Design storms are of paramount importance for hydrologic engineering and remain mainstream practice as they provide a simple and apparently appropriate tool for the design of hydraulic infrastructure. Design storms have been used for more than a century if we consider the block rainfall as input of the rational method (Watt and Marsalek, 2013). They experienced an important development during the 1970s and 1980s with more realistic approaches being implemented (Pilgrim and Cordery, 1975; Walesh et al., 1979; Hogg, 1980, 1982; Pilgrim, 1987).

The need for design storms in hydrologic engineering must be analysed according to the spatial scale of the problem, which might range from typical urban drainage designs to small and intermediate catchment basins. As reported by Watt and Marsalek (2013), one of the earliest applications of design storms to urban drainage took place in Rochester, New York (Kuichling, 1889). It followed the rational method which is still widely used today. In the urban context, the City of Los Angeles method (Hicks, 1944) and the Chicago Hydrograph Method (Keifer and Chu, 1957) represented an important step towards the development of hydrograph methods. On the watershed scale, design storms are needed to obtain design floods when streamflow data are scarce or do not exist (Watt and Marsalek, 2013) for the design of culverts, bridges and small dams, drainage systems, drainage planning, and flood management.

Design storms usually fall into two different categories. The first one considers models based on intensity–duration–frequency (IDF) relations. The second one corresponds to synthetic events where the temporal distribution is derived from observed storms.

Within the first category, the most widely used synthetic storms are probably the National Resource Conservation Service (NRCS, formerly SCS) dimensionless storms and the so-called alternating-block method storms. Standard rainfall patterns for 24 h storms are available for four different geographic regions of the United States (Froehlich, 2009). The NRCS design storms are appropriate for catchments smaller than 250 km2, and they are considered to be applicable to storms of any average return period. Temporal distributions within this method are based on depth–duration–frequency relations available for the US territory, divided into four different climatic regions (McCuen, 1989).

The alternating-block method (Chow et al., 1988) is solely based on an IDF curve. These design storms display a maximum intensity block in the centre of the event and a total rainfall depth at any time that coincides with the total depth given by the IDF relation. The method is simple but has also been widely criticized, because it does not represent any observed rainfall internal structure. Another noticeable weak point of the method, already pointed out by McPherson (1978), is the arbitrary selection of the storm duration, which causes total rainfall depth to also be arbitrarily selected. The Chicago design storm (Keifer and Chu, 1957) is a special case of an alternating-block storm. In Spain, the use of this method is still today concretized through local or regional IDF curves such as those proposed by Témez for all the Iberian Peninsula (Témez, 1978). Recent publications demonstrate that, generally, peak-flow calculations using these design storms tend to overestimate the results (Alfieri et al., 2008).

The second category of design storms corresponds to temporal patterns derived from observed records. One of the first temporal distributions using this approach was developed by Huff (1967) in Illinois (US). The method determines in which time quartile the maximum intensity occurs. This work eventually became the Illinois State Water Survey Design Storm (Huff and Angel, 1989), extensively used by state and local agencies in the US Midwest. Following the same methodology, Hogg (1980) presented his findings on temporal patterns depending on the storm duration for different regions in Canada. Results led to the AES design storm (Hogg, 1982), widely used in urban drainage design. The former design storm reproduces the maximum intensity, the time of this maximum and the rainfall depth that occurs before the peak on the basis of observed records. Other works into this category are those developed in Australia (Pilgrim, 1987; French and Jones, 2012) or the UK (Packman and Kidd, 1980). In Spain, García-Bartual and Marco (1990) studied hyetographs of extreme convective precipitation where the intensity resulting from the activity of each rainfall cell was represented by a gamma-type function with maximum intensity and volume as random variables.

Some authors point out that the design storm concept itself is fraught with conceptual error when used to simplify engineer analysis with unrealistic assumptions (Adams and Howard, 1986). Indeed, many of the concerns about classic design storms arise from the storm duration selection, the IDF concept limitations, the temporal distribution and the difficulties of relating the synthetic storm event to a specific return period.

The design storm duration is not a determining factor if the purpose is to determine a peak flow to design conveyance infrastructures. Consequently, it is common practice to fix it around the concentration time of the catchment basin. Nevertheless, when storage elements are to be analysed, the influence of storm duration and temporal pattern becomes critical (Ball, 1994).

As has been shown in the past (Watt and Marsalek, 2013), uncertainties arising from existing IDF relations have strong consequences. First, record series used to fit IDF expressions are usually short for low-frequency occurrences. Second, IDF curves are considered to represent worst maxima regardless of the physical nature of the storm. García-Bartual and Schneider (2001) exposed the inherent uncertainty in the process, which significantly affects the definition of the IDF curves' shape in the interval 0–10 min. Finally, there is enough reason to deem data acquisition insufficiently accurate in providing robust data for IDF analysis, especially in urban areas (Hoppe, 2008). Moreover, as is the case in Spain, outdated IDF curves are still regularly used, as they are still found in guidance and regulations. The above mentioned uncertainties in IDF curves' estimation can significantly affect the reliability of derived design storms, especially in the definition of its peak rainfall intensities, with undesirable consequences when used for hydrologic design purposes.

For the simplest applications (i.e. rational method), a temporal pattern is not required for the design storm. However, for most hydrologic engineering applications, a design hyetograph is necessary. Selecting this temporal trend is one of the most uncertain steps of the design storm definition, since the physical nature of the process cannot be disregarded.

A storm event presents many characteristics, so it cannot be fully described by the statistics of only one of them. For a return period definition, a common practice is to assign a given frequency to a specific event feature (i.e. its maximum intensity). But, given that a design storm is composed of many variables (depth, duration, temporal pattern, antecedent conditions), assigning a single return period may not be appropriate.

The objective herein is to formulate an analytical approach in order to describe rainfall intensities in time, as an alternative for practical design storm definition in Mediterranean areas. Another aim is to develop all required analytical properties to ensure their applicability under usual criteria and requirements of design storm approaches for hydrological design. These include a methodology for return period assignment based on both total depth and peak intensity of the storm. Also, a practical methodology to build the storm, applied to a given case study to validate it. For illustrative purposes, a comparison with most extended design storms in Mediterranean areas will be developed and discussed.

The temporal pattern of rainfall intensities representing the design storm is expressed in terms of a continuous analytical function of the form given as follows: it=i0f(t), where t ≥ 0 (min) is the time elapsed from the start of the rainfall episode (t= 0), i(t) (mm h-1) represents the rainfall intensity at instant t, i0 (mm h-1) is the instantaneous peak intensity of the storm and f(t) is a convenient non-dimensional, continuous and differentiable analytical function, which will be defined below.

The adopted function f(t) must reproduce the activity life cycle of a convective cell, i.e. an initial development until the maturity stage is reached, during which maximum intensities are attained, followed by a stage of dissipation in time, typified by a progressive attenuation of rainfall.

Several recent studies characterize the physical dynamics of convective cells from radar-provided data. More precisely, these data correspond to relevant characteristics such as duration, spatial extension or the importance of the above-mentioned stages, (Capsoni et al., 2009; Rigo and Llasat, 2005). On the basis of high-resolution rainfall data, some authors report statistical evidence of the predominance of temporal patterns where the attenuation or temporal dissipation stages tend to last longer than the initial growing and development stage (Brummer, 1984). This characteristic supports the use of relationships like the gamma function, successfully employed in previous mathematical models of rainfall (García-Bartual and Marco, 1990; Salsón and Garcia-Bartual, 2003) since it represents more accurately the patterns observed in the temporal registers of convective rainfall events in the east and south-east of the Iberian Peninsula. Nonetheless, there are other mathematical models where an analytic function f(t) is postulated, and where the maximum value is located precisely at half the total duration of the event produced by the convective cell (Northrop and Stone, 2005).

In terms of the proposed design storm, the adopted temporal pattern shows an evolution described in a parametrical way with a function f(t): a non-dimensional gamma-type function with a single parameter which describes a fast initial growing stage of intensities until reaching the maximum value, followed by a slower diminishing stage, asymptotic in time and tending towards a null value when time is growing towards infinity. ft=φte1-φt, where φ (min-1) is a parameter.

This model proved to be an acceptable and consistent representation of the rainfall intensities from convective Mediterranean storms (Andrés-Doménech et al., 2016)

Valencia is a Mediterranean city, located on the eastern coast of the Iberian Peninsula. It presents a typical temperate Mediterranean climate (Csa, according to Köppen climate classification). This type of climate is characterized by mild temperatures (annual average of 17 ∘C), without marked extremes and with a rainfall of about 450 mm yr-1. Rainfall is very unevenly distributed throughout the year, with very marked minima during the months of June, July and August and maxima happening during the months of September and October, these two months concentrating almost a third of the annual rainfall.

Another important characteristic of the rainfall regime is its irregularity, alternating dry and more humid intervals. These dry or humid periods tend to last several years due to the Mediterranean climatic inertia. The torrential character of storms is also a main feature of the rainfall regime of the region, with frequent convective rainfall mesoscale episodes, most widely known as cut-offs, characterized by very localized high-intensity storms.

The rainfall series used in this study were recorded by the Júcar River Basin Authority during the period 1990–2012. The rainfall gauge is installed in the city centre and the data time step is 5 min. Previous studies demonstrated the validity of this data set for similar purposes (Andrés-Doménech et al., 2010). The continuous rainfall series are processed to identify and extract convective storms. First, statistically independent rainfall events are identified. Then, amongst them, only convective events are extracted. Finally, convective storms are identified from convective events and finally selected to estimate model parameters.

If X1(T) is the quantile of the extreme value distribution corresponding to a given return period T, the two variables P and I10, which define the design storm for that given return period, are obtained by solving Eqs. (25) and (26) for each family i= 1, 2 and 3. That is, I10,i(T)=X1(T)βI+βPαiPi(T)=αiX1(T)βI+βPαi

In order to define, in practice, the design storm associated with I10,i and Pi values, and once chosen a convenient time level of aggregation (i.e. Δt= 10 min), it is necessary to obtain the two parameters, i0 and φ, which analytically define the design storm. To do so, Eqs. (20) and (24) are used, and they result as follows, for each i= 1, 2 and 3: Pi(T)=0.0443i0,iφi,I10,i(T)=i0ΔttL+1φie1-φitL-tU+1φie1-φitU, where tL and tU are calculated according to Eqs. (22) and (23).

After formulating the practical steps to build a synthetic storm, a comparison of the former with the most widely used storm (built with alternating blocks obtained from an IDF curve), is performed. In order to carry out this comparison, storms corresponding to a return period of 25 years are built. The choice of 25 years corresponds to the requirements set by the Municipality of Valencia regulations for the design of urban drainage hydraulic infrastructures.

Before obtaining the alternating-block design storm, an ID (intensity–duration) curve for 25 years must be determined, from the very same sample of storms previously used for the development of the Gamma storm and described in Sect. 3. To do this, the usual procedure for obtaining IDF curves is followed, adjusting the empirical sample to the following IDF relation: i(t)=ab+tc, where i (mm h-1) is the maximum intensity corresponding to a rainfall duration t (min), while a, b and c are the parameters of the IDF curve. Vaskova (2001) demonstrated the fitness of this expression for adjusting local IDF curves in Valencia. With the data employed in the present paper, the following coefficients result for the 25-year return period ID curve: a= 8198 mm h-1, b= 29.8 min and c= 1.06. Then, for each case, the alternating-block design storm is built from the ID curve defined by Eq. (31), following the usual methodology (Chow et al., 1988). To allow for a proper comparison with the Gamma storm, the same number of blocks is kept for every case.

To perform the comparison, first, the three synthetic storms corresponding to each of the families defined by α1 (short storms), α2 (medium duration storm) and α3 (long storms) are built. In order to do this, once the truncating level has been set η1 (0.05 in the present paper), the method summarized in Sect. 4 is followed. For a return period of 25 years, it results in a storm magnitude X1= 175.5 (Fig. 3). A continuous storm for each of the three families is obtained and, after being discretized in blocks of Δt= 10 min, generates, for each family, a storm of 2, 4 and 6 blocks respectively, as once the truncation criterion is selected, the storm duration is established (Eq. 9), so that, for a given time level of aggregation (ΔT), the number of blocks can be derived. Table 4 summarizes the essential parameters of each of the three storms.

Figure 4 represents, for each family, both the continuous and the aggregated Gamma storms, along with the alternating-block storm obtained from the ID curve.

Both methods lead to consistent and relatively similar results, those being particularly alike for the longer storms. However, for short- and medium-duration storms, it becomes clear that the classic method offers significantly more pessimistic results. In other words, the common method displays higher intensities. This result is coherent with the well-known process of defining the storm. Indeed, given that the alternating-block method assumes the simultaneous occurrence of maximum intensities for different durations, even when those values had not been encountered historically in the same rainfall event, overestimated intensities seem to be an unsurprising outcome. On the contrary, the Gamma storm is built directly from the temporal pattern observed in real episodes. That is, as demonstrated by Andrés-Doménech et al. (2016), the Gamma storm is coherent with the temporal structure of the rain process and that is why the proposed synthetic storm reproduces the observed rainfall more accurately. Table 5 gathers the quantitative differences found for each of the three storms.

As expected, the higher the duration of the storm, the lesser the difference between the maximum instant intensity of the continuous storm and the one of the maximum block. Furthermore, differences between the maximum block intensities between the aggregated Gamma storm and the alternating block storm are also reduced as the duration of the storm increases.

Nonetheless, the most remarkable differences lie on rainfall volumes. Given a return period, the alternating-block method combines in a single theoretical storm the most adverse statistics for several durations, which originally derive from different historical rainfall events. Conceptually, this is a worst-case-scenario storm ignoring actual rainfall patterns found in the rainfall registers, yielding to a volume overestimation (Di Baldassarre et al., 2006). For the aggregated Gamma storm, differences with regard to the continuous model are more limited, in all cases, which supports the conclusion of having generated a synthetic storm that not only reproduces peak intensities properly but also respects the observed temporal patterns and, consequently, reproduces better storm volumes. Concerning variable X1, results are very similar for both methods, as shown in Table 5.

The use of design storms has been a common worldwide practice for many years, employed to solve a range of hydrologic engineering problems in a direct way. These synthetic storms represent an appropriate statistical synthesis of historical rainfall records and therefore, are of maximal utility in their application to problems of urban drainage infrastructure design. In many European and North and South American countries, they are directly obtained from IDF curves, which are usually pre-established for a given area. This simplifies notably the setting of the design storm, making this a straightforward and fast process. Moreover, it presents the huge advantage of being applicable to places where is little or no rainfall information, inasmuch as it is possible to assume as a starting point certain IDF curves, deemed to be sufficiently reliable or representative of the maximum rainfall of that location.

One of the downsides of this process is the fact that it ignores, in its approach, aspects relative to the actual duration and structure – or inner pattern – of intensities of rain, visible in high-resolution rainfall registers. In some countries, automatic pluviometer networks have been working for decades and thus, detailed information is now available, allowing engineers to undertake such matters with statistical representativity (De Luca, 2014).

However, the diversity of hydraulic elements of current drainage systems (e.g. storm tanks, sustainable drainage systems) means that most conditioning storm parameters for the design are not only rain intensities but also duration, total cumulated rainfall and temporal structure of the storm. This makes the exploration of new strategies for building design storms particularly interesting, starting directly from the observed patterns in the high-resolution registers, instead of using IDF curves. This research explores the possibilities in this sense, for the case of convective-type Mediterranean storms and proposes a case study from the automatic pluviometer register of the city of Valencia.

The design storm is defined in an analytical way through a two-parameter function (i0 and φ), already substantiated by previous studies for the Mediterranean area. The former parameters are estimated directly from independent rainfall events, identified in the original temporal series. The assignment of a return period is done through an auxiliary variable which describes the magnitude of the event, and incorporates simultaneously both the total cumulated rainfall and the maximum intensity. In practice, this criterion leads to three different design storms for each return period, of a similar magnitude but with different temporal patterns and durations. Those storms, exclusively defined in terms of the two pointed parameters, are easily discretized in time intervals Δt, in view of their application to practical cases.

For illustrative purposes, the construction of these storms for Valencia is developed and then compared with the classical alternating-block storm, obtained by the usual methods from the same records. This enables the verification of the consistency of the proposed method, resulting in three storms for every return period, with temporal patterns derived from the observation and direct analysis of high-resolution rainfall series. Besides, they are exclusively defined through the value of their only two parameters in each case. While it is true that the process is clearly more laborious than the alternating-block method, the feasibility of the process in a real case is verified, starting from the principle of direct determination of the storm without using IDF curves. Naturally, it has the important limitation of being only applicable in geographic locations where there is high-resolution rainfall information of sufficient quality and appropriate length of historical record series. In the future, for a higher statistical representativity, it will become necessary to count with a longer register.

The proposed method herein, as well as other simple design storm approaches, present some inherent limitations for certain hydrological engineering applications, as they are not suitable for case studies where a more detailed or comprehensive description of the rainfall process is required. Some examples are continuous-time hydrological-system evaluation, hydrological applications in large catchments, or applications where ensembles or stochastic generation of events are needed to account for a number of possible scenarios (Frances et al., 2012).

Despite that, the proposed analytical definition defines a feasible work framework to provide the design storm with the spatio-temporal dimension of the event, through the addition of a component that considers the decline of intensities from the centre of the cell. By following the practical strategy contained in the present paper, the characterization and estimation of parameters of such a component must be founded on the direct observation of radar data for the most significant storms, with the goal of parametrizing the most characteristic spatial patterns (Barnolas et al., 2010).