## 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.