A nationwide investigation, carried out by the National Geological Survey, identified approximately 5 × 105 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 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 landslides – as 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 GASAKe (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 (CMSAKe – 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 shallow-slope 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 GASAKe for civil protection purposes. Moreover, a progressive, self-adaptive 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.

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 instruments 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, 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., 2007, 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 (zt), while they do happen above it. Alternatively, a range of conditions can be defined (Crozier, 1997), delimited by

a lower threshold (zlow), below which changes of state never occur, and

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” (tb) expresses a sort of memory with respect to rainfall: in classic rainfall–runoff modeling, tb 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.

GASAKe 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 ∫0∞htdt=1, 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 (tb). The integral of convolution is therefore expressed as zt=∫0tbht-τpτdτ=∫0tbhτpt-τdτ, 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 Δt and height hi, and Eq. (3) can be written as zu=∑i=1uhi⋅pu-i+1⋅Δt.

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 ht=atb, with b<0. Exporting the well-established 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 geo-environmental 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 (dt-from) to a final date (dt-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 tb are neglected. Suitable maximum and minimum values (tb-max and tb-min) have to be initialized to allow the model to determine optimal values. Commonly, tb 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).

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.

GASAKe 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.

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 GASAKe 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 self-adaptive 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 (Δzcr), 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 analyses was performed by considering again the Uncino case study. The experiments carried out are listed in Table 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 (Φmax), safety margin (Δzcr), number (ni) of iterations needed to first reach Φmax, and base time (tb) of the average kernel are shown in Fig. 19. If experiments with Φmax=1 are only taken into account, the minimum and maximum numbers of GA iterations needed to reach Φmax (Λmin⁡, Λmax⁡), the minimum and maximum base times of the average kernel (tb_min, tb_max), and the minimum and maximum safety margins of the average kernel (Δzcr_min, Δzcr_max), are listed in Tables 9, 10 and 11, respectively.

In the present paper, the GASAKe 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, Φmax, Δzcr and tb 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 Δzcr would imply short warning times. Consequently, a suitable kernel should be rather selected by privileging the shortest tb or the smallest μ0. 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 μ0, thus allowing for the most 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 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, Δzcr and tb of the 100 best kernels vary in a greater range (ca. 25 and 30.5 %, respectively), with Δzcr 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, tb increases 2.7 times with L, and then remains constant for L≥5;

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, Φmax⁡= 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 ne/N=10/20 or 8/15 allows for the fastest GA performances (Λi_min∼41 % of the reference value); nevertheless, for increasing both ne and N, this effect seems to vanish (e.g., ne/N=12/25);

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 Φmax⁡ 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 tb 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 Δzcr, 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.