Diversity of Rainfall Thresholds for early warning of hydrogeological disasters

For early warning of disasters induced by precipitation (such as floods and landslides), different kinds of rainfall thresholds are adopted, which vary from each other, on the basis on adopted hypotheses. In some cases, they represent the occurrence probability of an event (landslide or flood), in other cases the exceedance probability of a critical value for an assigned indicator I (a function of rainfall heights), and in further cases they only indicate the exceeding of a prefixed percentage a critical value for I , indicated as Icr. For each scheme, it is usual to define three different criticality levels (ordinary, moderate and severe), which are associated to warning levels, according to emergency plans. This work briefly discusses different schemes of rainfall thresholds, focusing attention on landslide prediction, with some applications to a real case study in Calabria region (southern Italy).

Focusing the attention on Italy, in February 2016 the Department of Civil Protection published the operational guidance to unify alert messages related to EWSs of all the Italian Regions (http://www.protezionecivile.gov.it/jcms/it/view_prov.wp?request_locale=it&contentId=LEG56184).The document unifies the criticality and alert levels: both kinds of levels are now characterized by color codes (yellow, orange and red) and, for each of them, the associated event and risk scenarios are described.Nevertheless, no indication is provided concerning the identification of rainfall thresholds, to be used for estimating the criticality levels; consequently, each Regional Administration could have its own threshold scheme inside its EWS, and this obviously implies heterogeneity at national scale.In details, the increase of the criticality levels can be due to: the occurrence of events (heavy rainfalls, floods or landslides) with increasing characteristics in intensity, persistence along time and/or spatial extension.The refer- Published by Copernicus Publications on behalf of the European Geosciences Union.
ence spatial scale is the alert zone (the Italian territory is subdivided in 137 alert zones) and the EWS generally includes rainfall forecasting based on meteorological models; an increase of probability associated to the event occurrence.The reference spatial scale is the municipal one, and the data recorded in real time by local monitoring networks are also considered.
Moreover, the concept of rainfall threshold is not associated with only one scheme; several approaches can be defined (Chacon et al., 2006;Guzzetti et al., 2008), with significant differences among them.This paper provides an overview of various schemes for rainfall thresholds, and a classification is proposed.

Schemes for rainfall thresholds
In general, it is possible to define an indicator I , which is a function of rainfall heights (observed and predicted) and can be correlated with the occurrence probability of an event E, which is indicated as P [E]: this is a conditional probability and it is also written as P [E|I ].Examples of I are: (1) discharge estimated with a Rainfall-Runoff model; (2) Safety Factor from a slope stability model; (3) the mean value of rainfall intensity on assigned durations, e.g. the classical Intensity-Duration (ID) schemes, adopted for landslide forecasting (Guzzetti et al., 2008).As a first aspect, the model adopted for defining the indicator can provide, for each forecasting interval, a single value of I (Fig. 1a) or a probability distribution F (I ), connected to the imperfect knowledge of model parameters (Fig. 1b).The second distinction regards the probability P [E|I ], that can be generally expressed as (Fig. 1c): where g(.) is a monotonic non decreasing function that assumes values between 0 and 1 into the interval [I 1 ; I 2 ]; for I < I 1 the event is impossible, while it is certain when I > I 2 .A simplification of Eq. ( 1) is represented by the step function (Eq.2), obtained by assuming a critical value I cr = I 1 = I 2 (Fig. 1d): The criterion for choosing I cr greatly varies: for landslide prediction, as examples Godt et al. (2006) assume a lower limit above to which all the historical landslide events occurred, Cannon et al. (2008) set the threshold equal to an upper limit for which no False Alarm (see Sect. 3 for its definition) occurred.Moreover, I cr can be chosen in order to minimize the sum of Missed and False Alarms (see Sect. 3 for their definition), by using specific optimization procedures (Bennet et al., 2013).A third and further distinction for the identification of rainfall thresholds regards the use of only the rainfall observed in real time (Fig. 1e), or also the predictions from meteorological and/or stochastic models (Fig. 1f), that usually provide an ensemble of realizations which can be or not equiprobable.Overall, eight different schemes of rainfall thresholds can be adopted into an EWS: each one enables to derive values that set three different criticality levels (ordinary, moderate and severe), denoted in the following as CL 1 , CL 2 and CL 3 .As examples, in this work authors analyze four schemes of threshold.
The simplest one, indicated as Scheme A, (Fig. 2a), considers a deterministic value of I , defines the probability P [E|I ] on the basis on Eq. ( 2) and uses only the rainfall heights observed in real time.In this case, only the comparison between I and I cr is allowed, and CL 1 , CL 2 , CL 3 can be assumed as prefixed percentages (indicated as ξ 1 , ξ 2 and ξ 3 ) of I cr .It should be observed that, on the basis of Eq. ( 2), the occurrence probability P [E|I ] associated to CL 1 , CL 2 , CL 3 is zero, and consequently this scheme can be used when the increase of a critical level indicates that the event is very close to occur.
The Scheme B (Fig. 2b) uses Eq. ( 1) for P [E|I ] and, like Scheme A, considers a deterministic value for I and only rainfall heights observed in real time.Specific values of P [E|I ], denoted as P 1 , P 2 , P 3 and associated to CL 1 , CL 2 , CL 3 , respectively, correspond to particular occurrence probabilities for a landslide.
Unlike the previous schemes of threshold, the Scheme C (Fig. 2c) takes into account the imperfect knowledge of parameters (which are assumed as random variables with assigned probability distributions, Biondi and De Luca, 2015) related to the adopted model for I , and then it provides a probability distribution F (I ) as output.The probability P [E|I ] is defined by Eq. ( 2), and, also in this case, only observed rainfall heights are used.This scheme provides as a result the exceedance probability for I cr , and therefore CL 1 , CL 2 , CL 3 can be associated to prefixed values of P [I > I cr ], which corresponds, on the basis of Eq. ( 2), to the occurrence probability of a landslide P [E].
Lastly, the Scheme D (Fig. 2d) constitutes a generalization of the previous Scheme C, because it uses (together with rainfall heights observed in real time) rainfall predictions derived from meteorological and/or stochastic models, consisting of an ensemble of realizations, which induces an ensemble of probability distributions for I .Consequently, P [E] is computed by using the theorem of total probability: where k is the total number of rainfall realizations, F i (I ) is the probability distribution associated to the ith realiza-   slide crown zone, and it produced saturation of pelitic terrains derived by weathered phyllite in some parts of the landslides body.
The rain gauge located at Tiriolo (see De Luca and Versace, 2017, for further details) was considered for rainfall analysis; authors used for application the daily time series, available from 1941, and the 20-min time series, available from 1992.All rainfall data were downloaded by the official website of the Multi Risk Center of Calabria region (www.cfd.calabria.it).
In details, FLaIR model (Forecasting of Landslides Induced by Rainfalls, Capparelli and Versace, 2011) was used as indicator, defined as a convolution between the rainfall Intensity R(.) and a filter function ψ (.), also named as transfer function: where M represents the temporal memory of the process, and it depends on parameters of adopted filter ψ (.).In Eq. ( 4) the dimensions are [L/T ] for R(.), [T −1 ] for ψ (.), and [L/T ] for I (.).Authors used a mixture of exponential filter as ψ (.) (De Luca and Versace, 2017), in order to better discriminate the influence of event and antecedent rainfall in the physical process of landslide trigger.
For this case study authors only considered rainfall heights observed in real time, and consequently results of FLaIR application regard Schemes A, B and C. For all schemes, a calibration period is related to the interval 1940-2006 (with 20  landslide events), while 2007-2011 (with 7 landslide events) is the validation period.Starting from the contingency table (Fawcett, 2006;Barnes et al., 2009;Staley et al., 2013), which is composed by the number of: -Correct Alarms (CA, also named True Positive, TP, i.e. historical landslides occurred when model output is above the threshold), -Missed Alarms (MA, also named False Negatives, FN, i.e. landslides occurred but not predicted by the model), -False Alarms (FA, also named False Positives, FP, i.e. landslides predicted by the model but not occurred), -True Negatives (TN, i.e. model outputs are below the threshold and no landslide happened), the metric Threat Score (TS, also named Critical Success Index, CSI, Gilbert, 1884;Schaefer, 1990): was adopted as index for evaluating the performance of the considered threshold schemes.Similar results were obtained by using the index named as True Skill Statistic (TSS, Hanssen and Kuipersm, 1965;Peres and Cancelliere, 2014;Gariano et al., 2015).Concerning calibration step (for further details, see Versace and De Luca, 2017): -I cr estimation provided a value equal to 17 mm day −1 for Scheme A; starting from results of Scheme A, the probability P [E|I ] for Scheme B was evaluated in the following way: for different values of I assumed as critical value, landslide probability is computed as ratio between CA and the number of model outputs above the threshold (i.e.CA + FA); www.adv-geosci.net/44/53/2017/Adv.Geosci., 44, 53-60, 2017 For validation period, the obtained results are shown in Fig. 3 and in Table 2.The temporal behavior of the dimensionless index I /I cr is reported in Fig. 3 (histograms related to the left vertical axis), which also shows P [E|I ] (Scheme B) and P [I > I cr ] (Scheme C) along the time (lines related to the right vertical axis).For Scheme A, the estimated threshold I cr allows for reproducing all the historical events as CA, with FA = 6 and TS = 0.54: in fact, from Fig. 3 it is evident that all the landslide events (red histograms) present a dimensionless index greater than one.From a comparison between results for Schemes A and B (by analyzing Table 2), it should be noted that the threshold I cr of Scheme A is not usually associated to high values of P [E|I ]; I cr typically assumes low values in order to reproduce many landslide events as CA, but it can also imply an increase of FA (and, consequently, a decrease of the associated probability of landslide occurrence).Conversely, the use of scheme B requires attention on the choice of P [E|I ] associated to the critical level CL 3 , which is the most severe: an increase of probability clearly induces a reduction of FA, but it also implies an increase of MA, which could be significant.The maximization of a metric like TS could help the choice of the optimal values for P 1 , P 2 , and P 3 (Fig. 2b).More specifically, I cr of Scheme A corresponds to a probability P [E|I ] of Scheme B equal to 0.2 (Table 2), and in validation period all the landslide events are associated to P [E|I ] values (represented by solid black lines in Fig. 3) which do not exceed 0.5.Concerning the differences among probability values in Schemes B and C (represented in Fig. 3 by solid black and dashed grey lines, respectively), the diverse meaning of probability in these approaches should be considered: (1) Scheme B uses a nonstep function (Eq.1); (2) Scheme C adopts a step function (Eq.2) and takes into account the imperfect knowledge of filter parameters.Therefore, in the former, P is the probability of landslide occurrence depending on the current value of I (by considering the unique parameters set of FLaIR model which is estimated in Scheme A), while in the latter it is the probability for which I > I cr (estimated with the procedure indicated in the third point of the previous bulleted list, i.e. by generating 2000 sets of filter parameters of FLaIR model and then estimating the specific I cr for each set).Consequently, it is expected that Scheme C provides high values of probability in correspondence of landslide events: for the selected case study P [I > I cr ] is greater than 0.9 (see dashed grey lines in Fig. 3).For both Schemes B and C, the optimal values for the threshold probabilities P 1 , P 2 , and P 3 (Fig. 2b and  c) can be carried out by analysis of the metric TS (Table 2).

Conclusions
A scheme of rainfall threshold has to be carefully defined, in order to avoid confusion in its use in EWSs.The main distinction regards deterministic and probabilistic schemes.For landslide prediction, examples of deterministic approaches (like Scheme A in this paper) are the classical Intensity-Duration (ID) models (Guzzetti et al., 2008) or FLaIR model (Capparelli and Versace, 2011).Instead, probabilistic thresholds consider a generic monotonic and non-decreasing function for modelling the occurrence probability of a landslide (Scheme B), or take into account the imperfect knowledge of parameters of the adopted model for the indicator I (like Schemes C and D), or use an ensemble of rainfall predictions from meteorological and/or stochastic models (Scheme D).In any case, the use of suitable metrics like Threat Score (TS) allows for evaluating performance in terms of Correct, Missed and False alarms, and consequently for choosing the optimal values for the criticality levels CL 1 , CL 2 and CL 3 .
For the selected case study, threshold value I cr of Scheme A corresponds to a low probability of landslide occurrence (0.2).The probability distribution, derived from Scheme B, provides much more information, because it allows to associate different values of I (t), assumed as I cr , to occurrence probability of a landslide.Finally, from Scheme C it is possible to evaluate the probability associated to the exceedance I (t) > I cr , by considering filter parameters as random variables: P [I > I cr ] is very different from that obtained in Scheme B, and then attention is necessary in making decisions on the basis of probability values.

Figure 1 .
Figure 1.Several aspects characterizing a scheme for rainfall thresholds.Definition of I : (a) deterministic value; (b) probability distribution F (I ).Modelling of P [E|I ]: (c) generic monotonic non-decreasing function; (d) step function.Use of rainfall input: (e) only data observed in real time; (f) also prediction from meteorological and/or stochastic models.

Figure 2 .
Figure 2. Examples of rainfall thresholds.(a) Scheme A: use of only rainfall data observed in real time, model providing a deterministic value for I , adoption of a step function for P [E|I ].(b) Scheme B: use of only rainfall data observed in real time, model providing a deterministic value for I , adoption of a generic monotonic non-decreasing function for P [E|I ].(c) Scheme C: use of only rainfall data observed in real time, model providing a probability distribution for I , adoption of a step function for P [E|I ].(d) Scheme D: use of rainfall predictions together with rainfall height observed in real time, model providing a probability distribution function for I , adoption of a step function for P [E|I ].

Figure 3 .
Figure 3. Application of FLaIR model -temporal behavior of I /I cr (Scheme A), P [E|I ] (Scheme B) and P [I >I cr ] (Scheme C) in 2007-2011 for Gimigliano municipality (southern Italy).

Table 1 .
Gimigliano: list of the selected 27 landslide events.

Table 2 .
Gimigliano municipality (southern Italy): validation results(period 2007-2011)for Schemes A, B and C. I cr was computed by adopting the same procedure used for Scheme A (Versace and De Luca 2017).For each day, P [I > I cr ] was set equal to the ratio between the number of simulations for which I > I cr and the total number of simulations.