GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus GmbHGöttingen, Germany10.5194/gmd-8-4027-2015r.randomwalk v1, a multi-functional conceptual tool for mass movement
routingMergiliM.martin.mergili@boku.ac.athttps://orcid.org/0000-0001-5085-4846KrennJ.ChuH.-J.Geomorphological Systems and Risk Research, Department of Geography and Regional Research, University of Vienna, Universitätsstraße 7, 1190 Vienna, AustriaInstitute of Applied Geology, University of Natural Resources and Life Sciences (BOKU), Peter-Jordan-Straße 70, 1190 Vienna, AustriaDepartment of Geological Sciences, University of Canterbury, Private Bag 4800, Christchurch, New ZealandDepartment of Geomatics, National Cheng Kung University, 1 University Road, Tainan 701, TaiwanM. Mergili (martin.mergili@boku.ac.at)16December2015812402740434August201525September201524November201530November2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/8/4027/2015/gmd-8-4027-2015.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/8/4027/2015/gmd-8-4027-2015.pdf
We introduce r.randomwalk, a flexible and multi-functional open-source tool
for backward and forward analyses of mass movement propagation.
r.randomwalk builds on GRASS GIS (Geographic Resources Analysis Support System – Geographic Information System), the R software for statistical computing
and the programming languages Python and C. Using constrained random walks,
mass points are routed from defined release pixels of one to many mass
movements through a digital elevation model until a defined break criterion
is reached. Compared to existing tools, the major innovative features of
r.randomwalk are (i) multiple break criteria can be combined to compute an
impact indicator score; (ii) the uncertainties of break criteria can be
included by performing multiple parallel computations with randomized
parameter sets, resulting in an impact indicator index in the range 0–1;
(iii) built-in functions for validation and visualization of the results are
provided; (iv) observed landslides can be back analysed to derive the
density distribution of the observed angles of reach. This distribution can
be employed to compute impact probabilities for each pixel. Further, impact
indicator scores and probabilities can be combined with release indicator
scores or probabilities, and with exposure indicator scores. We demonstrate
the key functionalities of r.randomwalk for (i) a single event, the Acheron rock avalanche in New Zealand; (ii) landslides in a
61.5 km2 study area in the Kao Ping Watershed, Taiwan; and
(iii) lake outburst floods in a 2106 km2 area in the
Gunt Valley, Tajikistan.
Introduction
Mass movement processes such as landslides, debris flows, rock avalanches, or
snow avalanches may lead to damages or even disasters when interacting with
society. Computer models predicting travel distances, hazardous areas,
impact energies, or travel times may help the society to mitigate the effects
of such processes and, consequently, to reduce the risk and the losses
(Hungr et al., 2005).
Physically based dynamic models are used for in-detailed analyses of specific
events or situations (e.g., Savage and Hutter, 1989; Takahashi et al., 1992;
Iverson, 1997; Pudasaini and Hutter, 2007; McDougall and Hungr, 2004, 2005;
Pitman and Le, 2005; Christen et al., 2010a, b; Mergili et al., 2012b;
Pudasaini, 2012; Hergarten and Robl, 2015; Mergili et al., 2015). Since the
processes are complex in detail and the input parameters are uncertain,
simplified conceptual models for the motion of mass flows are today used in
combination with GIS (Geographic Information System). These models may be
used for single events. However, they are particularly useful to indicate
potential impact areas at broader scales. Hypothetic mass points are routed
from a release pixel through a digital elevation model (DEM) until a defined
break criterion is reached. Monte Carlo techniques (random walks, Pearson,
1905; Gamma, 2000) or multiple flow direction algorithms (Horton et al.,
2013) are employed to simulate the lateral spreading of the flow.
The break criteria often consist in threshold values of the angle of reach
(i.e., the average slope of the path) or horizontal and vertical distances
(Lied and Bakkehøi, 1980; Vandre, 1985; McClung and Lied, 1987; Burton and
Bathurst, 1998; Corominas et al., 2003; Haeberli, 1983; Zimmermann et al.,
1997; Huggel et al., 2002, 2003, 2004a, b), sometimes related to volume
(Rickenmann, 1999; Scheidl and Rickenmann, 2010). However, those
relationships usually display a large degree of scatter. Further, key
parameters for design issues, such as impact pressures, are not provided
(Hungr et al., 2005).
Some approaches include simplified physically based models going back to the
mass flow model of Voellmy (1955), relating the shear traction to the square
of the velocity and assuming an additional Coulomb friction effect
(Pudasaini and Hutter, 2007). They consider only the centre of the flowing
mass, but not its deformation and the spatial distribution of the flow
variables. This type of models is mainly used for snow avalanches and debris
flows (Perla et al., 1980; Gamma, 2000; Wichmann and Becht, 2003; Mergili et al., 2012a; Horton et al., 2013).
Various – mostly open-source – software tools for conceptual modelling of
mass movements (mainly flows) at medium or broad scales are available (e.g.,
Gamma, 2000; Wichmann and Becht, 2003; Mergili et al., 2012a; Horton et al.,
2013). However, most of these tools lack substantial features: (i) they are limited to one single type of break criterion; (ii) they do not
allow one to directly account for the uncertainty of the break criteria; (iii) they do not allow one to back calculate the statistics of a set of observed mass
movements; and (iv) they do not offer built-in functionalities for evaluating
the model results against observations. Consequently, the key objectives of
the present study are
to introduce r.randomwalk, a freely available, comprehensive and flexible
tool for routing mass movements;
to demonstrate the various functionalities of r.randomwalk, particularly in
terms of overcoming the issues (i)–(iv);
to discuss the potentials and limitations of this tool.
Next, we will describe the r.randomwalk software tool (Sect. 2). Furthermore, we
will present the test areas and the results (Sect. 3). Finally, we will discuss the
findings (Sect. 4) and conclude with some key messages of the work
(Sect. 5).
The r.randomwalk applicationComputational implementation
r.randomwalk is implemented as a raster module of the open-source software
package GRASS (Geographic Resources Analysis Support System) GIS 7 (Neteler and Mitasova, 2007; GRASS Development Team,
2015). We use the Python programming language for data management,
pre-processing and post-processing tasks (module r.randomwalk). The routing
procedure (see Sect. 2.2–2.4) is written in the C programming language
(sub-module r.randomwalk.main). The R software environment for statistical
computing and graphics (R Core Team, 2015) is employed for built-in
validation and visualization functions (see Sect. 2.5). Parallelization of
multiple model runs is enabled. It allows for the exploitation of all
computational cores available, speeding up analysis processes. The
parallelization procedure is implemented at the Python level (analogous to
the way described in Mergili et al., 2014): the module r.randomwalk produces
a batch file for each model run. This batch file calls the Python-based
sub-module r.randomwalk.mult, which is then used to launch r.randomwalk.main
with the specific parameters for the associated model run. Thereby, the
Python library “Threading”, a higher-level threading interface, and the
Python module “Queue”, a class helping to block execution until all the
items in the queue have been processed, are exploited. Parallel processing
serves for reducing the computational time in the following contexts:
Analyses with multiple random subsets of the release areas or coordinates.
In each model run, one subset is used for back calculating the probability
density function (PDF) of the angle of reach, the other subset is employed
for validating the distribution of the impact probability derived with this
PDF against the observed deposition areas.
Analyses with multiple combinations of input parameters varied in a
controlled or randomized way, enabling one to consider parameter uncertainties
and to explore parameter sensitivity.
r.randomwalk was developed and tested with Ubuntu 12.04 LTS and is expected
to also work on other UNIX systems. A simple user interface is available.
However, the tool may be started more efficiently through command line
parameters, enabling a straightforward batching on the shell script level.
This feature facilitates model testing, the combination with other GRASS GIS
modules and the consideration of process chains (i.e., using the output of
one analysis as the input for the next one). The logical framework is
illustrated in Fig. 1, the key variables used in r.randomwalk are summarized
in Table 1.
Logical framework of r.randomwalk. Only those components covered
in the present article are shown.
All tests (see Sect. 3) are performed on an Intel® Core i7 975
with 3.33 GHz and 16 GB RAM (DDR3, PC3-1333 MHz), exploring a maximum of eight
cores through hyperthreading.
Random walk routing
The term random walk refers to a Monte Carlo approach for routing an object
through any type of space. The term was introduced by Pearson (1905).
Constrained random walk approaches are used for routing mass movements such
as debris flows through elevation maps (DEMs), e.g. by Gamma (2000),
Wichmann and Becht (2003), Mergili et al. (2012a), and Gruber and Mergili (2013). Such methods enable a certain degree of spreading of the movement by
also considering other routing directions than the steepest descent. It
avoids the concentration of flows – or any other types of mass movements –
to linear features, which would not be realistic for debris flows, snow
avalanches, or other types of mass movements. However, the routing is
constrained or weighted by factors such as the slope or the perpetuation of
the flow direction. An alternative to constrained random walk routing would
consist in a multiple flow direction algorithm (Horton et al., 2013).
Control length Lctrl and segment length Lseg.
(a) Application of Lctrl to avoid sharp bending of the flow. (b) Smoothing
of the flow path by introducing segments with maximum length of Lseg.
In the context of r.randomwalk, each random walk routes a hypothetic mass
point from a release pixel through the DEM until a break criterion is
reached (see Sect. 2.3). A large set of random walks is required for each
mass point in order to achieve a satisfactory cover of the possible impact
area. r.randomwalk is designed for
one set of random walks for one mass movement, starting from a defined set
of coordinates;
multiple sets of random walks for one mass movement, one set starting from
each pixel of the release area;
sets of random walks for multiple mass movements in a study area (either
starting from one set of coordinates per mass movement, or from all pixels
defined as release areas);
one set of random walks starting from each pixel in the study area.
Overlay rules for different random walks and sets of random walks are
applied (see Sect. 2.4).
Summary of the key variables used in r.randomwalk.
SymbolUnitNameDescription/remarks[range]Input nwalks-[≥ 0]Size of a set of walksLogarithm (base of 10) of the number of random walksLctrlm [> 0]Control lengthBackward distance of each step of a random walk over which the horizontal distance of motion has to increase (see Fig. 2a)Lsegm [> 0]Segment lengthLength of segments used for computing Lmax (see Fig. 2b)Rmaxm [≥ 0]Maximum vertical run-up height–fd-[≥ 1]Direction factorFactor for weighting the perpetuation of the movement direction during routingfβ-[≥ 0]Slope factorFactor for weighting the slope during routingVm3 [> 0]Movement volume–Qpm3 s-1 [> 0]Peak dischargeApplicable to lake outburst events (see Table 3)a,b,c– [ ]–Parameters needed by the rules and relationships applied (see Table 3)PR-[0–1]Release probabilitySpatial probability that a mass movement is released from a given pixel (Mergili and Chu, 2015)RIS-[≥ 0]Release indicator scoreOrdinal score denoting the tendency of a pixel to produce a mass movementEIS-[≥ 0]Exposure indicator scoreOrdinal score denoting the exposed values at a given pixelOutput Lmaxm [≥ 0]Travel distanceHorizontal distance between the release pixel and the most distant pixel reached by a set of random walks, measured along the segments of the path (see Fig. 2b)Zm [≥ 0]Elevation lossVertical distance between the release pixel and the most distant pixel reached by a set of random walksωT∘ [< |90|]Angle of reachAverage slope angle measured between the release pixel and the most distant pixel reached by a set of random walksIF-[≥ 0]Impact frequencyNumber of random walks impacting a given pixelIIS-[≥ 0]Impact indicator scoreNumber of rules and relationships predicting an impact on a given pixelIHIS-[≥ 0]Impact hazard indicator scoreOrdinal score serving as a qualitative surrogate for the hazard of an impact on a given pixelIII-[0–1]Impact indicator indexFraction of model runs impacting a given pixel out of all model runsPI-[0–1]Impact probabilitySpatial probability that a given pixel is impacted, building on user-defined release areas and a cumulative density functionPI,C-[0–1]Composite probabilitySpatial probability that a given pixel is impacted, building on PR and PI (Mergili and Chu, 2015)IRIS-[≥ 0]Impact risk indicator scoreOrdinal score denoting the expected/potential loss at a given pixel
During the pixel-to-pixel routing procedure, turns of > 90∘ are not supported. Neighbour pixels are further considered
invalid as target pixels in case they are out of the study area or conflict
with at least one of the following limitations:
In order to constrain upward movements, a user-defined maximum vertical
run-up height Rmax is introduced. It takes the lowest elevation the
random walk has passed through as reference.
Certain types of mass flows (i.e., those with high viscosity) hardly change
their flow direction sharply. The user-defined horizontal control distance
Lctrl defines the backward distance of each step over which the
horizontal distance of motion has to increase (Fig. 2a).
Possibilities to define the break criteria. The flags provided
through the command line or the user interface define the type of break
criterion. RC is release coordinates (release from highest points of
release areas), RP is release pixels (release from all pixels within
release areas), • is relevant for most applications, and ∘ is relevant for some applications.
FlagModeRelease OutputValidationMultiple coresRCRPqPublished relationships (see Table 3)•IIS∘mPublished relationship, multiple runs•III∘•pImpact probability∘∘PI∘p + nImpact probability, multiple runs∘∘PI••bReconstruction of observation∘∘CDF
The probability Ppx of any other neighbour pixel px to become the target
pixel is
Ppx=ppx∑qx=1qx=npqx,p=fdefβtanβ,
where n is the total number of valid neighbour pixels, and β is the
local slope between the current pixel and the considered neighbour pixel.
fd and fβ are weighting factors for the perpetuation of the
flow direction and for the slope. fd is governed by the input parameter
d: fd=d2 for the same flow direction as the previous one,
fd=d for a 45∘ turn and fd=1 for a 90∘
turn.
The break criteria for the random walks (see Sect. 2.3) are directly or
indirectly related to the travel distance Lmax i.e., the horizontal
length between the release pixel and the terminal pixel measured along the
flow path. Preliminary tests reveal that random walk routing through raster
maps may result in quite uneven flow paths (see Fig. 2b). Consequently, the
distance calculated by summing up all the pixel-to-pixel distances may be
significantly longer than the more relevant distance along the observed main
flow paths. Employing the sums of the pixel-to-pixel distances would lead to
an underestimation of the angle of reach and, consequently, of the predicted
travel distances and impact areas. We approach this problem by dividing the
flow paths into straight segments with a user-defined maximum length of
Lseg. The travel distance Lmax is defined as the sum of the length
of all segments (see Fig. 2b). Larger values of Lseg are expected to
result in shorter travel distances due to the more pronounced smoothing of
the path.
Break criteria
Each random walk continues until at least one neighbour pixel is outside the
study area, or until the user-defined break criterion is fulfilled. The
break criteria are the key parameters for estimating the mass flow impact
areas and can be defined in various ways (Table 2):
The angle of reach ωT or the maximum travel distance
Lmax is computed from empirical–statistical rules or relationships,
based on the analysis of observed events (Table 3). They usually refer to
the distance between the highest spot of the release area and the most
distant spot of the impact area along the flow path (the
Fahrböschung according to Heim, 1932). Consequently, random walks
using this type of break criterion have to start from the set of coordinates
defining the highest point of the observed or expected mass movement.
Alternatively, also a semi-deterministic model (Perla et al., 1980) can be
used.
Empirical–statistical relationships or the semi-deterministic model may be
applied in a large number of parallel computations with randomized values of
the parameters a, b and c (see Fig. 1 and Table 3). This allows one to explore the
effects of uncertainties in the relationships. Only one type of relationship
is considered at once, and the output consists in a raster map of the impact
indicator index III in the range 0–1, representing the fraction of tested
parameter combinations predicting an impact on the pixel (i.e., where
impact indicator score (IIS) = 1). Further, the results of all model runs are stored in a way ready to
be analysed with the parameter sensitivity and optimization tool AIMEC
(Automated Indicator-based Model Evaluation and Comparison; Fischer, 2013).
An impact probability raster map PI in the range 0–1 is computed from a
user-defined sample of observed values of tan(ωT), which is
employed to build a cumulative density function (CDF). The CDF represents
the probability that the movement reaches the pixel associated with each value
of tan(ωT). The sample of observed values may be divided into
one subset of mass movements for building the CDF, and another one for
computing PI. This ensures a clear separation between parameter
optimization and model validation (see Sect. 2.5). Parallel processing may
be used to repeat the analysis for many random subsets in order to achieve a
more robust result.
If an inventory of events is available, the observed impact areas may be
back calculated by routing each random walk until it leaves the observed
impact area of the corresponding mass movement. This mode can be used to
explore the statistical distribution of ωT. The resulting CDF
can be used as input to estimate PI.
Types of rules and relationships supported by r.randomwalk.
ωT is angle of reach, Lmax is travel distance,
V is volume of motion, Z is elevation loss, Qp is peak discharge at
release, and vT is velocity at termination.
IDEquationExamples ReferenceProcessabc1ωT=a(2)Haeberli (1983);Debris flow from GLOF11Huggel et al. (2002)2log10tanωT=alog10V+b(3)Scheidegger (1973)Rock avalanche-0.156660.624193Lmax=aVbZc(4)Rickenmann (1999)Debris flow1.90.160.834ωT=aQpb(5)Huggel (2004)GLOF18-0.075vT=0(6)Perla et al. (1980)Overlay of random walk results
The overlay of individual
random walks operates at two levels:
Random walks of the same mass point: impact frequency (IF) is increased by 1 for each random walk
predicting an impact. IIS is increased by 1 for each model where at least 1
random walk predicts an impact. The average angle of path – and therefore
also PI – is derived from the random walk with the shortest travel
distance (i.e., the straightest flow path and the highest value of ω) at the considered pixel.
Sets of random walks for different mass points: the values of IF for all
random walks impacting a pixel are just added up whilst the maximum of IIS is
applied to each pixel. The issue gets more complex when it comes to
PI: depending on the specific application, the maximum or the average
out of all sets of random walks is more appropriate.
The resulting maps of PI or IIS can be automatically overlaid with a
release probability (PR; result: composite probability PI,C; Mergili
and Chu, 2015) or a release indicator score (RIS; result: impact hazard
indicator score – IHIS), and with an exposure indicator score (EIS) derived from the
land cover (result: impact risk indicator score – IRIS; see Table 1). These steps
are not further considered in this article and are therefore not shown in
Fig. 1.
Validation
r.randomwalk includes three possibilities for validation of the model
results. All three build on the availability of a raster map of the observed
deposition area of the mass movement(s) under investigation. All parts of
the observed impact areas outside of the observed deposition areas are set
to no data (Fig. 3).
Model validation with an ROC plot, relating the false positive
rate rFP and the true positive rate rTP. This way of validation is
suitable for predictor raster maps in the range 0–1, such as III or PI. It
can also be used for binary predictor maps (0 or 1). In such a case
AUCROC is computed from two threshold levels only.
For IIS, the true positive (TP), true negative (TN), false positive (FP), and
false negative (FN) predictions are counted on the basis of pixels and put
in relation. All pixels with IIS ≥ 1 are considered as observed positives
(OP); all pixels with IIS = 0 are considered as observed negatives (ON).
ROC (receiver operating characteristics) plots are produced for III or
PI: the true positive rate rTP (TP/OP) is plotted against the false
positive rate rFP (FP/ON) for various levels of III or PI. The area
under the curve connecting the resulting points, AUCROC, is used as an
indicator for the quality of the prediction (see Fig. 3). If the CDF for
PI is derived from the same set of landslides, r.randomwalk includes the
option to randomly split the set of observed landslides into a set for
parameter optimization, and one for validation. This is done for a
user-defined number of times, exploiting multiple processors (see Sect. 2.3
and Fig. 1). It results in an ROC plot with multiple curves. Note that two
ROC plots are produced: one of them builds on the original number of TN
pixels. For the other one, the number of ON pixels is set to 5 times the
number of OP pixels. Whilst the number of FP pixels remains unchanged, the
number of TN pixels is modified accordingly. This procedure aims at
normalizing the ROC curves in order to enable a comparison of the prediction
qualities yielded for different study areas.
If only one mass movement is considered, a longitudinal profile may be
defined by a set of coordinates of the profile vertices. The observed and
predicted (IIS ≥ 1 or PI > 0) travel distances are
measured and compared along this profile.
Test cases and resultsAcheron rock avalanche, New ZealandArea description and model parameterization
The Acheron rock avalanche in Canterbury, New Zealand (Fig. 4), was
triggered approx. 1100 years BP (Smith et al., 2006). Within the present
study, the release volume, V=6.4 million m3, is
approximated from the reconstruction of the pre-failure topography and is
lower than the value of V=7.5 million m3 estimated by
Smith et al. (2006). We use a 10 m resolution DEM derived by stereo-matching
of aerial photographs. Impact, release and deposition areas are derived from
field and imagery interpretation as well as from data published by
Smith et al. (2006). All random walks start from the highest pixel of the
release area.
We use this case study for demonstrating how to compute the impact indicator
index III from an elevation map, the release area, and the release volume.
Before doing so, we have to analyse the influence of the pixel size and the
parameters nwalks, Rmax, Lctrl, Lseg, fβ, and
fd on the model result. Preliminary tests have shown that r.randomwalk
yields plausible results with the number of random walks:
nwalks=104, Rmax= 10 m, Lctrl= 1000 m,
Lseg= 100 m, fβ= 5, fd=2, and a pixel size of
20 m. These values are taken as a basis to explore the sensitivity of the
model results to the variation of each parameter and the best fit of the
parameters in terms of the travel distance, AUCROC, and the size of the
predicted impact area (Table 4). ωT=11.62∘ , the
angle of reach observed for the Acheron rock avalanche, is applied as the
break criterion for all tests. Some of the tests are run in the
back-calculation mode (flag b; see Tables 2 and 4).
III is computed by executing r.randomwalk 100 times, with the parameter
values optimized according to Table 4. We explore an empirical–statistical
relationship for ωT derived from a compilation of 127 case
studies (Fig. 5). The offset of the equation (b in Eq. 4 and Fig. 5) is
randomly sampled between the lower and upper envelopes of the regression.
The quality of the prediction is evaluated using the ROC plot (see Figs. 1
and 3). Note that the Acheron rock avalanche (not included in the
relationship developed in Fig. 5) is found close to the lower envelope,
meaning that it was very mobile compared to most of the other events.
Acheron rock avalanche. (a) Panoramic view; photo: M. Mergili,
28 February 2015. (b) Location and geometry.
Empirical–statistical relationship relating the angle of reach
ωT to the volume V of avalanching flows of rock or debris. The
data are compiled from Scheidegger (1973), Legros (2002), Jibson et al. (2006),
Evans et al. (2009), Sosio et al. (2012), and Guo et al. (2014).
Results
Figure 6 summarizes the findings of the test s 1–3 (see Table 4). Test 1
leads to the expected result that the predicted impact area increases with
the number of random walks. However, the predicted impact area is also a
function of the pixel size: with larger pixels, less random walks are needed
to cover an area of similar size than with smaller pixels. Figure 6a further
indicates that the possible impact area is not fully covered even at 105
random walks: no substantial flattening of the curves is observed. We
conclude that (i) a very high value of nwalks would be necessary
to fully cover the possible impact area, and (ii) this would lead to a
substantial overestimation of the observed impact area.
Tests of the parameters nwalks, Lctrl, Lseg,
Rmax, fβ, fd, and the pixel size. Where ranges of values are
given in bold, the model is run with 100 random samples constrained by the
minima and maxima indicated. Where values given in bold are separated by
commas, in these cases exactly these values are tested.
Results of the tests 1–3 (number of test indicated in the yellow
circle). Number of random walks plotted against (a) the impact area and
(b) the area under the ROC curve. (c) Computed travel distance Lmax as a
function of Lseg (in the legend, the corresponding value of Lctrl is
given in parentheses). (d) Computed Lmax as a function of Lctrl.
On the other hand, the quality of the prediction in terms of AUCROC
reaches a maximum at nwalks≈102 (pixel size 40 m) or
nwalks≈103 (pixel size 20 m), decreasing with higher
values of nwalks. At a pixel size of 10 m, AUCROC reaches a constant
level at nwalks≈104 (see Fig. 6b). We may conclude that
excessive numbers of random walks lead to an overestimation of the impact
area rather than to a better prediction quality. Coarser pixel sizes allow
one
to achieve the same level of coverage and the same prediction quality at
lower values of nwalks. However, the pixel size has to be fine enough to
account for the main geometric characteristics of the process under
investigation (see Sect. 4). All further tests are performed with
nwalks=104.
Sensitivity of impact area and AUCROC to selected input
parameters. The numbers of the corresponding tests (see Table 4) are
indicated in the yellow circles. (a) Control distance Lctrl; (b) maximum
run-up height Rmax; (c) slope factor fβ; (d) direction factor
fd.
Figure 6c illustrates that, at Lctrl=1000 m, the travel distance
computed within the observed impact area decreases with increasing values of
Lseg (tests 2 and 3 in Table 4). This pattern is well explained by
Fig. 2b. At short segment lengths, the effects of flow paths frequently
changing their direction are particularly evident for pixel sizes of 10 m
and 20 m. Lmax drops below the observed value of 3550 m (see Fig. 4b) at
75 ≤Lseg≤ 100 m. With Lseg≥3050 m, corresponding
to the Euclidean distance between the release point and the terminal point
of the Acheron rock avalanche, Lmax would also take a value of 3050 m.
At Lctrl=50 m (only shown for a pixel size of 20 m), r.randomwalk
tends to predict too long travel distances, compared to the observation.
This phenomenon occurs as flow directions are not well defined in the
relatively plane deposition zone of the Acheron rock avalanche; therefore, flow
paths may frequently change their direction or even go backwards or in a
circular way if such a behaviour is not impeded by sufficiently high values
of Lctrl (see Fig. 2a). Figure 6d indicates that this undesired behaviour
(visible in the area marked by the X in the gray circle) disappears at
Lctrl > 200 m.
On the other hand, the value of Lctrl should not be chosen too high as
this may negatively impact the model performance. In the case of the Acheron rock avalanche, a drop in AUCROC is observed between Lctrl≈2000 and Lctrl≈2500 m (Fig. 7a). This drop is explained by
an increasing number of false negative pixels in those areas, which cannot be
reached by the random walks due to the strict constraint of flow direction.
Within the tested ranges of parameter values, the quality of the prediction
is highest at values of Rmax≈5–10 m (see Fig. 7b) and
fβ≥5 (see Fig. 7c), whilst it reaches it maximum at
fd≈2–3 (see Fig. 7d). The predicted impact area increases
with increasing Rmax and fd whilst it decreases with increasing
fβ.
Impact indicator score for the Acheron rock avalanche.
(a) Classified III map. (b) ROC plot, building on normalized ON area (see
Sect. 2.5).
Figures 6 and 7 indicate that the initial values of nwalks, Lctrl,
LsegRmax, fβ, fd, and the pixel size suggested in
Sect. 3.1.1 and Table 4 are within the optimum range of values (see
Sect. 4). Therefore, they are used for computing the impact indicator index
for the Acheron rock avalanche (Fig. 8a). Concerning the break criteria,
this can be classified as a forward analysis. As expected from Fig. 5, where
the Acheron rock avalanche falls in between the envelopes of the
relationship employed, the upper part of the observed impact area displays a
value of III = 1, whilst the remaining part of the observed impact area
displays values of 1 > III > 0, decreasing towards the
terminus. As the event was comparatively mobile within the context of the
relationship used (see Sect. 3.1.1 and Fig. 5), the values of III are close to
zero in the terminal area, and the area with III > 0 does not reach
far beyond the observed terminus. Note that the maximum value of III is 0.8,
meaning that 20 % of all model runs did not even start due to very high
values of ωT yielded with the randomized values of b (see
Fig. 5). Evaluation against the observed deposit yields a value of
AUCROC=0.94 (see Fig. 8b). All values of AUCROC shown in Figs. 6
and 7 and the ROC plot of Fig. 8b build on normalized ON areas (see
Sect. 2.5).
III was generated within a computational time of 188 s.
Kao Ping Watershed, TaiwanArea description and model parameterization
Between 7 and 9 August 2009, Typhoon Morakot struck Taiwan and triggered
enormous landslides, causing significant land cover change (Fig. 9). More
than 22 000 landslides were recorded in southern Taiwan (Lin et al., 2011).
One of the hot spots of mass wasting was the Kao Ping Watershed (Wu et al.,
2011), where the extremely heavy rainfall (in total, more than 2000 mm depth
and 90 h duration) triggered a catastrophic landslide in the Hsiaolin
Village (Kuo et al., 2013).
Location, terrain and landslide inventory of the Kao Ping
Watershed, Taiwan. Comparison of the satellite images illustrates the
landslide-induced land cover changes associated with the Typhoon Morakot.
The landslide inventory builds on the interpretation of the FORMOSAT-2
imagery.
We consider a 61.5 km2 subset of the Kao Ping Watershed for
computing the landslide impact probability PI, based on the observed
landslide release areas. In all, 207 landslides are mapped in the shale,
sandstone,
and colluvium slopes (see Fig. 9). A 10 m DEM is used along with an
inventory of the landslide impact areas. Release and deposition areas are
extracted from the inventory. We employ the values of nwalks,
Rmax, fβ, fd, Lctrl, Lseg resulting from the
optimization procedure for the Acheron rock avalanche (see Sect. 3.1.1), and
a pixel size of 20 m. PI is computed as follows:
A set of random walks (nwalks=104) is started from each release
point (i.e., the highest pixel of each landslide). Each random walk stops as
soon as it would leave the impact area of the same landslide
(back calculation, flag b).
After completing all random walks for the study area, the statistical
distribution of ωT is analysed. All landslides with
Lmax < 100 m are excluded. A fraction of 20 % out of all
landslides (i.e., all values of ωT associated with those
landslides) is randomly selected and retained for validation. Using visual
comparison, we have identified the log-normal distribution as the most
suitable type of distribution for this purpose. Consequently, the log-normal
CDF stands for the probability that a moving
mass point leaves the observed impact area at or below the associated
threshold of ωT.
We perform a forward analysis of PI by starting a set of random walks
(nwalks=104) from the release points of the retained
landslides, and assigning the cumulative density associated with the average
angle of path to each pixel. The result is validated against the observed
deposition zones of the retained landslides by means of an ROC plot.
Steps 2. and 3. are repeated for 100 randomly selected subsets (parallel
processing is applied). The final map of PI is generated by applying for
each pixel the maximum of the values yielded by all the model runs.
We refer to this work flow as test 1 and repeat the analysis with starting
random walks not only from the release points but also from all the pixels
within the observed release areas (test 2). This means that the CDF is
derived from a much larger sample of data than when considering only one
point per landslide for starting random walks. We exclude all sets of random
walks yielding Lmax < 100 m, use a log-normal CDF and start a
set of only 103 random walks from each release pixel for
computing PI.
Results
Starting sets of 104 random walks from the highest points of all
landslides (test 1) results in a range of values of 16.0 ≤ωT≤43.5∘, an average of 30.4∘, and a
standard deviation of 5.2∘ (derived from n=132 landslides,
excluding those with Lmax < 100 m). Repeating the analysis with
104 random walks started from each pixel within the landslide release
areas (test 2), we observe a range of values 16.4 ≤ωT≤44.1∘, an average of 26.9∘, and a standard deviation
of 4.8∘ (n=1563). Figure 10 illustrates the histograms,
probability density, and cumulative density functions derived from both
analyses. Even though the ranges of values are similar in both tests, test 1
yields (i) a higher average of ωT and (ii) a broader range of
values than test 2; (i) is explained by the fact that those random walks
starting from lower parts of the release areas are expected to leave the
observed impact area at lower values of ωT; (ii) is most likely
the consequence of a number of rather small landslides with high or low
values of ωT strongly reflected in the statistics. Such
outliers are less prominent in the statistics of test 2 due to the much
higher number of cases, most of them related to the larger landslides.
Each of the impact probabilities shown in Fig. 11 represents the overlay of
100 analyses where random sets of 80 % of the landslides are used for
deriving the CDF and the remaining 20 % are used for computing the impact
probabilities. The maps illustrate the maximum values of PI out of the
overlay of the 100 results. Each of the results is derived using a slightly
different CDF. Both tests yield largely similar patterns of PI. We note
that (i) test 2 predicts larger impact areas and higher values of PI
than test 1, and (ii) some random walks take the wrong direction in
test 2 (indicated by “1” in the yellow circle in Fig. 11b), a phenomenon
not observed for test 1; (i) is explained by the higher number – and the
broader distribution – of release pixels in test 2, compared to test 1. The
reason for (ii) is that random walks starting from the highest point of an
observed landslide are forced to flow into the observed landslide area
(test 1), a constraint not applicable when starting random walks from each
release pixel (test 2). In this case it happens that pixels located at or
near a crest produce random walks in both directions. In test 1, the
computational time amounted to 63 s for deriving the CDF and
8613 s for calculating PI. In test 2, these times increased to
1719 and 9752 s, respectively. The relatively slight increase with
regard to PI results from the reduced value of nwalks in test 2.
Histograms, probability densities, and cumulative densities of
ωT of mass movements in the test area in the Kao Ping
Watershed. (a) Result for a set of 104 random walks started from the
highest point of each landslide (test 1). (b) Result for a set of 104
random walks started from each pixel within the release areas of all
landslides (test 2).
The prediction quality is tested for each of the 100 model runs for the two
tests, producing sets of 100 ROC curves (Fig. 12).
AUCROC=0.917±0.038 for test 1 and 0.920 ± 0.029 for
test 2, both computed with the original number of TN pixels (see Sect. 2.5).
In contrast, the procedures demonstrated in the two tests vary strongly in
their scope of applicability. We have demonstrated the methodologies by
back calculating observed landslides. As soon as this is done, one may go
one step further:
The methodology shown in test 1 can be employed to make forward predictions
for defined expected future landslides, given that a sufficient set of
observed landslides of similar behaviour is available to derive the CDF.
The methodology demonstrated in test 2 can be used in combination with maps
of landslide release probability to explore the composite probability of a
landslide impact (see Sects. 2.4 and 4).
Impact probability in the range 0–1. (a) Result of test 1
(random walks starting from the highest point of each landslide; cumulative
density according to Fig. 10a). (b) Result of test 2 (random walks starting
from all release pixels; cumulative density according to Fig. 10b).
ROC plots illustrating the prediction quality of (a) test 1 and
(b) test 2, using the original number of TN pixels (see Sect. 2.5).
In either case the statistics (see Fig. 10) have to be derived with the same
type of approach later used for producing the PI map.
Gunt Valley, TajikistanArea description and model parameterization
As most mountain areas worldwide, the Pamir of Tajikistan experiences a
significant retreat of the glaciers. One of the consequences thereof
consists in the formation and growth of lakes, some of which are subject to
glacial lake outburst floods (GLOFs), which may evolve into destructive
debris flows (Mergili and Schneider, 2011; Mergili et al., 2013; Gruber and
Mergili, 2013). No records of historic GLOFs in the test area are known to
the authors. However, in August 2002 a GLOF in the nearby Shakhdara Valley
evolved into a debris flow, which destroyed the village of Dasht, claiming
dozens of lives (Mergili et al., 2011).
The frequency of such events is low and historical data are sparse.
Consequently, possible travel distances of GLOFs may not be derived in a
purely statistical way. Instead, we have to use published
empirical–statistical relationships and simple rules to produce an impact
indicator score (IIS) map.
We compute IIS with regard to GLOFs for a 2106 km2 study area
in the Gunt Valley (Fig. 13). The analysis builds on the ASTER GDEM (Advanced Spaceborne Thermal Emission and Reflection Radiometer – Global Digital Elevation Model) V2 and
the coordinates and characteristics (estimates of V and Qp) of 113 lakes
in the area (Gruber and Mergili, 2013).
A set of random walks (nwalks=104) is routed from the outlet of
each lake through the DEM. Six break criteria are combined to compute IIS,
partly following Gruber and Mergili (2013). The relationships and rules
employed as break criteria are summarized in Table 5. Rule 1 is applied with
ωT=11∘ (test 1 – according to Haeberli, 1983 and
Huggel et al., 2003, 2004a, b for debris flows from glacier- or moraine-dammed
lakes, and Zimmermann et al., 1997 for coarse- and medium-grained debris flows)
and with ωT=7∘ (test 2 – Zimmermann et al., 1997
for fine-grained debris flows). All other rules and relationships are used
for both tests. For each pixel, IIS consists in the number of relationships or
rules predicting an impact (i.e., IIS takes values in the range 0–6).
Rmax, Lctrl, Lseg, fβ, and fd are set to the
optimum values found for the Acheron rock avalanche, the pixel size is set
to 60 m.
Results
Figure 14 illustrates the possible impact areas of GLOFs in the Gunt Valley
study area according to the relationships listed in Table 5.
Figure 14a shows the impact indicator score IIS i.e., the number of relationships
predicting an impact, resulting from test 1 (rule 1 applied with ωT=11∘). Except for one prominent exception,
IIS > 3 (possible debris flow impact) only for the largely
uninhabited upper portions of the tributaries to the Gunt Valley. In
contrast, a possible flood impact (1 ≤ IIS ≤3) is predicted for much
of the main valley. test 2 (rule 1 applied with ωT=7∘) predicts a possible debris flow impact also for
part of the main valleys (see Fig. 14b). The IF (per cent of
random walks impacting each pixel) for test 1 is shown in Fig. 14c for a
subsection of the test area, classified by quantiles. IF is strongly governed
by the width of the movement, i.e. by the local topography, and may serve as
a surrogate for the expected depth rather than as for the probability of an
impact.
The test area in the Gunt Valley, Tajikistan. (a) Location,
topography, glaciers and lakes. (b) Proglacial lake in the upper
Varshedzdara Valley; photo: M. Mergili, 18 August 2011.
Empirical–statistical relationships and simple rules used for
computing the IIS of GLOFs in the Gunt Valley (see Table 3).
IDtestRelationshipReferenceProcess11ωT=11∘Haeberli (1983); Zimmermann et al. (1997);Flood or debris flowHuggel et al. (2003, 2004a, b)12ωT=7∘Zimmermann et al. (1997)21,2ωT=18Qp-0.07Huggel (2004)31,2Lmax=1.9V0.16Z0.83Rickenmann (1999)341,2ωT=6∘Flood51,2ωT=4∘61,2ωT=2∘Haeberli (1983); Huggel et al. (2004a)
1,2 ID(s) of test(s) where the rule or relationship is applied. 3 A bulking
factor of 5 is applied to V (modified after Iverson, 1997).
Note that Fig. 14 only indicates the tendency of an already released GLOF to
impact certain pixels. It does not provide any information on the
susceptibility of a certain lake to produce a GLOF at all. Earlier,
Mergili and Schneider (2011) and Gruber and Mergili (2013) have attempted to
combine GLOF release indicators with impact indicators and land cover maps
to generate hazard and risk indicator maps. However, the results of their
studies may underestimate the possible impact areas as the travel distance
was computed on a pixel-to-pixel basis, possibly yielding too low values of
ωT (see Figs. 2 and 6).
The robustness and appropriateness of the rules and relationships for
low-frequency events, such as GLOFs (see Table 5), is questionable. The rules
building on a unique value of ωT overpredict the possible
impact areas for those lakes where not enough water is available to produce
a flood in downstream valleys. Applying the rules and relationships for
debris flows implies a blind assumption that enough entrainable sediment is
available to produce a debris flow. Whilst ωT≥11∘ is considered the worst case for debris flows of GLOFs from
glacier- or moraine-dammed lakes in the European Alps according to
Haeberli (1983) and Huggel et al. (2002), ωT=9.3∘
was measured for the 2002 Dasht Event, the only well-documented GLOF near
the test area (Mergili et al., 2011). Also the relationship proposed by
Rickenmann (1999) severely underestimates the travel distance of this event,
even when massive bulking is assumed. Applying ωT=7∘ as given by Zimmermann et al. (1997) for fine-grained
debris flows might be more suitable as worst-case assumption for debris
flows from GLOFs in the Pamir, even though this threshold leads to very
conservative predictions.
We have measured computational times of 1520 s for test 1 and
1556 s for test 2.
Possible GLOF impact areas in the Gunt Valley, Tajikistan.
(a) Impact indicator score derived with test 1. (b) Impact indicator score
derived with test 2. (c) Impact frequency derived with test 1, classified by
quantiles.
Discussion
Whilst conceptual tools are commonly applied for routing mass movements at
medium and broad scales, most of them use single values or rules as break
criteria, disregarding the high degree of uncertainty (e.g., Gamma, 2000;
Wichmann and Becht, 2003; Huggel et al., 2002; Horton et al., 2013;
Blahut et al., 2010). r.randomwalk introduces a set of tools to deal with
uncertain break criteria in a flexible way, depending on the quality of
rules or relationships available. In general, empirical–statistical
relationships represent rough simplifications as mass movement processes may
also stop when reaching valleys of higher order, run against opposite slopes
or lose energy when bending sharply. However, relatively robust rules or
relationships exist for the most common types of processes such as rock
avalanches (Scheidegger, 1973; see Fig. 5) or debris flows (Rickenmann,
1999). They build on data sets large enough to derive meaningful envelopes
and to compute impact indicator indices with r.randomwalk. Relationships for
less frequent types of processes are less robust as it was illustrated for
GLOFs (Haeberli, 1983; Zimmermann et al.; 1997; Huggel et al., 2002; Huggel, 2004;
see Sect. 3.3.2). In such cases we recommend to compute impact indicator
scores building on more than one model, as shown by Gruber and Mergili (2013) and in the present work. Impact indicator indices and scores are
mainly useful for anticipating the possible impact area of expected single
events (see Sect. 3.1.2), or for application at broader scales (see
Sect. 3.3.2).
The impact probability is useful for predicting possible impact areas of
mass movements in areas where many events are documented, but the volumes of
possible future events are not known. Whilst in the present paper it was
demonstrated how to compute impact probabilities related to observed release
areas, r.randomwalk also includes the option to combine the impact
probability with the release probability PR (see Table 1). Landslide
release probability (susceptibility) maps are often produced from a
landslide inventory and a set of environmental layers (e.g., Guzzetti,
2006). Starting random walks from each single pixel of a study area, and
combining the release probability of this pixel with the impact probability
allows one to produce a composite probability PI,C map. Doing this is
non-trivial and requires specific strategies. It is therefore covered in a
separate article (Mergili and Chu, 2015). Gruber and Mergili (2013) have
combined release and impact indicator scores for various types of
high-mountain hazards, and overlaid the results with a land cover data set
to produce a risk indicator score.
The sensitivity of r.randomwalk to variations of the parameters
nwalks, Rmax, fβ, fd, Lctrl, Lseg (see
Sect. 2.2) and the pixel size were tested for the Acheron rock avalanche.
Even though the optimized values are applied also to the other cases in the
present work, this issue requires further investigation, also with regard to
the scale of the processes. This is particularly true for the pixel size,
which has to be fine enough not to lose the geometrical characteristics
governing the motion (Blahut et al., 2010). Furthermore, coarser pixels and
a larger number of random walks make results more conservative. Rmax,
fβ, and fd control the degree of lateral spreading and therefore
influence the conservativeness of the results. In the future we plan to
compare the performance of r.randomwalk to software tools using multiple
flow direction algorithms (e.g., Flow-R; Horton et al., 2013) in terms of
computational times and prediction success.
Overestimating the travel distance at a certain pixel is avoided by choosing
sufficiently high values of Lseg (see Fig. 6c). Shorter travel distances
at a certain pixel are associated with higher values of ω and,
consequently, larger predicted impact areas, i.e. more conservative
results that are desirable for many applications. The values of Rmax
leading to the best prediction quality are considerably lower than run-up
height observed for the Acheron rock avalanche. This phenomenon is explained
by the facts that (i) the observed maximum run-up height refers to a limited
area, whilst r.randomwalk applies the run-up height defined by Rmax in
any place; and (ii) not all random walks reach the bottom of the valley
before running up.
We have demonstrated how to estimate the prediction quality of III and
PI maps. Where sufficient reference data are available to prove the
validity of the model, the results may be applied for hazard zoning. Where
data are
not available, the outcomes of r.randomwalk are suitable for broad-scale overviews of
possibly affected areas, which have to be considered as rough indicators
only. A suitable level of spatial aggregation may be necessary in such cases
(Gruber and Mergili, 2013).
r.randomwalk includes a break criterion building on the two-parameter
friction model of Perla et al. (1980) (see Sect. 1 and Table 3), which can be
used to compute flow velocities (e.g., Wichmann and Becht, 2013;
Mergili et al., 2012a; Horton et al., 2013). Evaluating this functionality
has to build on (i) specific strategies for the sensitivity analysis and
optimization of multiple parameters and (ii) a sound comparison with the
outcome of physically based models. This effort will be presented in a
separate article (Krenn et al., 2015). Further, the parameter
sensitivity and optimization code AIMEC (Fischer, 2013) can be directly
coupled to r.randomwalk.
Conclusions
We have introduced the open-source GIS tool r.randomwalk, designed for
conceptual modelling of the propagation of mass movements. r.randomwalk
offers built-in functions for considering uncertainties and for
validation. Employing a set of three contrasting test areas, we have
demonstrated (i) the possibility to combine results yielded with various
break criteria into one impact indicator score; (ii) the option to explore
multiple computational cores for combining the results obtained with many
randomized parameter combinations into an impact indicator index; (iii) the
possibility to back calculate the CDF of the angles of reach of observed
landslides, and to use this CDF to make forward predictions of the impact
probability; and (iv) integrated functions for the validation and visualization
of the results. This includes strategies to properly separate the data sets
for parameter optimization and model validation.
We have further shown that controls for smoothing of the flow path and the
avoidance of circular flows have to be introduced to avoid underestimating
travel distances and impact areas. The number of random walks executed for
each mass point and the pixel size influence the level of conservativeness
of the results rather than the quality of the prediction. The scope of
applicability of r.randomwalk strongly depends on the availability of robust
break criteria and on the availability of reference data for evaluation.
Code availability
The model codes, a user manual, the scripts used for starting the tests
presented in Sect. 3 and some of the test data are available at http://www.mergili.at/randomwalk.html.
Acknowledgements
The work was conducted as part of the international cooperation project “A
GIS simulation model for avalanche and debris flows” funded by the Austrian
Science Fund (FWF) and the German Research Foundation (DFG). Further, the
support of Massimiliano Alvioli, Matthias Benedikt, Yi-Chin Chen, Ivan Marchesini, and Tim Davies is acknowledged.
Edited by: T. Poulet
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