The present paper proposes a dimensionless analytical framework to investigate the impact of the rainfall event structure on the hydrograph peak. To this end a methodology to describe the rainfall event structure is proposed based on the similarity with the depth–duration–frequency (DDF) curves. The rainfall input consists of a constant hyetograph where all the possible outcomes in the sample space of the rainfall structures can be condensed. Soil abstractions are modelled using the Soil Conservation Service method and the instantaneous unit hydrograph theory is undertaken to determine the dimensionless form of the hydrograph; the two-parameter gamma distribution is selected to test the proposed methodology. The dimensionless approach is introduced in order to implement the analytical framework to any study case (i.e. natural catchment) for which the model assumptions are valid (i.e. linear causative and time-invariant system). A set of analytical expressions are derived in the case of a constant-intensity hyetograph to assess the maximum runoff peak with respect to a given rainfall event structure irrespective of the specific catchment (such as the return period associated with the reference rainfall event). Looking at the results, the curve of the maximum values of the runoff peak reveals a local minimum point corresponding to the design hyetograph derived according to the statistical DDF curve. A specific catchment application is discussed in order to point out the dimensionless procedure implications and to provide some numerical examples of the rainfall structures with respect to observed rainfall events; finally their effects on the hydrograph peak are examined.

The ability to predict the hydrologic response of a river basin is a central feature in hydrology. For a given rainfall event, estimating rainfall excess and transforming it to a runoff hydrograph is an important task for planning, design and operation of water resources systems. For these purposes, design storms based on the statistical analysis of the annual maximum series of rainfall depth are used in practice as input data to evaluate the corresponding hydrograph for a given catchment. Several models are documented in the literature to describe the hydrologic response (e.g. Chow et al., 1988; Beven, 2012): the simplest and most successful is the unit hydrograph concept first proposed by Sherman (1932). Due to a limited availability of observed streamflow data mainly in small catchment, the attempts in improving the peak flow predictions have been documented in the literature since the last century (e.g. Henderson, 1963; Meynink and Cordery, 1976) to date. Recently, Rigon et al. (2011) investigated the dependence of peak flows on the geomorphic properties of river basins. In the framework of flood frequency analysis, Robinson and Sivapalan (1997) presented an analytical description of the peak discharge irrespective of the functional form assumed to describe the hydrologic response. Goel et al. (2000) combine a stochastic rainfall model with a deterministic rainfall–runoff model to obtain a physically based probability distribution of flood discharges; results demonstrate that the positive correlation between rainfall intensity and duration impacts the flood flow quantiles. Vogel et al. (2011) developed a simple statistical model in order to simulate observed flood trends as well as the frequency of floods in a nonstationary context including changes in land use, climate and water uses. Iacobellis and Fiorentino (2000) proposed a derived distribution of flood frequency, identifying the combined role played by climatic and physical factors on the catchment scale. Bocchiola and Rosso (2009) developed a derived distribution approach for flood prediction in poorly gauged catchments to shift the statistical variability of a rainfall process into its counterpart in terms of statistical flood distribution. Baiamonte and Singh (2017) investigated the role of the antecedent soil moisture condition in the probability distribution of peak discharge and proposed a modification of the rational method in terms of a priori modification of the rational runoff coefficients.

In this framework, the present research study takes a different approach by
exploring the role of the rainfall event features on the peak flow rate
values. Therefore the main objective is to implement a dimensionless
analytical framework that can be applied to any study case (i.e. natural
catchment) in order to investigate the impact of the rainfall event
structure on hydrograph peak. Since the catchment hydrologic response and in
particular the hydrograph peak is subjected to a very broad range of
climatic, physical, geomorphic and anthropogenic factors, the focus is posed
on catchments where lumped rainfall–runoff models are suitable for
deterministic event-based analysis. In the proposed approach, the rainfall
event structure is described by investigating the maximum rainfall depths
for a given duration

The first specific objective is to define a structure relationship of the rainfall event able to describe the sample space of the rainfall event structures by means of a simple power function. The second specific objective is to implement a dimensionless approach that allows the generalization of the assessment of the hydrograph peak irrespective of the specific catchment characteristic (such as the hydrologic response time, the variability of the infiltration process, etc.), thus focusing on the impact of the rainfall event structure.

Finally a specific catchment application is discussed in order to point out the dimensionless procedure implications and to provide some numerical examples of the rainfall structures with respect to observed rainfall events; furthermore their effects on the hydrograph peak are examined.

A dimensionless approach is proposed in order to define an analytical framework that can be applied to any study case (i.e. natural catchment). It follows that both the rainfall depth and the rainfall–runoff relationship, which are strongly related to the climatic and morphologic characteristics of the catchment, are expressed through dimensionless forms. In this paper, [L] refers to length and [T] refers to time.

The rainfall event is then described as constant hyetographs of a given durations; this simplification is consistent with the use of deterministic lumped models based on the linear system theory (e.g. Bras, 1990). The proposed approach is therefore valid within a framework that assumes that the watershed is a linear causative and time-invariant system, where only the rainfall excess produces runoff. In detail, the rainfall–runoff processes are modelled using the Soil Conservation Service (SCS) method for soil abstractions and the instantaneous unit hydrograph (IUH) theory. Consistently with the assumptions of the UH theory, the proposed approach is strictly valid when the following conditions are maintained: the known excess rainfall and the uniform distribution of the rainfall over the whole catchment area.

Rainfall DDF curves are commonly used to describe the maximum rainfall depth
as a function of duration for given return periods. In particular for short
durations, rainfall intensity has often been considered rather than rainfall
depth, leading to intensity–duration–frequency (IDF) curves (Borga et al.,
2005). Power laws are commonly used to describe DDF curves in Italy (e.g.
Burlando and Rosso, 1996) and elsewhere (e.g. Koutsoyiannis et al., 1998).
The proposed approach describes the internal structure of rainfall events
based on the similarity with the DDF curves. Referring to a rainfall event,
the maximum rainfall depth observed for a given duration is described in
terms of a power function similarly to the DDF curve, as follows:

Rainfall event structure: the observed rainfall depth

In order to correlate the rainfall event structure function to the DDF
curve, a reference rainfall event has to be defined in terms of the maximum
rainfall depth,

Referring to a rainfall duration corresponding to

The dimensionless approach is then introduced since it allows an
analytical framework to be defined which can be applied to any study case (i.e. natural
catchment) for which the model assumptions are valid (i.e. linear causative
and time-invariant system). The reference values

Based on the proposed approach, the dimensionless form of the rainfall
depth,

The hydrologic response of a river basin is here predicted through a deterministic lumped model: the interaction between rainfall and runoff is analysed by viewing the catchment as a lumped linear system (Bras, 1990). The response of a linear system is uniquely characterized by its impulse response function, called the instantaneous unit hydrograph. For the IUH, the excess rainfall of unit amount is applied to the drainage area in zero time (Chow et al., 1988).

To determine the dimensionless form of the unit hydrograph a functional form
for the IUH and thus the S-hydrograph has to be assumed. In this paper the
IUH shape is described with the two-parameter gamma distribution (Nash,
1957):

By applying the change of variable

For a dimensionless unit of rainfall of a given dimensionless duration,

Based on the unit hydrograph theory and assuming a rectangular hyetograph of
duration

Note that the hypothesis of the rectangular hyetograph is not motivated in order
to simplify the methodology but in order to describe the rainfall event
structure. Based on such an approach, the rainfall event structure at a given
duration is represented throughout the

In the following sections the dimensionless hydrograph and the corresponding peak are examined in the case of constant and variable runoff coefficients.

By considering a constant runoff coefficient,

The variability of the infiltration process across the rainfall event as
well as the initial soil moisture conditions significantly affects the
hydrological response of the catchment. In order to take into account these
elements a variable runoff coefficient,

The dimensionless excess rainfall depth,

The corresponding dimensionless excess rainfall intensity becomes

According to the dimensionless approach proposed in the present paper,
different initial moisture conditions can be analysed by considering
different

The ratio

From Eqs. (13), (14) and (23), the dimensionless hydrograph and the
corresponding peak may be expressed by

The proposed dimensionless approach is derived using the two-parameter gamma
distribution for the shape parameter equal to 3. Such an assumption is derived
by using the Nash model relation proposed by Rosso (1984) to estimate the
shape parameter based on Horton order ratios according to which the

Dimensionless rainfall duration vs. dimensionless time to peak;
dimensionless instantaneous unit hydrograph and the corresponding
dimensionless unit hydrographs for

The dimensionless UH is evaluated, varying the dimensionless rainfall
duration in the range between 0.5 and 2 in accordance with the

Finally the dimensionless procedure is applied to a small Mediterranean catchment. In the catchment application the dimensionless procedure is fully specified as from the evaluation of the rainfall structures associated with three observed rainfall events with regard to the determination of the reference peak flow and consequently of the dimensionless hydrograph peaks for the three observed rainfall structures.

The dimensionless form of the hydrograph is shown in Fig. 3 with variation
of the rainfall structure exponents,

Dimensionless flow rates obtained for excess rainfall intensities
characterized by constant runoff coefficient and different rainfall
structure exponents,

The impact of the rainfall structure exponents on the hydrograph form
depends on the rainfall duration: for

3-D mesh plot

Maximum dimensionless hydrograph peak and the corresponding
rainfall structure exponent vs. dimensionless time to peak in the case of a constant runoff coefficient; dimensionless instantaneous unit hydrograph and
the corresponding dimensionless unit hydrographs for

In Fig. 5, the maximum dimensionless hydrograph peak and the corresponding
rainfall structure exponent are plotted vs. the dimensionless time to peak.
Further, the dimensionless IUH and the corresponding dimensionless UH for

The excess rainfall depth, in the case of variable runoff coefficient, is
evaluated by assigning a value to the reference runoff coefficient. In
particular, the reference runoff coefficient is defined as follows, utilizing
Eq. (21):

Similarly to the results presented for the case of constant runoff
coefficient, Fig. 6 illustrates the dimensionless hydrographs obtained for
excess rainfall intensities characterized by variable runoff coefficient and

Dimensionless flow rates obtained for excess rainfall intensities
characterized by a variable runoff coefficient and different rainfall
structure exponents,

3-D mesh plot

Figure 7 shows the 3-D mesh plot and the contour plot of the dimensionless
runoff peak as a function of the rainfall structure exponent and the
dimensionless rainfall duration in the case of a variable runoff coefficient. By
comparing Figs. 7 and 4, it emerges that the contour lines observed in the case of a variable runoff coefficient reveal a steeper trend with respect to
constant runoff coefficient trends; indeed, the impact of the

Maximum dimensionless hydrograph peak and the corresponding
rainfall structure exponent vs. dimensionless time to peak in the case of a variable runoff coefficient; dimensionless instantaneous unit hydrograph and
the corresponding dimensionless unit hydrographs for

In Fig. 8, the maximum dimensionless hydrograph peak and the corresponding
rainfall structure exponent are plotted vs. the dimensionless time to peak
in the case of a variable runoff coefficient. Results plotted in Fig. 8 confirm
that the maximum runoff peak curve reveals the local minimum point at

Maximum dimensionless hydrograph peak and the corresponding
rainfall structure exponent vs. dimensionless time to peak in the case of variable runoff coefficients with respect to dimensionless maximum retention

In order to point out the dimensionless procedure implications and to provide some numerical examples of the rainfall event structures, the proposed methodology has been implemented for the Bisagno catchment at La Presa station, located at the centre of Liguria region (Genoa, Italy).

The Bisagno–La Presa catchment has a drainage area of 34 km

In this application, three rainfall events observed in the catchment area
have been selected in order to analyse the different runoff peaks that occurred
for the three rainfall event structures. For comparison purposes, the
selected events are characterized by an analogous magnitude of the maximum
rainfall depth observed for the duration equal to the reference time (i.e.

Rainfall event structure of three events observed in Genoa (Italy):
the observed rainfall depths

Figure 10 illustrates the rainfall event structure curves derived for the
three selected rainfall events. The graphs at the top report the observed
rainfall depths while the central graphs show the estimated rainfall
structure exponents. At the bottom of Fig. 10, by considering the three
structure exponents corresponding to the Bisagno–La Presa reference time
(i.e.

The excess rainfall hyetographs, the corresponding hydrographs and the reference value of the hydrograph peak flow evaluated for three rainfall structure exponents applied to the Bisagno–La Presa catchment. Note that each graph includes four rainfall durations (i.e. 0.5, 1.0, 1.5 and 2.0 times the reference time).

Contour plot of the dimensionless hydrograph peak evaluated for
the Bisagno–La Presa catchment in the case of a variable runoff coefficient
(

Figure 12 shows the contour plot of the dimensionless hydrograph peak in the case of a variable runoff coefficient (

The proposed analytical dimensionless approach allows the investigation of the
impact of the rainfall event structure on the hydrograph peak. To this end a
methodology to describe the rainfall event structure is proposed based on
the similarity with the depth–duration–frequency curves. The rainfall
input consists of a constant hyetograph where all the possible outcomes in
the sample space of the rainfall structures can be condensed through the

The proposed dimensionless approach allows an analytical framework
to be defined which can be applied to any study case for which the model assumptions are
valid; the site-specific characteristics (such as the morphologic and
climatic characteristics of the catchment) are no more relevant, as they are
included within the parameters of the dimensionless procedure (i.e.

The curve of the maximum values of the runoff peak reveals a local minimum point (saddle point).

Different combinations of

Analogous behaviour of the maximum dimensionless runoff peak curve is observed for different runoff coefficients although wider range of variations are observed with increasing soil abstraction values.

Referring to the Bisagno–La Presa catchment application (

The developed approach, besides suggesting remarkable issues for further researches and being unlike the merely analytical exercise, succeeds in highlighting once more the complexity in the assessment of the maximum runoff peak.

The rainfall data used in the catchment application are freely available for download
(

The authors declare that they have no conflict of interest.

We thank Federico Fenicia, Giorgio Baiamonte and the anonymous reviewer for having contributed to the improvement of the original manuscript with their valuable comments. Edited by: Fabrizio Fenicia Reviewed by: Giorgio Baiamonte and one anonymous referee