Hydrol. Earth Syst. Sci., 21, 2377–2387, 2017
www.hydrol-earth-syst-sci.net/21/2377/2017/
doi:10.5194/hess-21-2377-2017
© Author(s) 2017. CC Attribution 3.0 License.
A two-parameter design storm for Mediterranean
convective rainfall
Rafael García-Bartual and Ignacio Andrés-Doménech
Universitat Politècnica de València, Instituto Universitario de Investigación de Ingeniería del Agua y Medio Ambiente
(IIAMA), Camí de Vera s/n, 46022 Valencia, Spain
Correspondence to: Ignacio Andrés-Doménech (igando@hma.upv.es)
Received: 2 December 2016 – Discussion started: 12 December 2016
Revised: 28 March 2017 – Accepted: 15 April 2017 – Published: 9 May 2017
Abstract. The following research explores the feasibility
of building effective design storms for extreme hydrologi-
cal regimes, such as the one which characterizes the rain-
fall regime of the east and south-east of the Iberian Penin-
sula, without employing intensity–duration–frequency (IDF)
curves as a starting point. Nowadays, after decades of func-
tioning hydrological automatic networks, there is an abun-
dance of high-resolution rainfall data with a reasonable
statistic representation, which enable the direct research of
temporal patterns and inner structures of rainfall events at a
given geographic location, with the aim of establishing a sta-
tistical synthesis directly based on those observed patterns.
The authors propose a temporal design storm defined in ana-
lytical terms, through a two-parameter gamma-type function.
The two parameters are directly estimated from 73 indepen-
dent storms identified from rainfall records of high tempo-
ral resolution in Valencia (Spain). All the relevant analytical
properties derived from that function are developed in order
to use this storm in real applications. In particular, in order
to assign a probability to the design storm (return period),
an auxiliary variable combining maximum intensity and to-
tal cumulated rainfall is introduced. As a result, for a given
return period, a set of three storms with different duration,
depth and peak intensity are defined. The consistency of the
results is verified by means of comparison with the classic
method of alternating blocks based on an IDF curve, for the
above mentioned study case.
1 Introduction
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 ex-
perienced 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 meth-
ods. On the watershed scale, design storms are needed to ob-
tain 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 plan-
ning, 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.
Published by Copernicus Publications on behalf of the European Geosciences Union.
2378 R. García-Bartual and I. Andrés-Doménech: Design storm for Mediterranean convective rainfall
Within the first category, the most widely used synthetic
storms are probably the National Resource Conservation Ser-
vice (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 geo-
graphic regions of the United States (Froehlich, 2009). The
NRCS design storms are appropriate for catchments smaller
than 250 km
2
, 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 dif-
ferent climatic regions (McCuen, 1989).
The alternating-block method (Chow et al., 1988) is solely
based on an IDF curve. These design storms display a maxi-
mum intensity block in the centre of the event and a total rain-
fall 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 to-
tal 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, gener-
ally, peak-flow calculations using these design storms tend to
overestimate the results (Alfieri et al., 2008).
The second category of design storms corresponds to tem-
poral patterns derived from observed records. One of the
first temporal distributions using this approach was devel-
oped 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 De-
sign 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 tempo-
ral patterns depending on the storm duration for different re-
gions in Canada. Results led to the AES design storm (Hogg,
1982), widely used in urban drainage design. The former de-
sign 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 inten-
sity and volume as random variables.
Some authors point out that the design storm concept it-
self is fraught with conceptual error when used to simplify
engineer analysis with unrealistic assumptions (Adams and
Howard, 1986). Indeed, many of the concerns about clas-
sic 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. Never-
theless, when storage elements are to be analysed, the influ-
ence 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 expres-
sions are usually short for low-frequency occurrences. Sec-
ond, 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 accu-
rate 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 un-
certainties in IDF curves’ estimation can significantly affect
the reliability of derived design storms, especially in the def-
inition of its peak rainfall intensities, with undesirable con-
sequences when used for hydrologic design purposes.
For the simplest applications (i.e. rational method), a tem-
poral pattern is not required for the design storm. However,
for most hydrologic engineering applications, a design hyeto-
graph 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 ap-
propriate.
The objective herein is to formulate an analytical approach
in order to describe rainfall intensities in time, as an alterna-
tive for practical design storm definition in Mediterranean
areas. Another aim is to develop all required analytical prop-
erties to ensure their applicability under usual criteria and
requirements of design storm approaches for hydrological
design. These include a methodology for return period as-
signment 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.
Hydrol. Earth Syst. Sci., 21, 2377–2387, 2017 www.hydrol-earth-syst-sci.net/21/2377/2017/
R. García-Bartual and I. Andrés-Doménech: Design storm for Mediterranean convective rainfall 2379
2 Design storm
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:
i
(
t
)
= i
0
f (t), (1)
where t ≥ 0 (min) is the time elapsed from the start of the
rainfall episode (t = 0), i(t) (mm h
−1
) represents the rain-
fall intensity at instant t, i
0
(mm h
−1
) is the instantaneous
peak intensity of the storm and f (t) is a convenient non-
dimensional, continuous and differentiable analytical func-
tion, 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 intensi-
ties are attained, followed by a stage of dissipation in time,
typified by a progressive attenuation of rainfall.
Several recent studies characterize the physical dynam-
ics of convective cells from radar-provided data. More pre-
cisely, 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 au-
thors report statistical evidence of the predominance of tem-
poral patterns where the attenuation or temporal dissipation
stages tend to last longer than the initial growing and de-
velopment stage (Brummer, 1984). This characteristic sup-
ports the use of relationships like the gamma function, suc-
cessfully employed in previous mathematical models of rain-
fall (García-Bartual and Marco, 1990; Salsón and Garcia-
Bartual, 2003) since it represents more accurately the pat-
terns 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 tem-
poral 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 ini-
tial 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.
f
(
t
)
= ϕte
1−ϕt
, (2)
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)
Table 1. Parameters η
1
and η
2
for different truncation criteria.
Truncation criterion η
1
η
2
as a % of the intensity
peak value
1 % 0.01 7.6386
5 % 0.05 5.7439
10 % 0.10 4.8897
2.1 Analytical properties
Some interesting analytical properties of the f (t ) function
are revised, which will prove useful in subsequent develop-
ment. The following can be deduced from Eq. (2):
f
(
0
)
= 0, (3)
lim
t→∞
f (t) = 0. (4)
In addition, as
f
0
(
t
)
= ϕ
(
1 − ϕt
)
e
1−ϕt
, (5)
function f (t) displays a relative maximum at point t = t
0
=
ϕ
−1
. The corresponding value of this maximum is as follows:
f
(
t
0
)
= 1. (6)
Given that the duration, t
C
, of the cell is finite, and in order to
establish a finite duration of the process, a simple truncating
criteria is adopted for the asymptote of this function. To do
so, a final or residual value is established as a fraction η
1
of
the maximum so that
f
(
t
C
)
= η
1
, (7)
where t
C
(min) represents the total storm duration, with
t
C
> t
0
and 0 < η
1
< 1. Convenient η
1
values are shown in Ta-
ble 1. Introducing conditions given in Eq. (7) into Eq. (2), we
obtain the following:
f
(
t
C
)
= ϕt
C
e
1−ϕt
C
= η
1
. (8)
Equation (8) admits the following solution:
t
C
=
η
2
ϕ
, (9)
and thus verifies the condition
η
2
e
1−η
2
= η
1
. (10)
Table 1 shows some of the solution values for this equation,
for chosen values of the parameter η
1
.
In other words, once the truncating criteria is defined, for
example 5 %, the duration of the rainfall event is automat-
ically defined as a function of parameter ϕ through Eq. (9)
with η
2
= 5.7439.
www.hydrol-earth-syst-sci.net/21/2377/2017/ Hydrol. Earth Syst. Sci., 21, 2377–2387, 2017
2380 R. García-Bartual and I. Andrés-Doménech: Design storm for Mediterranean convective rainfall
2.2 Properties of the aggregated process
The suggested analytical function can be integrated, yielding
to the following result:
F
[
t
1
;t
2
]
=
Z
t
2
t
1
f
(
t
)
dt =
Z
t
2
t
1
ϕte
1−ϕt
dt =
t
1
+
1
ϕ
e
1−ϕt
1
−
t
2
+
1
ϕ
e
1−ϕt
2
, (11)
where 0 ≤ t
1
< t
2
≤ t
C
. In this way, the integrated value of
F
[
t
1
;t
2
]
is expressed in minutes. By applying Eqs. (9) and
(11), the following particular results are easily obtained:
F
[
0;t
C
]
=
e
ϕ
−
t
C
+
1
ϕ
e
1−ϕt
C
=
e
ϕ
1 −
(
1 + η
2
)
e
−η
2
, (12)
F
[
0;∞
]
=
e
ϕ
, (13)
F
[
0;t
C
]
F
[
0;∞
]
= 1 −
(
1 + η
2
)
e
−η
2
. (14)
It must be noted that the result of Eq. (14) is independent
of parameter ϕ. For instance, if a truncating value of 5 % is
adopted (η
1
= 0.05), it automatically leads to η
2
= 5.7439 as
shown in Table 1, and therefore
F
[
0;t
C
]
F
[
0;∞
]
= 0.98. (15)
That is, the truncating criteria of 5 % for f (t) is equivalent
to establishing the total duration of the cell when 98 % of the
cumulative rainfall has already taken place with respect to
the hypothetical 100 % linked to a cell whose intensities are
asymptotic to 0 and have infinite duration, according to the
known analytical properties of the tail of f (t).
From Eqs. (1) and (11), the total cumulative rainfall (mm)
can be obtained, for a given time interval, [t
1
; t
2
], as follows:
P
[
t
1
;t
2
]
=
Z
t
2
t
1
i
(
t
)
dt =
i
0
60
Z
t
2
t
1
f
(
t
)
dt
=
i
0
60
t
1
+
1
ϕ
e
1−ϕt
1
−
t
2
+
1
ϕ
e
1−ϕt
2
. (16)
The average rainfall intensity (mm h
−1
) during such a given
time interval can be calculated as follows:
i
[
t
1
;t
2
]
=
i
0
t
2
− t
1
t
1
+
1
ϕ
e
1−ϕt
1
−
t
2
+
1
ϕ
e
1−ϕt
2
.
(17)
In the same manner, the total cumulative rainfall for the time
interval [0;t] results in the following:
P
[
0;t
]
=
i
0
60
e
ϕ
−
t +
1
ϕ
e
1−ϕt
. (18)
Replacing t = t
C
in Eq. (18) and substituting Eq. (9), we ob-
tain the total rainfall for the theoretical storm, given by the
following expression:
P
[
0;t
C
]
=
i
0
60
e
ϕ
−
η
2
ϕ
+
1
ϕ
e
1−η
2
. (19)
If we assume a truncating criteria of 5 % (η
1
= 0.05) a
straightforward expression is obtained for the total cumula-
tive rainfall associated with the analytical storm:
P
[
0;t
C
]
= 0.0443
i
0
ϕ
. (20)
2.3 Maximum intensity for a given 1t
For practical applications, a given time interval of aggrega-
tion 1t is used, conveniently chosen depending on the type
of hydrological application, the rainfall–runoff model to be
used, and the characteristics of the urban hydrology applica-
tion to be carried out.
Once a given 1t (in minutes) is selected, it is convenient to
locate the most intense rainfall interval along the time axes,
so that
I
1t
=
i
0
60
max
F
[
t;t+1t
]
, (21)
where t < t
0
< t +1t and I
1t
is the maximum rainfall inten-
sity (mm h
−1
), for the most intense interval of the storm, as
shown in Fig. 1.
If the above-mentioned central interval is
[
t
L
; t
U
]
=
1
ϕ
− ξ 1t;
1
ϕ
+
(
1 − ξ
)
1t
, (22)
as indicated in Fig. 1, the optimization problem has a solution
in terms of the auxiliary variable ξ , being 0 < ξ < 1. Such a
solution is given by the following:
ξ =
1
ϕ1t
−
e
−ϕ1t
1 − e
−ϕ1t
. (23)
Consequently, according to Eq. (17), the maximum intensity
of the storm, once it has been discretized in time intervals of
1t minutes, can be calculated as follows:
I
1t
=
i
0
1t
t
L
+
1
ϕ
e
1−ϕt
L
−
t
U
+
1
ϕ
e
1−ϕt
U
. (24)
In summary, the main derived properties of the chosen an-
alytical shape of the storm are total duration of the storm
given a truncation criterion (Eq. 9), total cumulative rainfall
(Eq. 20) and maximum intensity for a given time level of
aggregation 1t (Eq. 24). All these relations are uniquely ex-
pressed as functions of the two parameters of the storm, i
0
and ϕ.
Hydrol. Earth Syst. Sci., 21, 2377–2387, 2017 www.hydrol-earth-syst-sci.net/21/2377/2017/
R. García-Bartual and I. Andrés-Doménech: Design storm for Mediterranean convective rainfall 2381
i
0
t
0
t
L
t
U
i(t)
1
i
0
tt
C
t
Figure 1. Most intense interval of the storm defined by [t
L
; t
U
] for
a 1t time interval of aggregation.
3 Rainfall data processing
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 ex-
tremes 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 local-
ized 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 iden-
tify and extract convective storms. First, statistically indepen-
dent 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.
3.1 Convective storms set
3.1.1 Identification of statistically independent rainfall
episodes
Before undertaking the storm analysis, a preliminary step
is required in order to separate the original continuous se-
ries of rainfall records in statistically independent rainfall
events. There is no universal method for identifying the min-
imum inter-event time of a rainfall regime and, thus, inde-
pendent storms. Dunkerley (2008) presents an interesting re-
view of the range of approaches used in the recognition of
main events. Early works by Restrepo-Posada and Eagle-
son (1982) are still in force, and according to them the identi-
fication of independent events is based on considering events
such as statistically independent events, so that the minimum
inter-event time must be an outcome of a Poisson process.
Bonta and Rao (1988) bore out this theory, studying some
other aspects in depth. Andrés-Doménech et al. (2010) com-
pleted the original methodology based on the coefficient of
variation analysis and established for Valencia a minimum
inter-event time equal to 22 h. The latter implies that if two
rainfall pulses are separated by more than 22 h, then, they be-
long to different events. Under this premise, 987 statistically
independent events are identified for the period 1990–2012.
3.1.2 Identification of convective episodes
The required rainfall episodes must have a certain convec-
tive character. Therefore, only storms that verify the follow-
ing conditions can be taken into account: maximum intensity
over 35 mm h
−1
and convectivity index β
∗
> 0.3. The con-
vectivity index introduced by Llasat (2001) reflects in an
objective way the greater or lesser convectivity degree of
a rainfall episode, on the sole basis of the registered 5 min
data, with no additional meteorological information being
required. The value β
∗
depends on a convectivity threshold
which depends itself on the record time step. This convec-
tivity threshold was estimated for the Spanish Mediterranean
coastline by Llasat (2001). For a 5 min resolution data series,
the threshold was set to 35 mm h
−1
. Consequently, this in-
dex represents the proportion of total rainfall fallen with an
intensity higher than 35 mm h
−1
. Events with β
∗
> 0.3 rep-
resent convective storms at this location. Thus, according to
this additional criterion, only 64 convective events from the
complete set are selected.
3.1.3 Selection of convective storms
Some of the independent convective events selected above
can correspond to long or very long episodes with important
dry intra-periods (always less than 22 h). Concatenation of
some convective cells can lead to this situation, resulting in
long episodes on some days.
Often, these rainfall cells (storms) can be linked by very
slight background intensity (around 2 mm h
−1
). Usually,
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