Multivariate assessment of droughts: Frequency analysis and dynamic return period



[1] Droughts, like floods, are extreme expressions of the river flow dynamics. Here, droughts are intended as episodes during which the streamflow is below a given threshold, and are described as multivariate events characterized by two variables: average intensity and duration. In this work, we introduce the new concept of Dynamic Return Period, formulated using the theory of Copulas, and calculated via a Survival Kendall's approach. We show how it can be used (i) to monitor the temporal evolution of a drought event, and (ii) to perform real time assessment. In addition, a randomization strategy is introduced, in order to get rid of repeated measurements, which may adversely affect the statistical analysis of the available data, as well as the calculation of the return periods of interest: a practical example is shown, involving the fit of the drought duration distribution. The case study of the Po river basin (Northern Italy) is used as an illustration.

1. Introduction

[2] Droughts are a natural phenomenon related to a below average water availability, perhaps lasting a few weeks, or even months, over a region, in terms of rainfall, discharge or groundwater, and they can occur in any region of the planet.

[3] Droughts can be classified into [Dracup et al., 1980a; Beran and Rodier, 1985; Wilhite and Glantz, 1987]: (i) hydrological, if they involve periods of below normal flow and depleted reservoir storage; (ii) meteorological, relatively, to below normal precipitation; (iii) agricultural, if the soil moisture is not sufficient to support crop growth; and (iv) socio-economical, when the low water supply affects society's productive and consumptive activities. Operational definitions are based on the main features of a drought, like onset, termination, duration, severity, and intensity [Wilhite and Glantz, 1987].

[4] Yevjevich [1967] introduced a threshold procedure, denominated run method, to identify a hydrological drought: accordingly, a drought (negative run) is characterized by a duration, i.e., the time interval during which the discharge is below a fixed threshold, and a severity, defined as the cumulative volume deficit with respect to the fixed threshold. The ratio between severity and duration is the average drought intensity. A positive run, conversely, is a period during which the discharge is above the fixed threshold. The run method has been widely applied to daily, as well as to yearly, data series. The threshold is assumed to be equal to a given percentile of the flow-duration curve (generally, the 95-, 90-, 80-, 70-, or 60- percentile). Among others, see Sen [1977]; Dracup et al. [1980b]; and Clausen and Pearson [1995] for the application of the run method at a yearly time scale, and Zelenhasić and Salvai [1987]; Tallaksen et al. [1997]; Demuth and Külls [1997]; Demuth and Heinrich [1997]; Kjeldsen et al. [2000]; Engeland et al. [2004]; Byzedi and Saghafian [2009]; and Byzedi [2010] at a daily time scale.

[5] However, the application of the run method to subyearly time scales (e.g., daily) requires a particular attention, due to the possible presence of mutually dependent droughts, i.e., to the possibility that a long negative run is interrupted by short positive ones, and thereby a long drought turns out to be split into a number of shorter ones.

[6] Zelenhasić and Salvai [1987] have modified the run method, to cope with the possible dependence of drought periods, introducing two additional conditions: (i) two consecutive drought events, separated by a relatively short interval (interevent time) during which the flow is slightly above the threshold, have to be considered as just one drought event with duration and severity equal to the sum of the two events duration and severity, respectively; (ii) all droughts with severity less than 0.5% of maximum severity are neglected. Similar investigations are carried out by Tallaksen et al. [1997]; Demuth and Külls [1997]; Kjeldsen et al. [2000]; Engeland et al. [2004]; Byzedi and Saghafian [2009]; and Byzedi [2010].

[7] Until the end of the 1990s, the frequency analysis of drought variables, like the calculation of the return period (hereinafter, RP), has been principally approached using a univariate frame, by calculating the probability distributions of duration, severity, and intensity, and by considering these variables as independent. However, the drought variables are generally dependent on one other, and a proper frequency analysis of droughts should take into account such dependencies within a suitable multivariate framework. Since the introduction of Copulas in hydrological practice [see e.g., Salvadori et al., 2007, for a list of references], multivariate frequency analyses of droughts have been proposed in several works: among others, Shiau [2006]; Serinaldi et al. [2009]; Shiau and Modarres [2009]; Kao and Govindaraju [2010]; Wong et al. [2010]; Song and Singh [2010]; Mirabbasi et al. [2012]; Reddy and Ganguli [2012]; Ganguli and Reddy [2012], and also Mishra and Singh [2011] for a review. With the exception of Song and Singh [2010], these papers investigate the frequency of droughts by referring to meteorological droughts, as identified principally via the Standardized Precipitation Index. Instead, Song and Singh [2010] have addressed hydrological droughts with a trivariate analysis of duration, severity, and interarrival time, as identified by means of a threshold on the flow discharge. A collection of recent papers dealing with different approaches to droughts can be found in VV. AA. [2013], notably the paper by Maity et al. [2013].

[8] In this paper, we focus the attention on hydrological droughts, since the flow discharge at a river section allows to quantify the available water resource, while the water deficiency allows to evaluate the drought impact on human activities. In particular, we present a multivariate frequency analysis, based on the theory of Copulas, characterizing the drought episodes by means of two variables calculated via the run method, with a daily time resolution: viz., the drought duration and the drought average intensity.

[9] In addition, we introduce the new concept of Dynamic Return Period—based on the Copula formalism, and calculated via a Survival Kendall's approach—and we show how it can be used (i) to monitor the temporal evolution of a drought event, and (ii) to perform real time assessment of a drought. The relative importance of the two main variables (i.e., average intensity and duration) in ruling the drought dynamics is also stressed and discussed.

[10] Furthermore, a randomization strategy is introduced, in order to get rid of repeated measurements, which may adversely affect the statistical analysis of the available data, as well as the calculation of the RP's of interest: a practical example is shown, involving the fit of the drought duration distribution.

[11] The paper is organized as follows. In Section 2, we present the database and the case study. In Section 3, a randomization procedure is introduced, in order to get rid of the repeated measurements, possibly affecting both the univariate and the multivariate analyses. In Section 4, the univariate frequency analysis is presented, including the discussion of the outcomes of the randomizations. In Section 5, the multivariate frequency analysis is sketched, and the notion of Survival Kendall's Return Period is introduced. In Section 6, the concept of Dynamic Return Period is presented, and practical applications are shown. Finally, in Section 7, a discussion of the results is presented, and suitable conclusions are drawn.

2. The Database

[12] The Po river is the largest Italian basin, covering an area of 74,000 km2 (70,000 km2 in Italy and 4000 km2 in Switzerland and France), characterized by a main river length of 650 km. The Po river basin is a strategic area for the Italian economy, producing 40% of the national gross domestic product, and having a population of more than 16 million. Water uses concern industrial activities (principally, the electricity sector, with 48% of the national hydroelectric production, and 31% of the thermo-electric production), agricultural productions, livestock, and inland navigation. As a consequence, drought events can provoke serious economic damages.

[13] In this work, we use the time series of daily discharge data collected at Pontelagoscuro, the section closest to the outlet into the Northern Adriatic Sea: data are available from 1923 to 2007. The reason for selecting this particular station is that the water management strategies of the “Control Room” (CR) of the Watershed Council are principally based upon the data collected at Pontelagoscuro [AA. VV., 2011]. The mean daily discharge at Pontelagoscuro is inline image, the minimum is inline image, and the maximum is inline image.

[14] In the following, we consider a hydrological drought as an episode, or a period of time, during which the discharge values are smaller than a given threshold [Yevjevich, 1967; Zelenhasić and Salvai, 1987]—for other approaches see e.g., the literature cited in the Introduction. Each episode is characterized by two variables: the Duration D (in days), and the average intensity I (in m3/s), defined as the average of all the volumes below the threshold (i.e., the average of all the differences between the threshold and the actual daily streamflows). Note that all the analyses illustrated below were also carried out by considering both the maximum drought severity and the maximum drought intensity. As a result, the average intensity turned out to be the variable of most interest: on the one hand, it provides valuable information about the overall drought features, useful for the CR purposes (in fact, both the maximum severity and the maximum intensity only provide restricted information associated with a particular time instant of the drought); on the other hand, it represents a variable consistent with the one used for the calculation of the Dynamic Return Period (see Section 6 later).

[15] For the sake of coherency with the policy of the CR [AA. VV., 2011], for the selection of the drought events of interest, we use here the discharge threshold “Q300,” equal to 824 m3/s, indicated by the CR as the “alarm threshold.” In addition, following Zelenhasić and Salvai [1987] and AA. VV. [2011], we use a minimum interevent time of 3 days, and a minimum drought duration of 5 days. The total number of drought episodes extracted is N = 276.

3. Randomization

[16] Before proceeding with the analyses, it is important to realize that, in principle, both I and D describe continuous phenomena (viz., a discharge and a duration). Unfortunately, due to the data collection procedure, both the values of I and D are often a discretized version of the actual continuous measurements of these variables: viz., they are rounded to an integer number. As an illustration, Figure 1 (right) shows a zoom of the Pontelagoscuro series in the (I, D) region inline image: the discretized observations are well visible (viz., the straight horizontal and vertical lines of markers). Similarly, straight vertical lines of markers are also evident in both the panels of Figure 2, showing the empirical univariate distributions of the data. The repeated values are called “Ties” in the statistical jargon. A quick survey of the whole database shows that only a few average intensity values are repeated, whereas about 75% of the duration data are ties. In this work, we consider both I and D as continuous variables: in fact, such an assumption makes both the univariate and multivariate analyses less troublesome (see Sections 4 and 5 below). Therefore, the ties must be accounted for in a reasonable way.

Figure 1.

Original Pontelagoscuro data—see text. (left) Bivariate plot of average intensity and duration data; the empty markers (circle, triangle, and square) correspond, respectively, to the 1943s, 2003s, and 2006s drought episodes discussed in Section 6. (right) Zoom of the (I, D) region inline image.

Figure 2.

Univariate fits of the Pontelagoscuro data—see text. The dashed lines indicate the distributions chosen via the AIC by considering the original (nonrandomized) data, and the corresponding ML fits. The continuous lines indicate the distributions chosen via the AIC by considering the randomized data, and the corresponding ML fits. (left) Average intensity I. (right) Duration D.

[17] Now, let X be the factual value of the phenomenon under investigation (e.g., the drought duration), and let Δ be the resolution adopted. In the present case, the choice of Δ is quite a natural one: in fact, observations of D are available on a daily basis, and thus inline image day, while observations of I are available with a resolution of 1 m3/s, and thus inline image.

[18] Then, X belongs to a unique interval inline image for a suitable integer kX, but the true value X is incorrectly stored in the database as, say, inline image. Clearly, all the actual (and almost surely different) occurrences falling into the interval inline image are wrongly recorded as X*, and thus artificial repetitions (viz., the ties) are introduced in the database. Not only this may adversely affect the statistical analysis of the data (see the discussion of Figure 2 later), but, even worse, it can make the multivariate analysis ambiguous and questionable, since the ranks of the data cannot be defined uniquely [Genest and Favre, 2007; Genest and Neslehová, 2007].

[19] A possible way to circumvent the problem is to use suitable randomization techniques (also known as “Jittering” [Chambers et al., 1983]): for a viable solution and a hydrological application see Vandenberghe et al. [2010]. Here, we adopt a practical and physically based strategy to reintroduce in inline image all the repeated common values X*, by transforming each available inline image as inline image, where the Ui's are i.i.d. r.v.'s Uniform over (0,1): clearly, inline image. Thus, inline image may represent a “copy” of the original continuous observation Xi, wrongly recorded as a (repeated) discrete random variable inline image. Note that the Uniform law is “noninformative,” and may represent the best choice when, as in the present case, no information is available about the true distributions of the drought variables.

[20] Obviously, inline image cannot be expected to have the same (unknown) probability law as of Xi. However, if Δ is small with respect to the range of X, we may hope that inline image be “statistically equivalent” to Xi. We shall see later in Sections 4 and 5 the possible ways to quantify how the strategy proposed here affects the distribution of the randomized inline image's.

[21] The randomization procedure described above poses interesting questions. In fact, for the sake of statistical consistency, several independent randomized samples should be generated and considered: in this work, we use NR = 400 independent randomizations of the pairs (I, D). Clearly, any simulated sample corresponds to a possible drought “scenario.” In turn, the parameters of the distributions (univariate and multivariate) fitted on the randomized data would almost surely vary by considering different samples, giving the possibility (i) to properly investigate the variability of the phenomenon, and (ii) to provide, e.g., mean estimates, as well as corresponding (empirical/Monte Carlo) confidence intervals, of the quantities of interest: this may help in quantifying the uncertainties associated with the procedure (see Sections 4 and 5 below).

4. Univariate Frequency Analysis

[22] In this work, we test 6 standard probability distributions as univariate marginals for the average intensity I and the duration D (see Table 1). The corresponding parameters are estimated via the maximum likelihood (ML) method: this in order to be coherent with the fitting strategy of the copula modeling the joint behavior of the bivariate data—see Section 5 below. Here, the background assumption is that the drought phenomenon under study is, to some extent, stationary, otherwise a much more involved statistical approach would be needed.

Table 1. List of the 6 Univariate Distributions (1-D) and 12 Bivariate Copulas (2-D) Fitted Over the Available Data—See Texta
  1. a

    The “Clayton,” “Gumbel,” and “Frank” labels denote the corresponding families of Archimedean 2-copulas [Nelsen, 2006; Salvadori et al., 2007]; the labels “Mix[AB]” denote a convex mixture C of the two families A and B indicated (i.e., inline image with inline image); the labels “X[AB]” denote the Khoudraji-Liebscher extraparametrization C of the two families A and B indicated [Durante and Salvadori, 2010; Salvadori and De Michele, 2010].

Generalized ParetoLogNormalWeibull

[23] As a model selection strategy, we adopt the (corrected) Akaike Information Criterion (AIC) [Burnham and Anderson, 2002; Claeskens and Hjort, 2008]. In addition, the Kolmogorov-Smirnov (KS) goodness-of-fit test is used to check the appropriateness of the chosen distributions: here we calculate approximate p-Values using simulation/Monte Carlo procedures [Davison and Hinkley, 1997], since the parameters of the probability laws of interest are estimated using the available data.

[24] Considering the average intensity I, the Gamma distribution is always selected as the best AIC model, over all the NR randomizations. Note that the Bayesian information criterion yields the same results as of the AIC. Actually, Figure 2 (left) shows that the Gamma law fitted over the original (nonrandomized) data is practically the same as the ones fitted over the randomized data: apparently, the few ties present in the database do not affect the fitting procedure. Finally, the KS test indicates that the fitted Gamma distributions can always be accepted, since the approximate p-Values are larger than 10% in all cases.

[25] Similarly, considering the duration D, the Weibull distribution is always selected as the best AIC model, over all the NR randomizations (see Figure 2 (right)). Note that the Bayesian Information Criterion yields the same results as of the AIC. However, if the original (nonrandomized) data were considered (instead of the randomized ones), the best AIC model turns out to be of a Gamma (instead of a Weibull) type. Yet, the fitting performance of the Gamma distribution over the original data is quite poor, as shown in Figure 2 (right): evidently, the presence of quite a few ties may definitely affect the fitting procedure. As a consequence, if the “wrong” Gamma distribution were used for the calculation of the univariate RP's of D, then unreliable estimates were almost surely generated. In turn, apparently, this supports the use of suitable randomization techniques in order to circumvent the problem of ties. Finally, the KS test indicates that the fitted Weibull distributions can always be accepted, since the approximate p-Values are larger than 10% in all cases.

[26] As already mentioned in Section 3, it is interesting to study how the randomization procedure affects the statistics of the data. For this purpose, we use an indirect approach, since neither the probability distributions of the factual observations, nor the ones of the discretized data, are known a priori. Thus, it may be more appropriate to simply study the statistical behavior of the estimates of the fitted distributional parameters, calculated over all the NR randomizations. Such a study may provide a hint of how the statistics of the data are affected by the artificial reintroduction of the repeated measurements. As a result, in all cases, the ML estimates of the parameters are always well centered around specific sample means, with very small sample variances. In other words, apparently, the randomization procedure does not significantly spoil the univariate statistics of the data. As a consequence, from a practical point of view, we may consider the randomized samples as valuable “copies” of the true (but unknown) nondiscretized data, to be used for performing further analyses.

5. Multivariate Frequency Analysis

[27] As is customary in hydrological applications, here we consider the notion of RP as the one used for the identification of dangerous events: thus, the “failure region” depends upon the chosen RP, and we use the prescribed RP as a regulation parameter to discriminate between “safe” and “dangerous” occurrences. Below we show how, using a suitable notion of multivariate RP, it is possible to split a multidimensional sample space into a “safe” region, an “alert” region (i.e., the one with prescribed RP), and a “dangerous” region. In particular, the safe region turns out to be bounded, and contains multivariate events with limited marginals (an intuitive request for nondangerous events).

[28] A consistent definition of a multivariate RP has a probabilistic distributional ground, related to the notion of copula: for a thorough review, see Gräler et al. [2013], and references therein. Since the introduction of copulas in hydrology [De Michele and Salvadori, 2003], many works show the capability and flexibility of copulas to describe the joint behavior of dependent hydrological variables: a thorough list of papers is available at the STAHY website (—see also the references mentioned in the “Introduction” section for some specific papers concerning droughts. For a theoretical introduction to copulas, see Joe [1997] and Nelsen [2006], and for a practical approach see Salvadori et al. [2007]. A free software package written for “R” [Kojadinovic and Yan, 2010] is available online for working with copulas.

[29] As a difference with the modeling proposed in Kao and Govindaraju [2010], which exploits the approach outlined in Salvadori and De Michele [2004], here we use the one recently sketched in Salvadori et al. [2013] (to which we make reference for all the details), which uses the joint survival function inline image—instead of the joint distribution function inline image—as a fundamental statistical tool. The following survival version of Sklar's Theorem for copulas [Nelsen, 2006] provides the link between univariate and joint survival functions: inline image for all inline image, where the inline image's are the marginal survival functions of the random variables (hereinafter, r.v.) Xi's with copula C (i.e., inline image), and inline image is the survival copula [Joe, 1997; Nelsen, 2006] of the Xi's. In particular, inline image can be written in terms of C, since inline image for all inline image, and inline image is the survival function associated with C given by inline image, where inline image is the set of all (nonempty) subsets of inline image, the sum is over all the inline image elements S of inline image denotes the cardinality of S, and inline image represents the marginal copula of C, with dimension equal to #(S), involving only those indices i's belonging to S. Note that the software implementation is rather straightforward: for instance, in the bivariate case, inline image.

[30] Now, let inline image. The associated survival critical layer inline image of level t, which plays the role as of a multivariate threshold, is defined as inline image. Then, inline image partitions inline image into three nonoverlapping and exhaustive sets: either the “dangerous” region inline image, or the “alert” region inline image, or the “safe” region inline image. Here, we consider inline image as the region of interest (which is unbounded), whereas inline image is bounded whenever the variables at play are lower bounded (as in the present case). Then, the following definition of multivariate RP is well-founded—see also an analogous approach in Salvadori et al. [2011], and Salvadori and De Michele [2013] for a discussion.

[31] Definition 1. Let inline image be a multivariate r.v., and let inline image be the survival critical layer supporting a realization x of X, with inline image. Then, the Survival Kendall's RP inline image of x (hereinafter, SKRP) is defined as

display math(1)

where inline image is the average interarrival time between successive drought occurrences, and inline image is the Kendall's survival function associated with inline image, given by

display math(2)

[32] For the numerical estimate of inline image, see the algorithms outlined in Salvadori et al. [2011], and references therein. In turn, given a drought state inline image, it is immediate to calculate the associated SKRP inline image via equation (1). Furthermore, a notion of survival criticality preorderinline image”can be introduced for ranking the droughts states. In formulas, inline image: viz., the drought state inline image is “less critical” than the drought state inline image if its SKRP is smaller than the one of x2.

[33] There exist several coefficients to objectively quantify the degree of association between I and D [see e.g., Nelsen, 2006; Salvadori et al., 2007]: among others, the most commonly used are the Kendall's τ, the Spearman's ρ, and the Blomqvist's β. In the present case, the estimated value of the Kendall's τ using the original (nonrandomized) data, properly corrected for the ties [see e.g., Press et al. 1992], is inline image. The estimate is significantly different from zero, since the corresponding p-Value is negligible. In turn, statistically speaking, this means that I and D are definitely concordant, and hence dependent. In addition, the effect of the randomization procedure on the amount of association between I and D appears to be insignificant: the estimated values of the Kendall's inline image's over all the NR randomizations show a Gaussian pattern of variation, with a sample mean of about 0.45 and a very small sample variance, as shown in Figure 3.

Figure 3.

Empirical distribution (marked line) and corresponding Normal fit (line) of the Kendall's inline image's calculated over all the NR randomized samples, by considering the Pontelagoscuro station—see text. Also shown are the mean and the standard deviation of the fitting Normal law, as well as a 95% Confidence band (dotted lines). The p-Value indicated is the one provided by the Lilliefors' test of normality [Lilliefors, 1967].

[34] Concerning the choice of a suitable bivariate copula for modeling the joint stochastic behavior of the pair (I, D), we adopt the following strategy. First, NR randomizations of the available data are generated. Then, for each simulation, several survival copulas inline image's (see Table 1) are fitted on the corresponding survival ranks using the ML procedure, and the (corrected) AIC is used to select among the competing bivariate models: the survival copula that, on average, yields the smallest AIC coefficient is eventually chosen as the preferred model.

[35] In the present case, the bivariate survival model selected is a versatile four parameters dependence structure described in Durante and Salvadori [2010] and Salvadori and De Michele [2010], belonging to the so-called extraparametrized Khoudraji-Liebscher's family (here labeled as “XClaytonFrank”). The expression of inline image is

display math(3)

with inline image, where A and B are one-parameter copulas belonging, respectively, to the well-known Clayton and Frank families [see e.g., Nelsen, 2006; Salvadori et al., 2007]. Clearly, other models could be more indicated for describing the random behavior of data sets different from the one investigated here. An algorithm for simulating inline image is outlined in Durante and Salvadori [2010] and Salvadori and De Michele [2010].

[36] From a statistical point of view, it is then essential to verify whether or not the chosen 2-copula could be accepted as a model compatible with the available (survival) data. For this purpose, suitable multivariate goodness-of-fit tests could be used [Genest et al., 2009, Appendix A]. As a result, the goodness-of-fit approximate p-Values calculated over all the NR randomized samples are always larger than 10%, and therefore the selected dependence model cannot be rejected at the standard levels.

[37] Furthermore, it is also interesting to study the effect of the randomization technique on the fit of the bivariate dependence structure considered here. As a matter of fact, fitting the four parameters of inline image turns out to be a numerically ill-conditioned problem, since it involves the maximization, in a four dimensional space, of the highly nonlinear likelihood function resulting from equation (3). However, as for the univariate case, the ML estimates of the copula parameters are always centered around sample mean values, with small sample variances. Therefore, empirically, we may conclude that the randomization procedure only weakly spoils the survival copula model.

[38] Finally, Figure 4 shows the comparison between the theoretical Kendall's survival function, associated with the fitted “XClaytonFrank” 2-copula, and the empirical estimates calculated using the available data: the agreement is valuable, indicating that the selected bivariate model may consistently approximate the RP's of the events of interest.

Figure 4.

Comparison between the theoretical Kendall's survival function, associated with the fitted 2-copula “XClaytonFrank” (line), and the empirical estimates calculated using the available data (markers)—see text.

6. The Dynamic Return Period

[39] For illustrative purposes, let's fix a daily temporal scale. The evolution of a drought can be monitored by calculating the SKRPs associated with the pairs (I, D)'s observed on each day during the drought episode. More specifically, let Ik denote the running average intensity measured on the kth day since the beginning of the drought, with inline image By using equation (1), it is possible to characterize the state of the drought on the kth day via the daily SKRP inline image, with inline image. In turn, using the concept of RP, it is possible to quantify, on a daily basis, the dangerousness of the drought during its evolution, and possibly adopt suitable real time strategies for the mitigation of its effects. The calculation of the RP along with the evolution of a drought will be denominated Dynamic Return Period procedure (hereinafter, DRP): it provides an operative tool for river authorities to monitor in real time the development of the drought event, in order to make decisions and interventions to weaken its consequences.

[40] As an illustration, in Figures 5-7 we show the calculation of the DRP for three selected drought episodes at Pontelagoscuro (see also Figure 1 (left)): the corresponding years are 1943, 2003, and 2006. In particular, the 2003s event is the one showing the largest duration (166 days), while the 2006s event is the one showing the largest running average intensity inline image. The plots show both the temporal evolution of the drought state (in terms of the daily pairs (Ik, k)'s), and the corresponding SKRPs inline image's in years; also indicated are the univariate RP's of Ik and D. The results shown in Figures 5-7 deserve due comments.

Figure 5.

Illustration of the Dynamic Return Period procedure for the drought episode occurred at Pontelagoscuro during the year 1943—see text. (left) Daily drought state given in terms of the running average intensity I and duration D. (right) Median values of the univariate and Survival Kendall RP's (DRP) calculated over all the NR randomizations; 95% confidence bands are also shown (dotted lines).

[41] 1. Plotting the running average intensity, instead of the daily intensity, has several advantages. On the one hand, the resulting behavior is more smooth, and less prone to abrupt changes, which could induce early interventions of the CR. On the other hand, the running average has a sort of “memory feature”: in fact, all the observations from the beginning of the drought till a given day are considered.

[42] 2. In all the figures, the trend of the DRP is never monotone. In particular, when the DRP is decreasing (probably, due to the effectiveness of the CR intervention, or to the occurrence of some precipitation) indicates that the drought state is getting less and less dangerous.

[43] 3. In all the figures, the univariate RP of D is monotonically increasing, whereas the one of Ik follows the behavior of the running average intensity: this is obvious, since the drought duration always increases, whereas the intensity may fluctuate (usually, it first increases, then it attains a maximum, and then it decreases). Instead, the DRP is a function of both Ik and D, and may provide useful indications to the CR: in fact, it takes into account the joint behavior of Ik and D (instead of the marginal ones of the two variables).

[44] 4. In the 1943s event (see Figure 5), neither Ik nor D are really too extreme, with maximum RP's smaller than 40 and 25 years, respectively. Around the 80th day, the running average intensity of the drought decreases for the occurrence of a precipitation event, and also the DRP decreases. However, in about 2 weeks, the DRP starts increasing again: in fact, although the intensity is getting monotonically smaller and smaller, the drought is still “active,” and the combined action of the prolonged duration indicates that some further intervention is required. Note that this latter information cannot be conveyed by the analysis of the marginal RP's of Ik and D only: indeed, by the end of the drought episode, both these univariate RP's are less than 25 years, whereas the DRP shows an abrupt raise toward the very dangerous 50 years level.

[45] 5. In the 2003s event (see Figure 6), the running average intensity is always moderate, with a maximum RP of about 15 years, whereas the duration is really extreme (more than 5 months, with a maximum RP of about 250 years). Apparently, the intervention of the CR has been successful, since the DRP is always small (about 5 years), and the running average intensity starts decreasing around the third month. However, the combined action of the very prolonged duration yields a raise of the DRP by the beginning of the fifth month, indicating that some further intervention is required. Practically, the DRP is sensitive to both the intensity and the duration (viz., to their joint behavior), and may provide a tool to decide if, and when, even small intensities may become dangerous because too a prolonged duration.

Figure 6.

Same as Figure 5, for the episode of the year 2003.

Figure 7.

Same as Figure 5, for the episode of the year 2006.

[46] 6. In the 2006s event (see Figure 7), the duration is not truly extreme, with a maximum RP of less than 40 years, whereas the running average intensities are unusually large, with centenary RP's. The analysis of the corresponding DRP's is particularly interesting: in fact, the DRP increases rapidly, attaining the level 10 years in less than 5 weeks, the level 20 years in only 6 weeks, and the very dangerous level 40 years in only 10 weeks. Apparently, the intervention of the CR has not been successful in this case, and probably should have been somewhat anticipated. In fact, even if the DRP decreases after the intervention around the 10th week, its levels remain large, and the abrupt raise around the 15th week (a large, but not extreme, duration) indicates that more drastic countermeasures should have been taken in the previous weeks.

7. Discussion and Conclusions

[47] The advantages of using a multivariate assessment strategy are evident. In fact, by using only the univariate information provided by either Ik or D, may yield under or overestimates of the actual drought state, and, in turn, of the corresponding risk. As a general recommendation, apparently, a DRP of about 10–20 years seems to indicate that the drought is entering an alert or a dangerous state, and that some immediate intervention may be advisable. Thus, the real time evaluation of the SKRP may help the CR to plan effective mitigation strategies in due time.

[48] Furthermore, by using short-term meteorological forecasting techniques and rainfall-runoff schemes, it might be possible to draw drought scenarios for the successive 2 or 3 days. In turn, the calculation of the corresponding DRP's may provide valuable information to the CR about the possible evolution of the drought episode, and suggest adequate countermeasures. It is worth pointing out that, not only the DRP is a statistical index, with a strong probabilistic distributional base, but it also carries over valuable information about the known history of the drought process as occurred in the basin of interest.

[49] As a conclusion, in this paper a multivariate frequency analysis of hydrological droughts is addressed using the theory of Copulas, and the new concept of Survival Kendall's Return Period. Such a latter approach may offer several advantages over previous definitions of multivariate RP's [Salvadori et al., 2013]. In addition, a randomization strategy is introduced in order to get rid of repeated measurements, which may adversely affect the statistical analysis of the data, make the multivariate analysis ambiguous and questionable, and spoil the calculation of the RP's of interest. The randomization is a good practice that should be considered in hydrological applications once ties occur systematically in a database. Finally, the temporal evolution of a drought event is monitored by introducing the Dynamic Return Period approach: operatively, it provides a measure, based on the notion of multivariate RP, of the dangerousness of the drought along with its development. The outcomes of such a procedure can be used by the river authorities for planning real time interventions to mitigate the consequences of low flows.


[50] The authors thank C. Sempi (Università del Salento, Lecce, Italy) and F. Durante (Free University of Bozen-Bolzano, Bolzano, Italy) for invaluable helpful discussions and suggestions. [G. S.] The research was partially supported by the Italian Ministry of Education, University and Research via the project “Metodi stocastici in finanza matematica”. The support of “Centro Euro-Mediterraneo sui Cambiamenti Climatici” (CMCC—Lecce, Italy) is acknowledged. [R. V.] The support of the Italian Ministry of Education, University and Research and the Italian Ministry of Environment, Land and Sea via the project “GEMINA” is acknowledged.