HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus GmbHGöttingen, Germany10.5194/hess-18-4913-2014Derivation and evaluation of landslide-triggering thresholds by a Monte Carlo approachPeresD. J.djperes@dica.unict.ithttps://orcid.org/0000-0003-4387-6291CancelliereA.Department of Civil Engineering and Architecture, University of
Catania, Catania, ItalyD. J. Peres (djperes@dica.unict.it)8December20141812491349315February20147March201411August201420October2014This 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://www.hydrol-earth-syst-sci.net/18/4913/2014/hess-18-4913-2014.htmlThe full text article is available as a PDF file from https://www.hydrol-earth-syst-sci.net/18/4913/2014/hess-18-4913-2014.pdf
Assessment of landslide-triggering rainfall thresholds is useful for early
warning in prone areas.
In this paper, it is shown how stochastic rainfall models and hydrological
and slope stability physically based models can be advantageously combined in
a Monte Carlo simulation framework to generate virtually unlimited-length
synthetic rainfall and related slope stability factor of safety data,
exploiting the information contained in observed rainfall records and
field-measurements of soil hydraulic and geotechnical parameters. The
synthetic data set, dichotomized in triggering and non-triggering rainfall
events, is analyzed by receiver operating characteristics (ROC) analysis to
derive stochastic-input physically based thresholds that optimize the
trade-off between correct and wrong predictions. Moreover, the specific
modeling framework implemented in this work, based on hourly analysis,
enables one to analyze the uncertainty related to variability of rainfall
intensity within events and to past rainfall (antecedent rainfall). A
specific focus is dedicated to the widely used power-law rainfall
intensity–duration (I–D) thresholds.
Results indicate that variability of intensity during rainfall events
influences significantly rainfall intensity and duration associated with
landslide triggering. Remarkably, when a time-variable rainfall-rate event is
considered, the simulated triggering points may be separated with a very good
approximation from the non-triggering ones by a I–D power-law equation,
while a representation of rainfall as constant–intensity hyetographs
globally leads to non-conservative results. This indicates that the I–D
power-law equation is adequate to represent the triggering part due to
transient infiltration produced by rainfall events of variable intensity and
thus gives a physically based justification for this widely used threshold
form, which provides results that are valid when landslide occurrence is
mostly due to that part. These conditions are more likely to occur in
hillslopes of low specific upslope contributing area, relatively high
hydraulic conductivity and high critical wetness ratio. Otherwise, rainfall
time history occurring before single rainfall events influences landslide
triggering, determining whether a threshold based only on rainfall intensity
and duration may be sufficient or it needs to be improved by the introduction
of antecedent rainfall variables. Further analyses show that predictability
of landslides decreases with soil depth, critical wetness ratio and the
increase of vertical basal drainage (leakage) that occurs in the presence of
a fractured bedrock.
Introduction
Rainfall thresholds indicating landslide triggering are useful for the
development of early warning systems in prone areas cf.,
e.g.,. Commonly, such
thresholds are derived by the analysis of historical rainfall and landslide
data, and identified by drawing a lower-bound envelope curve of the
triggering event characteristics
e.g.,
or by enhanced methods which consider curves associated with a given
frequency of non-exceedance by triggering events cf.
. A review by
indicated the prevailing use in literature of so-called power-law rainfall intensity–duration (I–D)
thresholds, which are of the form I=a1Da2, where D is rain
duration to triggering and I is rain intensity I=W/D, W being rainfall
accumulated over duration D. The a1 and a2 parameters have been
derived by different researchers, for specific sites, regions or the whole
globe.
Many factors of uncertainty affect the reliability of empirical thresholds,
such as rainfall temporal and spatial variability, uncertain knowledge of the
triggering instants, simplicity of threshold equation that does not include
all control variables and statistical issues as well
. Nonetheless, it can be argued that most of the
uncertainty stems from the availability and quality of the data used to
derive the thresholds .
In fact, adequate historical data on landslides and simultaneous rainfall are
in most cases available for a relatively short period, which may not be
sufficiently significant from a statistical point of view. Moreover, the
identification of the triggering instant is in many cases significantly
uncertain and landslide archives are seldom complete (i.e., all landslide
events occurred in the historical period are not known). This has a direct
consequence on threshold derivation, because critical (where critical here
means corresponding to landslide triggering) duration D, assumed as the
time interval from rainfall event start and the triggering instant, cannot be
computed accurately. Another key factor is the criterion used for rainfall
identification, and in particular how the beginning of a rainfall event is
identified. Many authors either do not specify the criteria used for rainfall
identification or apply qualitative criteria, and indeed only few works in
literature
explicitly addressed this problem. This makes thresholds subjective and
impairs comparisons of results obtained by different researchers, as in
analyzing the data the criterion may have been modified from one rainfall
event to another. Another point is that in many countries automatic rain
gauge networks have been installed only quite recently, and one has to rely
on analysis based on rainfall records at the daily aggregation timescale
cf.and references therein. Since many
landslides, especially the most devastating shallow rapidly moving ones, may
be triggered by rainfall events of a few hours cf., e.g.,,
use of daily rainfall for threshold derivation in these cases is quite
questionable.
Apart from data quality issues, it can be pointed out that use of
characteristic variables for the representation of rainfall events, and in
particular of their intensity and duration, introduces an intrinsic
uncertainty factor, because these variables may not be adequate to represent
all the rainfall characteristics that affect landslide triggering. In fact,
rainfall events represented by the same values of duration and intensity may
correspond to totally different event time histories (hyetographs) that thus
may or may not result in triggering. proposed an
empirical method based on the use of convolution between rainfall time series
and a filter function, which attempts to overcome this uncertainty. Also, use
of solely the duration and intensity pair (D,I) in threshold formulation
implies that the effect of initial wetness on triggering rainfall is
neglected. Regarding this issue, several authors have added to D and I
antecedent rainfall as a control parameter, though the empirical analyses
have not yet provided unequivocal indications on the role of antecedent
rainfall and different researchers used diverse temporal horizons for its
computation .
Another important point is that many thresholds have been derived by
analyzing triggering events only and thus neglecting the non-triggering ones.
This may lead to an underestimation of the triggering conditions, i.e., to
thresholds that implemented in a early warning system tend to produce an
unacceptable degree of false alarms, causing populations to no longer rely on
early warnings (so-called cry-wolf effect; cf., e.g.,
). In fact, thresholds always should be provided with
a measure of their reliability. To this end, proposed
Bayesian probabilistic analysis to evaluate landslide-triggering thresholds
in the presence of uncertainty. Receiver operating characteristics (ROC) analysis cf., e.g.,,
based on the analysis of correct and wrong predictions, may be advantageously
applied as well cf., e.g.,.
Alternatively to empirical models, physically based models that couple
hydrological and slope stability analysis have been proposed to
assess landslide triggering by rainfall, with the advantage that they take
explicitly into account the meteorological, hydrological and geomechanical
processes and variables that determine landslide triggering. From such models
physically based thresholds may be derived cf.,
e.g.,. Such thresholds generally deviate from
a straight line in the log(D)–log(I) plane; this casts some doubts on
the use of the power-law as a proper functional form for deriving rainfall
thresholds. In other words, because such thresholds were derived from a
physically based model, this may be interpreted as an evidence that the use
of the power-law form is not supported from a physically based standpoint.
Nevertheless, in such studies the meteorological aspects were analyzed in a
simplistic way, because the thresholds do not consider variability of
rainfall intensity during events and the initial conditions are not computed
as a function of rainfall time history preceding the current event.
In spite of the limitations that we have put into evidence above, I–D rainfall
thresholds are widely applied for landslide early warning systems. Perhaps
their success is due to the fact that simple forms of the threshold are more
easily understood by stakeholders and decision makers than the more complex,
albeit more accurate, physically based models.
In this paper, a Monte Carlo-based methodology to derive and evaluate
rainfall landslide-triggering thresholds is proposed, which makes use of an
existing body of stochastic and physically based models. The approach
combines research findings in the fields of rainfall and landslide
hydrological modeling to provide an output that can be easily implemented in
a early warning system, i.e., a landslide-triggering threshold, based on
rainfall monitoring, of the same type that is commonly derived by direct
empirical analysis of observed rainfall and landslide data. In particular,
from the Monte Carlo simulations synthetic rainfall series are generated by
a stochastic model and corresponding triggering/non-triggering conditions are
identified by an hydrological and slope stability model. The generated data
set is then analyzed to derive and evaluate I–D thresholds that take
into account the variability of both rainfall intensity within events and
initial conditions determined by past rainfall, as well as
triggering/non-triggering events to measure uncertainty by ROC analysis.
Furthermore, the derived stochastic-input physically based thresholds are
compared with the constant-intensity physically based thresholds, which
result from the simplistic assumption mentioned above (uniform hyetographs
and prefixed initial conditions) in order to assess the effect on landslide
triggering of rain intensity variability during events and variable initial
conditions, computed as dependent by past rainfall time history. This
analysis is related to the one by , in which the effect
of rain intensity variability within events is studied by considering
beta-shaped hyetographs inputs to the model of for
derivation of hillslope response. Nevertheless, in their work, the
variability of initial conditions as dependent from antecedent rainfall has
not been considered because the steady-state asymptotic solution of
is utilized for computation of initial conditions.
From their study they conclude that beta-shaped non-uniform hyetographs have
a stronger destabilizing effect than uniform hyetographs of the same volume,
since the associated return period of slope instability resulted higher in
this last case. In this study we instead use hyetographs generated by
a Neyman–Scott rectangular pulses (NSRP) stochastic model
and account for
variability of initial conditions using a water table recession model to
derive the initial water table height from the response to rainfall events
preceding the current one, based on a linear reservoir mass-conservation
equation with similar assumptions adopted by . The transient
response to rainfall events is computed by a model based on the transient
rainfall infiltration and grid-based regional slope-stability (TRIGRS)
program . An application of the proposed methodology is
carried out to the highly landslide-prone area of the Peloritani mountains,
north-eastern Sicily, Italy. A sensitivity analysis on some of the most
important control variables is carried out to analyze their effect on
landslide-triggering thresholds and the associated uncertainty.
Monte Carlo synthetic data generation
The Monte Carlo simulation procedure for synthetic rainfall–landslide data
generation consists of the following steps:
A stochastic rainfall model, calibrated on observations at
a selected site, is used to generate a 1000-year long hourly
rainfall time series. In particular we use a NSRP model (see Appendix )
The synthetic rainfall time series is pre-processed in order to
identify rainfall events and their inter-arrival durations. In
particular, when two wet spells are separated by a dry time interval
less than Δtmin, these are considered to belong to
the same rainfall event; otherwise two separate rainfall events are
considered. Details on the choice of the Δtmin simulation parameter are given at the end of this section
Some of the generated rainfall events are removed from the analysis because, according to
the hydrological model, they will produce no significant variation of pressure
head distribution, as their instantaneous (hourly) intensity is too low. In
particular the events having maximum intensity less than imin are
removed from the analysis. We assume imin equal to the leakage flux
limit, given by cdKS(1-cos2δ), cd being
the vertical leakage ratio, KS the saturated hydraulic conductivity
and δ the slope of the hillslope (see Appendix )
Application of previous steps leads to the generation of NRE individual rainfall events
An initial value of the water table height is fixed to start
simulations of the hydrological response for the whole rainfall time
series. For the analyzed case-study area and many similar cases, it
may be assumed that at the beginning of each hydrological year the
water table is at the basal boundary, because an almost totally dry
season had come prior to it (this may be a slightly conservative assumption, since pressure head at
the soil–bedrock interface may assume negative values after a long dry season). As this is
valid also for the first year, simulation for first event is conducted considering the water table
at the soil–bedrock interface
The response to the generated rainfall events is simulated by the models described in
Appendix and the following procedure to be applied for
i=1,2,…,NRE:
Response in terms of pressure head ψ within rainfall events
is computed using the TRIGRS model (see Sect. ).
As pressure head rise may continue after the end of rainfall, the TRIGRS transient response
simulation interval is prolonged Δta=Δtmin-1 h
after the ending time tend,i of rainfall events.
The instant tf,i=max(tend,i,tmax,i) is looked for, where tmax being the time
instant at which maximum transient pressure head occurs. It follows
that the final response to rainfall event i, in terms of water
table height, is ψ(dLZ,tf,i)/β,
where β=cos2δ (slope parallel flow is assumed), and
dLZ is the soil depth. Moreover, the time interval
Δti+1=ti+1(in)-tf,i is computed, with
ti+1(in) being the instant at which rainfall event i+1
begins.
The water table height at the beginning of rainfall event i+1
is computed by a sub-horizontal drainage model (see Sect. ) which uses
ψ(dLZ,tf,i)/β and Δti+1.
The result is a series of maximum pressure head, or minimum
factor of safety FS responses (computed by infinite-slope stability analysis; see
Sect. ), corresponding to the NRE rainfall events. The
rainfall and the FS series are together analyzed to derive and evaluate landslide-triggering thresholds via a ROC-based approach (see Sect. ).
Regarding the choice of the inter-event time Δtmin – an
issue that is the focus of some works in literature
e.g., – we have followed an approach
analogous to that used by and
– for which the inter-event time may be assumed as
the minimum time needed to avoid overlapping of the response produced by two
subsequent rainfall events. To this end we considered that the temporal peak
of pressure head due to an individual rainfall event may be reached, as
mentioned above, at an instant significantly after rainfall ceases. Hence, a
criterion for selecting the inter-event time has been that of choosing a
value that approximates the dry time interval that contains the peak pressure
head response relatively to all the NRE simulated rainfall events.
In our case, from preliminary simulations a Δtmin=24h appeared suitable for the hydraulic and geotechnical soil
properties which are considered in this work (see Sect. ).
Figure summarizes the main steps of the described Monte
Carlo methodology.
Scheme of the Monte Carlo methodology for derivation of landslide-triggering thresholds. Text in red indicates briefly the input required,
while text in black indicates models or modeling phases.
Threshold derivation and evaluationTriggering and non-triggering rainfall identification
For a hillslope of given properties, Monte Carlo simulations lead to a series
of time instants at which the factor of safety drops below the value of 1 (a
FS = 1 down crossing).
A triggering rainfall may be associated with each down crossing,
though it is noteworthy to point out that some uncertainty is present in
the link between the actual failure of the slope and its theoretical
instability. Nevertheless, following several works in literature
e.g.,
this uncertainty has been not taken into account here, though it may affect at a certain degree the
way that rainfall events are classified as triggering and
non-triggering and the subsequent ROC-based analysis (Sect. ).
We investigate thresholds that are based on rainfall intensity I and
duration D.
Various procedures have been used to identify and compute I and D,
as discussed in the “Introduction” section of this paper and by .
From a general standpoint, this procedure may be disconnected from the way
event separation has been performed to compute the triggering instants with
the methods described in Sect. .
Nevertheless, for consistency with the event separation criterion that is
considered in the Monte Carlo simulation methodology, it is preferable to base the
procedure for identification of triggering and non-triggering events on
the same inter-event time Δtmin used in Monte Carlo simulations.
Based on the considerations above, we adopt the following procedure for
triggering and non-triggering rainfall identification. First, rainfall events
are separated when their dry inter-arrival is longer than Δtmin. Rainfall events then have a total duration Dtot
and mean intensity Itot=Wtot/Dtot, where
Wtot is the total event cumulative rainfall. For a triggering
event, triggering may occur before or after the end of the rainfall event. In
the first case, the critical duration DCR is the time interval
that starts at the beginning of the rainfall event and finishes at the
triggering instant, and critical intensity is given by
ICR=WCR/DCR, where WCR is
rainfall accumulated over duration DCR. In the second case it is
instead characterized with Dtot and Itot. Moreover,
the P0 events that have at their beginning a water table height hi≥dLZζCR, ζCR being the critical wetness
ratio (corresponding to FS≤1; see Eq. ), are
removed from the analysis, as the triggering is due to the preceding events,
which have already been included in the set of triggering points.
Non-triggering events are represented by Dtot and
Itot.
In our case (see Sect. ) Δtmin=24h. It
is worthwhile to write that with this choice the procedure of triggering
rainfall identification happens to be equal to the one that was applied by
in analyzing empirical data (observed landslides instead
of simulated).
Uncertainty and ROC-based evaluation and optimization of thresholds
The analysis of the Monte Carlo simulations produces two sets: the set of
positives P, i.e., of triggering events, and the set of negatives N,
i.e.,
of non-triggering events. These sets may be represented as scatterplots in
a double-logarithmic (D,I) plot, and in general there is a region where
both sets are present – lets say, an intersection region P∩N. In our
framework this is due to two separate factors:
To a given (D,I) pair there may be corresponding diverse variable
NSRP-simulated hyetographs, because I is the mean intensity
I=W/D (rain-intensity variability within events)
To a given (D,I) pair there may be corresponding diverse initial
conditions (variability of initial conditions, due to variability of
rainfall before the current event).
The first uncertainty factor is analyzed by letting the initial water table
height hi=0 for i=1,…NRE in performing the Monte Carlo
experiments (indicated in the ensuing text as ψ0=0 case; see also
Appendix ). To investigate the second uncertainty factor,
those experiments are compared with the complete ones, where the effect of
initial conditions depending on past rainfall time history is taken into
account by Eq. (), and considering different levels of
memory, by varying the parameters that appear in the water table recession
constant τM (again see Eq. ).
Moreover, we compare the results with thresholds derived from the model by
assuming uniform hyetographs as input and a prefixed initial condition
(constant-intensity physically based thresholds) cf.,
e.g.,. In this case a univocal
triggering threshold exists I=f(D), for given hillslope properties,
and the two factors of uncertainty illustrated above are not taken into account.
Due to the analytical complexity of the TRIGRS (see
Sect. ) unsaturated model it is possible to determine
these thresholds only numerically (not in closed form). Hence, we have
derived these thresholds by simulation of infiltration and slope stability
using constant-intensity hyetographs in the (D,I) domain discretized at
a sufficient level, and searching the triggering curve by interpolation of
the results. In doing this we have assumed an initial water table height at
the soil–bedrock interface in order to properly compare results with the
stochastic-input physically based thresholds of the ψ0=0 case.
Confusion matrix for the possible success and failure cases of
a warning process based on a landslide-triggering threshold I=f(D).
As a consequence of the presence of the region P∩N, when a triggering
rainfall threshold is fixed – e.g., a power-law one I=a1Da2 – the
four cases of true positives, true negatives (correct predictions), false
positives and false negatives (wrong predictions) can occur, as illustrated
in Table . In general, to each pair of parameters
a1 and a2 corresponds a prediction performance that may be measured by
indices based on the number of occurrences in the four cases, denoted
respectively as TP (true positive), FN (false negative), TN (true negative)
and FP (false positive) (or ROC-based indices). In order to derive optimal
thresholds one may maximize an objective function based on these quantities.
Several indexes do exist and their advantages and drawbacks have been
discussed by different researchers
cf..
Among the various possibilities, we consider for threshold evaluation the use
of the true skill statistic (TSS) (also known as Hanssen–Kuipers
discriminant; ), which was introduced by
, and is
given by the difference between the true positive rate TPR=TPP=TPTP+FN (also known as sensitivity or
hit rate, or recall or probability of detection) and the false
positive rate FPR=FPN=FPTN+FP
(also known as probability of false detection or 1 – specificity):
TSS=TPR-FPR.
It is TSS=0 for TPR=FPR (random guess) and
TSS=1 for a perfect prediction (TPR=1 and FPR=0). In fact this index TSS is bounded in the interval [-1,1],
but negative values are fictitious as an inversion of the triggering
threshold use brings TSS to its absolute value, which is always in
the interval [0,1] (i.e., saying that values below the threshold trigger
landslides and, vice versa, values above the threshold do not trigger
landslides). Different weights may be given to the TP, TN, FP and FN, as
pointed out by , in order to account for the fact that a FN
is more harmful than a FP see also. Since data
on the possible weights to assume are usually scarce, in this paper we prefer
to proceed in a more simple and standard manner, where this different
weighting is not considered.
We estimate the best performing power-law threshold I=a1Da2 as the
one that gives the maximum value of TSS=TSS(a1,a2).
At the same time the simulation-optimization methodology enables one to evaluate
the use of I–D power-law thresholds, as the value of the objective function is
a measure of the maximum performances that can be expected from the adopted
functional form for the threshold, and thus a measure of its validity.
It is noteworthy to highlight that the real uncertainty associated with this
threshold generally yields different – likely worse – performances of that
assessed here, since uncertainty factors are more than the ones related to
the stochastic nature of rainfall listed at the beginning of this section.
Map showing the location of landslide-prone study area of Peloritani
mountains, Italy. The area may be subdivided into seven catchments as
highlighted in the map. Relief is shown by a 5 m×5m
resolution digital terrain model based on lidar measurements in the year
2007. Location of Fiumedinisi raingauge is shown as well. The inner box
contains the area surrounding the Giampilieri municipality, where a
widespread rapidly moving landslide event killed 37 people on 1 October 2009
(see Fig. 3).
Investigated area and dataGeological setting and soil properties
An application of the described methodology is carried out to the Peloritani
mountains near the Ionian coastal area,
in north-eastern Sicily, Italy (Fig. ). The mountain ridge extends longitudinally
for about 50 km, with a SW–NE orientation,
resulting in peaks higher than 1200 m. This area can be subdivided into
seven basins: (1) minor basins between Alcantara and
Agrò (70.0 km2), (2) Agrò basin (81.8 km2), (3) Savoca
basin (44.2 km2), (4) Pagliara basin (27.1 km2), (5) minor basins
between Pagliara and Fiumedinisi (27.1 km2), (6) Fiumedinisi basin
(49 km2) and (7) minor basins between Fiumedinisi and Cape Peloro
(172.9 km2).
The study area belongs geologically to the Calabrian arc and
represents the inner
chain of the Apennin–Maghrebian mountain belt; moreover, since the middle
Pleistocene, the entire Calabrian arc has undergone strong tectonic uplift.
Present-day activity is testified by the strong historical seismicity
and references therein. In the area outcropping
lithologies consist of (1) phyllites of the Mandanici
units; (2) paragneiss and mica schists of the Mela units;
and (3) gneiss and paragneiss metamorphic rocks (Aspromonte units).
Further information on the geological setting, is given in , and .
In the last decade, this area has been hit by highly damaging diffused
shallow landslides. Precisely, widespread landslide events occurred in this
area on (I) 15 September 2006 (areas 4, 5, 6 and 7), (II) 25 October
2007 (area 7), (III) 24 September 2009 (areas 1, 2, 3, 4, 5) and
(IV) 1 October 2009 (area 7). The associated areas indicated in parentheses
have been derived from newspapers archives
(cf., e.g., http://gazzettadelsud.virtualnewspaper.it/gdsstorico/), which
also present further information on the events. Among these events, the one
that
occurred on 1 October 2009 was the most severe. On that date, landslides were
triggered by a rainfall event of more that 220 mm
in less than 7 h, causing 37 deaths and innumerable injured people, most of them in the municipality of Giampilieri
.
(a) Enlargement of inner box of the map of
Fig. showing slides occurred on 1 October 2009. Polygons
were derived from slide (red) areas reported in map of Civil Protection
Department of the Sicilian Region (source:
http://www.regione.sicilia.it/presidenza/protezionecivile/documenti/rischi/r_idrogeologico/documenti/20091001_CartaDissesti.jpg).
In all, 643 slides, occupying a total area of 2.30 km2, are present in the
map. (b) Statistical distribution, within the slide areas, of slope
δ and of upslope specific contributing area A/B.
A map of the landslides which occurred on 1 October 2009, derived from
interpretation of orthophotos, is available from website
http://www.regione.sicilia.it/presidenza/protezionecivile/documenti/rischi/r_idrogeologico/documenti/20091001_CartaDissesti.jpg.
This map presents in red the slide/erosion areas and in orange the
propagation/deposition areas. Figure shows the
slides derived from red areas of that map. The analysis of the specific
upslope contributing area A/B (ratio between the upslope draining area A
and the contour length B; see Appendix B) and the slope δ within the
slide areas, based on a pre-event DTM at a 5 m resolution, shows that the
most populated class of A/B is centered on the value of 10 m, while the
mean slope within the range of theoretical potentially unstable slopes 29∘≤δ≤47∘ is slightly lower than 40∘.
Moreover, the values of A/B=10 m and δ=40∘ correspond to
a portion of the Peloritani mountains for which it starts to be worthy of
issuing landslide early warnings. Hence these values may be adopted for the
successive derivation of a threshold for the area (see
Sect. ).
Parameters of the NSRP rainfall model resulting from calibration on
Fiumedinisi rainfall data, for the four homogeneous rainy seasons (the
Weibull shape parameter has been fixed to b=0.6).
Varied soil properties considered for sensitivity analysis.
KSD0cddLZτM(A/B=10m)[m s-1][m2 s-1][-][m][days]1×10-5 (36 mm h-1)2.5×10-50.11, 1.5, 25.52×10-5 (72 mm h-1)5×10-50.05, 0.1, 0.21, 1.5, 22.73×10-5 (108 mm h-1)7.5×10-50.11, 1.5, 21.8
Core samples collected in the area indicate the presence of surficial
debris material dLZ=2m deep, covering a fractured bedrock
strata.
The debris cover consists of a sandy loam
with a significant proportion of gravel (up to 50 %) which corresponds to
a gravelly sandy loam according to USDA soil classification.
The assumption of a leaking basal boundary, characterized by a given cd
ratio (see Sect. ) is realistic, given the fractured bedrock strata.
We assume that the hydraulic and geotechnical properties of
Table may represent the natural heterogeneity within the
study area.
Spatial variability of each of the soil properties could be included in our
model simulations. Nonetheless, detailed information on how the properties
are distributed spatially is unavailable. Hence we preferred to carry out a
sensitivity analysis, by varying the hydraulic conductivity KS, the
leakage ratio cd and the soil depth dLZ according to
Table , and the critical wetness ratio in the range 0<ζCR<1. To proceed in this way enables one to better analyze
the way model results are influenced by these variables rather than assuming
that they are distributed spatially with interpolating laws of difficult
validation. Since slope mainly affects slope stability
(Eq. ) rather than the infiltration process, variation of
slope is indirectly taken into account by variation of ζCR. It
is noteworthy to mention that an alternative approach may be to consider
model parameters generated according to a probability distribution, as
proposed by the TRIGRS-P modification of the TRIGRS code, developed by
.
Rainfall data and NSRP model calibration
Climate in the Peloritani area is Mediterranean with with hot and dry
summers, and precipitation – mainly convective – falling mostly in the
period from October to January.
For calibration of the NSRP model, the rainfall series measured at the
Fiumedinisi rain gauge from 21 February 2002 to 9 February 2011 (almost
9 years) has been used (see Fig. ). Based on
a preliminary analysis of monthly statistics, six homogeneous rainfall
seasons have been identified: (1) September and October, (2) November,
(3) December, (4) January–March, (5) April and (6) May–August (see
Fig. and its caption). Separate sets of parameters of the
NSRP model have been determined for each one of the four rainy seasons (in
total 5⋅4=20 parameter values), while the last two seasons have
been considered to have negligible rainfall. The Weibull shape parameter b
has been fixed to 0.6 for all seasons, based on different trials.
Parameters obtained from calibration are shown in Table .
Moments for each month for Fiumedinisi SIAS hourly rainfall data. In
particular μ denotes the mean, γ the variance, ρ(1,1) the
linear autocorrelation coefficient at lag = 1, ϕ the probability of a
dry interval, ϕDD the probability that a given interval is dry
after another dry one, ϕWW the probability that a given interval
is wet after another wet one. These moments have been used in calibration of
the NSRP model via the method of moments. It can be seen that there are low
differences of most of the moments within the following groups of months:
Sept–Oct, Nov, Dec, Jan–Mar, Apr, May–Aug. A separate set of NSRP model
parameters was calibrated for each of the first four of these seasons, while
the period April–August has been neglected from the successive analyses
because precipitation rates are so low that it is very unlikely that a
triggering event occurs in such period.
From the assumed inter-event time Δtmin=24h and soil properties of Table the resulting number of
rainfall events is NRE=19 826 (in average 19.83 events per year).
This number derives from the initial 28 751
events from which the events with hourly intensities below
imin=cdKs(1-cos2δ)=2.975mmh-1 were cut. These values are statistically comparable
to the ones on the observed series (19.18 events/year from 28.91 events/year
before the cut of under-leakage events).
Derivation of thresholds from ROC optimization of Monte Carlo
simulations. Red points represent triggering simulated rainfall,
while green ones represent the non-triggering. The best power-law
stochastic-input physically based thresholds (S) are derived by maximization of the
TSS ROC-based index. The constant-intensity input physically based threshold
(C) is determined considering the response to uniform hyetographs and water
table initially at the basal boundary. (a) zero memory case
ψ0=0, (b)A/B=10m (threshold derived by
for Sicily is shown as well, thin dotted line, for
comparison with the one derived) and (c)A/B=20m.
Validation of threshold-derivation procedure with observed
rainfall events. Red lines indicate mean
intensity I(D)=W(D)/D time histories that exceed the derived
threshold. Green lines represent observed events that do not exceed
the threshold. Threshold is exceeded for all and only the four dates in which
landslides occurred in the Peloritani area: (I) 15 September 2006, (II) 25 October 2007,
(III) 24 September 2009 and (IV) 1 October 2009.
Results and discussionDerivation and evaluation of rainfall thresholds
In Fig. the scatterplots of triggering and non-triggering
events in the log(D)–log(I) plane, derived from analysis of Monte Carlo
simulations, are shown for the ψ0=0 case and for specific upslope
contributing areas A/B= 10, 20 m (τM=2.75,5.49days). Related results are also shown in
Table . In the figure, red points represent triggering
rainfall events, or the set of positives P, while green points represent
the non-triggering ones, or the set of negatives N.
ROC-based indices for the derived best power-law stochastic-input
physically based thresholds
(S) and comparison with constant-intensity input physically based ones (C).
Optimal thresholds have been derived by maximization of the TSS
index (see Eq. ), preliminarily by considering both the
power-law coefficient a1 and exponent a2 as variable parameters.
Inspection of the results revealed minimal changes of the exponent a2 with
changing ratio A/B, and so a second optimization has been carried out only
with reference to the a1 parameter, fixing the exponent a2 to its mean
value of a2=-0.8. Fixing the exponents forces the different thresholds
corresponding to different A/B ratios to be parallel and therefore to not
intersect each other; this is somehow consistent with the fact that as the
A/B increases, landslides are generally more likely to occur for less
severe rainfall events because of increased past-rainfall memory.
From the case of ψ0=0, i.e., of an initial water table
at the soil–bedrock interface
for all events (Fig. a),
where simulated uncertainty of triggering is due only to the variation
of rain intensity within events,
it is seen that in this case the region
in which triggering and non-triggering events coexist is quite narrow;
moreover, a power-law relation between I and D dichotomizes well
between triggering and non-triggering conditions. In
fact, the optimal power-law threshold in this case has a reliability
of TSS=0.991, practically equal to the ideal value of 1.
Additional insights on the effect of variability of rainfall intensity within
events
may be derived comparing the scatterplots for this ψ0=0 case with the
constant-intensity
physically based threshold (also determined considering ψ0=0) represented in the
plots of Fig. as a dashed black line. Figure a reveals that
the deterministic threshold approximates the lower envelope curve of
critical events (red dots) for short durations – that, for the analyzed
data, corresponds to durations less than about 12 h. For higher durations,
this is no longer true and variable–intensity hyetographs start to have
a higher destabilizing effect than the constant-intensity ones of same
rainfall volume. The variability of rainfall intensity within events leads to
a deviation from the deterministic line of the triggering NSRP rainfall-event
points, making the scatter of triggering points more similar to a straight
line than to the curved deterministic threshold. This behavior is essentially
due to the presence of the leakage term ql=min{cdKs(1-cos2δ),q(du,t)}, whose effect is
stronger for uniform hyetographs than for variable ones, since in the former
there are no peaks of intensity. In particular, a uniform hyetograph produces
no water table rise if intensity is below a rate slightly greater than
cdKs(1-cos2δ), because all infiltrating
water, after percolating through the unsaturated-zone, goes to basal loss.
The same does not generally occur for a variable intensity hyetograph of the
same volume, because instantaneous intensity may be significantly higher than
the event mean intensity Wtot/Dtot, and consequently
a water table rise is produced. The opposite behavior for short durations is
due to the fact that in this case variable hyetographs may have peaks of
intensity higher than infiltration capacity, and thus not all rainfall
infiltrates into the soil. Due to these reasons, the model deterministic
threshold results in a poor erformance (TSS=0.642).
The above results lead to conclude that it is important to account for
variability of intensity during events and that landslide occurrence related
to the transient part of the response to rainfall events can be represented
with good approximation by a I–D power-law equation. This provides a
physically based justification for such a widely used threshold form, which
turns out to be valid when landslide occurrence is mostly due to the
transient part of the hillslope response to rainfall.
For the A/B=10m case (Fig. b), which may
represent prevalent conditions for the Peloritani
mountains area (see Sect. ), scattering of the red dots increases due to the
introduced variability of initial conditions. Consequently,
performances of predictions based only on intensity and duration of
rainfall events become worse. Simulations for larger values of the specific catchment area (e.g., A/B=20m, Fig. c) confirm this conclusion.
Based on these results, it may be stated that, for a given climatic input,
performances of thresholds which do not account for past rainfall time
history (antecedent rainfall) are expected to decrease as the water table
recession time constant τM increases. Rainfall time history occurring
before single rainfall events generally influences landslide triggering,
determining whether a threshold based only on rainfall intensity and duration
may be sufficient or the I–D threshold needs to be improved by the
introduction of antecedent rainfall variables.
Finally, the threshold
I=71.52D-0.8
may be a reasonable choice for the Peloritani mountains area since
performances are still high, since TSS=0.862.
Validation of the threshold using observed data and comparison with other thresholds
The Monte Carlo simulation technique provides a framework that is useful for
exploiting the information contained in the observed rainfall series and the
physics of the modeled phenomenon. Nonetheless, it remains important to
validate the results against observed data, to check if the models are
capable of reproducing characteristics of interest which are not directly
taken into account in model calibration and development.
Sensitivity of triggering thresholds and of the relative performances to
hydraulic conductivity KS (cf. Table ). Plots show, for the
two cases of zero memory (ψ0=0) and A/B=10m, how the a1
power-law coefficient of optimized stochastic-input physically based
thresholds and relative TSS performance index vary with the critical wetness
ratio ζCR. Performances of the constant-intensity
physically based thresholds are shown as well (in green). Different soil
depths are considered: (a)dLZ=1m,
(b)dLZ=1.5m and (c)dLZ=2m.
Here we perform a global validation by comparing the derived threshold
(Eq. ) with the triggering and non-triggering observed
rainfall events.
In particular we have derived from the series the rainfall events with the
same criterion adopted in Monte Carlo simulations. Yet the events in the
months neglected there (April–August) and the events with intensities below
the leakage flow cdKS(1-cos2(δ)) were not removed
here in the observed record, for the test to be unbiased to this
preprocessing of data. This resulted in 190 events, whose temporal
evolution of accumulated intensities I(D)=W(D)/D has been compared with
the derived threshold, as shown in Fig. . The
figure
indicates positive validation of the methodology,
as the events in the I–D plane that exceed the threshold are all and
only the four events that have triggered landslides in the considered period
(red-line time histories). This is the best result one can obtain from this
test, but it is perhaps noteworthy to clarify that it is expected that in the
long period the same test will not perform without errors, consistently with
the Monte Carlo simulations and the way that the threshold was derived.
Comparison with other thresholds may also help in understanding how reliable
the performed analysis is. proposed for Sicily the
threshold E=10.4D0.22, where E=I×D is cumulative event
rainfall, and hence threshold is equivalent to I=10.4D-0.78. This
threshold has been derived considering only observed triggering events and it
is corresponding to an exceedance frequency of 1%. It is firstly
interesting to notice that the exponent is practically equal to the one that
results from our analyses (a2=-0.8). Furthermore, as can be seen from
Fig. b this threshold exceeds one triggering event of the
Monte Carlo simulated data, which constitutes the 1% of the
triggering-rainfall data set (see Table :
0.01×(TP+FN)=0.01×(104+11)=1.15). This result is
a further support to the validity of the performed Monte Carlo analysis and
highlights the importance to take into account non-triggering rainfall in
assessing threshold performance.
Sensitivity of triggering thresholds and of the relative performances to the
leakage ratio cd (fractured bedrock) (cf. Table ). Plots show, for
the two cases of zero memory (ψ0=0) and A/B=10m, how the a1
power-law coefficient of optimized stochastic-input physically based thresholds and
relative TSS performance index vary with the critical wetness ratio ζCR .
Performances of the constant-intensity physically based thresholds are shown as well
(in green). Different soil depths are considered: (a)dLZ=1m,
(b)dLZ=1.5m and (c)dLZ=2m.
Sensitivity analysis
A sensitivity analysis has been conducted with respect to the following
variables (Sect. and Table ): the hydraulic
conductivity KS, the leakage ratio cd, the soil depth
dLZ and the critical wetness ratio ζCR. Plots similar
to those of Fig. 5 can be derived for each set of values of such variables.
For brevity those plots are not shown here and the analysis is performed
considering the plots of the optimal threshold coefficient a1 (again the
exponent has been fixed to a2=-0.80) and the maximum value of the
objective function TSS as functions of ζCR.
The results of this analysis are shown in
Figs. –, which can be commented on as follows:
Sensitivity to hydraulic conductivity: in the ψ0=0 case the
variation of KS induces relevant changes neither in the threshold
nor in the performance TSS. It can however be hypothesized that considering
more low values of KS, infiltrating rates more strongly depend on
how rainfall is distributed within the event and thus uncertainty increases.
Conversely, to an higher KS variation neither of the threshold nor
of the performance may be observed, since infiltration capacity will always
be higher than rainfall intensity, which then infiltrates totally. In the
AB=10m case the variations of a1 and TSS are
relevant and due to increased memory with decreasing KS; the
threshold decreases with KS and the associated uncertainty
increases (lower TSS).
Sensitivity to leakage ratio: both in the ψ0=0 and
the AB=10 m cases, the increase of the cd ratio
induces an increase of the threshold and in the relative uncertainty. Such a
variation is of comparable magnitude in the two cases. This happens because
the variation of cd affects only pressure head response during
rainfall events, but does not affect significantly memory due to antecedent
rainfall. Sensitivity to cd increases with soil depth, because the
increased water absorption in the unsaturated strata and the consequent
increased damping and smoothing effect induces an increase of the portion of
infiltrating water that goes to leakage. Indeed this affects more the
threshold (a1) than the relative performance (TSS).
Variation of ζCR and of soil depth dLZ: with increasing
ζCR the threshold increases, while the associated uncertainty
decreases. The threshold and relative performances decrease with soil depth.
This indicates that landslides become less predictable as soil depth
dLZ and ζCR diminish.
Generally, antecedent rainfall has to be taken into account to improve
performance of landslide-triggering thresholds based on rainfall.
Nonetheless, the use of only the I and D variables may still lead to good
performing thresholds when memory is relatively low, soil thickness is not
too shallow and hillslope is naturally not close to instability
(ζCR is relatively high). In fact I–D power-law thresholds
resulted in good performance (TSS > 0.8) when τM≤3days, ζCR>0.5 and dLZ≥1.5 m.
The constant-rainfall physically based thresholds always perform poorly. This
confirms that variability of intensity during rainfall events influences
significantly rainfall intensity and duration associated with landslide
triggering.
Conclusions
In this work it has been shown how stochastic rainfall models and
hydrological and slope stability physically based models can be
advantageously combined in a Monte Carlo simulation framework to derive and
evaluate landslide-triggering thresholds. The approach synthesizes research
findings in the fields of rainfall and landslide hydrological modeling to
provide an output that is easily implemented in a early warning system, i.e.,
a landslide-triggering threshold, based on rainfall monitoring, of the same
type that is commonly derived by direct empirical analysis of observed
rainfall and landslide data. The advantages of the approach consist in a
better exploitation of the information contained in observed rainfall series
and measurements of hydraulic, geotechnical soil properties and
geomorphological analysis. Because both triggering and non-triggering
rainfall events are taken into account, the approach enables a more correct
derivation and evaluation of thresholds, for which well-known
prediction-skill receiver operating characteristic (ROC) analysis may be
advantageously used to reduce subjectivity in the identification of
thresholds and to estimate the convenience of the use of the threshold within
a landslide early warning system.
Furthermore, the specific modeling framework implemented in this work enabled
to analyze some general issues on landslide-triggering phenomena regarding
its controlling factors and uncertainty related to variability of rainfall
intensity within events and past rainfall (antecedent rainfall), with a
particular focus on the widely used power-law rainfall intensity–duration
threshold form. In particular, from the application to the Peloritani
mountains area in north-eastern Sicily (Italy) and the conducted sensitivity
analysis on various controlling parameters, the following conclusions can be
drawn: (1) variability of intensity during rainfall events significantly
influences rainfall intensity and duration associated with landslide
triggering. In particular constant-intensity input thresholds perform
conservatively only for low rainfall durations, while the opposite occurs for
events of longer duration. On the other hand, when a time-variable
rainfall-rate event is considered, the simulated triggering points may be
separated with a very good approximation (i.e., true skill statistic is close
to 1) from the non-triggering ones by a I–D power-law equation. This
indicates that this widely used model is adequate to represent the triggering
part due to transient infiltration produced by rainfall events. Thus, this
gives a physically based justification for such a widely used threshold form,
which turns out to be valid when landslide occurrence is mostly due to that
part. This depends, for a given rainfall climate, mostly on the timing of
recession of the saturated zone occurring during dry inter-event intervals
(in our model represented by the constant τM), but also on the
other soil hydraulic and geotechnical parameters, and in particular on soil
depth dLZ, which must not be too shallow, and critical wetness
ratio ζCR, that must be not too low. For instance, for the
case-study area, the I–D power-law threshold performs with a
TSS>0.80 when it is τM≤3days and
dLZ≥1.5m and ζCR>0.50. (2) In general,
rainfall time history occurring before single rainfall events influences
landslide triggering, determining whether a threshold based only on rainfall
intensity and duration may be sufficient or the I–D threshold needs to
be improved by the introduction of antecedent rainfall variables.
(3) Sensitivity analysis indicated that in general threshold performance is
affected by the depth of the basal boundary and the critical wetness ratio
that represents the natural degree of stability of the hillslope. In
particular, uncertainty of landslide-triggering prediction increases as the
soil depth or the critical wetness decrease. Hence, it is more difficult to
predict landslides the more an hillslope is shallow and the more it is naturally
close to instability.
A decrease of performance is obtained as basal drainage (leakage) increases as well.
Hence the I–D power-law may not be performing adequately in the case that the bedrock
is significantly fractured and the soil cover is very shallow.
Results also indicate that hydraulic conductivity mainly influences memory of
past rainfall and only slightly affects the uncertainty related to
variability of rainfall intensity within events.
Further ongoing research is oriented to introduce additional information in
the derivation of the thresholds, such as antecedent precipitation as well as
indexes representative of the shape of the hyetograph.
Stochastic rainfall model
Stochastic rainfall point models are aimed at the generation realistic
synthetic time series of (virtually) unlimited length, by calibration based
on a observed rainfall series. The uninitiated reader is invited to read
, ,
, ,
and . Here we give some
specific details on the NSRP model, for it being the one used in this work to
model rainfall at a site.
NSRP process belongs to the so-called class of cluster models
cf., e.g., .
The NSRP cluster model consists in a two-level mechanism process. This process is
related to a conceptualization of meteorological processes that originate rainfall
events , and it is obtained by the following steps (see Fig. ):
First, clusters – also known as storm-generating systems, or
simply storms – arrive governed by a Poisson process of parameter
λt (this is the first-level mechanism)
For each cluster origin, rectangular pulses (rain cells) are
generated (this is the second-level mechanism). The number of pulses C associated with each storm is
extracted from another separate Poisson distribution. In order to have
realizations of C not less than one, it is assumed that
C′=C-1, with c′=0,1,2,… (which implies c=1,2,3,…), is Poisson distributed with mean ν-1
Each cell has origin at time τi,j with j=1,2,…,ci measured from ti, according to an exponential random
variable of parameter β
A rectangular pulse of duration di,j and intensity xi,j is associated with each rain cell. Pulses have duration
exponentially distributed with parameter η while intensities
X are extracted from a Weibull distribution
cf., which has cumulative distribution function F(x;ξ,b)=1-exp(-ξxb)
Finally, the total intensity at any point in time is given by
the sum of the intensities of all active cells at that point.
We have calibrated the NSRP model by the method of moments, i.e., using the
properties of the aggregated NSRP process Y(τ) at different timescales
of aggregation τcf.,
e.g.,. According
to this method, model parameters, i.e., λ, ν, β, η and
ξ (b is typically fixed, in the range 0.6≤b≤0.9; see
) are estimated using at least as many moments as
the parameters of the model, considering different statistics (moments) at
various time aggregations, and solving the related equation system, where the
theoretical expressions, containing the parameters, are equated to the sample
moments. Theoretical moments of Y(τ), such as the mean μ(τ),
variance γ(τ) and autocorrelation at lag k, ρ(τ,k), are
given by formulas derived by . Transition probabilities
were derived as well, by , and have been included in
the calibration process. We have solved the nonlinear equation system by
numerical minimization of an objective function
S(λ,ν,β,η,ξ), that measures the global relative error
between theoretical and sample moments cf..
Representation of the NSRP
stochastic process for at-site rainfall modeling (adapted from ).
Though calibration is conducted taking into account seasonality, by
calibrating the model separately for the various homogeneous seasons within
the year (Sect. ), it is noteworthy to point out that the
generated
series is globally stationary, and consequently eventual annual non-stationarity due to climate
change is not taken into account. In other words, the generated series
represents possible realizations of the rainfall events under current climate
conditions, the final aim being of deriving thresholds suitable under the
present climate and not to assess how climate change may affect them.
Hillslope hydrological and stability model
The total pressure head response ψ of an hillslope soil to a rainfall
event is given by the sum of a transient part ψ1 and an initial part
ψ0cf.. The transient part is due
to infiltration of event rainfall, while the initial part depends on rainfall
time history before the current rainfall event. As pointed out by
, for soils that are relatively shallow, i.e., when the
ratio between soil depth and the square root of the upslope contributing area
is small, ε=dLZ/A≪1, the prevailing
process that determines ψ1 is 1-D vertical infiltration, while in the
dry periods in between events, the prevailing process is sub-horizontal
drainage.
Soil 1-D vertical scheme used to model infiltration and slope
stability based on the TRIGRS unsaturated model adapted
from.
Based on these considerations, we use a vertical infiltration model for
computing ψ1, the TRIGRS unsaturated model ,
and a linear reservoir sub-horizontal drainage model to compute the initial
conditions from the water table height at the end of the rainfall event
preceding the current one (which in turn depends on past rainfall time
history). This latter model is derived from a mass-conservation equation of
soil water coupled with the Darcy's law used to describe seepage flow, where
for simplicity the soil volumetric strain is neglected (the variation of
porosity with pressure head is assumed null). A similar conceptualization is
the basis of the model proposed by .
In order to understand the controlling factors of landslide triggering, it is
useful to separate the analysis of the response to rainfall in terms of the
transient part only. This may be done by performing the simulations assuming
ψ0=0 at the beginning of events.
From the pressure head response, the factor of safety FS for slope stability is
computed, using a infinite slope model.
The description of these model components follows.
Initial conditions model
The initial condition to rainfall event i is computed from the response at
the end of rainfall event i-1, using a water table height h recession
model between storms based on the following mass-conservation equation
:
BhKssinδ=-A(θs-θr)dhdt,
where A is the contributing area draining across the contour length B of
the lower boundary of the hillslope, δ is the inclination of the
hillslope, Ks is the saturated hydraulic conductivity, and
θs-θr is soil porosity,
θs and θr being the saturated and
residual soil water contents, respectively. The ratio A/B, which can be
computed based on a digital terrain model (DTM), is the well-known
specific upslope contributing area, an important variable on which
the topographic control on shallow landslide triggering depends
.
For instance, it is A/B=BNd, where Nd is the number
of cells draining into the local one, if one determines flow paths via
the non-dispersive single direction (D8) method . Other
methods consider multiple flow directions cf. .
The solution to Eq. () is used to compute the water table
height at the beginning of rainfall event i:
hi=ψ(dLZ,tf,i-1)cos2δexp-KssinδAB(θs-θr)Δti=ψ(dLZ,tf,i-1)cos2δexp-ΔtiτM,
where ψ(dLZ, tf,i-1) and the inter-arrival time
Δti are defined in Sect. . The time constant τM
regulates the pressure head memory from one event to another.
The initial pressure head distribution above the water table is computed
accordingly with assumptions of the transient vertical infiltration model
(see next Sect. ), letting the steady (initial) surface
flux IZLT=0, which yields the following equation
see:
ψ(Z,t=0)=-(du-Z)cosδ-1α,
for depths Z≤du, du being the depth to the top of the capillary
fringe and α the parameter of 's
() exponential soil–water characteristic curve (cf.
Fig. ).
Transient infiltration model
Reference scheme for a simulated hillslope is shown in Fig. .
Infiltration in the unsaturated zone is modeled through
' () vertical-infiltration
equation for a sloping surface particularized for the
's () exponential soil–water
characteristic curve K(ψ)=Ksexp{α(ψ-ψcf)}:
α1(θs-θr)KS∂K∂t=∂2K∂Z2-α1∂K∂Z,
where Ks is the saturated hydraulic conductivity, α is
the SWCC parameter, ψcf=-1/α is the pressure head at the
top of the capillary fringe, θr is the residual water
content, θs is the water content at saturation and
α1=αcos2δ.
A closed-form solution to this equation for δ=0 has been provided by
and extended to a sloping surface by
, and used in the TRIGRS unsaturated model
.
The solution to ' equation provides the pore pressure
profile in the unsaturated zone, and a flux to the saturated zone
q(du,t). Because of the partial absorption of water within the
unsaturated zone, this flux turns out to be damped and smoothed with respect
to the infiltrating flux at the ground surface cf. Fig. 8 of
. The TRIGRS model then computes water table rise using
q(du,t) and subtracting from it a leakage flow rate given by
ql=min{cdKs(1-cos2δ),q(du,t)} (vertical drainage at the basal boundary, which is
not assumed perfectly impervious), where cd represents the
ratio between saturated hydraulic conductivities of the basal boundary layer
and of the regolith surficial layer. In the case that no specific information
on cd ratio is available, a reasonable value may be
cd=0.1cf., which means that hydraulic
conductivity of the layer below depth Z=dLZ is of one order of
magnitude less than the regolith surficial layer.
The resulting water table rise is computed by comparing this excess flux
accumulating at the top of the capillary fringe to the available pore space
directly above it.
Pressure head rise is assumed transient in the saturated zone as well, and
computed by formulas adapted from analogous heat-flow problems
see.
Slope stability model
For analysis of hillslope stability we assume an infinite slope scheme, and
compute minimum factor of safety FS with the following formula
:
FS(dLZ,t)=tanϕ′tanδ+c′-ψ(dLZ,t)γwtanϕ′γsdLZsinδcosδ,
where c′ is soil cohesion for effective stress, ϕ′ is
the soil friction angle for effective stress, γw is the
unit weight of groundwater, γs is the soil unit weight and
δ is the slope angle. In this scheme the failure occurs at the basal
boundary Z=dLZ, because the pressure head results maximum at that
depth.
It is useful to consider the critical wetness ratio, derived from
Eq. () letting FS=1, which is a parameter that for
a given hillslope (given slope δ and soil depth dLZ)
depends only on the geotechnical characteristics of the soil:
ζCR=γsγwc′γsdLZsinδcosδ-1tanδtanϕ′+1.
The ζCR varies from 0 to 1, respectively, for an
unconditionally unstable and a unconditionally stable hillslope
, and hence it indicates the natural degree of
stability of the hillslope.
Acknowledgements
This research was partially funded by the Italian Education,
University and Research Ministry (MIUR), PON project no. 01_01503
Integrated Systems for Hydrogeological Risk Monitoring,
Early Warning and Mitigation Along the Main Lifelines, CUP
B31H11000370005. Rex L. Baum of United States Geological Survey and Jorge A. Ramirez of
Colorado State University are acknowledged for their useful suggestions. The authors would
like to thank the four anonymous reviewers and S. L. Gariano for their useful
comments.
Edited by: T. Bogaard
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