Water Resources Management (2021) 35:99–111
https://doi.org/10.1007/s11269-020-02712-3
Performance of Green Roofs for Rainwater Control
A. Raimondi 1
& G. Becciu
1
Received: 10 January 2020 / Accepted: 2 November 2020 /
Published online: 16 November 2020
# The Author(s) 2020
Abstract
Green roofs can be an effective tool for sustainable urban drainage, since they reduce and
retain runoff by delaying its peak. Most studies analysing the retention capacity of green
roofs are usually referred to a specific place and roof condition and do not consider the
possibility that the roof could be partially pre-filled from previous rainfalls at the
beginning of the given event. The aim of this paper is to develop an analytical probabilistic approach to evaluate green roof performance for stormwater control in terms of
runoff that could be applied for different sites and climate conditions. To this end, the
possibility that the green roof retention capacity could not be completely available owing
to pre-filling from previous rainfall events has been considered and equations for an
optimum green roof design, relating the runoff average return interval to the water
retention capacity, have been proposed. The influence of parameters affecting the runoff
process has been examined in depth and a case study to test the goodness of fit of the
resulting equations has been developed.
Keywords Green roofs . Stormwater control . Probability . Runoff . Retention capacity . Pre-filling
1 Introduction
The urban population of the world has grown rapidly from 751 million in 1950 to 4.2 billion in
2018. Today, 55% of the world’s population lives in urban areas, a proportion that is expected
to increase to 68% by 2050 (United Nations 2018). The cover of natural surfaces causes many
problems as higher temperatures, poorer air quality, increased noise levels, loss of biodiversity
and greater runoffs. In particular, the reduction of infiltration into the groundwater system
increases the possibility of floods and of surface waters contamination. In this context, the
implementation of strategies for a sustainable urban drainage to restore the natural hydrological
cycle and to minimize the environmental impact is urgent (Lee 2019; Lee et al. 2019; Li et al.
2017; Wang et al. 2019). Among these, green roofs that do not require additional space beyond
* A. Raimondi
anita.raimondi@polimi.it
1
D.I.C.A., Politecnico di Milano, Milano, Italy
100
Raimondi A., Becciu G.
a building’s footprint could be an effective tool in densely built urban areas where rooftops
constitute from 30 to 50% of impervious surfaces (Carter and Rasmussen 2007). The installation of green roofs involves numerous environmental and economic benefits, as stormwater
management, energy conservation, reduction of heat islands, improvement of water quality,
protection of biodiversity. Focusing on stormwater management, green roofs allow: the local
disposal of runoffs; the reduction of runoff volumes, through evapotranspiration from vegetation and exposed surfaces; the delay of runoff, triggered by soil saturation; the reduction and
delay of runoff peak rates, for the infiltration of rainwaters into the soil and their temporary
storage in the substrate and in the drainage layer; the improvement of stormwater quality for
effect of its percolation into the soil. The first green roofs were installed in Germany in the
1970s (Getter et al. 2007) and since then they have spread in all major modern countries
where, in some cases, incentive programs to encourage or even impose their installation have
been undertaken.
Hydrological performances of green roofs have been investigated by several studies and
models in the last decades (Li and Babcock 2014, 2015; Palla et al. 2012; Locatelli et al. 2014;
Stovin et al. 2017), but often these studies refer to a specific place or climate (Palermo et al.
2019; Piro et al. 2018; Herrera et al. 2018; Peng and Jim 2015) having a limited perspective in
terms of space and time (Hakimdavar et al. 2014; Nawaz et al. 2015). Moreover, especially for
full-scale installations, scientific evidence provided is not sufficient yet to demonstrate hydrological benefits of green roofs (Berndtsson 2010; Gnecco et al. 2013; Zhang and Guo 2013).
Many authors studied the retention capacity of green roofs, focusing their attention on the
relation between storage volume and rainfall depth (Getter et al. 2007; Carter and Rasmussen
2007), antecedent dry period (Chowdhury and Beecham 2012; Voyde et al. 2010), evapotranspiration (Bengtsson 2005).
The aim of this paper is to develop an analytical probabilistic approach to evaluate green
roof performances for stormwater control by estimating the runoff probability distribution
function. The use of this kind of methodology for the modelling of urban drainage systems was
first proposed in the Nineties by Adams and Papa (2000). Once we define the relation that
links the output variables to the input variables and to the parameters characterising the
examined process, the methodology allows the estimation of the probability distribution
functions of output variables starting from the probability distribution functions of the input
variables. One of the main advantages of analytical probabilistic approaches is that they
combine the simplicity of “design storm” methods with the probabilistic reliability of continuous simulations. Equations resulting from the application of this methodology are generally
easy to implement and provide useful information to designers. Moreover, they are not
influenced by a specific place, but can be applied to different basins and climates all over
the world. In recent years, different applications of the analytical probabilistic approach have
been proposed in literature for the modelling of stormwater detention facilities (Raimondi and
Becciu 2017; Becciu and Raimondi 2014, 2015); rainwater tanks (Raimondi and Becciu 2011,
2014; Becciu et al. 2018); infiltration trenches (Guo and Gao 2016); permeable pavements
(Zhang and Guo 2014a); bioretention systems (Zhang and Guo 2014b), and green roofs (Guo
2016; Guo and Zhang 2014; Zhang and Guo 2013). In these works, the modelling of green
roofs, does not take into consideration the possibility that the retention capacity could be
partially pre-filled from more than one previous event at the beginning of the considered
rainfall; this aspect is not negligible for this kind of systems, since outflows from green roofs
are limited to evapotranspiration from vegetation and soil.
Performance of Green Roofs for Rainwater Control
101
This paper tries to fill the gap considering the possibility that retention volume is not
completely available because partially filled by previous rainfall events; in particular, the
runoff probability has been calculated considering the possibility of pre-filling from a chain of
N rainfall events. Moreover, the proposed method allows for an optimum green roof design
since resulting equations relate the maximum retention capacity to the runoff average return
interval. The modelling accurately defines the different variables involved in the process,
analysing their single effects on retention capacity and runoff. Final equations have been
validated comparing the results of their application with those from the continuous simulation
of rainfall data recorded at the gauge station of Milano-Monviso.
2 Methodology
Green roofs are engineered multi-layered structures, with a vegetated upper surface, that work in
very shallow systems without connections to the natural ground. A typical green roof is composed
of: vegetation; growing media, a blend of mineral material enriched with organic material where
water is retained and in which vegetation is anchored; filter fabric; drainage layer, generally
constituted of plastic profiled elements, that stores water for plants sustainment during dry periods,
evacuating excess water in roof drains; root resistant membrane, mechanic protection geotextile.
Figure 1 shows the scheme of reference for the modeling of green roofs used in this paper.
The input to the system is rainfall depth, h; output variables are respectively runoff volume,
v and evapotranspiration volume ET. Three different layers have been considered: vegetation
layer, of thickness zv; growing medium layer, of thickness zg; drainage layer of thickness zd.
Volumes must be intended as specific for unit of area. The parameter z0 represents the height
of the overflow threshold into the drainage layer to evacuate excess rainwater when the
retention capacity of the roof becomes null.
Fig. 1 Green roof schematization
used in the modeling
102
Raimondi A., Becciu G.
Rainfall is first intercepted by plants into the vegetation layer and then infiltrated into the
growing medium where it is retained, used by vegetation, and released back into the atmosphere through evapotranspiration. The excess is infiltrated into the drainage layer equipped
with an overflow to drain heavy rainfalls. The amount of water stored in a green roof w, can be
estimated as follows:
8
0 < h ≤wv;max
wv
>
>
<
wv;max þ wg
wv;max < h≤wv;max þ wg;max
ð1Þ
w
þ wg;max þ wd
wv;max þ wg;max < h≤ wv;max þ wg;max þ wd;max
>
>
: v;max
wv;max þ wg;max þ wd;max
h > wv;max þ wg;max þ wd;max
The parameters wv, wg and wd represent the volume of water stored respectively into the
vegetation layer, the growing medium layer and the drainage layer, while wv, max, wg, max
and wd, max represent their maximum (always for unit of area). The amount of water stored
in a green roof can vary between zero, during dry periods when rainfall depth is null and the
volume is not partially pre-filled from previous events, and wmax = wv, max + wg, max + wd, max,
when all the three layers reach their maximum capacity (0 ≤ w ≤ wmax). The amount of water
volume intercepted by vegetation can be estimated by the following expression:
wv ¼ cv wv;max
ð2Þ
where, cv is a reduction coefficient ranging between zero and one. The amount of water into
the growing medium layer can be calculated as follows:
wg ¼ cg wg;max ¼ cg ϕ f zg ¼ ϕ hg
ð3Þ
that is, either as a percentage of the growing medium storage capacity, calculated by means of
a reduction coefficient cg varying between zero and one, or multiplying the growing medium
moisture content by the water depth into the layer. In particular, ϕ represents the growing
medium moisture content, variable is between zero and the growing medium field capacity ϕf
and hg is the water depth into the growing medium layer that can range between zero and the
growing medium thickness zg. The amount of water into the drainage layer can be estimated
with the following formula:
wd ¼ cd wd;max ¼ cd pd zd ¼ pd hd
ð4Þ
that is either as a percentage of the maximum storage capacity of the drainage layer, calculated by
means of a reduction coefficient cd varying between zero and one, or multiplying the drainage
layer porosity pd by the water depth hd. In particular, the drainage layer porosity pd can vary
between zero and one, while the water depth into the drainage layer hd can range between zero and
the height of overflow threshold z0, obviously lower than the drainage layer thickness zd.
The evapotranspiration volume, that is the amount of water released to the atmosphere from
plants transpiration and soil evaporation can be estimated by the following equation:
ET P
h≥ET P or h < ET P and h þ ∆w ≥ET P
ET ¼
ð5Þ
h−∆w
h < ET P and h−∆w < ET P
The actual evapotranspiration ET equals the potential evapotranspiration ETP, when the rainfall
depth is higher than the potential evapotranspiration or when the rainfall depth is lower than
the potential evapotranspiration, but the rainfall depth added to the water content already
present into the green roof resulting from previous rainfalls ∆w exceeds the potential
Performance of Green Roofs for Rainwater Control
103
evapotranspiration. The water content into green roofs resulting from previous rainfalls ∆w can
vary between zero, if the green roof is completely dry, and wmax,if the water content into the
green roof is at its maximum, as in the case of two very close heavy rainfall events. In this
paper, the actual evapotranspiration has been assumed equal to the potential one. Meteorological variables that most affect runoff from green roofs are rainfall depth h, rainfall duration θ
and interevent time d; in the modeling they have been assumed as independent and exponentially distributed. Numerous studies on different basins concluded that this hypothesis could be
considered acceptable to reduce the complexity of the analytical derivation (Adams et al. 1986;
Eagleson 1978; Bedient and Huber 1992).
Bacchi et al. (2008) tested that for most Italian basins the Weibull probability distribution
function fits the frequency distribution of meteorological input variables better than the exponential probability distribution function; however, its use involves a considerable complication in
the equation’s integration. Becciu and Raimondi (2014) verified that the double-exponential
probability distribution function complies with the frequency distribution of observed data for
the main rainfall characteristic parameters; such distribution may be easily integrated but derived
expressions are quite complex. Moreover, its application to a case study highlighted the fact that
using the double-exponential probability distribution function does not improve so much the
accuracy of results and that the bias due to the use of the exponential probability distribution
function is negligible when compared to the simplicity of equations integration.
To isolate independent events from a continuous record of rainfalls, a minimum interevent
time, the so-called Inter Event Time Definition (IETD) (USEPA 1986), must be defined. If the
interevent time between two consecutive rainfall events is smaller than IETD, the two rainfalls
are joined into a single event, otherwise they are assumed as independent. The probability
distribution function of rainfall depth, rainfall duration and interevent time, considered exponentially distributed, results:
f h ¼ ξ e−ξh
ð6Þ
f θ ¼ λ e−λθ
ð7Þ
f d ¼ ψ e−ψðd−IETDÞ
ð8Þ
where: ξ = 1/μh; λ = 1/μθ; ψ = 1/(μd − IETD). The parameters μh, μθ and μd are respectively the
mean values of rainfall depth, rainfall duration and interevent time.
To evaluate green roof performance in terms of stormwater control, a chain of rainfall
events has been considered (Fig. 2).
With reference to runoff, it generally occurs when the water content at the end of a rainfall
event exceeds the green roof retention capacity; the possibility that the retention capacity is not
completely available at the beginning of a rainfall event, because it may be partially filled from
previous rainfalls has been considered. The water content in a green roof at the end of a generic
rainfall event wi can be expressed by:
8
Condition1
wi−1 −Et d i þ hi −Et θi
>
>
<
Condition2
hi −Et θi
ð9Þ
wi ¼
Condition3 ; Condition4
wmax
>
>
:
Otherwise
0
104
Raimondi A., Becciu G.
Condition1
Condition2
Condition3
Condition4
: wi−1 −Et d i > 0; 0 < wi−1 −Et d i þ hi −Et θi < wmax
: wi−1 −Et d i ≤0; 0 < hi −Et θi < wmax
: wi−1 −Et d i ≤0; hi −Et θi ≥wmax
: wi−1 −Et d i > 0; wi−1 −Et d i þ hi −Et θi ≥wmax
for i = 1, …, N where N is the number of given rainfall events.
The water content for i = 0, that is w0, results:
(
0 < hi −Et θi < wmax
h0 −Et θ0
hi −Et θi ≥wmax
w0 ¼
wmax
0
hi −Et θi ≤0
ð10Þ
With reference to eq. (9): Condition1 expresses the case when there is pre-filling resulting from
previous rainfall events at the beginning of the given event and the given event does not
produce runoff; Condition2 expresses the case when there is no pre-filling resulting from
previous rainfall events at the beginning of the given event and the given event does not
produce runoff; Condition3 expresses the case when there is no pre-filling resulting from
previous rainfall events at the beginning of the given event and the given event produces
runoff; Condition4 expresses the case when there is pre-filling resulting from previous rainfall
events at the end of the given event and the given event produces runoff. The variable Et in
eqs. (9) and (10) represents the evapotranspiration rate. The runoff at the end of a generic event
vi can be calculated by the following expression:
8
wi−1 −Et d i þ hi −Et θi −wmax
Condition1
>
>
<
hi −Et θi −wmax
Condition2 ; Condition3
vi ¼
ð11Þ
w −Et d i þ hi −Et θi −wmax
Condition4
>
>
: max
0
Otherwise
Condition1
Condition2
Condition3
Condition4
:
:
:
:
wi−1 ≤wmax ; wi−1 > Et d i ; wi−1 −Et d i þ hi −Et θi > wmax
wi−1 ≤wmax ; wi−1 ≤ Et d i ; hi −Et θi > wmax
wi−1 > wmax ; wmax ≤Et d i ; hi −Et θi > wmax
wi−1 > wmax ; wmax > Et d i ; wmax −Et d i þ hi −Et θi > wmax
for i = 1, …, N where N is the number of given rainfall events.
The runoff for i = 0, that is v0, results:
h0 −Et θ0 −wmax
hi −Et θi > wmax
v0 ¼
0
Otherwise
ð12Þ
With reference to the eq. (11): Condition1 expresses the case when there is no runoff at the end
of event i − 1, there is pre-filling from event i − 1 at the beginning of event i and there is runoff
from the green roof at the end of event i; Condition2 expresses the case when there is no runoff
at the end of event i − 1, there is no pre-filling from event i − 1 at the beginning of event i and
there is runoff from the green roof at the end of event i; Condition3 expresses the case when
there is runoff at the end of event i − 1, there is no pre-filling from event i − 1 at the beginning
of event i and there is runoff from the green roof at the end of event i; Condition4 expresses the
case when there is runoff at the end of event i − 1, there is pre-filling from event i − 1 at the
beginning of event i and there is runoff from the green roof at the end of event i. The
probability that the runoff volume exceeds a given threshold v has been estimated, setting
Performance of Green Roofs for Rainwater Control
105
h = hi = hi + 1, θ = θi = θi + 1, d = di = di + 1 in eq. (11); this leads to the deletion of Condition2.
Raimondi and Becciu (2014) maintained that to consider two chained rainfall events may be
acceptable only for a long IETD and high outflow rates; for rainwater detention facilities with
low outflow rates (e.g., infiltration basins, green roofs..) or when strict limitations on discharges in the downstream drainage system are imposed, the Authors suggested to assume
three chained rainfall events. For green roofs, outflows rates are very low since they are only
due to evapotranspiration, so it is reasonable to consider a chain of N rainfall events in the
estimation of the runoff probability. In the calculation two different conditions have been
recognized: maximum emptying time, which is the time needed to empty the retention capacity
when it is full, lower and higher than minimum interevent time IETD. For Case 1, the prefilling from previous rainfalls at the beginning of the given event has been excluded and the
full storage capacity has been considered available; for Case 2, the possibility that the retention
volume is partially filled from previous rainfalls has been analyzed. Moreover, a threshold
runoff volume v has been used.
Case 1: wmax/Et ≤ IETD:
∞
Pv ¼ P v > v ¼ ∫
h¼wmax þvþEtθ
fh
∞
dh∫θ¼0 f θ
dθ ¼ γ e
−ξ wmax þv
ð13Þ
λ
whit: γ ¼ λþEtξ
Case 2: wmax/Et > IETD:
∞
Pv ¼ P v > v ¼ ∫ f θ dθ
θ¼0
2
∞
∞
∫
d¼IETD
∫
f d dd
v
v
wmax þ þði−2ÞEtd
þEtθ
i−1
wmax þ
Et
∞
N 6
∫ f d dd
þ ∑6
4 ∫ f θ dθ d¼IETD
i¼2 θ¼0
f
¼γ e
h
−ξ wmax þv
þ ψ ∑Ni¼2
i
vþwmax
− ξi EtIETDði−1Þþ
f h dh
h¼wmax þvþEtθ
∫
h¼
v
wmax þ þði−1ÞEtd
þEtθ
i
3
7
f h dh7
5¼
ð14Þ
ξ
−ξEtIETDð i−2
i−1 Þ− i−1 vþwmax
−ði−1Þ β i e
−i β*i e
½
−ξ Et β i
β *i
e
ψIETD−
vþwmax ð Etψ þξÞ
g
1
Where: β i ¼ ξEt1ði−2Þ þψ ði−1Þ; β *i ¼ − iψþði−1
ÞξEt .
Equation (14), that considers the possibility of pre-filling resulting from previous rainfall
events, depends on the expected values of rainfall depth, rainfall duration, interevent time, as
well as on the evapotranspiration rate, interevent time definition, maximum retention capacity,
runoff threshold and the number of chained events. If the possibility of pre-filling resulting
from previous events is excluded, then the whole retention capacity is available at the
beginning of the given rainfall event (eq. 13), in this case the runoff probability distribution
function only depends on the expected values of rainfall depth and rainfall duration,
106
Raimondi A., Becciu G.
Fig. 2 Chain of rainfall events
evapotranspiration rate, maximum retention capacity and runoff threshold. Two characteristic
variables to analyze green roof performance for stormwater control are the water retention
capacity wmax and the evapotranspiration rate Et; in the following pages, eqs. (13) and (14)
have been studied with respect to these variables. If the water retention capacity tends to
infinite, the probability of runoff from green roofs obviously tends to zero since the storage
volume is unlimited (for wmax → ∞, it results Pv → 0); for the limit case in which the water
retention capacity tends to zero, a single rainfall event is considered (eq. 13) and the runoff
probability depends on the runoff threshold, the evapotranspiration rate and the expected
values of rainfall depth and rainfall duration (for wmax → 0, it results Pv →γ e−ξv ). If the
evapotranspiration rate tends to infinite, the runoff probability obviously tends to zero, since no
rainwater is stored into the green roof (for Et → ∞, it results Pv → 0); if the evaporation rate
tends to zero, the runoff probability is a function of the maximum retention capacity, the runoff
threshold, the mean rainfall depth and the number of given events: that is for Et → 0, it results
h
i
ξ
ξ
in Pv →e−ξðwmax þvÞ þ ∑Ni¼2 −e−i−1ðvþwmax Þ þ e− i ðvþwmax Þ .
3 Application
Resulting formulas have been tested using the rainfall series recorded from Milano-Monviso
rain gauge station during a period of twenty years. An IETD = 10 [hours] has been selected
identifying N = 979 independent rainfall events. Mean μ, standard deviation σ and a coefficient of variation V of the three hydrologic parameters used in the modeling rainfall depth h,
rainfall duration θ and interevent time d have been reported in Table 1.
The hypothesis of characteristic rainfall variables as exponentially distributed, while well
suited for rainfall duration, is not perfectly respected for rainfall depth and interevent time,
however the bias on results due to its use can be considered negligible, as discussed and tested
by Becciu and Raimondi (2014). Table 2 contains the correlation coefficients with the three
hydrological parameters.
Performance of Green Roofs for Rainwater Control
107
Table 1 Mean, standard deviation and coefficient of variation of hydrologic parameters
μ [mm]
σ [mm]
V [−]
h [mm]
θ [hour]
d [hour]
18,49
21,33
1,15
14,37
14,81
1,03
172,81
223,89
1,30
Interevent time results only weakly correlated to the other two variables, while the
correlation between rainfall depth and duration is not negligible. However, to overcome the
correlation with rainfall variables, copula functions have been recently introduced in the
hydrologic research in order to broaden the multivariate inference capability (Abdollahi
et al. 2019); for simplicity they have not been considered in this work. The runoff probability
has been estimated by varying the maximum retention capacity wmax; it has been calculated
summing up the maximum retention capacity of the three layers composing the green roof
(vegetation layer, growing medium and drainage layer): wmax = wv,max + wg,max + wd,max. The
maximum retention capacity of the vegetation layer is generally of a few millimeters. The
maximum retention capacity of the growing medium layer can be estimated considering both
the case of extensive green roofs covered with grass (with a thickness of a few centimeters)
and the case of intensive green roofs covered with shrubbery and small trees (their thickness
could be of some hundreds of centimeters). Considering a soil moisture content being variable
between 0 and 100 [%], the maximum retention capacity of the growing medium layer can
vary between 0 and 1000 [mm]. The maximum retention capacity of the drainage layer usually
varies from 0 to 150 [mm], therefore the maximum retention capacity of the whole roof can
vary between 0 and 1250 [mm]. In the calculation an extensive green roof has been considered
and the maximum retention capacity underwent a variation between 0 and 200 [mm]. Different
studies in literature tried to define the value of the evapotranspiration rate on green roofs:
Lazzarin et al. (2005) estimated that evapotranspiration rates range from 0,69 to 6–9 [mm day]
with typical values of 1–6 [mm day] using a Penman-Monteith model; Wolf and Lundholm
(2008) found evapotranspiration rates varying between 0 to 5 [mm/day] analysing wet,
intermediate, and dry conditions; Voyde et al. (2010) averaged evapotranspiration rates of
about 2 [mm day] for the seven days following saturation. In the calculation two different
values of evapotranspiration rates have been considered: Et = 0,125 [mm/hour] and Et =
0,25 [mm/hour], respectively corresponding to daily values of Et = 3 [mm/day] and Et =
6 [mm/day]. The runoff probability estimated by the application of eqs. (13) and (14) have
been compared with the runoff frequency calculated by the continuous simulation of recorded
data. For both cases, the runoff threshold v has been set equal to zero. Figure 3 shows results
for Et = 0,125 [mm/hour]; in this case the runoff probability estimated from the application of
eqs. (13) and (14) adequately fits the runoff frequency calculated by the continuous simulation
of recorded data, if four chained rainfall events are considered. If only two or three chained
events are considered in eqs. (13) and (14), formulas underestimate the possibility of runoff
from green roofs; this underestimation is due to the fact that for low outflow rates, the retention
volume results partially pre-filled from previous rainfalls and it does not completely empty
during interevent time.
Figure 4 reports results for Et = 0, 25 [mm/hour]; in this case the runoff probability
estimated from the application of eqs. (13) and (14) fits the runoff frequency calculated by
the continuous simulation of recorded data, if two chained rainfall events are considered. If
three or four chained events are considered in eqs. (13) and (14), formulas overestimate the
108
Raimondi A., Becciu G.
Table 2 Correlation coefficients among hydrologic parameters
ρθ,d [−]
ρh,d [−]
ρθ,h [−]
0,11
0,11
0,62
possibility of runoff from green roofs; when increasing outflow rate, the possibility that the
retention volume results partially pre-filled from more than one previous rainfall events
quickly decreases.
Equation (13) and (14) enable to obtain full stormwater control, since they relate the
average return interval of runoff to the maximum retention capacity. For example, for an
average return interval equal to 10 [years], it results: wmax = 65 [mm] for Et = 0,125 [mm/hour]
and wmax = 50 [mm] for Et = 0, 25 [mm/hour].
4 Conclusions
Results from the application of final equations for the estimation of runoff probability to a case study
and their comparison with results obtained from the continuous simulation of recorded data have
shown the efficiency of the proposed method. To consider more than two chained events in the
estimation of the runoff probability is a very effective tool in the analysis of green roofs characterized
by low outflow rates: in this case the possibility that the retention volume is partially pre-filled from
previous events cannot be neglected, since it strongly influences the runoff probability. Moreover,
the proposed method allows an optimum design of green roofs for stormwater control, since the
retention capacity for different average return intervals can be estimated from the runoff probability
distribution function. The key point that the Authors aim to develop in future studies is the
estimation of the number of chained rainfall events that characterize runoff from green roofs,
considering the variation of the evapotranspiration rate, and the runoff threshold.
Fig. 3 Runoff probability and frequency for Et = 0,125 [mm/hour] and N = 2 − 3 − 4
Performance of Green Roofs for Rainwater Control
109
Fig. 4 Runoff probability and frequency for Et = 0, 25 [mm/hour] and N = 2 − 3 − 4
Acknowledgements Open access funding provided by Politecnico di Milano within the CRUI-CARE Agreement.
Availability of Data and Materials Authors agree with data transparency and undertake to provide any required
data and material.
Author’s Contributions Conceptualization: A. Raimondi; G. Becciu. Methodology: A. Raimondi; G. Becciu.
Formal analysis and investigation: A. Raimondi. Writing - original draft preparation: A. Raimondi. Writing review and editing: A. Raimondi. Supervision: G. Becciu.
Compliance with Ethical Standards
Conflict of Interest The authors declare that they have no conflict of interest.
Ethical Approval Not applicable.
Consent to Participate Not applicable.
Consent to Publish Not applicable.
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110
Raimondi A., Becciu G.
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