Interactive comment on “ GA SAKe : forecasting landslide activations by a genetic-algorithms based hydrological model ” by O . G . Terranova

Authors are grateful to the Referee #1 for his/her precious suggestions and comments. Below, please find the changes to the text and replies for each issue raised. A more readable version of our replies is attached. 1) runoff for unsaturated soils (pag. 1227, line 23), a discussion could be useful with reference to the suction-runoff dependence highlighted by Cuomo and Della Sala (2013, Eng. Geol. Journal) and similar contributions. Following referee’s suggestion, we propose to modify the text as specified below. At


Introduction
A nationwide investigation, carried out by the National Geological Survey, identified approximately 5 × 10 5 slope movements in Italy, with an average of 1.6 failures per square kilometer (Trigila, 2007).According to other investigations, this figure would rather be a low estimate (cf.Servizio Geologico, Sismico dei Suoli, 1999;Guzzetti et al., 2008).In the period 1950-2009, at least 6349 persons were killed, went missing, or were injured by landslides, with an average of 16 harmful O. G. Terranova et al.: GA SAKe events per year, thus confirming the notable risk posed to the population (Guzzetti, 2000;Salvati et al., 2010).Petley (2008) estimated that about 90 % of worldwide casualties can be attributed to landslides triggered by rainfall.With reference to the Italian territory, about 70 % of landslides result from being triggered by rainfall (cf.CNR-GNDCI AVI Project, Alfieri et al., 2012).Slope instability conditions are in fact influenced by rainfall that, infiltrating into the slopes, causes temporary changes in groundwater dynamics (Van Asch et al., 1999).The combination of infiltration and runoff may cause different types of mass movements (either slope failure or erosion processes) depending on the intensity and duration of the rainfall and the values of soil suction (Cuomo and Della Sala, 2013).Concentration of water deriving from either contemporary or antecedent storms at specific sites plays a major role in triggering landslidesas testified by slope instabilities that commonly follow the heaviest phases of rainfall events.
To model the relationships between rainfall and landslide occurrence, two distinct approaches are generally adopted in the literature.The first, named "complete" or "physically based", attempts to determine the influence of rainfall on slope stability by modeling its effects in terms of overland flow, groundwater infiltration, pore pressure and related balance of shear stress and resistance (cf.e.g., Montgomery and Dietrich, 1994;Wilson and Wieczorek, 1995;Crosta, 1998;Terlien, 1998;Crosta et al., 2003;Pisani et al., 2010).With regard to this latter purpose, numerical models are employed, and a notable (and expensive) number of detailed data are commonly required to define the geological scheme of the slope in litho-structural, hydrogeological, morphologic and geotechnical terms.The second approach (adopted in the present study), named "empirical" or "hydrological" (Cascini and Versace, 1988), is based on a statistical-probabilistic analysis of rainfall series and of dates of occurrence of slope movements (see, among others, Campbell, 1975;Caine, 1980;UNDRO, 1991;Sirangelo and Versace, 1996;Guzzetti et al., 2007, 2008, Brunetti et al., 2010;Gariano et al., 2015).In the literature, methodological examples generally focus on thresholds obtained for (i) single phenomena or (ii) given types of landslides within a homogeneous geo-environmental setting (cf.e.g., Jakob and Weatherly, 2003).
In this study, hydrological model GA SAKe (i.e., the genetic-algorithms-based release of the Self Adaptive Kernel) model, developed to forecast the triggering of slope movements, is described.The model can be applied to either single landslides or to a set of similar phenomena within a homogeneous study area.Model calibration is performed by means of genetic algorithms: in this way, a family of optimal, discretized kernels can iteratively be obtained from initial tentative solutions.In a different release of the model ( CM SAKe -i.e., cluster model SAKe), the calibration is instead performed through an iterative procedure (Terranova et al., 2013).
Examples of application of the model to a medium-size landslide (the Uncino landslide at San Fili) and to shallowslope movements in the Sorrento Peninsula are discussed in the following sections.Temporal validation is discussed for both cases, in view of early-warning applications of GA SAKe for civil protection purposes.Moreover, a progressive, selfadaptive procedure of calibration and validation is discussed, by considering the Uncino case study, to verify changes in fitness, predictive ability and base time when an increasing number of dates of activation is employed.Finally, the results of preliminary parametric analyses are presented, aimed at investigating the role of the main parameters of the model.

Background
Physical systems evolve in time due to their own inner dynamics and/or as a consequence of external causes.Suitable observational tools can be employed to monitor their evolution, and arranged to promptly send reports or warnings to authorities of civil protection to support the management of emergencies (Cauvin et al., 1998; for applications to landslides, cf. also Keefer et al., 1987;Iovine et al., 2009;Capparelli and Versace, 2011;Pradhan and Buchroithner, 2012).
In the case of complex systems (e.g., nuclear power stations, telecommunication networks), many parameters, in part interdependent, have to be monitored.Missing an automated phase of analysis and proper filtering, a great number of reports may be delivered by the monitoring apparatus in a few seconds.For this purpose, the concepts of threshold (Carter, 2010), event and warning must therefore be suitably defined.
Regarding slope movements, the notions of threshold and warning have long been investigated.In particular, a threshold constitutes a condition -generally expressed in quantitative terms or through a mathematical law -whose occurrence implies a change of state (White et al., 1996).According to the ALARM study group (Cauvin et al., 1998), an event is (i) a portion of information extracted from either continuous or discrete signals (i.e., a significant variation), transmitted by a component of the monitoring network; or (ii) a set of data concerning the considered context (e.g., restorations, actions, observations).According to such a definition, an event must be instantaneous and dated.As for warning, its definition derives from that of the event: it is a discrete indicator aimed at triggering a human or an automated reaction.The warning can be classified into distinct levels (e.g., in terms of security) or by type (e.g., related to a distinct component of the dynamic system under consideration), to be transmitted by the monitoring system.
In complex systems, causal factors responsible for emergency conditions may be difficult to identify.Therefore, warnings may be issued according to pre-fixed thresholds related to suitable physical properties of the system.In these cases, the timing of data sampling of the monitoring instru-ments should be progressively adapted to the evolution of the phenomenon.A further issue concerns the chances of missing and false alarms, as well as the camouflage of an alarm among simultaneous others.
In physical terms, slope instability can occur when the shear strength gets lower than a given threshold (Terzaghi, 1962).Rain infiltration may temporarily change the dynamics of groundwater (Van Asch et al., 1999): due to an increase in pore water pressure, the effective shear strength of the material decreases, and a slope movement can be triggered.Groundwater may reach a given location within the slope by different paths.The main natural mechanisms include (i) surface flow, strongly influenced by morphology; (ii) direct infiltration from the surface; (iii) flow within the soil mantle (throughflow) from upslope and sideslopes; and (iv) seepage from the bedrock toward the overlying colluvium.The length of the different paths may be quite different, and is characterized by distinct velocities: as a consequence, aliquots of the same rainfall event may reach a given site at different times, variously combining with other groundwater amounts (Ellen, 1988).
To apply a hydrological approach, empirical relations have to be determined by means of thresholds to distinguish among conditions that likely correspond to landslide occurrence or not.To this aim, different hydrological parameters can be selected (Guzzetti et al., 2007(Guzzetti et al., , 2008, and http: //rainfallthresholds.irpi.cnr.it/): the cumulative rain recorded in a given temporal window (hours/days/months) before landslide activation; the average rain intensity in the same temporal window; and rains normalized to reference values (e.g., annual averages).Simplified hydrological balances can also be adopted in empirical approaches, by considering losses of aliquots of rains by runoff, evapo-transpiration, etc.
Regarding superficial landslides, triggering thresholds can be derived from relations between the "triggering" rain (daily, hourly or shorter), corresponding to the onset of the slope movement, and the rain cumulated over an "antecedent period" (usually, a few days to 2 weeks before landslide activation) (e.g., Campbell, 1975;Cannon and Ellen, 1985;Wieczorek, 1987;Terlien, 1996;Crosta, 1998;Zêzere and Rodrigues, 2002).In other cases, thresholds refer to relations between rain intensity, I , and duration, D (e.g., Brunetti et al., 2010;Berti et al., 2012;Peres and Cancelliere, 2014).In some studies, antecedent rains are also considered, allowing one to obtain better results (e.g., Campbell, 1975).Larger amounts of antecedent rain should allow slope movements to be activated by less severe triggering storms.In general, a direct relationship between antecedent rain and landslide dimension can be observed (Cascini and Versace, 1986), though, under some peculiar conditions (e.g., Hong Kong case studies, caused by suction reduction - Brand et al., 1984), this is not the case, and the role of antecedent rains looks less important.In addition, as underlined by Cuomo and Della Sala (2013), time to runoff, time to failure and runoff rates strongly depend on soil water characteristic curves, soil initial conditions, rainfall intensity and slope angle in unsaturated shallow deposits.Moreover, soil mechanical parameters affect the time to failure, which can be either shorter or longer than time to runoff.
Due to physical and economic issues, difficulties in hydrological modeling of landslides generally increase when dealing with deeper and larger phenomena (Cascini and Versace, 1986).In such cases, landslide activation depends on the dynamics of deeper groundwater bodies.By the way, it is not by chance that most studies do refer to small and superficial slope movements.Large landslides usually show complex relationships with rains, as different groundwater aliquots may combine and reach the site of triggering.Depending on type (cf. dimension, material, kinematics, etc.), different hydrological mechanisms should be considered, thus limiting the possibility of generalization of the thresholds (Dikau and Schrott, 1999;Corominas, 2001;Marques et al., 2008).Again, the mobilization of deeper phenomena commonly requires greater rainfall amounts, spanned over longer periods, with respect to shallow landslides (Aleotti, 2004;Terranova et al., 2004;Guzzetti et al., 2007Guzzetti et al., , 2008)).In these cases, rain durations responsible for landslide activations commonly range from ca. 30 days to several months, even beyond a single rainy season (Brunsden, 1984;Van Asch et al., 1999;Gullà et al., 2004;Trigo et al., 2005).
To analyze the triggering conditions of slope movements -either shallow or deep-seated -a threshold-based modeling approach can be employed.Empirical thresholds (e.g., Aleotti, 2004;Wieczorek and Glade, 2005;Terranova et al., 2004;Vennari et al., 2014) can be expressed in terms of curves, delimiting the portion of the Cartesian plane that contains "all and only" the hydrological conditions related to known activations (cf.e.g., the I -D chart proposed by Caine, 1980).A further improvement to this approach can be obtained by considering hydrological conditions not related to landslide activations (Crozier, 1997;Sengupta et al., 2010;Gariano et al., 2015).In general, no changes of state are assumed to occur below the threshold (z t ), while they do happen above it.Alternatively, a range of conditions can be defined (Crozier, 1997), delimited by -a lower threshold (z low ), below which changes of state never occur, and -an upper threshold (z upp ), above which changes always happen.
For values between z upp and z low , the probability that the state changes can be defined, essentially depending on (i) the incompleteness of knowledge of the physical process under investigation, and (ii) the incapacity of the model to fully replicate the behavior of the same process.tively; and G[z(t)] is a probability function, monotonically increasing with t in the range ]0, 1[.In hydrological models, to express the influence of rainfall on runoff and groundwater dynamics, a "kernel" (also named a "filter function") can be employed, usually defined in terms of a simple, continuous analytical function (Chow et al., 1988).In such a way, suitable weights can be assigned to the precipitations that occurred in the last hours/days before a given geo-hydrological process (e.g., discharge, measured at a generic river cross section, landslide activation), as well as to earlier rains recorded weeks/months before.The most employed types of kernels are beta, gamma, Nash, and negative exponential distribution.Furthermore, the "base time" (t b ) expresses a sort of memory with respect to rainfall: in classic rainfall-runoff modeling, t b defines the time of concentration, while in slope stability analyses, it represents the time interval, measured backward from landslide activation, during which rainfall is deemed to effectively affect groundwater dynamics, and contributes to destabilization.
To model slope stability, both the shape and the base time of the kernel must be properly selected depending on the type and dimension of the investigated phenomena, as well as geo-structural and hydrogeological characteristics.Unfortunately, in several real cases, the abovementioned analytical functions may fail in properly capturing the complexity of groundwater dynamics, as well as the related landslide activations.In this respect, the adoption of discretized kernels, automatically calibrated through iterative computational techniques, may offer effective solutions.
3 The GA SAKe model GA SAKe is an empirical-hydrological model for predicting the activation of slope movements of different types.It is based on a classic threshold scheme: the exceedance of the threshold determines a change of state, i.e., the triggering of the landslide.The scheme is inspired by the FLaIR (Forecasting Landslides Induced by Rainfall) model, proposed by Sirangelo and Versace (1996): through changes of state in time, the variable z(t) assumes the meaning of a "mobility function".In other terms, the values of z(t) depend on the amount of rain stored in the aquifer.
In hydrology, rainfall-runoff modeling is commonly performed by adopting a linear, steady scheme (Chow et al., 1988).Such an approach implies that the transformation of rainfall in runoff can be described by an integral of convolution between a unitary impulsive response of the basin -the kernel, h(t) -and the rainfall, p(t).
The kernel (filter function) represents the unitary volume influx in an infinitesimal period, and is defined as in which h(t) = h(−t), h(t) ≥ 0, ∀t.
In practical applications, the lower bound (t = 0) corresponds to the beginning of the flood-wave rising, and the kernel assumes a finite duration (t b ).The integral of convolution is therefore expressed as in which z(t) represents the discharge at time t.For a specific case study, the kernel can be determined by means of calibration procedures, by relating discharge measurements to rains.
In discretized terms, the elements of the kernel are characterized by width t and height h i , and Eq. ( 3) can be written as (4) Sirangelo and Versace (1996) proved that the same approach may turn out to be promising also in slope-stability modeling.Capparelli and Versace (2011) stressed that the I -D chart of Caine (1980) corresponds to a kernel defined by a power function h (t) = a t b , with b < 0. Exporting the wellestablished knowledge of rainfall-runoff modeling (usually based on many measurements) to rainfall-landslide modeling is not trivial, due to the scarcity of adequate information for proper calibration.Only a few dates of activation are, in fact, commonly available in rainfall-landslide modeling (often with unsatisfactory details on location and phenomena), and the values of z(t) are unknown.From a mathematical point of view, such a problem can be handled by assuming that the timing of the maxima of z(t) corresponds to the dates of landslide activation.When studying the triggering conditions of landslides, calibration can therefore be performed by maximizing the mobility function in correspondence to the dates of activation.
Scarcity of information inevitably reflects on the resulting kernel, whose shape may turn out highly indeterminate: different functions, or different parameters of the same function, can in fact maximize z(t) in correspondence to the dates of mobilization.Model optimization -and its reliable utilization for early-warning purposes -can turn out to be an awkward issue.
In this work, an innovative modeling approach -based on discretized kernels, automatically calibrated through iterative computational techniques -is proposed, which may help in facing the above-cited difficulties.For modeling purposes, the rainfall series and a coherent set of dates of landslide occurrence -either related to a given slope movement, or to a set of landslides of the same type in a homogeneous geoenvironmental zone -must be given as input.
Unfortunately, when dealing with the timing of occurrence, historical notices may refer either to portions of the considered phenomena or to entire landslide bodies.Therefore, dates should be properly selected to consider only consistent cases.Moreover, dates of activation are usually known with only a broad approximation: with respect to the reports, the actual timing of occurrence may be located backward (documents may assign a later date) or forward (in the case of later, more relevant movements).For modeling purposes, it is then useful to specify a temporal window, lasting from an initial (d t−from ) to a final date (d t−to ), containing the presumable timing of occurrence.
Rainfall series are commonly reconstructed from data recorded at rain gauges located within a reasonable proximity of the study site.The temporal window of the hydrological analysis is defined by the intersection of (i) the period of observation of the rains and (ii) the period delimited by the most ancient and most recent dates of activation of the landslide.A potential source of uncertainty lies in the fact that, occasionally, the recorded rainfall amounts notably differ from those actually experienced at landslide location.Furthermore, landslide triggering may also be due to other causes (e.g., human activity, earthquakes): a thorough preliminary analysis has always to be performed to verify the significance of rainfall preceding landslide activation, to detect cases not to be considered in the hydrological study.
In the model, rains older than t b are neglected.Suitable maximum and minimum values (t b-max and t b-min ) have to be initialized to allow the model to determine optimal values.Commonly, t b ranging from a few hours to some weeks are suggested for shallow landslides, while greater values (up to several months) sound suitable for deep-seated phenomena.
Based on the geological knowledge of the phenomenon under investigation, the initial shape of the kernel can be assumed among a set of basic types.Among these, (i) a "rectangular" shape can be adopted if older precipitations have the same weight of more recent rains; (ii) a "decreasing triangular", if older precipitations have a progressively smaller weight than more recent rains; and (iii) "increasing triangular", if older precipitations have a progressively greater weight than more recent rains.A casual shape or any other function can also be implemented in the model (e.g., beta, gamma, Nash, negative exponential distribution).

Model calibration
In GA SAKe, model calibration is performed against real case studies through genetic algorithms (GAs).These latter ones are general-purpose, iterative search algorithms inspired by natural selection and genetics (Holland, 1975).Since the 1970s, GAs have been applied to several fields of research, from applied mathematics (Poon and Sparks, 1992), to evolution of learning (Hinton and Nowlan, 1987), evolutionary robotics (Nolfi and Marocco, 2001), and debris-flow modeling (Iovine et al., 2005;D'Ambrosio et al., 2006).GAs simulate the evolution of a population of candidate solutions to a given problem by favoring the reproduction of the best individuals.The candidate solutions are codified by genotypes, typically using strings, whose elements are called genes.
GAs explore the solution space, defined as the set of possible values of the genes.At the beginning of a given optimization experiment, the members of the initial population of genotypes (in this study, the kernels) are usually generated at random.The performance of each solution, in terms of phenotype (i.e., the mobility function), is evaluated by applying a suitable fitness function, thus determining its "adaptability", i.e., the measure of its goodness in resolving the problem.
The sequence of random genetic operators selection, crossover and mutation, constrained by prefixed probabilities, constitutes a single GA iteration that generates a new population of candidate solutions.At each iteration, the best individuals are in fact chosen by applying the selection operator.To form a new population of offspring, crossover is employed by combining parents' genes.Mutation is successively applied to each gene, by randomly changing its value within the allowed range.Thanks to the GA approach, better individuals (i.e., those characterized by higher fitness values) can be obtained over time.In fact, according to individual probabilities of selection, any change that increases the fitness tends to be preserved over GA iterations (Holland, 1975).For further details on GAs, cf.Goldberg (1989) and Mitchell (1996).
In the present study, a steady-state and elitist GA (cf.De Jong, 1975) was employed to obtain the family of optimal kernels that maximize the mobility function in correspondence to known dates of landslide activations.The procedure employed for calibration of GA SAKe is schematized in Fig. 1.
At the beginning of an optimization experiment, the initial population of N kernels is generated at random, and the fitness of the related mobility functions is evaluated (cf.below).In order to evolve the initial population of candidate solutions, and to progressively obtain better solutions, a total number of GA iterations follow.
At each iteration of the GA, the operator selection, crossover and mutation are applied as follows (Fig. 2): -selection i. n e "elitist" individuals are merely copied in a "mating pool" from the previous generation, by choosing the best ones; and ii. the remaining N −n e candidate solutions are chosen by applying the "tournament without replacement" selection operator.More in detail, a series of tournaments are performed by selecting two individuals at random from the previous generation: the winner (i.e., the one characterized by the highest fitness) is copied into the mating pool, according to a prefixed surviving probability (p s ), which is set greater for the fittest individual.Note that, when choosing the N − n e candidate solutions, a given individual cannot be selected more than once.
-crossover After the mating pool is filled with N individuals, the crossover operator is applied, according to a prefixed probability (p c) : i. two parent individuals are chosen from the mating pool at random; ii. a cutting point (crossover point) is then selected at random in the range ]t b-min , t b-max [; iii. the obtained portions of parents' strings are exchanged, thus mixing the genetic information and resulting in two children (Fig. 3).
When the crossover is not applied, the two parents are merely copied into P new .
-mutation Based on a prefixed probability (p m ), a random number of elements of the kernel (p me , expressed as a percentage of t b ) is mutated, by adding to each element an amount dh that is randomly obtained in the range [p mh1 , p mh2 ], as a function of the maximum value of the kernel (h max ).Then dh ranges from dh 1 to dh 2 : Furthermore, the base time is also mutated (increased or decreased) within the bounds [t b-min , t b-max ], according to a random factor dt b selected in the range [1/p mtb , p mtb ] (Fig. 4).
Children obtained by either crossover or mutation must be normalized before being included in the population P new , by properly scaling the elements of the kernels to ensure the validity of Eq. ( 2).During calibration, the shape of the kernel and its t b are iteratively refined.Note that the shape is not subject to any constraint, while t b is allowed to vary in the range [t b-min − t b-max ].The fitness is computed for each examined mobility function, and new populations of kernels are generated as described above.
As for the fitness function, in GA SAKe it is defined as follows: -the L available dates of landslide activation -as derived from the historical analyses -are arranged in a vector S = {S 1 , S 2 , . .., S i , . .., S L }; -the vector of the relative maxima of the mobility function, Z = {z 1 , z 2 , . .., z j , . .., z M }, is sorted in decreasing order (M = number of relative maxima); and -the vector of the partial fitness is ϕ = {ϕ 1 , ϕ 2 , . .., ϕ i , . ..ϕL }, where ϕ i = k −1 depends on the rank k of the relative maxima of z j that coincide with known dates of activation, S i .In case S i does not correspond to any relative maximum, it is ϕ i = 0.
With reference to a given kernel, the resulting fitness is expressed by u = L i=1 ϕ i .To generalize the results for an easier comparison with other study cases, a normalized fitness index is adopted, = u / max , defined in the range [0,1], For instance, if two dates of activation are available and both are well captured by the mobility function (i.e., they correspond to the highest peaks), the obtained fitness is u = 1+1/2 = 1.5.On the other hand, in case only one of the dates is captured and the remaining one ranks fifth, u = 1+1/5 = 1.2.
Thanks to the above procedure, a family of "optimal kernels" that maximizes the fitness can be determined.The mobility function is, in fact, forced toward a shape characterized by relative maxima (z j ) coinciding with the dates of landslide occurrence (S i ).An optimal solution leads to a mobility function having the highest peaks in correspondence to such dates; further peaks may also be present, characterized by lower values.Nevertheless, kernel solutions generally determine mobility functions whose highest peaks only partly match the dates of landslide occurrence (i.e., some dates may neither correspond to the highest peaks nor to any peak at all).
To select the most suitable kernel from a given family of optimal ones, let us define z j −min as the lowest of the peaks of the mobility function in correspondence to one of the dates of activation (S i ); z cr as the "critical threshold", i.e., the highest peak of the mobility function just below z j −min ; and -the "safety margin", z cr = (z j −min − z cr )/z j −min .
When applying the fitness function to evaluate a given kernel, either incompleteness or low accuracy of input data may lead to "false alarms" -i.e., peaks of the mobility function (z j ) that are greater than the threshold z cr , but that do not correspond to any of the known dates of activation.Such alarms can actually be of two different types: (1) "untrue false", due to an informative gap in the archive (i.e., correct prediction); and (2) "true false", in the case of real misprediction of the model.In such cases, further historical investigations may help to discriminate between the mentioned types of false alarms.Also depending on the specific purpose of the analysis, the most suitable kernel can therefore be selected by one or more of the following criteria: (i) the greatest z cr ; (ii) the shortest t b ; and (iii) the smallest µ 0 = i≤t b (i − 0.5) h i t, i.e., the first-order momentum of the kernel with respect to the vertical axis.The first criterion allows for the activation of earlywarning procedures with the greatest advance; the remaining ones (to be employed when z cr is too small) generally correspond to more impulsive responses to rainfall.Differently from what is usually experienced in rainfallrunoff models, GA SAKe therefore provides multiple equivalent solutions -i.e., a number of optimal kernels with same fitness, u , despite different shapes.This may depend on the limited number of available dates of activations, and on other noises in input data (e.g., rain gauges located too far from the site of landslide activation, inaccurate information on dates of activation or on the phenomenon).The adoption of synthetic kernels -e.g., obtained by averaging a suitable set of optimal kernels -permits one to synthesize the family of results for successive practical applications: in this work, the 100 best kernels obtained for each case study were in fact utilized to synthesize "average kernels" (see below) to be employed for validation purposes.

Case studies
The case studies considered in this paper are (i) a set of shallow landslides in the Sorrento Peninsula between Gragnano and Castellammare di Stabia (Campania, southern Italy) and (ii) the Uncino landslide at San Fili (Calabria, southern Italy).
Note that, as the numbers of known historical activations in the study areas were adequate, some dates could be excluded from calibration, and were successively employed for validation purposes.In particular, the most recent dates of landslide activation (cf.Tables 1 and 2) were employed to validate the average kernels (these latter ones obtained from the families of optimal solutions defined through calibration).The procedure employed for validation is schematized in Fig. 5.

Shallow landslides in the Sorrento Peninsula -Campania
The Sorrento Peninsula is located in western Campania, southern Italy (Fig. 6).In the area, Mesozoic limestone mainly crops out, covered by Miocene flysch, Pleistocene volcanic deposits (pyroclastic fall, ignimbrite), and Pleistocene detrital-alluvional deposits (Di Crescenzo and Santo, 1999).The carbonate bedrock constitutes a monocline, gently dipping towards WNW, mantled by sedimentary and volcanoclastic deposits, with thicknesses ranging from a few decimeters to tens of meters.Rainfall-induced shallow landslides are widespread in the pyroclastic soils covering the slopes of the study area.Among the various factors affecting the spatial distribution and the type of slope instabilities, Cascini et al. (2014) pointed out that both the rainfall conditions and the conse- or few (single) activation; site: municipality including the affected location; period employed: dates used for calibration (except for no.11); rank: relative position of the corresponding maximum of the mobility function obtained by calibration.An asterisk marks the date employed for validation.In italics, the activation date (no.0) excluded due to hydrological constraints.quent seasonal variations of soil suction play a significant role.In particular, when suction is low and frontal rainfall occurs (from November to May), first time shallow landslides are triggered; when suction is high or very high and convective or hurricane-type rainfall occurs (from June to October), mostly erosion phenomena occur, often turning into hyperconcentrated flows.
The study area is characterized by hot, dry summers and moderately cold and rainy winters.Consequently, its climate can be classified as Mediterranean (Csa in the Köppen-Geiger classification).In particular, the mean annual temperature ranges from 8-9 • C, at the highest elevations of M. Faito and M. Cerreto, to 17-18 • C along coasts and valleys.Average annual rainfall varies from 900 mm west of Sorrento to 1500 mm at M. Faito; moving inland to the east, it reaches 1600 mm at M. Cerreto and 1700 mm at the Chiunzi pass (Ducci and Tranfaglia, 2005).On average, annual totals are concentrated on about 95 rainy days.During the driest 6 months (from April to September), only 30 % of the annual rainfall is recorded in about 30 rainy days.During the three wettest months (November, October, and December), a similar amount is recorded in about 34 rainy days (Servizio Idrografico, 1948-1999).In the area, convective rainstorms may occur, characterized by a very high intensity, at the beginning of the rainy season (from September to October).In autumn-winter, either high intensity or long duration rainfall are usually recorded, while uniformly distributed rains generally occur in spring (Fiorillo and Wilson, 2004).As for annual maxima of daily rainfall recorded at sea level, the Amalfi coast (southern border of the Sorrento Peninsula) is characterized by smaller values (59 mm) of average annual maxima of daily rainfall than the Sorrento coast (86 mm), on the northern border.Such a difference seems to persist even at higher elevations (up to 1000 m a.s.l.), with 84 mm vs. 116 mm for the southern and northern mountain slopes, respectively (Rossi and Villani, 1994).
Severe storms frequently affect the study area, triggering shallow landslides that propagate seaward, often causing casualties and serious damage to urbanized areas and trans- portation facilities (Mele and Del Prete, 1999;Calcaterra and Santo, 2004;Di Crescenzo and Santo, 2005).In the second half of the twentieth century, several shallow landslides activated nearby Castellammare di Stabia: in Table 1, the major events recorded between Vico Equense and Gragnano are listed, with details on types of events, affected sites and references.The shallow landslides listed in Table 1 occurred between November and March, a period characterized by a medium to low suction range and included in the rainy season (October to April), according to Cascini et al. (2014).The same authors pointed out that, in this period, frontal rainfall typically occurs and may trigger widespread first-time shallow landslides, later propagating as debris flow or debris avalanches.
Rainfall responsible for landslide occurrences in the Sorrento Peninsula is shown in Fig. 7, in terms of cumulated antecedent rains, extracted from the records of the nearest gauges (Tramonti, Castellammare, and Tramonti-Chiunzicf.Fig. 6).The trends of antecedent rains look quite different, ranging from abrupt (cf.curves 5, 6, 7) to progressive increases (cf.2, 4, 10).On the other hand, curve 0 does not highlight significant amounts of rainfall in the 14 days preceding landslide activation: therefore, the occurrence recorded on 14 April 1967 was excluded from the hydrological analysis.Quite moderate amounts of cases 6 and 7 (that occurred on 4 November 1980 and 14 November 1982, respectively) were instead recorded in short periods, thus resulting in high-intensity events that could be considered as the triggering factor of the observed landslides.
As a result, the dates of activation from no. 1 to no. 10 were selected for calibration, whilst no.11 was employed for validation.As shallow landslides were being considered, the rainfall period employed for calibration spanned from 17 January 1963 to 10 December 1996; for validation, the rainfall series extended from 11 December 1996 to 10 February 1997 -i.e., to the validation date +t b (this latter as obtained from calibration).

The Uncino landslide -San Fili (northern Calabria)
San Fili (Fig. 8) is located on the western margin of the Crati graben, a tectonic depression along the active Calabrian-Sicilian rift zone (Monaco and Tortorici, 2000).In the area, vicarious, N-S trending normal faults mark the base of the coastal chain, at the transition between Palaeozoic metamorphic rocks, to the west, and Pliocene-Quaternary sediments, to the east (Amodio Morelli et al., 1976).Nearby San Fili, Palaeozoic migmatitic gneiss and biotitic schist, generally weathered, are mantled by a late Miocene sedimentary cover of reddish continental conglomerate, followed by marine sandstone and clays (CASMEZ, 1967).In particular, the village lies in the intermediate sector between two faults, marked by a NE-SW trending connection fault, delimiting Miocene sediments, to the north, from gneissic rocks, to the south.
In Calabria, the Tyrrhenian sector (including the study area) results are rainier than the Ionian sector (about 1200-2000 mm vs. 500 mm).Nevertheless, the most severe storms occur more frequently in the Ionian sector (Terranova, 2004).The average annual temperature is about 15 • C: the coldest months are January and February (on average, 5 • C), fol-Table 2. Dates of activation of the Uncino landslide.Periods (instead of singular dates) were considered in case of uncertain timing of activation.Key = no.: identification number of the date (in bold, used for calibration); dates/periods derived from the literature; dates/periods employed for calibration or validation; references: sources of information on activation dates; rank: relative position and dates of the maxima of the mobility function during calibration.An asterisk marks the activation employed for validation.In italics, the activation date (no.0) excluded due to hydrological constraints.As in most of the region, the climate at San Fili is Mediterranean (Csa, according to Köppen, 1948).Being located on the eastern side of a ridge, the area is subject to Föhn conditions with respect to perturbations coming from the Tyrrhenian Sea.It is characterized by heavy and frequent winter rainfall, caused by cold fronts mainly approaching from the northwest, and autumn rains, determined by cold air masses from the northeast.In spring, rains show lower intensities than in autumn, whilst strong convective storms are common at the end of summer.The average monthly rains recorded at the Montalto Uffugo gauge (the closest to San Fili) are listed in Table 3. From October to March (i.e., the wet semester), 77 % of the annual rainfall is totalized in about 77 rainy days; 36 % of the annual rainfall is recorded in 38 days during the three wettest months; finally, from June to August (i.e., the three driest months), 6 % of the annual rains fall in 11 days.
The Uncino landslide is located at the western margin of San Fili (Fig. 8).It is a medium-size rock slide (maximum width = 200 m, length > 650 m, estimated maximum vertical depth = 25 m), with a deep-seatedness factor (sensu Hutchinson, 1995) that may be classified as "intermediate".
The slope movement involves a late Miocene conglomerate, arenite and marly clay overlaying Palaeozoic gneiss and biotitic schist.It repeatedly affected the village, damaging the railway and the local road network, besides some buildings: the most ancient known activation dates back to the beginning of the twentieth century (Sorriso-Valvo et al., 1996); from 1960 to 1990, seven dates of mobilization are known (as listed in Table 2).In such events, the railroad connecting Cosenza to Paola was damaged or even interrupted.By the way, on 28 April 1987, the railway was put out of service; hence, the relevance of the infrastructure decreased, together with media attention.Usually, such information is collected from archives not compiled by landslide experts, and is therefore affected by intrinsic uncertainty (e.g., concerning the dates of activity, and the partial or total activation of the phenomenon), with unavoidable problems of homogeneity of the data employed for model calibration.
The informative content of the Uncino case study is quite high, and allows for a more accurate calibration of the kernel with respect to the Sorrento Peninsula case: consequently, a smaller family of optimal solutions is expected.Nevertheless, the known activations still suffer from uncertainties related to dates and affected volumes.
Cumulated antecedent rains, corresponding to the Uncino landslide occurrences, are shown in Fig. 9. Rainfall data were extracted from the records of the Montalto Uffugo rain gauge (cf.Fig. 8).The trends of antecedent rains may be distin- guished into 3 main patterns: the curve 2 shows a constant increase of rainfall in time, totalizing the greatest amounts from ca. 90 to 180 days.On the other hand, the case 0 shows the lowest values throughout the considered accumulation period.The curves 1, 3, 4, and 5 totalize intermediate values, with abrupt increases from 120 to 180 days for curves 3 and 5. Finally, the case 6 looks similar to case 2 between 30 and 90 days, but shows no more increases in the remaining period (analogously to 1 and 4).As curve 0 does not highlight significant amounts of rainfall in the 30-180 days preceding the landslide activation, the occurrence recorded on 23 November 1988 was excluded from the hydrological analysis.Of the remaining curves, case 1 generally shows the lowest amounts, from ca. 40 to 180 days.Consequently, the dates of activation from no. 1 to no. 5 were selected for calibration, whilst no.6 was employed for validation.Since a medium-size landslide was being considered, the rainfall period employed for calibration spans

Results
GA SAKe was applied to shallow-landslide occurrences in the Sorrento Peninsula and to a medium-size slope movement at San Fili, by considering the dates of activation and the daily rainfall series mentioned in Sects.4.1 and 4.2, and adopting the values of parameters listed in Table 4.
Among the kernels obtained from calibration, several provided similar fitness values.Thus, "average kernels" were computed for the considered case studies, by averaging the 100 best kernels.

Application to shallow landslides in the Sorrento Peninsula
In Table 5, the statistics related to the 100 best filter functions obtained from calibration (optimal kernels) are summarized.
From such values, a low variability of , t b and µ 0 can be appreciated; instead, z cr shows a greater range of values.The average kernel is shown in Fig. 10: it is characterized by fitness = 0.806, z cr = 0.00282, and t b = 28 days.From such a kernel, antecedent rainfall mostly affecting landslide instability ranges from 1 to 12 days, and subordinately from 25 to 26 days (negligible weights refer to rains that occurred in the remaining period).
The mobility function related to the average kernel is shown in Fig. 11.In this case, 4 out of 10 dates of landslide activation are well captured by the model (being ranked at the first 7 positions of the mobility function maxima); the remaining 6 dates do also correspond to relative maxima of the function, but are ranked from the 43rd to the 151st position.When considering the remaining relative maxima, several false positives can be recognized, mainly up to 1979.
During calibration, the best fitness ( = 0.807) was first reached after 1749 iterations (at 6th individual), with   z cr = 0.00441 and t b = 26 days.The kernel corresponding to such individual looks similar to the best one in terms of t b , z cr , and µ 0 (Fig. 12).The pattern of the best kernel is only slightly dissimilar from the average one: significant weights can, in fact, be appreciated up to 14 days, and then between 20-22 and 25-26 days.
By applying the average kernel, a validation was performed against the remaining date of activation (cf.Table 1, no. 11; multiple events occurred on 10 January 1997).The validation that resulted was fully satisfied, as shown in Fig. 13: the value of the mobility function for event no.11 is well above the z cr threshold (49.01 vs. 18.05), and is ranked as the second highest value among the function maxima (Fig. 13a).The same peak can also be appreciated as the maximum of the period ±t b (Fig. 13b).Accordingly, when adopting the average kernel, event no.11 of landslide activation could properly be predicted by the model.

Application to the Uncino landslide
In Table 6, the statistics related to the family of optimal kernels are summarized.From such values, a low variability of t b and z cr can be appreciated.The average kernel (Fig. 14) is characterized by fitness = 1, z cr = 0.0644, and t b = 66 days.Based on such a kernel, antecedent rains from  1 to 17 days, and from 27 to 45 days, mainly affect landslide instability.Relatively smaller weights pertain to the rains that occurred more than 53 days before the triggering; for periods older than 66 days, the weights are negligible.In Fig. 15, the mobility function related to the average kernel highlights that all the 5 dates of activation are well captured by the model (they are ranked at the first 5 positions among the function maxima).When considering the remain-Table 7. Uncino landslide case study.Results of progressive calibration.Key: L, t b , z j −min , z cr , z cr ): model parameters concerning calibration (for explanation, cf.text); v ) fitness obtained by validating the "average kernel", obtained in calibration, against the 6 dates of activation.In italics, results obtained when calibrating the model by using all the six available dates (no validation performed).quasi-critical situations (between 1965 and 1966, and subordinately in 1970 and 1977).

L
During calibration, the best fitness ( = 1) was first reached after 684 iterations (at 13th individual) with z cr = 0.0595.The best kernel (Fig. 16) was obtained at iteration 993, at 8th individual, with z cr = 0.0631.Its pattern results very similar to the average one, with a t b of 66 days.
By applying the average kernel, a validation was performed against the last known date of activation (cf.Table 2, no.6, which occurred in December 1980).The validation that resulted was fully satisfied, as shown in Fig. 17: the value of the mobility function for event no.6, in fact, is well above the z cr threshold (17.49 vs. 16.87), and is ranked as the sixth highest value among the function maxima (Fig. 17a).The same peak can be appreciated as the maximum of the period ±t b (Fig. 17b).Accordingly, when adopting the average kernel, event no.6 could properly be predicted by the model.

Self-adaptive procedure and sensitivity analyses
The capability of the model to react and self-adapt to input changes, such as new dates of landslide activation, was evaluated by a progressive, self-adaptive procedure of calibration and validation, using the information available for the Uncino case study.To simulate the adoption of GA SAKe in a landslide warning system, the model was iteratively calibrated by the first 2, 3, 4, and 5 dates of activation (L), and validated against the remaining 4, 3, 2, 1 dates, respectively.In each experiment, the GA parameters listed in Table 4 were adopted.Finally, the model was merely calibrated by considering all the 6 dates of activation.The results of the selfadaptive procedure are listed in Table 7.The related kernels are shown in Fig. 18.As a result, a progressive increase in fitness and predictive ability ( z cr ), together with the base time (ranging from 30 to 80 days), can be appreciated when employing a greater number of dates of activation.
Furthermore, aiming at evaluating the sensitivity of the model with respect to the GA parameters, a series of anal-   8.Each simulation stopped after 1500 iterations: GA parameters were initialized by considering the "benchmark experiment" (cf.values in Table 4), except for the parameter that was in turn varied, as indicated in Table 8.The obtained maximum fitness ( max ), safety margin ( z cr ), number (n i ) of iterations needed to first reach max , and base time (t b ) of the average kernel are shown in Fig. 19.If experiments with max = 1 are only taken into account, the minimum and maximum numbers of GA iterations needed to reach max ( min , max ), the minimum and maximum base times of the average kernel (t b_min , t b_max ), and the minimum and maximum safety margins of the average kernel ( z cr_min , z cr_max ), are listed in Tables 9, 10 and 11, respectively.

Discussion and conclusions
In the present paper, the GA SAKe model is presented with examples of application to shallow landslides in the Sorrento Peninsula (Campania), and to the medium-size Uncino landslide at San Fili (Calabria).Furthermore, the capability of the model to simulate the occurrence of known landslide activations was evaluated by a progressive, self-adaptive procedure of calibration and validation against the Uncino case study.Finally, the sensitivity of the model with respect to the GA parameters was analyzed by a series of experiments, performed again by considering the latter landslide.
Regarding the Sorrento Peninsula case study, the maximum fitness obtained during calibration is smaller than unity.For the 100 best kernels, max , z cr and t b vary in a small range (ca.0.1, 4.8, and 13 %, respectively).Furthermore, as mentioned above, for specific types of application (e.g., civil protection), the observed small values of z cr would imply short warning times.Consequently, a suitable kernel should be rather selected by privileging the shortest t b or the smallest µ 0 .From Fig. 12, it can be noticed that the greatest weights for the first 12-15 days are obtained by selecting the kernel characterized by the smallest µ 0 , thus allowing for the most Table 9. Minimum ( i_min ) and maximum ( i_max ) numbers of GA iterations needed to reach max (only experiments with max = 1 are considered).In the first column, the letters refer to Fig. 19.In bold, the best and worst experiments.An asterisk marks the experiment e, in which max was reached only for p c = 75.In italics, the combinations of parameters of the benchmark experiment (cf.Table 4).§ N Parameter timely advice if used within an early-warning system.In the average kernel, the greatest weight can be attributable to the first 12 days, with a maximum base time of about 4 weeks, reflecting the general shape of the curves in Fig. 7, and in good agreement with the shallow type of slope instability considered.Furthermore, the validation of the average kernel is satisfactory, as the validation date (no.11 in Table 1) corresponds to the second highest peak of the mobility function.
In addition, no missing alarms and only four false alarms in about 5 years are to be found (i.e., in the period from the last date used for calibration to the one for validation).The peaks Table 10.Minimum (t b_min ) and maximum (t b_max ) base times of the average kernel (only experiments with max = 1 are considered).In the first column, the letters refer to Fig. 19.In bold, the best and worst experiments.An asterisk marks the experiment e, in which max was reached only for p c = 75.In italics, the combinations of parameters of the benchmark experiment (cf.   of the mobility function corresponding to the activation dates can roughly be grouped into two sets, characterized by distinct values: a first set, with z(t) > 40, generally includes the most ancient plus the validation dates (no. 1, no. 2, no. 4, no. 5, no.6, and no.11); a second set, with 18 < z(t) < 25, includes nos.3, 7, 8, 9, and 10.False alarms result more frequently and higher in the first period (from 1963 to 1980), presumably due to a lack of information on landslide activations.
Regarding the Uncino case study, the maximum fitness in calibration reaches unity.With respect to the Sorrento Peninsula case study, z cr and t b of the 100 best kernels vary in a greater range (ca. 25 and 30.5 %, respectively), with z cr 1 order of magnitude greater.In this case, the kernel would in fact allow for a safety margin of ca. 5 %.In the average kernel, three main periods can be recognized with heavier weights, attributable to (i) the first 17 days, (ii) 27-45 days, and (iii) 54-58 days.The base time ranges from about 8 to 12 weeks, in good agreement with the medium-size type of the considered slope instability.Furthermore, the validation of the average kernel was performed successfully: in fact, the validation date (no.6 in Table 2) corresponds to the third highest peak of the mobility function; even in this case, neither missing alarms nor false alarms in about 2 years (from the last date calibration date to the validation one) are to be found.The peaks of the mobility function corresponding to the activation dates are characterized by z(t) > 18.
In the self-adaptive procedure applied to the Uncino case study, values for L = 6 merely refer to calibration, whilst the ones for 2 ≤ L ≤ 5 concern validation.With regard to Table 7 and Fig. 20, it can be noticed that -for 2 ≤ L ≤ 5, t b increases 2.7 times with L, and then remains constant for L ≥ 5; -from L = 2 to L = 4, z j −min and z cr slightly decrease, and then abruptly increase for L ≥ 5; -for L ≥ 4, z cr monotonically increases 72 times with L (being almost constant in the 2-4 transition); v monotonically increases 1.7 times with L.
As a whole, a satisfying performance is obtained starting from three dates (i.e., correct predictions in more than three out of four times).For L = 5, only one false alarm is observed.Finally, the calibration performed by considering all six dates of activation provided fully satisfying results.Accordingly, the results of the progressive procedure underlined how GA SAKe can easily self-adapt to external changes by optimizing its performances, providing increasing fitness values.
The average kernels obtained by considering two to six dates of landslide activation show increasing base times, with significant weights for the most ancient rains of the temporal range (Fig. 18).Such a result is in good accordance with the extent of the slope movement and, therefore, with the expected prolonged travel times of the groundwater affecting landslide activation.
In the sensitivity analyses, again performed by considering the Uncino landslide, max = 1 was obtained in 60 % of the experiments (cf.Table 8).The results (cf.Fig. 19 and Tables 9, 10, and 11) permit one to select the set of parameters that allow for faster GA performances.More in detail, -a ratio between the number of elitist individuals and the whole population of n e /N = 10/20 or 8/15 allows for the fastest GA performances ( i_min ∼ 41 % of the reference value); nevertheless, for increasing both n e and N, this effect seems to vanish (e.g., n e /N = 12/25); -with respect to the benchmark experiment, the explored changes in p c , p m , p mh1 , p me , and p mtb do not substantially affect the GA performances with respect to  8.
-with respect to the benchmark experiment, the explored changes of parameters determine the variation of t b from 66 to 219 %; -in case of civil protection applications, the combination of parameters with p mh1 = 55 allows for activating early-warning procedures with the greatest advance; and -concerning z cr_max , the best result (increase by 10 times) is obtained when reducing N to 15.
The calibration experiments discussed in this paper were performed on a standard PC platform (3 GHz CPU, 4 GB RAM, SQL stand-alone system and application process).For the study cases of the Sorrento Peninsula and the Uncino landslide, 2.5 and 1.1 GA iterations were, respectively, performed per minute, reaching max in 11 h 40 m and 10 h 20 m.Depending on the availability of high-performance computing clusters, the mentioned durations may strongly be reduced, thus allowing for prompt civil protection applications, e.g., based on short-term weather forecasts.By the way, the time needed to calibrate the model can profitably be shortened by properly initializing the kernel, based on expected characteristics of the phenomena under consideration (e.g., the range of t b strongly depends on landslide size).
In this study, a two-step efficiency criterion was employed: the relative position of the peaks of the mobility function with respect to the dates of landslide activation was first considered, and the fitness computed.Based on the value of z cr , the obtained solutions were further ranked.Average, synthetic filter functions could then be computed by selecting the 100 best kernels for successive validation purposes.Alternative metrics (cf.among others, Krause et al., 2005) for the fitness function are being tested.However, due to uncertainties concerning input data (i.e., rainfall and dates of landslide activation), the adoption of sophisticated techniques does not sound very promising.In addition, problems of over-fitting may depend on both data uncertainties and the number of parameters.Commonly, kernels characterized by a complex pattern (and then by many parameters) are needed for simulating groundwater dynamics (Pinault et al., 2001).Nevertheless, more complex kernels do not necessarily imply higher predictive uncertainties (Fienen et al., 2010;Long, 2015).Still, the adopted discrete approach allows one to focus only on the timing of the peaks of the mobility function, thus somewhat relieving the computational effort.Due to the cited uncertainties in input data, a "temporal window" was in fact employed to help in matching dates of activation with the peaks of the mobility function.Further attempts at defining the fitness function by different metrics, and the analysis of its effects on calibration and validation, are being considered against another case study (San Benedetto Ullano, in Calabria, southern Italy), whose mobility phases have been recently monitored by the same authors (Iovine et al., 2010;Capparelli et al., 2012).
As mentioned above, model calibration may be hampered by either quality or completeness of input data.Commonly, missing dates of activation (mainly in remote periods or in isolated areas) and unsuitability of the rain gauge network (e.g., due to excessive distance of gauges from the landslides) negatively affect model results.Depending on the availability of new dates of activation, stemming from further mobilizations or improvement of historical investigations, the predictive capability of the model can be increased through additional calibrations, hence providing new families of optimal solutions, constituted by fewer, highly significant kernels.
The above considerations suggest an indirect link between the model -despite being empirical in type -and the physical characteristics of the slope movements (e.g., dimensions, permeability, initial water content of the slope, length of subsurface water paths).In general, to select the kernel to be applied, it is rather preferable to consider a set of optimal kernels or the average one, instead of a single solution.
Further efforts are in progress to improve the model and its chances of practical application, mainly concerning the implementation of different GA techniques of optimization (in addition to the elitist one, employed here), the parallelization of the model, and the adoption of a genetic programming approach.Finally, through the analytical study of the optimal kernels, a mathematical formulation of discrete filter functions is presently being attempted, aiming at synthesizing optimal and average kernels for an easier comparison with the results of other models available in the literature.funzionale per la previsione meteorologica e il monitoraggio meteoidro-pluviometrico e delle frane".
Finally, we thank the Editors and two anonymous Referees for their constructive comments and insights that allowed us to considerably improve the manuscript.
Edited by: J. Neal

Figure 1 .
Figure 1.Scheme of the calibration procedure of the GA SAKe model.

Figure 2 .
Figure 2. Scheme of the adopted genetic algorithm.

Figure 3 .
Figure 3. Example of crossover.The genetic codes of the parents (elements in orange and green) are first mixed; then, the children are normalized (black elements) to ensure the validity of Eq. (2).

Figure 4 .
Figure 4. Examples of mutation.On the left, the genetic code of the parent individual (elements in blue).In the second histogram, mutation is applied to some elements of the parent (in red, added amounts; in grey, subtracted amounts).Then, the base time can either be decreased (upper sequence) or increased (lower sequence).Finally, the children are normalized (black elements) to ensure the validity of Eq. (2).

Figure 5 .
Figure 5. Scheme of the validation procedure of the GA SAKe model.
from 01 September 1959 to 31 August 1980; for validation, it ranges from 01 September 1980 to 31 March 1981 -i.e., including the validation date by ±t b (this latter as obtained from calibration).

Figure 10 .
Figure 10.Sorrento Peninsula case study.Average kernel obtained from the 100 best filter functions.

Figure 11 .
Figure 11.Sorrento Peninsula case study.Mobility function, z(t), of the average kernel.The red line (z cr = 22.53) shows the maximum value of the mobility function (critical condition) that is unrelated to known landslide activations.The green line (z j −min = 22.63) -almost overlapping with the red line in this case -shows the minimum value of the mobility function related to known landslide activations.When the mobility function exceeds the threshold marked by the red line, landslide activation may occur.The red dots represent the maxima of the mobility function corresponding to the dates of landslide activation considered for calibration.

Figure 13 .
Figure 13.Sorrento Peninsula case study.(a) Validation of the average kernel against the no.11 event.(b) Particular of (a), limited to the period ±t b , including the date of validation.Key as in Fig. 11.The blue label indicates the date of validation.Grey background marks the period after the event that may be employed for re-calibration.

Figure 14 .
Figure 14.Uncino landslide case study.Average kernel obtained from the 100 best filter functions.

Figure 15 .
Figure 15.Uncino landslide case study.Mobility function, z(t), of the average kernel.The red line (z cr = 17.85) shows the maximum value of the mobility function (critical condition) that is unrelated to known activations.The green line (z j −min = 18.98) shows the minimum value of the mobility function related to known activations.When the mobility function exceeds the threshold marked by the red line, landslide activation may occur.The red dots represent the maxima of the mobility function corresponding to dates of landslide activation considered for calibration.

Figure 16 .
Figure 16.Uncino landslide case study.Kernel providing the best fitness.

Figure 17 .
Figure 17.Uncino landslide case study.(a) Validation of the average kernel against the no.6 event.(b) Particular of (a), limited to the period ±t b including the date of validation.Key as in Fig. 15.The blue label indicates the date of validation.Grey background marks the period after the event that may be employed for re-calibration.
) and maximum ( z cr_max ) safety margins of the average kernel (only experiments with max = 1 are considered).In the first column, the letters refer to Fig.19.In bold, the best and worst experiments.An asterisk marks the experiment e, in which max was reached only for p c = 75.In italics, the combinations of parameters of the benchmark experiment (cf.

Figure 19 .
Figure 19.Maximum fitness ( max ), safety margin ( z cr ), number (n i ) of iterations needed to first reach max , and base time (t b ) of the average kernel, based on GA parameters listed in Table8.

Figure 20 .
Figure 20.Uncino landslide case study.Results of progressive calibration.Variation of z cr and v for L increasing from 2 to 6.

Table 1 .
Dates of activation of the shallow landslides in the Sorrento Peninsula.Key: date: day of occurrence; type: widespread (multiple)

Table 3 .
Average monthly rainfall and number of rainy days at the Montalto Uffugo rain gauge (468 m a.s.l.).

Table 4 .
Values of the parameters of GA SAKe adopted in the calibration procedure (benchmark experiment).

Table 5 .
Sorrento Peninsula case study.Statistics for the 100 best kernels.

Table 6 .
Uncino landslide case study.Statistics for the 100 best kernels.

Table 8 .
Uncino landslide case study.Values of the parameters adopted in the sensitivity analyses.In bold, the experiments with max = 1.In italics, the worst experiment.Underlined, the best one.Reference values (i.e., those of the benchmark experiment -cf.Table4). *