E3S Web of Conferences 415, 05021 (2023)
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https://doi.org/10.1051/e3sconf/202341505021

Using integrated growth to delineate debris-flow inundation
Mark E. Reid1*, Dianne L. Brien1, Collin Cronkite-Ratcliff2, and Jonathan P. Perkins2
1U.S. Geological Survey, Volcano Science Center, PO Box 158, Moffett Field, CA, 94035, USA
U.S. Geological Survey, Geology, Minerals, Energy, and Geophysics Science Center, PO Box 158, Moffett Field, CA, 94035, USA

2Abstract. Debris-flow volume is fundamental to mobility, yet many debris flows change volume as theytravel. Growth can occur through diverse processes such as channel-bed entrainment, bank failures,
aggregation of landslides, and coalescence of multiple flows. Integrating growth, either over upslope areaor stream length, combines the effects of these growth processes and requires specification of only thegrowth zone extent and a growth factor. To delineate potential debris-flow inundation, we implement
integrated growth factors and simple volume-area relations in a new USGS software package, Grfin Tools.
We present two examples of forecasting debris-flow inundation – one using an area growth factor in Puerto
Rico and another using a channel-length growth factor in Oregon, USA. The use of growth zones and growth
factors enables scenario-based hazard assessments for geomorphic settings with debris-flow growth.

1 Introduction
Debris-flow volume exerts a fundamental control onmobility, speed, momentum, inundation area, andhazard [1-3]. As debris flows travel, they may growvolumetrically and become more destructive in manysettings, including steep forested slopes, alpine terrain,
post-wildfire burned landscapes, breached glacial lakes,
and volcano flanks [4,5]. Growth can vary substantially,sometimes increasing an initial volume by more than anorder in magnitude [5,6]. A wide variety of processes
can promote growth, including channel-bed erosion andentrainment, incorporation of channel bank failures,
aggregation of multiple landslide sources, and hillslope
rilling on adjacent slopes [5-8]. Various growthmechanisms may modify a single flow event. Moreover,flow volumes can increase abruptly when multiple flows
or surges coalesce from merging channels [8-10].
Explicitly embedding these diverse processes into
computational models for debris-flow routing andhazard assessment is an on-going challenge [11-13]. As
an alternative, we pursue an empirical modelingapproach that allows us to represent the combined
effects of various growth processes through growth
factors. Here, we use the term growth factor to denotespatial growth rates rather than temporal rates. Using
pre- and post-debris flow event measurements,
integrated growth factors can be measured or estimatedover contributing areas or stream-channel lengths wheregrowth occurs [4,14].
Integrated area-growth factors (m3/m2), similar tobasin erosion or lowering rates, can represent thecombined effects of debris-flow growth-inducing
processes such as shallow landsliding, rilling, or surface
erosion [14,15]. Contributing area can be the entire
upslope area or just the region susceptible to providing

debris-flow material. Channel-length growth factors(m3/m), sometimes termed bulking or yield rates, arecommonly determined with processes scaled to stream
length, such as channel bed entrainment or bank erosion[4,5,16,17]. Growth factors reflect the many conditions
promoting debris-flow growth, including steep slopes,
the availability of entrainable sediment, and the effectsof coalescing flows.

Explanation
Initial volume
Growth in volume
Maximum volume
Stream channel - growth
Stream channel - no growth
Transition from growth tono growth
Debris-flow inundation

Fig. 1. Perspective view of drainage network showing
estimated debris-flow inundation areas and stream-channelzones of volumetric growth and no growth. Dots indicate
relative debris-flow volumes. In growth zones, coalescence offlows can abruptly increase flow volume downstream oftributary junctions (e.g. location A). Multiple growth zones
can exist in a given channel. In our approach, volume does notchange in the no-growth zones. Long axis of area shown isroughly 500 m.

* Corresponding author: mreid@usgs.gov© 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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𝑉 = 𝑐$ 𝑈
𝑉 = 𝑐& 𝐿

Zones of debris-flow growth typically occur in theupper part of a basin with steep stream slopes andconfined channels [7,18,19] (Fig. 1). As stream slopes
decrease farther downstream, growth halts anddeposition begins. The transition from growth todeposition commonly ensues in a zone of changing
growth, deposition, and sediment transport [16,18,20].
Thus, understanding growth-zone extent and growthfactor magnitude is crucial to forecasting inundation.

At each channel network cell in the growth zones,
we use computed volume from the selected growthequations (3 and/or 4) to construct a cross section ofinundation roughly perpendicular to the local stream
flow direction. To provide smoother plan-view
coverage, we also construct two additional crosssections 45º to either side of the perpendicular section.
Each section has an area, A, based on eq. 1. We thenintersect the digital topography with a plan-view circle
centered on the channel and having a diameter equal tothe average width and a maximum elevation of the threecross sections. The part of this circle with an elevation
above the topography is considered inundated. The
series of these overlapping circles defines the lateralextent of inundation.
In the no-growth zones, debris-flow volumes at eachchannel network cell are determined at the downstream
end of the adjacent growth zones. We compute crosssections and circles of inundation, as in the growthzones. In addition, we use eq. 2 with the specified
volume to define a target total planimetric area, B, to beinundated. The total inundation area is delineated byprogressing down the channel network, determining
cumulative inundation area at each incremental step, andhalting when the target area is reached.
We use the USGS software package Grfin Tools (anacronym of growth + flow + inundation), currently
under development, to implement this growth/nogrowth approach. Grfin Tools is computationallyefficient and does not rely on specific GIS software. The
above described circle-fitting method minimizes spikey
plan-view inundation artifacts in unconfined regions, as
can occur with Laharz [4,29].

2 Inundation using growth factors
To portray the spatial extent of potential debris-flow
inundation using integrated growth factors, we use thesegrowth factors within the framework of a simple
statistical-empirical method. The inundation mapping
approach uses two volume-area power-law equations,
one relating debris-flow volume, V, to cross-sectionalarea, A (eq. 1), and one to planimetric area, B (eq. 2), as
used in the Laharz and DFLOWZ models [21-23].
&𝐴 = 𝛼$ 𝑉 '&𝐵 = 𝛼& 𝑉 '(3)
(4)

(1)
(2)

Multiple studies have determined various
coefficients, 𝛼$ and 𝛼& , for the volume-area relations(eqs. 1 and 2) using empirical observations from bothglobal and region-specific debris-flow events [24-27].
Some analyses compiled statistics for overallplanimetric inundation from the start of a debris flow,
thus including potential growth zones [27], even thoughmany of the flows used in these analyses grew as theytraveled. Other analyses have emphasized downstream
depositional areas after growth had ceased and others
have mixed the two approaches [24,25,28].
Several researchers [24,28] have found that volumearea relations derived from these various debris-flow
inundation cases have similar power-law coefficients
when fit to area with a 2/3 exponent (eqs. 1 and 2).
Given these observations, we used the general debrisflow volume-area relations of [27] to determine crosssectional inundation (𝛼$ = 0.1) in the growth zones andboth cross-sectional and planimeteric (𝛼& = 20)
inundation in the no-growth zones.
A fundamental difference in our approach, however,is that we divide the drainage network into regions ofvolumetric growth and regions of no growth (Fig. 1),
rather than specifying a maximum volume. To map
inundation, we first define a complete topographic
curvature-based drainage network within a digital
elevation model. Debris-flow growth zones can then bedelineated within the channel network using
topographic metrics such as local stream slope and/or
stream order (Fig. 1). At each channel network cellwithin the growth zones, volume (V) is computed as theproduct of either the integrated area-growth factor (c1)by the upslope area (U) or the channel-length growth
factor (c2) by the upstream length (L), using either eq. 3
or 4 as described in [4].

3 Examples of computed inundation
We present two examples using integrated growth
factors for storm-induced debris flows that grew involume as they traveled through mountainous terrain.
Table 1 shows both area and channel-length growth
factors for two study areas estimated using detailed
mapping of debris flows and DEM differences between
repeat aerial photogrammetry or lidar, supplemented byfield observations. These growth factors were
determined in small areas heavily impacted by debris
flows – thus applying these in regional contexts provides
relatively “worst cases.” Additional measured growth
factors for other regions are summarized in [4].
Table 1. Ranges of integrated debris-flow growth factors fortwo study areas.

Coast Range,Oregon, USA [4]Puerto Rico, USA[14]2

Area-growth
factor, c1(m3/m2)Channel-length
growth factor, c2(m3/m)

0.12 to 0.2

11.4 to 24.2

0.01 to 0.13

0.7 to 30.4

E3S Web of Conferences 415, 05021 (2023)
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from growth throughout the entire stream network.
However, the delineated areas may be affected in futureevents.

3.1 Use of area growth
In 2017, Hurricane Maria lashed the island of Puerto
Rico in the Caribbean, triggering over 70,000 rainfallinduced landslides; thousands transformed into debris
flows and traveled down drainage channels causingwidespread damage [30,31]. Debris-flow growthcommonly ceased on stream slopes between 3º and 10º[14]. Most flows initiated from shallow landslides andmany formed from multiple landslide sources [8,14].
Post-event detailed mapping [32] revealed that landslide
transport on hillslopes was commonly extensive, before
transitioning into a channelized debris flow (Fig. 2a).

a3.2 Use of channel-length growth
During a large storm in 1996, numerous and widespreaddebris flows were triggered in the steep and highly
dissected terrain of the Oregon Coast Range, USA.
Here, debris flows typically originated as shallow
landslides in hollows at the heads of drainages before
moving down the channel network [33-35]. They grewby entrainment of channel sediment and coalescence ofmultiple flows [4,9].

landslide source
123º59.4ꞌWinundation & runout

123º58.8ꞌW00100 mbRiver

43º41.4ꞌN43º41.4ꞌNUmpqua

100 m

potential source area
43º40.8ꞌN43º40.8ꞌNdebris-flow inundation

observed debris flows
computed inundation
123º59.4ꞌW

123º58.8ꞌWFig. 3. Example using a channel-length growth factor in theOregon Coast Range near the Umpqua River. Observed
debris-flow inundation from [35]. Computed debris-flow
inundation shown for two small basins, as also analyzed in [4].
Portrayed on hillshade from a 0.9-m lidar base.
0100 m

In this case, channel-dependent growth processes(e.g., bed entrainment) provide an opportunity toexamine the use of a channel-length growth factor. We
use a channel-length growth factor of 11 m3/m (Table 1)
and growth zones defined by channels with slopes ≥10º(Fig. 3). As in the area-growth example, computed
inundation zones are slightly more extensive thanobservations, due in part to aggregation of growth.

Fig. 2. Example using an area-growth factor in Utuado
Municipality, Puerto Rico. (a) Observed landslide sources anddebris-flow inundation areas following Hurricane Maria fromdetailed mapping [32]. (b) Potential shallow-landslide sourceareas (≥30º slope) and computed debris-flow inundation.
Portrayed on hillshade from a 1-m lidar base.

In this example, area-dependent growth processes(e.g., multiple landslides) support the use of an areagrowth factor. Fig. 2b portrays debris-flow inundation
using a growth factor of 0.01 m3/m2 with growth zones
defined by channels having stream slopes ≥6º. Area
growth is applied to hillslopes ≥30º, similar to observed
median slopes for landslide initiation areas [8].
Without calibrating volume-area or growth-factor
parameters to match runout observations, our approachis able to delineate reasonable stream-channel
inundation areas. The computed runout is slightly larger
than observed, due to the use of growth factors fromhighly impacted areas and relatively extensive growthzones. These conditions tend to over predict basin-widerunout for a specific event due to aggregated effects

4 SummaryUnderstanding the magnitude and location of debrisflow growth is crucial to delineating inundation. Many
diverse processes can promote growth, and the effects
can be integrated over either upslope area or stream
length. Using simple growth relations, combined withvolume-area relations for defining inundation area (asimplemented in Grfin Tools), we illustrate the use ofdifferent integrated growth factors in two settings. The
combination of relatively simple growth factors andgrowth zones can generate reasonable inundation zones,
suitable for multiple scenario-based hazard assessments
in geographic regions having debris-flow growth.

3

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