Rainfall threshold determination is a pressing issue in
the landslide scientific community. While major improvements have been made towards more reproducible techniques for the identification of triggering
conditions for landsliding, the now well-established rainfall intensity or
event-duration thresholds for landsliding suffer from several
limitations. Here, we propose a new approach of the frequentist method for
threshold definition based on satellite-derived antecedent rainfall
estimates directly coupled with landslide susceptibility data. Adopting a
bootstrap statistical technique for the identification of threshold
uncertainties at different exceedance probability levels, it results in
thresholds expressed as AR =(α±Δα)⋅S(β±Δβ), where AR is antecedent rainfall (mm), S is
landslide susceptibility, α and β are scaling parameters, and
Δα and Δβ are their uncertainties. The main
improvements of this approach consist in (1) using spatially continuous
satellite rainfall data, (2) giving equal weight to rainfall characteristics
and ground susceptibility factors in the definition of spatially varying
rainfall thresholds, (3) proposing an exponential antecedent rainfall
function that involves past daily rainfall in the exponent to account for
the different lasting effect of large versus small rainfall,
(4) quantitatively exploiting the lower parts of the cloud of data points, most
meaningful for threshold estimation, and (5) merging the uncertainty on
landslide date with the fit uncertainty in a single error estimation. We
apply our approach in the western branch of the East African Rift based on
landslides that occurred between 2001 and 2018, satellite rainfall estimates
from the Tropical Rainfall Measurement Mission Multi-satellite Precipitation
Analysis (TMPA 3B42 RT), and the continental-scale map of landslide
susceptibility of Broeckx et al. (2018) and provide the first regional rainfall
thresholds for landsliding in tropical Africa.
Introduction
Rainfall is widely recognized as an important trigger for landslides (Sidle
and Bogaard, 2016), posing an increased threat to people and economies
worldwide under climate change conditions (Gariano and Guzzetti, 2016).
Rainfall thresholds, defined as the best separators for triggering and
non-triggering known rainfall conditions (Crozier, 1997), are the most used
instrument in landslide hazard assessment and early warning tools (Segoni et
al., 2018). Whereas physically based models require detailed geotechnical,
hydrological, and environmental parameters, which is achievable only on
a hillslope to small-basin scale, the empirical approach is adopted for local
to global scales (Guzzetti et al., 2007).
The most common parameters used to define empirical thresholds are the
combinations of rainfall intensity-duration, rainfall event-duration,
and antecedent rainfall-conditions (Guzzetti et al., 2007). Standard
approaches for the definition of the first two combinations of parameters
are on the rise (e.g. Segoni et al., 2014; Vessia et al., 2014, 2016;
Robbins, 2016; Rossi et al., 2017; Melillo et al., 2018) as substitutes for
the formerly used subjective expert-judgement approaches (Aleotti, 2004;
Brunetti et al., 2010). Conversely, no unanimous definition of
triggering antecedent rainfall (AR) conditions is currently achieved. This is
related to the complexity and process dependence of environmental factors
that influence the impact of AR on a slope (Sidle and Bogaard, 2016), yet
it is regrettable because of AR physical relation with soil shear strength and thus
landslide potential (Ma et al., 2014; Hong et al., 2018). AR has been taken into
account by combining the rainfall accumulation periods identified as most
significant for landslide triggering in the study area, varying up to 120 d depending on landslide type (Guzzetti et al., 2007). In some cases, an AR function
convoluting rainfall over the selected period is defined with the aim of
reflecting the decaying effect of rainfall on soil moisture status (e.g.
Crozier, 1999; Glade et al., 2000; Capparelli and Versace, 2011; Ma et al., 2014).
Once the triggering rainfall conditions of landslides have been
quantitatively described, thresholds are determined through more and more
refined techniques claiming objectivity and reproducibility (Segoni et al.,
2018). Because the transition between triggering and non-triggering
conditions for landslides cannot be sharply devised (Berti et al., 2012;
Nikolopoulos et al., 2014), statistical approaches including probabilistic
and frequentist methods have replaced a deterministic approach of the
threshold definition. Probabilistic methods such as Bayesian inference
(Guzzetti et al., 2007; Berti et al., 2012; Robbins, 2016) are based on
relative frequencies, considering information on triggering and
non-triggering rainfall conditions. Criticisms of this method are based on the biased
prior and marginal probabilities related to the incompleteness of the
landslide input data (Berti et al., 2012).
Brunetti et al. (2010) proposed a frequentist method allowing threshold definition at different exceedance
probability levels, a method improved by Peruccacci et al. (2012) for the
estimation of uncertainties associated with the threshold through a
bootstrap statistical technique (Gariano et al., 2015; Melillo et al., 2016,
2018; Piciullo et al., 2017). A limitation of the frequentist approach is
the dependency on a large and well-spread data set in order to attain
significant results (Brunetti et al., 2010; Peruccacci et al., 2012). Other,
less influential, threshold identification approaches are reviewed by Segoni
et al. (2018).
Regional ground conditions, but also the progressive adjustment of
landscapes to the governing climatic parameters, affect the meteorological
conditions required for landsliding (Ritter, 1988; Guzzetti et al., 2008;
Peruccacci et al., 2012; Parker et al., 2016). For this reason, thresholds
gain in efficiency when rainfall regimes are accounted for through rainfall
normalization (e.g. Guzzetti et al., 2008; Postance et al., 2018) and when
the input data are partitioned according to homogeneous predisposing ground
conditions or failure processes (Crosta, 1998; Crosta and Frattini, 2001;
Peruccacci et al., 2012; Sidle and Bogaard, 2016). Yet, to the authors'
knowledge no threshold mapping involving landslide susceptibility as a proxy
integrating the causative ground factors has been proposed to date beyond
local-scale physically based models (e.g. Aristizábal et al., 2015;
Napolitano et al., 2016). Conversely, landslide early warning tools
aim at coupling primary landslide susceptibility data and thresholds based
on rainfall characteristics, demonstrating the importance of their
combination for landslide prediction at regional to global scales (Piciullo
et al., 2017; Kirschbaum and Stanley, 2018).
Though being identified as a pressing issue in the scientific community,
rainfall threshold research is almost nonexistent in Africa (Segoni et al.,
2018) despite high levels of landslide susceptibility and hazard, especially
in mountainous tropical Africa, characterized by intense rainfall, deep
weathering profiles, and high demographic pressure on the environment
(Aristizábal et al., 2015; Jacobs et al., 2018; Migoń and
Alcántara-Ayala, 2008; Monsieurs et al., 2018a). The lack of scientific
investigation in this area is most likely related to the dearth of data on
timing and location of landslides (Kirschbaum and Stanley, 2018). However,
the other fundamental data for threshold analysis, namely rainfall data, have
been
globally freely available through satellite rainfall estimates (SREs) since
the 1990s. Even if their use in threshold analysis remains limited (Brunetti
et al., 2018; Segoni et al., 2018), SREs have many advantages in sparsely
gauged areas such as tropical Africa. A review paper by Brunetti et al. (2018)
reveals that, to date, the most recurring SRE products used for
research on landslide triggering conditions come from the Tropical Rainfall
Measuring Mission (TRMM) (e.g. Liao et al., 2010; Kirschbaum et al., 2015;
Cullen et al., 2016; Robbins, 2016; Nikolopoulos et al., 2017; Rossi et al., 2017).
The main objective of this paper is to devise an improved version of the
frequentist method of rainfall threshold definition that goes beyond the
sole aspect of rainfall characteristics and will be applicable in regions
with limited rainfall gauge data such as, for example, tropical Africa.
Consequently, it will rely on the use of TRMM satellite rainfall data.
Directly operational thresholds and threshold maps are expected from several
methodological improvements regarding the definition of an elaborate AR function,
the integration of climatic and ground characteristics (through landslide
susceptibility) into a 2-D trigger–cause graph, and a better focus on the
information delivered by landslide events associated with low AR values. The
western branch of the East African Rift (WEAR, Fig. 1) serves as a suitable
study area prone to landsliding (Maki Mateso and Dewitte, 2014; Jacobs et
al., 2016; Monsieurs et al., 2018a; Nobile et al., 2018), in which recent
efforts have been made to collect information on landslide occurrence
(Monsieurs et al., 2018a) and validate TRMM products (Monsieurs et al., 2018b).
(a) Landslide susceptibility at 0.25∘ resolution,
derived from the map of Broeckx et al. (2018), and distribution of landslides
in the western branch of the East African Rift, comprising 29 landslides in
mining areas (triangles) and 145 landslides outside mining areas (dots) of
which the red dots are landslides associated with antecedent rainfall less than
5 mm. Only the black dots (143 landslides) are used for calibrating the
rainfall thresholds. (b) Spatial pattern of mean annual precipitation (MAP)
based on 18 years (2000–2018) of TMPA (3B42 RT) data, and thus affected by
significant underestimation (Monsieurs et al., 2018b). Numbers in the lakes
are as follows.
1: Lake Albert; 2: Lake Edward; 3: Lake Kivu; 4: Lake
Tanganyika. Background hillshade Shuttle Radar Topography Mission (90 m).
Setting and dataLandslides in the WEAR
The study area extends over ∼350000 km2 in
the WEAR (Fig. 1). High seismicity (Delvaux et al., 2017), intense rainfall
(Monsieurs et al., 2018b), deeply weathered substrates (e.g. Moeyersons et
al., 2004), and steep slopes with an elevation range of 600 m at Lake Albert
to 5109 m in the Rwenzori Mountains (Jacobs et al., 2016) are all
predisposing factors rendering the area highly prone to landsliding (Maki Mateso
and Dewitte, 2014; Broeckx et al., 2018; Monsieurs et al., 2018a; Nobile et al., 2018).
We updated the currently most extensive database existing over the WEAR from
Monsieurs et al. (2018a), which formerly contained information on the
location and date of 143 landslide events that occurred between 1968
and 2016. New information on landslide occurrence was added through an extensive
search of online media reports and to a lesser extent information from local
partners. Only landslides with location accuracy better than 25 km and for
which the date of occurrence is known with daily accuracy are included, with
Monsieurs et al. (2018a) stressing that a residual uncertainty on landslide
date especially affects landslides having occurred overnight. Information on
the timing of the landslide within the day of occurrence is rarely reported.
Omitting pre-2000 events so as to adjust to the temporal coverage of the
satellite rainfall data, the updated inventory comprises a total of
174 landslide events that occurred between 2001 and 2018 located with a mean
accuracy of 6.7 km. Their spatial distribution is limited in the longitude
axis (Fig. 1) because of data collection constraints related to the remote
and unstable security conditions (Monsieurs et al., 2017). The landslide
temporal pattern shows that most of them occurred after the second rainy
season from March to May, with almost no landslides being reported in the
following dry season (June–August) (Fig. 2). Daily rainfall distributions
per month are provided in the Supplement.
Monthly distribution of 174 landslide events (LS) in the WEAR and mean
monthly rainfall based on 20 years (1998–2018) of TMPA (3B42 RT) daily data,
downloaded from http://giovanni.sci.gsfc.nasa.gov (last access: 14 April 2019).
A distinction is made for landslides mapped in mining areas, counting 29 out
of the 174 events. As media reports generally lack scientific background and
insights into the landslide process, we discard these events because of the
possibility of anthropogenic interference in their occurrence. We also
acknowledge that the rest of the inventory may encompass a wide range of
landslide processes, from shallow to deep-seated landsliding (Monsieurs et
al., 2018a), and that another important bias in the WEAR data set highlighted
by field observations is the non-recording of many landslide events
(Monsieurs et al., 2017, 2018a). Therefore we claim neither catalogue
completeness nor ascertained identification of the conditions determinant for landsliding.
Rainfall data
Owing to the absence of a dense rain gauge network in the WEAR over the
study period (Monsieurs et al., 2018b), we use SRE from the TRMM
Multisatellite Precipitation Analysis 3B42 Real-Time product, version 7
(hereafter spelled TMPA-RT). While the TRMM satellite is no longer
operating, the multisatellite TMPA product has continued to be produced by
combining both passive microwave and infrared sensor data (Huffman et al.,
2007). TMPA-RT is available at a spatio-temporal resolution of
0.25∘× 0.25∘ and 3 h for the period from 2000 to present,
over 50∘ N–50∘ S, provided by NASA with 8 h latency.
Compared to the TMPA Research Version product, TMPA-RT shows lower absolute
errors and was found to overall perform better in the WEAR for higher
rainfall intensities (Monsieurs et al., 2018b). Yet, average rainfall
underestimations of the order of ∼40 % and a low
probability of detecting high rainfall intensities as such have to be taken
into account. We maintain TMPA-RT's native spatial resolution, while
aggregating the 3-hourly data to daily resolution, in accordance with the
landslide inventory temporal resolution. Based on 18 years (2000–2018) of
TMPA-RT data, Fig. 1 shows the spatial pattern of mean annual precipitation
in the study area, which results from a complex system of climate drivers in
equatorial Africa (Dezfuli, 2017).
Susceptibility data
As we want to introduce ground factors directly within the frequentist
estimation of rainfall thresholds, we make use of susceptibility data as a
proxy for the joint effect of ground characteristics on spatial variations
in thresholds. We adopt here the landslide susceptibility model from Broeckx
et al. (2018). Calibrated for all landslides regardless of type and covering
the African continent at a spatial resolution of 0.0033∘, this
model has been produced through logistic regression based on three
environmental factors, namely topography, lithology and peak ground
acceleration. Susceptibility is expressed as the spatial probability of
landslide occurrence in each pixel. As the values of these probability
estimates scale with the ratio between the numbers of landslide (L) and
no-landslide (NL) pixels used in the model calibration (King and Zeng,
2001), we stress that Broeckx et al. (2018) applied a ∼4:1L/ NL
ratio. Interestingly, as their susceptibility map covers the whole of
Africa, this model characteristic will not contribute to mar potential
extrapolations of our calculated thresholds to similar analyses elsewhere in
the continent. Finally, when resampling the susceptibility data to the
coarser 0.25∘ resolution of the SRE used in the threshold
analyses, we assigned to each TMPA pixel a value corresponding to the
95th percentile of the original values in order to reflect the behaviour of the
most landslide-prone sub-areas within the pixel.
A novel approach of the frequentist methodConceptual framework
In order to overcome the limitations of the current frequentist approach of
rainfall thresholds related to, for example, the variable definition of triggering
rainfall events and non-consideration of ground conditions, we feel that the
generally used rainfall characteristics (intensity-duration or cumulative
rainfall event-duration) should be lumped into a single metric, thus
allowing space for introducing other parameters in the frequentist analysis.
This has also been suggested by Bogaard and Greco (2018), who advocate a
combination of meteorological and hydrological conditions into a
“trigger–cause” framework of threshold definition where short-term rainfall
intensity would represent the meteorological trigger. The main limitation
thereof is however the limited availability of information about the other
variable, namely the causative hydrological status of slopes. Here, we
propose an alternative concept where an AR function describes the progressive
build-up of the landslide trigger, and the set of determining causes, mostly
related to ground conditions, is accounted for by landslide susceptibility.
Said otherwise, we substitute for the trigger–cause framework proposed by
Bogaard and Greco (2018) the coupling of a dynamic meteorologically based
variable (trigger) and a static indicator of the spatially varying
predisposing ground conditions (cause). In this way, we obtain rainfall (AR)
thresholds as functions of susceptibility, which enables us to associate
threshold mapping with susceptibility maps. We show below how this new
approach furthermore includes the definition of a more meaningful AR function
and the use of subsets of the landslide data set in the threshold function
estimation. Analyses are performed in the R open-source software, release 3.4.3
(http://www.r-project.org, last access: 14 April 2019). The source code
is provided in the Supplement.
A new antecedent rainfall function
Though various expressions of the AR function have been proposed (see an
overview in Capparelli and Versace, 2011), most authors calculate AR for any
day i by convolving the time series of daily (or any other length)
rainfall rk with a filter function in the form of an exponential function
of time t, over a period empirically fixed to the preceding n d (e.g. Langbein et
al., 1990; Crozier, 1999; Glade et al., 2000; Melillo et al., 2018):
ARi=∑k=ii-ne-a⋅ti-tk⋅rk.
Such a function, which attempts to reflect the time-decaying effect of past
rainfall on the soil water status, has two weaknesses. The first one is that,
a being a constant, AR does not vary the time constant for decay with daily
rainfall amount. Yet, one may expect that, even though higher percentages
are drained for larger rainfall (Dunne and Dietrich, 1980), the quantity of
water infiltrating deeper and remaining in the soil from a large rainfall
will most frequently be higher than that from a small rainfall. Observation
that interception by the canopy, transpiration and evaporation rapidly
increase with diminishing rainfall intensity, especially in equatorial
areas, also supports this assumption (Schellekens et al., 2000). We take
this into account by also introducing daily rainfall rk in the filter
function and expressing AR as
ARi=∑k=ii-ne-a⋅ti-tkrkb⋅rk,
where the dimension of a is T-(1+b)Lb, with T here expressed in
days and L in millimetres. The b power of rk in the exponential function
allows us to
tune the gradient of decay time with daily rainfall. We empirically
determined that a=1.2 and b=1.2 provide decay curves that
realistically contrast the decay rate of different rainfall in the soil and
comply with the duration of their effect on soil moisture expected in the
WEAR (e.g. McGuire et al., 2002) (Fig. 3).
Decay curves for three daily rainfalls of 1, 10 and 25 mm according
to the expression of the exponential filter function in Eq. (2), with a=1.2
and b=1.2. The black dotted lines show that 0 %, 4.2 % and 34.7 % of
the respective original rainfall values are still contributing to the accumulated
antecedent rainfall function (Eq. 2) after 42 d (6 weeks).
As for the second weakness of the usual AR formulation, related to the length
of the period of time to be used for AR calculation, we stick so far to the
simplest solution of relying on expert knowledge to select it depending on
the regional environmental conditions. Two observations from the landslide
temporal distribution are taken into account for the choice of an
appropriate accumulation period: (1) landslide frequency progressively
increases during the long rainy season (hardly interrupted by a short drier
period centred on January) and peaks at its end in May, suggesting that the
length of the preceding period of wet conditions indeed controls landslide
frequency, and (2) the abruptly decreasing number of landslides as soon as
the dry season starts in June indicates that the period of useful AR should not
exceed a few weeks (Fig. 2). As a trade-off, we choose to calculate AR over a
period of 6 weeks, or 42 d. Using such a fairly long period is also
required because all landslide types are included in the data set, including
large-scale and deep rotational slope failures that often occur only after a
long rainy period (Zêzere et al., 2005; Fuhrmann et al., 2008; Robbins,
2016). A 6-week period is also consistent with studies having estimated
the soil water mean residence time to about 2 months for two watersheds in
the mid-Appalachians of central Pennsylvania, USA (McGuire et al., 2002),
and they showed that the best fit between creep rate on the Parkfield segment of
the San Andreas Fault (California, USA) and rainfall is obtained for a time
constant of about 1 month (Roeloffs, 2001), the latter being however
probably affected by specific conditions of infiltration in the damage zone
of the fault. Finally, we note in passing that another advantage of basing AR on
a long period of time is that the effect of rainfall events missed by the
satellite TMPA-RT data due to time gaps between satellite microwave
observations (Monsieurs et al., 2018b) is reduced.
Log–log plot of antecedent rain (mm) vs. ground susceptibility to
landsliding for the recorded landslides, with their associated sampling
probability: 0.67 at the reported landslide date; 0.17 at the days prior to and
after the reported landslide date. The black curve is the regression curve
obtained from the whole data set; the green and red curves are the AR thresholds
at 5 % and 10 % exceedance probability levels respectively, along with their
uncertainties shown as shaded areas.
Definition of AR thresholds for landslides
Owing to the variables we employ to construct the frequentist graph, the
rainfall thresholds will be given as AR values in function of susceptibility,
or landslide-predisposing ground factors. Hereby we avoid regionalizing the
input data according to individual variables such as lithology, land cover,
or topography and getting into problems of data subsetting in regions with
limited data (Peruccacci et al., 2012). However, the use of these variables
brings us to change the statistical way of threshold calculation, which
leads to conceptual improvements in the threshold definition and might also
be fruitfully applied to threshold estimation based on rainfall
intensity-duration or event-duration data.
The first steps of the analysis follow the procedure devised by Brunetti et
al. (2010) and Peruccacci et al. (2012). We first plot the landslide data
points in the 2-D AR susceptibility (S) space (Fig. 4). Only the 145 landslides
unrelated to mining conditions are considered, from which two landslides
associated with AR < 5 mm are further discarded, as they can barely be
said to have been triggered by rainfall in such conditions. While such low
AR values cannot be ascribed to errors in the TMPA-RT rainfall estimates, due
to the length of the period of AR calculation, they might result from gross
errors of landslide location or date identification, or possibly the
intrinsic evolution of hillslopes with time-dependent strength degradation
of the slope material resulting in slope failures without apparent trigger
(Dille et al., 2019). The retained 143 landslides occurred between 2001
and 2018 and are located in 58 different TMPA and susceptibility pixels.
The threshold function is then approached through a power-law regression of AR
against S (equivalent to a linear regression in the log–log space) in the form
AR=(α±Δα)⋅S(β±Δβ),
which appeared to work best.
The uncertainties Δα and Δβ associated with
the α and β scaling parameters are obtained using a bootstrap
statistical technique (Efron, 1979) where we generate 5000 series of randomly
selected events (with replacement) from the data set. The parameter values
and their uncertainties correspond to the mean and the standard deviation,
respectively, of their 5000 estimates. Peruccacci et al. (2012) applied this
technique in order to get the fit uncertainty on the estimated
parameters α and β. Here, we first produce a derived data set that must
allow the merge of the fit uncertainties with those upon the data themselves into
the error estimates provided by the bootstrap process. Data uncertainties
relate to the accuracy of landslide location and date identification. As the
mean location accuracy of 6.7 km is much better than the ∼28 km
pixel size, we decided to neglect this type of uncertainty. However, the
dating uncertainty is more of an issue. Uncertainty on the date most
frequently arises from landslides having occurred during the night. Beyond
the fact that reports do not always mention it, it is also generally unsure
whether any nightly landslide has been assigned to the day before or after
the night. In terms of uncertainty, this implies that a reported landslide
may have occurred randomly at any time over a 36 h period centred on the
reported day. To account for this randomness, we associate each landslide
with three weighted dates, the reported day having a weight of 24/36
(∼0.67) and the previous and next days each a weight of 6/36
(∼0.17) corresponding to the first half of the preceding
night and the second half of the following night, respectively. We then
simply expand, according to the day weights, the original 143-event data set
to a set of 858 derived events of the same 0.17 probability (in which, for each
landslide, day 0 is represented four times whereas only one occurrence of
days -1 and +1 is present). The date uncertainty is therefore incorporated
in the expanded data set and thus will also be included in the Δα and
Δβ uncertainty estimates from the
bootstrapping, with each bootstrap iteration randomly sampling 858 independent
events out of this data set, for a probability sum of (∼0.17)⋅858=143.
Note that a close but less practical alternative might
have consisted in using the intermediate data set of 429 (=3⋅143) weighted
events and requiring every bootstrap iteration to randomly sample a variable
number of events for a total sum of weights of 143. In order to satisfy the
requirement of the frequentist method for the largest possible data set, we
used the entire set of landslide events for the calibration of the new
method, leaving aside a validation of our results based on updates of the
WEAR data set or on landslide sets of neighbouring regions for the near future.
In each iteration of the bootstrap procedure, once the best fit parameters
of Eq. (3) have been obtained by least-square approximation, the
residuals of the regression are calculated and subsets of their largest
negative values are selected according to the exceedance probabilities of
the thresholds we want to calculate. Brunetti et al. (2010) use for this a
Gaussian fit to the probability density function of the population of
residuals and take the residual value δ that limits the lowest x %
of this fit to define the x % exceedance probability threshold as a line
parallel to the global regression line (in the log(AR)-log(S) space), i.e.
with an unchanged β parameter, and simply translated toward lower α
by a distance δ (see their Fig. 2). Here, we prefer to
put more weight on the distribution of the data points in the lower part of
the cloud of points as the most meaningful part of the data set for
threshold identification. Once the residuals have been computed, we take the
subset of their x % largest negative values and regress AR against S only for
the corresponding data points, obtaining a new regression line with not only
a lower α but also a modified β parameter that better follows
the lower limit of the cloud of points. Running through the middle of the
x % lowest data points, this new curve is thus taken as the threshold
curve for the (x/2) % exceedance probability. In this way, the whole data
set is used to calculate the trend that allows meaningful sampling of
subsets of low-AR points before the emphasis is put on the subsets to obtain
curves better reflecting the actual threshold information contained in the
data set. Note also that we select a subset of actual data points to
estimate the threshold, whereas the approach of Brunetti et al. (2010)
relies on a Gaussian approximation of the residual distribution.
AR threshold estimates
The range of AR values associated with the landslide events extends from
5.7 to 164.4 mm for landslides that occurred in areas displaying a range of
susceptibility (expressed as probabilities) from 0.38 to 0.97. As a first
result, the fit to the (AR, S) pairs of the whole set of landslide events was
expressed as (Fig. 4)
AR=(36.5±1.2)⋅S(-0.41±0.09),
showing a stable solution of the bootstrap with fairly small (date + fit)
uncertainties on α and β but a rather small β value
indicating a weak dependence of AR on S, confirmed by the very low
determination coefficient R2=0.02 even though the fitted
trend is significant (df = 856, r=0.14>rcrit (95 %) = 0.08).
We then selected two subsets of 10 % and 20 % of all data
points with the most negative residuals with respect to the above-calculated
trend in order to obtain the threshold curves for the 5 % and 10 %
probabilities of exceedance, respectively, on which power-law regressions yielded (Fig. 4)
5AR(5%)=(9.2±0.6)⋅S(-0.95±0.14),6AR(10%)=(13.1±0.7)⋅S(-0.66±0.15).
A 5 % exceedance probability, for instance, means that any landslide
occurring in the field has a 0.05 probability of being triggered by an
antecedent rainfall AR lower than that defined by the threshold curve, with
about weighted 5 % of the data points effectively lying below the curve. A
first observation is that the two threshold curves present significantly
higher β values than the previously calculated general trend, thus
enhancing the susceptibility-dependent gradient of AR threshold. The maximum
β value is obtained for the lowest threshold, which targets most
sharply the data points of interest, while larger subsets yield values
progressively closer to that of the general trend and thus less meaningful.
Again, the bootstrap-derived uncertainties are rather low, even though the
β uncertainties appear slightly higher than previously, owing to
smaller sample size and narrower range of represented S values. At the 5 %
exceedance probability, the AR threshold amounts to 22 and 9.2 mm for
susceptibilities of 0.4 and 1, respectively, making an AR difference of
∼13 mm between weakly and highly susceptible ground
conditions. At the same time, the goodness of fit of the regressions on the
subsets of data significantly increases, with an average R2=0.27
(r=0.52>rcrit (95 %) = 0.18) for the 5 % curve and
0.13 (r=0.36>rcrit (95 %) = 0.12) for the 10 %
curve and with quasi all single bootstrap iterations providing significant
α and β parameters for both thresholds, showing that there
exists a true correlation between susceptibility and rainfall threshold as
soon as one focuses on the data points really pointing to the minimum AR
required for landsliding to start. Though fairly small, these R2 values
have proved best among not very different linear, exponential and power-law
fits. Better coefficients are probably hampered mainly by inhomogeneities in
the subset data distribution within the susceptibility range, with very poor
information for S<0.7 (Fig. 4). This is also why the threshold
curves of Fig. 4 have not been extrapolated over the entire possible range
of susceptibility because the relation between susceptibility lower than 0.38
and triggering AR conditions is uncertain as long as it cannot be
empirically tested. At the continental scale, pixels with a
susceptibility ≤ 0.38 are ranked in any case as low- and very low-susceptibility areas
(Broeckx, oral communication). In the WEAR, the only landslides that were
recorded in areas with S<0.38 are all related to mining activity.
Discussion
Having proposed a new approach of the frequentist method of rainfall
threshold determination for landsliding, we have applied it with encouraging
results in the WEAR despite the difficulties of the context (limited size of
the landslide set, heterogeneity of the study area with respect to ground
conditions, coarse spatial resolution of the rainfall data, coarse temporal
resolution of the landslide inventory). We want now to review every new
element of the method and have a look at their advantages, implications and
limitations, especially from the point of view of the added value for the
landslide scientific community.
A key point of our approach is the introduction of ground
characteristics as one variable of the 2-D frequentist graph, with the aim of
directly associating the climatic trigger of landslides and the determining
(static) ground conditions in the threshold analysis. Indeed, in the current
way of treating this problem, separately examining the effect on thresholds
of various ground variables (e.g. lithology, topography) has the drawbacks
that (1) partitioning the study area on the basis of categories of any
variable may entail that some subsets of data become too small for a
significant analysis (Peruccacci et al., 2012) and (2) the combined effect
of the variables cannot be investigated. In order to put everything at once
in the frequentist graph, we thus needed to use two variables synthesizing
the climatic and ground characteristics. While the climatic
trigger issue was fixed by proposing a refined AR function (see point 2
below), we found that susceptibility to landsliding is an ideal single
indicator integrating all ground characteristics that significantly
determine the hillslope sensitivity to rainfall accumulation. Conversely, we could cite the fact that no single raw variable is
explicitly stated in this approach, and especially the soil water status of
slopes. But the main point is that using susceptibility values from
pre-existing studies implies knowing how they were estimated. This is not a
real issue for a regional study but becomes relevant if thresholds obtained
somewhere were to be compared with those of other regions where the
susceptibility data would have been calculated in a different way. In the
frequent case that susceptibility is modelled through logistic regression
for example, the probabilities that quantify susceptibility have no absolute
meaning, depending on the ratio between the landslide and no-landslide
sample sizes used in the modelling. Such information should thus always be
specified when susceptibility data are exploited for threshold determination.
AR functions are a common tool to lump daily rainfall and antecedent
rainfall into a single measure. In general, they either simply use cumulated
rainfall over empirically determined significant periods or take into
account the decaying effect of rainfall on the soil water status. With
respect to the intensity-duration or cumulated rainfall event-duration descriptions
of rain characteristics, they replace the difficulty of objectively defining
rainfall events by that of choosing a relevant period of meaningful
antecedent rainfall and, if a filter function is used, of parameterizing it.
The latter also offers a better proxy for the time-varying soil moisture
content (Hong et al., 2018; Melillo et al., 2018). However, no AR function has
so far considered a non-linear dependence of the decay time constant on daily
rainfall and, thus, soil wetting. Here, we have applied this idea by
introducing daily rainfall in the filter function of AR as a scaling factor of
the time constant (Eq. 2). In addition to the usual virtue of this AR function
type of assigning full weight to the rainfall of the current day, this
allows a better contrast between the intensity-dependent lasting effect of
different past rainfall, with more weight put on high-intensity rainfall.
Another facet of the AR issue is that we used remotely sensed rainfall data from
the TMPA-RT products (e.g. Hong et al., 2006; Robbins, 2016). In the WEAR
case, this was required anyway because the existing rain gauge network in
the area is neither dense nor installed soon enough to adequately cover
the study area and period. Moreover, using TMPA-RT data is advantageous in
that the information is spatially continuous (Rossi et al., 2017; Postance
et al., 2018) and freely available with a global coverage in near-real time
(Hong et al., 2006; Kirschbaum and Stanley, 2018). The rather coarse spatial
resolution of TMPA-RT data may also turn into an advantage because of their
higher spatial representativeness compared to gauge point observations of
very local meaning in areas with pronounced topography (Marra et al., 2017;
Monsieurs et al., 2018b). However, one has to cope with the typical bias of
SRE, which systematically underestimates rainfall amounts with respect to
ground observations (Brunetti et al., 2018; Monsieurs et al., 2018b). As
stated by Brunetti et al. (2018), this does not affect the performance of
threshold determination as long as the bias is spatially and temporally
homogeneous, which is to some extent the case in the WEAR. Based on the
estimation by Monsieurs et al. (2018b) that average SRE underestimation
amounts to ∼40 % in this area, we calculate an approximate
first-order correction of the AR thresholds. For instance the calculated
∼13 mm difference in 5 % AR threshold between low- and very
high-susceptibility areas of the WEAR becomes ∼21 mm after
correction for SRE underestimation, with corrected 5 % AR thresholds of
36.6 and 15.3 mm in areas with S=0.4 and 1, respectively. However,
Monsieurs et al. (2018b) also highlight how SRE underestimation increases
with rainfall intensity, reaching an average 80 % for daily
rainfall around 30 mm. This means that, even after correction, the
thresholds, in which high daily rainfall have the highest weight, are still
underestimated, and thus only indicative.
Another characteristic of our approach lies in the fashion of
determining thresholds by focusing on the data points with the lowest AR. Though
this is not quite new (Althuwaynee et al., 2015; Lainas et al., 2016; Segoni
et al., 2018), it is carried out here in a statistically rigorous manner so
as to exploit the part of the data most meaningful for threshold
appreciation. This methodological change was needed initially because,
contrary to the obvious strong relation between rainfall intensity or event
rainfall and duration (Guzzetti et al., 2007), the intuitively expected
relation between ground susceptibility and rainfall threshold was hardly
expressed in the data, with a bare 2 % of the variance of
landslide-triggering AR explained by susceptibility. Many reasons potentially
contribute to the noise that obscures such a relation among the (AR, S)
landslide data, relating to (i) probably chiefly the mixing of all types
of landslides in our data set (Flageollet et al., 1999; Sidle and Bogaard,
2016; Monsieurs et al., 2018a); (ii) the spatial, temporal and
rain-intensity-dependent inhomogeneity of TMPA-RT underestimation, with
local bias caused, for example, by high percentages of water areas within a pixel
or by topographic rainfall (Monsieurs et al., 2018b); (iii) determining
factors of landsliding important in the WEAR region but not accounted for in
the continental-scale prediction of susceptibility by Broeckx et al. (2018),
such as slope aspect, thickness of the weathering mantle, deforestation and
other human-related factors; (iv) the occurrence of landslides in less
susceptible areas of a pixel classified as highly susceptible; (v) the
probability of landslides having occurred in the very first hours of a day
with 24 h long high rainfall, inducing artificially swollen pre-landslide AR. By
contrast, focusing on subsets of landslides with low-AR residuals leads to a
better data fit and thresholds with higher β values more closely
reflecting the visually outstanding lower bound of the cloud of data points
and the AR threshold dependence on susceptibility. Working with independent
regressions on subsets strongly reduces the data noise and thus better
captures the true threshold shape. In this scheme, many of the actual
landslide events associated with AR much higher than the calculated threshold
might be viewed as “quasi false positives” that, for any reason, required
much more rainfall than predicted before at last occurring.
Another issue of this new method (and of most studies based on the
frequentist approach) is that it explicitly deals only with “false
negatives” (hereafter FNs, i.e. landslides having occurred for AR values
below the defined threshold) and thus evaluates only the type II error.
However, using thresholds that minimize this error, the amplitude of the
type I error (“false positives” – FPs, i.e. AR values above the threshold
that nevertheless did not lead to landsliding) is proportionately increased.
For example, for a randomly chosen pixel of the study area, which underwent
six landslide events during the 2000–2018 period, the 5 % and 10 %
thresholds involve no FN but cause 4715 (70 % of the AR time series of the
pixel) and 4242 (63 %) FPs, respectively. There are however several
reasons why these high numbers of FPs are neither reliable nor really
problematic. Firstly, it is important to note that the landslide data set
used for threshold calculation is far from complete and that a lot of
landslides occurring in remote areas are de facto unreported and may even
go easily unnoticed on satellite imagery if they occurred in regions with
fast vegetation regrowth or land reclamation or in places with poor temporal
satellite coverage. Most of these unreported events, if associated with
above-threshold AR values, imply many “false false positives”, i.e.
ignored true positives. Moreover, once one or several consecutive high daily
rainfalls have occurred and AR has jumped to values largely above the
threshold, causing one landslide event, the subsequent
construction-dependent slow return of the index to below-threshold values
frequently lasts for days or weeks without further landsliding. Said
otherwise, the tail of a period of above-threshold AR (after a landslide
event) is generally much longer than its head (before the event) and one can
barely call false positives all these days with high AR that follow the
event. The true rate of FPs is therefore much lower than it may seem at
first glance. Furthermore, even the large number of remaining FPs
should not necessarily be deemed an issue because it actually constitutes
the essence of early warning. By definition, flagging a day as hazardous
does not mean that a landslide will occur on that day but only that there is
an increased probability of an event and people should be prepared to face
it. Even more, when a landslide prediction turns into a true positive, with
a few landslides occurring in the pixel, most people living in this
∼28× 28 km area may nonetheless consider it a false
positive because the landslides have affected only a tiny part of the total
pixel area. Finally, it is also worth noting that, in contrast to most
other methods, this susceptibility-based approach allows the distribution of the
warnings (and possible false alarms) temporally and spatially, thus reducing
the number of warnings in any individual pixel.
A main requirement for a widely usable method of threshold calculation
is an automated threshold procedure, ideally made available online, in order
to enhance reproducibility of analysis and promote worldwide comparison of
results (Segoni et al., 2018). Steps towards this goal are achieved through
the following.
Using TMPA-RT data, a freely available, spatially homogeneous product covering the 50∘ N–S latitude range. This ensures that the results
of other regional analyses using the same data may be safely compared with
ours. The RT (Real-Time Version of the product has intentionally been
preferred to the more elaborated Research Version calibrated against
gauge-based Global Precipitation Climatology Centre rainfall data (Huffman et al., 2007) because the
inhomogeneous distribution of the reference gauges worldwide, and especially
in the tropics, introduces a spatially variable bias into the residual
underestimation of the latter data (Monsieurs et al., 2018b).
Reduction of the number of adjustable parameters in the definition of the climatic characteristics leading to landsliding. Here, only the constant
coefficient and the exponent on daily rainfall in the filter function have
to be fixed, along with the length of the period over which AR is calculated.
A dedicated statistical study of their best values (e.g. Stewart and
McDonnell, 1991; McGuire et al., 2002) might perhaps somewhat improve those
we empirically defined but, in any case, the AR time series calculated from
our empirical tests have shown that this formulation of AR is not much
sensitive to moderate changes in the parameter values.
Drawing attention onto the effect on the calculated thresholds of the way the used susceptibility data have been obtained. In particular, it is
possible to correct the threshold results for differences in the ratio
between the landslide and no-landslide sample sizes used with the widely
recognized logistic regression model of susceptibility.
Improving the evaluation of uncertainty. All sources of uncertainties
(here, there are uncertainties in date and fit but location uncertainty
may be treated as similar) are merged into a single error estimation in a bootstrap
procedure randomly taking from a weighted data set samples that have the
same size in terms of the sum of the weights (or probabilities) of the selected
events rather than in the number of events.
Providing our source code in the Supplement.
In addition to method development, this study has yielded new valuable regional
information in the form of AR threshold-susceptibility relations and a
threshold map at 0.25∘× 0.25∘ resolution
(Fig. 5). These results are immediately usable for early warning of landslide
hazard in the WEAR. Depending on the local susceptibility, thresholds at
5 % exceedance probability, which we consider the best operational
measure, range from AR =∼15.3 mm (corrected for SRE
underestimation) in the highest-susceptibility areas to 38.4 mm in the least
susceptible pixels (S=0.38) that recorded landslides during the
2001–2018 period. While this, as a matter of fact, is unquestionable, its
geomorphic meaning is hard to discuss because a single
AR value may cover very different 6-week-long time series of daily rainfall,
from more or less continuous moderate- to high-intensity rainfall over weeks
causing deep rotational landslides to very high-intensity rainfall of short
duration just before the occurrence of extended shallow landsliding and
debris flow. We also observed that a significant percentage
(∼40 %) of the landslide events did not occur on the day when then highest
rainfall was recorded but 1 or 2 d later. As it seems unlikely that
all of these landslides would have been wrongly dated, this fact might
betray a particular hydrological behaviour of slopes or be related to
specific landslide processes (Montgomery and William, 2002; Lollino et al.,
2006). Meaningful hypotheses about the interplay between slope physics and
rainfall characteristics in this setting will however require in-depth
analysis of the 6-week rainfall time series associated with the landslide
events. Meanwhile, although this is not straightforward, we can at least
attempt a comparison with the results of the many studies based on
intensity-duration (ID) or event-duration (ED) analysis of rain gauge data.
Extrapolating their ED or ID curves towards a duration of 42 d, many
published 5% exceedance probability thresholds fall in the range of 75–150 mm
over this time length in, for example, NW Italy (Melillo et al., 2018), NE Italy
(Marra et al., 2017), central Italy (Perucacci et al., 2012; Rossi et al.,
2017), Sicily (Gariano et al., 2015) and the NW USA (Seattle area, Chleborad et
al., 2006). Moreover, many landslides that actually occurred after rainfall
events of shorter duration were associated with lower cumulated rainfall.
The 75–150 mm range is thus an upper bound in these areas and we tentatively
suggest an average 50–75 mm cumulated rainfall as representative for
antecedent rainfall of landslide events. The reasons why these figures are
still significantly, though not irreducibly, higher than those we obtained
in the WEAR are as follows. On the one hand most of these studies (except that
of Melillo et al., 2018) do not apply a decay function to past rainfall.
On the other hand, this might also be related to the poor approximation of SRE
underestimation in the WEAR by Monsieurs et al. (2018b) due to the
non-linear dependence of underestimation on rainfall intensity and the weak
representativeness of limited gauge data of very local significance with
respect to ∼28 km × 28 km pixels. Another reason might
be related to the specific conditions of this tropical climate that could
influence the weathering conditions of the hillslope material and increase
the sensitivity to failure; this however remains hypothetical at this stage
and calls for further analysis (beyond the scope of this research).
Interestingly, in a tropical region similar to the WEAR, namely Papua New
Guinea, Robbins (2016) used cumulative rainfall to calculate thresholds
based on TMPA data and selected event durations. For an antecedent time
length of 40 d, she derived thresholds amounting to ∼25 and
∼175 mm for short- and long-duration landslide events,
respectively. Taking into account that no decay function was involved in her
antecedent rainfall calculation, these values are fully consistent with our
data, where short-duration events of shallow landsliding probably prevail in
determining the 5 % threshold of ∼9–22 mm (uncorrected for
SRE underestimation) in the WEAR whereas long-duration events triggering
larger and deeper landslides would make the bulk of noisy high-AR
(∼40–120 mm) data points. Likewise, 5 % thresholds
estimated in central Italy by Rossi et al. (2017) based on SRE data are of
the order of 30 mm of cumulative rainfall over an extrapolated duration of
42 d, again fairly similar to our uncorrected ∼9–22 mm 5 %
thresholds if we take account of the absence of decay function in their
calculations. We also note that our AR values are in the range of observed
values compiled by Bogaard and Greco (2018), while Guzzetti et al. (2007)
even reported extreme values as low as <10 mm. However, Bogaard and
Greco (2018) point to the difficulty of interpreting long-duration rainfall
measures in terms of average rainfall intensity and their trigger role for
shallow landslides and debris flows. To this extent, another added value of
our approach lies in the complex decay filter function used in AR, which mixes
triggering recent rain and predisposing rain of the past weeks in such a way
that the index is meaningful for both shallow and deep-seated landslides.
Although our results offer first insights into rainfall thresholds in the
WEAR, they still need refinement before becoming transposable into an
operational early warning system. Significant improvement is expected in the
near future from more regionally focused susceptibility maps and higher-resolution SREs coming soon with the IMERG product, which shows better
performance for rainfall detection (Gebregiorgis et al., 2018; Xu et al.,
2017). Together, higher-resolution, better-quality rainfall and
susceptibility data should produce a more robust correlation between both
variables for landslide events and, as a corollary, predictions should
involve fewer false positives. In parallel, the number of false positives
will have to be further reduced through appropriate filtering of
above-threshold AR data following landslide events. A larger database of
correctly described and dated landslide events will also allow threshold
validation and, once sufficiently large subsets of data are available
for particular landslide types, the calculation of adapted thresholds.
Antecedent rainfall (AR) threshold map (0.25∘ resolution) at
5 % exceedance probability (see Eq. 5). Depending on the local landslide
susceptibility (from Broeckx et al., 2018, Fig. 1) threshold values range from
AR = 9.5 mm in the highest-susceptibility areas (S=0.97) to AR = 23.1 mm
in the least susceptible pixels (S=0.38), having recorded landslides during
the 2001–2018 period. Numbers in the lakes are as follows. 1: Lake Albert; 2: Lake
Edward; 3: Lake Kivu; 4: Lake Tanganyika. Background hillshade SRTM (90 m).
Conclusion
In this study, we propose a new rainfall threshold approach fundamentally
different from previous research and based on the relation between
antecedent rainfall and landslide susceptibility through a modified
frequentist approach with bootstrapping. This method has the main advantage
of directly mappable susceptibility-dependent rainfall thresholds. The 6-week-long antecedent rainfall is calculated based on satellite rainfall estimates
from TMPA 3B42 RT. It uses an exponential filter function with a time
constant scaled by a power of daily rainfall accounting for the dependence
on rainfall intensity of the decaying effect of rain water in the soil.
Susceptibility data come from a study by Broeckx et al. (2018) based on
logistic regression and a continental-scale data set of landslides in
Africa. Using this method, we identify the first rainfall thresholds for
landsliding in the western branch of the East African Rift, based on a
landslide inventory of 143 landslide events over the 2001–2018 period. The
obtained AR thresholds are physically meaningful and range, without correction
for SRE underestimation, from 9.5 mm for the most susceptible areas of the
WEAR (S=0.97) to 23.1 mm in the least susceptible areas (S=0.38)
where landslides have been reported, for an exceedance probability of 5 %.
We conclude that the proposed new threshold approach forms an added value to
the landslide scientific community, while future improvements are expected
from applying the method to larger data sets and using satellite rainfall
estimates with higher spatial (and temporal) resolution and increased rain
detection efficiency.
Data availability
Data sets can be accessed at https://disc-beta.gsfc.nasa.gov/ (Huffman, 2016).
The supplement related to this article is available online at: https://doi.org/10.5194/nhess-19-775-2019-supplement.
Author contributions
AD conceived the new aspects of the method, with input
from EM and OD to its development. EM collected the data, implemented the source
code and made all calculations. AD, EM and OD contributed to the discussion
of the results. EM and AD jointly wrote the paper, with contribution from OD.
OD coordinated and designed this collaborative study in the frame of the RESIST
project.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We thank the Investigative Reporting Project Italy research institute and
their code developers for publically sharing open-source codes for landslide
hazard studies. Special thanks go to our partners at Centre de Recherche en
Sciences Naturelles de Lwiro (DR Congo) and Université Officielle de
Bukavu (DR Congo), who facilitated fieldwork in the study area and provided
information on landsliding. We are grateful for the fruitful discussions
with Thom Bogaard with regard to the antecedent rainfall equation and
constraints in threshold calibration. The authors acknowledge the NASA Goddard
Earth Sciences Data and Information Services Center for providing full
access to the precipitation data sets exploited in this study. We also thank
Jente Broeckx, who provided the landslide susceptibility model for Africa.
Financial support came from BELSPO for the RESIST (SR/00/305) and AfReSlide
(BR/121/A2/AfReSlide) research projects (http://resist.africamuseum.be/,
last access: 14 April 2019; http://afreslide.africamuseum.be/, last access:
14 April 2019), and an F.R.S. – FNRS PhD scholarship for Elise Monsieurs.
Review statement
This paper was edited by Mario Parise and reviewed by two anonymous referees.
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