E3S Web of Conferences 415, 03029 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341503029

Objective definition of discharge thresholds for post-fire debrisflows
Hui Tang 1, Luke McGuire 2, and Ann Youberg 3
1Section 4.7: Earth Surface Process Modelling, German Research Centre for Geosciences (GFZ), Potsdam, Germany
2Department of Geosciences, The University of Arizona, Tucson, USA
3

Arizona Geological Survey, The University of Arizona, Tucson, USA

Abstract. Runoff-generated debris flows are a common post-fire hazard in the western United States anda growing number of regions around the world. As wildfire continues to emerge across a broader range of
geographic regions and plant communities, there is an increasing need for generalizable methods to predict
post-fire debris-flow initiation. The prediction of post-fire debris flow during intense rainstorms has
traditionally relied upon empirical rainfall thresholds. Rainfall intensity-duration thresholds are often
developed based on rainfall data and the hydrologic response to those rainstorms. They are most applicableto the specific regions where data are collected. Here, we present a new predictive approach that utilises
processes-based models with fundamental physics and machine learning methods to estimate discharge
thresholds for runoff-generated debris-flow initiation in four recently burned areas in the western UnitedStates. We assess the performance of the objectively defined discharge threshold-based predictions for postfire debris-flow initiation from our hybrid framework, which utilises debris-flow timing within rainstorms,
physically based numerical simulations of runoff, and the support vector machines method. The proposed
thresholds have a good balance between true and false predictions for debris flow and floods. Importantly,
our method permits the direct estimation of rainfall intensity-duration thresholds for areas where post-firedebris flow observations are limited.

1 Introduction
Wildfire significantly changes vegetation and soilhydrologic properties, making burned areas more likely
to produce increased runoff and debris flows thanunburned areas [1, 2]. Specifically, wildfires usually
decrease canopy interception and reduce surface water
storage, allowing more rain to reach the ground andgenerate runoff [3]. Flow may be further enhanced by
reduced infiltration due to soil sealing, waterrepellency, and hyper-dry soil conditions induced by
wildfire [4]. The increased surface water flow following
fires can initiate post-wildfire debris flows in manyburned drainage basins in the Western United States[e.g., 4, 5].
Post-fire debris-flow hazard assessments often
employ regionally specific rainfall thresholds. However,many different site- or region-specific variables control
rainfall thresholds, such as antecedent conditions,
topography, lithology, soil properties, initiation
mechanism, sediment source, and land cover [6].
Rainfall intensity-duration (ID) thresholds are the mostcommon method for predicting debris flow andlandslide occurrence [e.g., 5-7]. Most studies havetraditionally defined thresholds using a subjective
approach. Some attempts have been made to define
rainfall ID thresholds objectively using statisticalmethods [e.g., 6], machine learning models [e.g., 7], or
theoretically process-based models [e.g., 8]. However,defining thresholds with these methods is stillchallenging when direct observations and data
constraining the precise timing of debris-flow activity
are unavailable. Another general challenge for rainfall
thresholds derived from statistic or machine learning
methods is that they usually can not reveal the physicalmechanisms that control the initiation of debris flow andthe hydrologic response of the drainage basin after a fire.
Past studies have suggested that debris-flow
initiation is related to discharge in the channel, which isslope-dependent [10-12]. Runoff-generated debris flows
initiate once discharges exceed slope-dependent
dimensionless discharge thresholds during debris-flowproducing storms on a catchment scale [11-12]. [12]tested the applicability of dimensionless discharge forpredicting debris-flow activity at the basin scale using a
post-fire debris flow dataset in the San Gabriel
Mountains of southern California. They also showedthat these rainfall intensity-duration thresholds derived
from discharge thresholds are comparable to empiricalrainfall thresholds in different catchments with less than
a 20% difference. Processes-based models offer an
opportunity to explore the links between rainfall
intensity, discharge, and debris-flow initiation [13].
However, these physics-based models usually require
meteorological, hydrological, and topographic data formodel calibration. Consequently, a combination of amachine learning model and a process-based model for
predicting the temporal occurrence of debris flows could

Corresponding author: Hui Tang, htang@gfz-potsdam.de© 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, 03029 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341503029

overcome the drawbacks of these two types of models
and potentially improve post-fire debris-flow
predictions.
This study proposes and tests a new hybrid
framework combining process-based models withmachine learning methods to objectively determine
discharge thresholds that can be applied to areas with noestablished empirical thresholds. Specifically, our new
framework derives discharge thresholds for post-firedebris flows from two process-based models based on
the Kinematic wave equations (KWE) and shallow
water equations (SWEHR) with the support vector
machines method. We use numerical models andmonitoring data that constrains the timing of 135 postfire debris flow or flood events from 28 monitored
basins to estimate the discharges associated with debrisflow initiation and flood-dominated events. The debris
flow and flood are identified based on monitoring data
and field surveys. We then apply the support vector
machine method to derive dimensionless discharge
thresholds for debris-flow initiation. We also use
receiver operating characteristics analysis to evaluate
the performance of our derived thresholds.

found in [19] and [20], which include components
representing the fluid flow, size-selective sedimenttransport, infiltration, rainfall interception, and mass
failure of bed material. Runoff and sediment transport
are simulated for all runoff-generating rainstorms at Las
Lomas and Arroyo Seco to compute the unit discharge
for every channel location in the basin by
𝑄𝑠 = ℎ√𝑢2 + 𝑣 2 ,(1)

where ℎ is the water depth, and 𝑢 and 𝑣 are the flow
velocity in the x and y-direction, respectively.
For the other study sites affected by the Station Fire,
Pinal Fire, and Buzzard Fire, we employed thekinematic wave (KW) model described in [22], whichapproximates the full Shallow Water equations for
simulated discharge. The water flux was determined by
the Manning equations:
𝑄𝑥 =
𝑄𝑦 =

1⁄ 2

ℎ 5⁄3 𝑆0𝑥𝑛1⁄ 2
ℎ 5⁄3 𝑆0𝑦

𝑛

,

(2)

,

(3)

where 𝑆0𝑥 and 𝑆0𝑦 are the ground surface slope in the x
and y-direction, respectively. 𝑛 is the Manningcoefficient. Then, the discharge is:
𝑄𝑘 = √𝑄𝑥2 + 𝑄𝑦2 .

(4)To estimate dimensionless discharge thresholds, wequantify dimensionless discharge when debris flows are
initiated or when there is a flood response that does notinclude any debris-flow activity. We calculate thedimensionless discharge following [11] and [23]:Fig 1: Study sites: a: Station Fire, 2009; b: Fish Fire, 2016; c:
Pinal Fire, 2017; and Buzzard Fire, 2018.

𝑄∗ =2 Methods

𝑄
𝜌𝑠 −𝜌
3
√ 𝜌 𝑔𝐷50,(5)where 𝑄 is the discharge from the SWEHR model (𝑄𝑠 )or KW model (𝑄𝑘 ). 𝐷50 is the median grain size. 𝜌 and𝜌𝑠 are the density of water and sediment, respectively. 𝑔is the gravity acceleration. Here, we constrain 𝐷50 basedon the grain size distribution from debris-flow deposits
at each study site. We follow the approach from [12] andcompute time-averaged values of 𝑄∗ immediately prior
to the time a debris flow was observed at the basin outlet
if it is known. In cases where debris-flow timing withina debris-flow-producing storm is unknown, we use thepeak discharge. In the case of a flood response to arainstorm, we also use the modelled peak discharge.

2.1 Monitoring and Field Measurements
Our study sites include 28 watersheds in the western
USA, burned by four different wildfires (Station Fire,
California, 2009; Fish Fire, California, 2016; Pinal Fire,
Arizona, 2017; Buzzard Fire, New Mexico, 2018; see
Figure 1). Study basins from the 2009 Fish Fire and the2016 Station Fire have similar climate, vegetation, burnseverity, and topographic characteristics [19, 20]. Table
1 summarise some basic features of wildfires and basinsin this study. For details, we refer to the respective
papers provided in the notes to Table 1.

2.3 Receiver
Operatinganalysis (ROC)

2.2 Process-based Models

Characteristics

We used a receiver operating characteristic (ROC)
analysis to objectively analyse discharge thresholds.
ROC is an optimisation method that assesses
classification model performance and has been widely
applied for rainfall intensity-duration thresholds [6, 17].
This method generally maximises the number of correct
predictions while minimising the amount of falsepositive and false-negative predictions [24-25]. Eachdischarge for a given bed angle was classified based on

This study uses two process-based models with basic
physics to estimate discharges. For study sites at Las
Lomas (burned by the Fish Fire) and Arroyo Seco(burned by the Station Fire), where data exist to calibratea sediment transport model, we applied a model basedon the shallow water equations with advection equations
for sediment transport and the Hairsine and Rose soil
erosion model (SWEHR). The details of SWEHR can be2

E3S Web of Conferences 415, 03029 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341503029

2.4 Support Vector Machines (SVM)

whether a debris flow was recorded from monitoringdata and field observations. The boundary between
debris-flow events and water-dominated flow events isthe optimal discharge threshold to distinguish debrisflow-producing events from non-debris-flow-producing
events. Here, we use the ROC analysis to objectively
define a slope-dependent discharge threshold based on
numerical simulations and compare it with discharge
thresholds for the San Gabriel Mountains from [12]. We
further evaluate the performance of the discharge
thresholds from support vector machines. The ROC
method can determine the best threshold for a given bed
𝑇𝑃angle by using the true-positive rate (, where TP
𝑇𝑃+𝐹𝑁
is the number of true-positive cases and FN is thenumber of false-negative cases) and false-positive rate𝐹𝑃(
, where FP is the number of false-positive cases,
𝐹𝑃+𝑇𝑁
and TN is the number of true-negative cases). The AUC
value is defined as the area below the ROC curve. The
perfect prediction model will yield an AUC value ofone. The discharge threshold derived from the ROC
method will subsequently be referred to as the ROC
threshold.

The Support Vector (SV) algorithm is a class ofnonlinear search algorithms based on a statisticallearning theory developed by [26] and [27]. The SV
algorithm has been successfully developed as aclassification tool, such as the Support Vector Machines(SVM) based on a supervised learning system. Sample
classification is determined by finding a hyperplane that
produces the optimal separation between classes. The
SVM differs from other approaches because thehyperplane is constructed from the training dataset
points for linearly separable binary problems. If two
categories are not linearly separable in a dataset, kernel
functions are used to map the data into a higherdimensional space to separate the classes with a linear
hyperplane. The best hyperplane for an SVM is the one
with the most significant margin between the twocategories. Margin means the maximum slab widthparallel to the hyperplane with no interior data points.
This study applies the support vector machine with akernel function to define the boundary to separate thedimensionless discharges that produce debris flows andwater-dominated floods as two categories. The defined
threshold model from this method will subsequently bereferred to as the SVM threshold.

Table 1 Basic features for monitoring sites
Fish Fire,
2016 (1)

Pinal
BuzzardFire,Fire,
2017 (2) 2018 (3)Station Fire,2009 (4)3 Results and Discussion
Study Site

DuarteGlobe

ApacheCreek

DuarteNumber of
Basins

1

5

12

10

Basin Sizes,
𝑘𝑚2

0.12

0.0480.188

0.060.61

0.06

Storm
Events (a)

25

6

8

2

Debris Flow/ 11/3
Flood Events

2/14

17/68

42/202Channel bed
angle,
degree

0-45

15-40

15-40

0-45

Grain Size
(D50), 𝑚𝑚

1

2.8

15

15

Model

SWEHR (b)KW (c)KW

SWEHR

Figure 2 shows the preliminary results from both the
ROC and SVM methods. For comparison, we added theoriginal discharge thresholds derived from processesbased model simulations in the San Gabriel Mountains[12]. There are two types of thresholds in [12]: upper
limit thresholds (the averaged value of discharge during
debris flows, UL, red dash line) and lower limit
thresholds (the lowest value of discharge during debris
flows, LL, blue dash line). See [12] for details. The ROC
thresholds are very close to these LL thresholds fromnumerical model simulation. In contrast, the SVM
thresholds are generally higher than ROC thresholds but
smaller than the UL thresholds from [12]. The
discrepancy between SVM thresholds and thosepreviously derived from the processes-based modelling
may arise because the previous study only focused on
one study site (Fish Fire). Meanwhile, the significant
difference in the grain size of debris-flow deposits fromdifferent sites (1 to 15 mm) increases variability in thedimensionless discharges.
The AUC value for the SVM threshold is about
0.870-0.940. The AUC value for the ROC threshold isslightly smaller than the best SVM threshold with aradial basis kernel, which is about 0.904. Since only
limited data points are available, we cannot directlycompare the AUC value for the UL and LL thresholds
from [12]. The UL usually has lower true-positive rates,
but LL usually has higher false-positive rates. Moregenerally, the SVM threshold makes a good balance
between the true predictions and false predictions for all
channel bed angles. This is because the SVMs need tosimultaneously consider all data from different channel
bed angle. However, ROC, LL, and UL get the best
performance for each channel bed angles individually.

(a): This is the number of rainfall events; (b): SWEHR
indicates the model based on Shallow Water equations; (c):KW indicates the model based on Kinematic Wave equations;(1): Reference 12 and 19; (2) : Reference 17; (3): Reference18; (4): Reference 14-16

3

E3S Web of Conferences 415, 03029 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341503029

Numerical simulations of debris flow for rainstorms
with different durations and intensities from monitored
basins demonstrate that dimensionless discharges
needed to initiate debris flows are similar across four
recently burned areas. Therefore, an advantage ofestablishing dimensionless discharge thresholds fordebris-flow initiation is that they are not strongly basedon the hydrologic properties of the catchment. Thus,
generating rainfall ID thresholds based on
dimensionless discharge thresholds with an approachsimilar to that in [12] can provide a promisingalternative to empirical methods for assessing thepotential of runoff-generated debris flows in areas withlimited or no historical data on debris flows after awildfire.

5.

S. Cannon, E. Boldt, J. Laber, J. Kean, D. Staley,
Natural Hazards, 59(1), 209–236 (2011)
6. D. Staley, J. Kean, S. Cannon, K. Schmidt, J. Laber,Landslides, 10(5), 547–562 (2013)
7. D. Staley, J. Negri, J. Kean, J. Laber, A. Tillery, A.
Youberg, Geomorphology, 278, 149-162 (2017)
8. E. Nikolopoulos, E. Destro, M.Bhuiyan, M. Borga,
E. Anagnostou, NHESS, 18(9), 2331-2343 (2018)
9. M. Berti, M. Bernard, C. Gregoretti, A. Simoni,
JGR-ES, 125(6) (2020)
10. C. Gregoretti, JHR, 38(2), 83-88 (2000)
11. C. Gregoretti and G. Fontana, Hydrological
Processes, 22(13), 2248-2263 (2008)
12. H. Tang, L. McGuire, F. Rengers, J. Kean, D.
Staley, J. Smith, GRL, 46(15), 8830–8839,(2019)13. J. Hoch, L. McGuire, A. Youberg, and F. Rengers,
JGR-Earth Surface, 126(12), (2021)
14. J. Kean, D. Staley, S. Cannon, JGR-ES, 116(F4)(2011)
15. J. Kean, D. Staley, R. Leeper, K. Schmidt, J.
Gartner, WRR, 18(5) (2012)
16. D.
Staley.,T.
Wasklewicz,J.
Kean,
Geomorphology, 214, 324-338 (2014)
17. C. Raymond, L. McGuire, A. Youberg, D. Staley,
J. Kean, ESPL, 45 (6), 1349-1360 (2020)
18. L. McGuire., and A. Youberg, Landslides, 17(10),2385-2399 (2020)
19. H. Tang, L. McGuire, F. Rengers, J. Kean, D.
Staley, J. Smith, JGR-ES, 124(6), 1572-1595(2019)
20. L. McGuire, J. Kean, D. Staley, F. Rengers, T.
Wasklewicz, JGR-ES, 121(11), 2211–2237 (2016)
21. L. McGuire, F. Rengers, J. Kean, D. Staley, GRL,
44(14), 7310–7319 (2017)
22. F. Rengers, L. McGuire, J. Kean, D. Staley, D. E.
Hobley, WRR, 52(8), 6041–6061 (2015)
23. G. Parker, P. Klingeman, D. McLean, JHE,
108(HY4), 544–571 (1982)
24. J. Swets, Science, 240(4857), 1285-1293 (1988)
25. T. Fawcett, PRL, 27(8), 861-874 (2006)
26. V. Vapnik, A note one class of perceptrons,
Automation and remote control (1964)
27. A. Smola, and B. Schölkopf, Statistics andcomputing, 14(3), 199–222 (2004)

Fig.2: Dimensionless discharge thresholds (𝑄∗ ) as a functionof bed angle (𝜃): ROC thresholds (green solid line); SVM
thresholds (yellow solid line); UL threshold (red dash line);
LL threshold (blue dash line). Circles indicate support vector
using in SVM thresholds. Red dots are debris-flow events;
blue dots represent no debris-flow events or flood events.

In summary, our results indicate that slopedependent dimensionless discharge thresholds canpredict runoff-generated debris flow following fire, inagreement with prior work that utilised a more limited
dataset. This study proposes a new physics-informed
machine learning approach to discriminate debris-flow
events from water-dominated flood events. Thresholds
derived from support vector machines (SVMs) fall
between the UL and LL thresholds from [12] and show
good performance relative to those derived with receiver
operating characteristics analysis.

References
1.
2.

3.

4.

J. Moody and B. Ebel, Earth-Science Reviews, 122,
10-37 (2013)
F. Rengers, L. McGuire, N. Oakley, J. Kean, D.
Staley, H. Tang, Landslides, 17 (11), 2631-2641(2020)
C. Stoof, R.Vervoort, J. Iwema, E. van den Elsen,
A. Ferreira, C. Ritsema, HESS, 16(2), 267–285(2012)
S. Cannon, J. Gartner, M. Rupert, J. Michael, A.
Rea, C. Parrett, GSA Bulletin, 122(1-2), 127–144(2010)

4