzed field data set did not include any recently burned
metamorphic and plutonic basins; however, we would antici-
pate runout behavior similar to the unburned basins. The ACS
model provides an average runout length based on the ranges of
material properties represented within the field data set. Data
scatter around the predictive model (Fig. 5) is likely due to
different material properties and event volumes between the
individual events.
For the field data set, Eq. (3) approximately quantifies the
relationship between landscape degradation and construction
through the process of debris-flow deposition. The wedgeshaped region on Fig. 5 bounded by the lines α = β and α = 0.80β
shows that runout (α ) angles within the field data set become
more variable as basins become steeper. This does not
necessarily mean that actual runout lengths vary more widely,
though, because a variation in α for steep basins produces less
of a change in the runout length than the same variation in α for
shallow basins (Fig. 8). For example, the runout lengths for the
two debris-flow events on Fig. 5 near β = 7° differ by over
2500 m, while the runout lengths for the three events shown on
Fig. 5 at β = 23 only vary by less than 600 m. These runout
differences between individual basins are not the predictive
error of the ACS model, as the conversion of an α angle to a
runout distance will also depend on the site-specific topography.
6.2. ACS model applicability and limitations
The ACS model was developed from modest-sized debris
flows (Size Class 3 and 4, (Jakob, 2005)) that emanated from
confined channels with simple geometries and deposited on welldefined alluvial fans, and thus it would be applicable to only
Table 6
Power law regression results of debris-flow mobility versus event volume for the
field data set
R2 (%) P-value Eq.
Dependent variable (y)
versus Volume (V) (m3)
Best-fit Power
Law Equation
Actual runout length (m)
Actual α (degrees)
β (degrees)
Runout predicted by Eq. (3)/actual
runout length (%)
44
y = 36.6*V0.257
y = 24.7*V− 0.057 7
y = 26.6*V− 0.052 6
y = 131*V− 0.020
1
0.000
0.374
0.416
0.813
(6)
(7)
(8)
(9)
Fig. 7. α versus β for flume events.
A.B. Prochaska et al. / Engineering Geology 98 (2008) 29–40
37
since the angle α must be smaller than the angle β and can not
be negative. The geometry of the ACS model makes it sensitive
to small changes between α and β, especially for low-gradient
channels, which may result in the variability of predicted runout
lengths. Uncertainty in β (and thus in the calculated α angle)
will depend on the certainty with which the fanhead point can be
identified and the steepness of the basin profile, which can not
easily be quantified. Since the ACS model is empirical rather
than physically based, variability will also be introduced
through the inherent complexity of debris-flow processes.
6.3. Confidence in prediction
Fig. 8. Illustration of how a variation in α produces less of a runout change in
steep basins (a) than in shallow basins (b).
events matching these criteria. The model would not be applicable
to more complex situations such as canyons that have been
glacially oversteepened or where a fanhead location can not be
clearly identified. Depending on fan morphology, the fanhead
point may not be easily identifiable and large errors may results. In
cases such as these, application of the ACS model would be much
more subjective and could result in anomalous results. The ACS
model provides an average runout length based on the ranges of
material properties represented within the field data set. However,
only a modest data set was available for development of the ACS
model. If this model is applied to a more diverse population of
debris-flow events, certain other geographic, geologic, or material
property exclusions may appear.
The ACS model does not account for debris-flow volume,
and thus does not consider the frequency–magnitude relationship of debris flows. It has been shown that debris-flow travel
lengths and inundation areas increase with event volume (Heim,
1932; Corominas, 1996; Schilling and Iverson, 1997; Iverson
et al., 1998; Griswold, 2004), and for a given topography a large
debris-flow event would be expected to runout farther than a
small event. Therefore, the ACS model may significantly
underestimate the runout length for events larger than those
used in its development.
The ACS model also does not account for 3-D topography.
The model was developed based only on the maximum runout
length for each event. For alluvial fans where lateral debris
spreading or channel avulsions would be suspected, a conservative estimate of the hazard zone could be delineated by
swinging an arc across the fan with a radius equal to the
predicted runout length and its center located at the fanhead
point. This hazard zone would be applicable for large events
where fan inundation is expected. For smaller events where
local topography and confinement will control runout, a more
sophisticated model that accounts for 3-D topography may
provide more accurate runout estimations.
The ACS model was based on a limited data set that does not
contain the variability inherent to debris-flow processes. Thus,
scatter could be greater than that shown within the available
data. However, the data are also constrained physically by an
upper-bound line of α = β and a lower-bound of (β, α) = (0, 0)
Fig. 6 was based on the actual field data used to develop and
test the ACS model. The results shown in the graph are dependent on site-specific geometries, because the conversions
between runout (α) angles and actual runout lengths are dependent on basin profile shapes. The information shown on
Fig. 6 would be applicable to drainage basins within the range
of geometries used to develop the data. The field data set had β
angles that ranged from 7.1 to 26.5° and alluvial fan slopes that
were less than 12°. The use of a small data set to develop this
information is accounted for statistically by increasing the width
of the confidence interval band.
The user of Fig. 6 is free to choose whichever confidence
level and percentage of the data set is deemed appropriate. This
choice would likely result from a cost-benefit analysis depending on the hazard and risk associated with the site in question.
As an alternative to Fig. 6, the relationship α = 0.80β may be
used, which represents an upper-bound to the runout distances
observed in the field data set (Fig. 5). Due to the scatter present
in the field data set, more rigorous investigations and analyses
may be warranted to provide improved runout predictions on a
site-by-site basis.
6.4. Comparison to existing runout models
The angles of reach predicted by Eqs. (4) and (5) were
steeper than, and thus fell below, the basin profile for three of
the debris flows listed in Table 5. This can be explained by the
physical attributes of these three flows and their drainage
basins. The gradient of Hummingbird Creek is very low for a
debris-flow-transporting basin (Jakob et al., 2000), and thus the
predicted angle of reach could likely be steeper than it. The
small volume of the Pilot Ridge debris flow (Table 4) would
have resulted in the prediction of a steep angle of reach.
For the three debris flows in Table 5 for which the angles of
reach predicted by Eqs. (4) and (5) could be used to estimate a
runout length, the predicted runout lengths ranged from 184 to
408% of the actual runout lengths. This over-estimation can be
explained by the sources of the event volumes. During angle of
reach calculations using Eqs. (4) and (5), the entire event volume
was assumed to have originated at the initiating landslide.
Although these events were initiated by discrete landslide masses,
considerable material was scoured from the channel along the
debris-flow path (Nasmith and Mercer, 1979) and multiple
initiation points were present (Curry, 1966). Thus, applying the
38
A.B. Prochaska et al. / Engineering Geology 98 (2008) 29–40
entire event volume to a single initiation point would overestimate the actual volume present at that location, and a lower
(more conservative) angle of reach would result. From a physical
sense, this error results from the large discrepancies between the
heads of the initiating landslides and the centers of mass of the
source areas, making angle-of-reach estimations questionable.
For the six debris flows listed in Table 5, the ACS model
predicted runouts that were 64 to 130% of the actual runouts, with
an average prediction of 91%. For these six events, the ACS
model provides more accurate runout lengths than do Eqs. (4) and
(5). This should be expected, since many of the events listed in
Table 5 were used to develop the ACS model.
6.5. Volume effects on debris mobility
The low P-value for Eq. (6) (Table 6) shows that there is a
statistically significant relationship between increasing runout
lengths and increasing event volumes for the field data set.
However, this trend is not present when the runout lengths are
converted to their α and β components. The high P-values for
Eqs. (7) and (8) (Table 6) indicate that there are not statistically
significant relationships between event volumes and either the
angles of reach (as defined by the ACS model) or channel
slopes, respectively, for events within the field data set. The
high P-value for Eq. (9) (Table 6) shows that no statistically
significant relationship exists between the predictive capability
of Eq (3) and event volume for events within the field data set.
Thus, we concluded that Eq (3) could not be improved upon by
including event volume.
6.6. Flume tests
Fig. 7 shows that the field data are comparably tight with
respect to the flume data within the same range of β angles. The
flume tests by Shieh and Tsai (1997) were conducted in a
roughened flume at slope angles within the range of slopes
included in the field data set; these results also fit well with the
ACS model best-fit regression of α = 0.88β. The flume tests
conducted by the USGS and Chau et al. (2000) were performed
at β angles higher than most of those found in the field data set.
Many of these tests also had runout (α) angles that were
considerably lower than the trend indicated by the field data,
which means the runouts for these tests were unexpectedly long.
It is presumed that these longer runouts are caused by increased
velocities due to travel paths that are steeper than those found in
the field data set. Additionally, flume tests are subjected to
unnatural boundary conditions and limitations to the maximum
particle size that may be tested. It is believed that the ACS model
would not be applicable to these steeper flume inclinations, since
it was developed from events with shallower β angles.
7. Summary and conclusions
A model has been developed to provide preliminary runout
predictions from the average channel slope (ACS model) for
modest-sized debris flows that emanate from confined channels
and have unobstructed deposition on well-defined alluvial fans.
A debris flow's runout angle, α, is predicted from the fanhead
angle (the average slope of the travel path), β, through the
relationship α = 0.88β. Both of these angles are referenced from
the location along the basin’s main channel that is at an
elevation half-way between the fan apex (representing the onset
of deposition) and the drainage divide. This model has been
developed from and tested on debris-flow events in basins
underlain by non-volcanic rocks across western North America.
The model appears to work equally well for both fire-related
and non-fire-related debris-flow events. The model was tested
using three debris-flow events from locations independent of
those used in its development; the predicted runout lengths
ranged from 82 to 131% of the actual runout lengths. The model
also compared well with laboratory flume tests within the same
range of channel slopes. The user of the model is provided with
prediction interval multipliers so that predicted runout can be
calculated within specified confidence limits.
The ACS model does not account for material properties, event
volume, or 3-D topography, and thus other rigorous analyses may
be required to provide more accurate estimates of runout distance.
The ACS model should not be applied to lahars or to drainage
basins other than those where confined channels have unobstructed debris-flow deposition on well-defined alluvial fans. The
variability of ACS model results will depend on the accuracy of
the chosen fanhead location, small changes between the α and β
angles (especially for low-gradient channels), and physical
debris-flow processes.
Acknowledgments
This work has been funded in part by the U.S. Department of
Education through a Graduate Assistance in Areas of National
Need (GAANN) Fellowship, award #P200A060133. Also thanks
to Richard Giraud from the Utah Geological Survey for the data
he provided and to Dr. A.K. Turner for his assistance with
statistical analyses. Improvements to earlier drafts of this paper
resulted from fruitful comments by Rex Baum, Rob Ferguson,
Oldrich Hungr, Richard Iverson, Dieter Rickenmann, Bill Schulz,
and three anonymous reviewers.
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