# Uncertainty Analysis in the Evaluation of Extreme Rainfall Trends and Its Implications on Urban Drainage System Design

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Proposed Methodology

_{0}is that there is no trend in the population from which the data are drawn and hypothesis H

_{1}is that there is a trend in the records (for test description, see [30]).

_{l}is the l-th observation.

_{T}is the rainfall depth at the specified return period T and duration d and where a

_{T}and n

_{T}are parameters.

_{T}is characterized by the property of scale invariance [34]. Denoting a scale factor by λ > 0, the property of scale invariance is valid if the random variable Z(λd) and λ

^{n}·Z(d) have the same probability distribution for t

_{in}≤ t ≤ t

_{out}and t

_{in}≤ λt ≤ t

_{out}, where t

_{in}and t

_{out}represent the physical bounds for which the scale invariance property is valid. This property also implies that the quantiles and raw moments of any order are scale-invariant, i.e.,

_{T}can be expressed as

_{T}is the Tth quantile of the annual maximum storm depth normalized by its mean for any duration in the range of the existence of a scaling behavior (also referred as the growth factor). The product E[${H}_{60}^{1}$]·w

_{T}represents parameter a

_{T}in Equation (2).

_{T}quantiles can be obtained by means of different probability distribution laws, such as Gumbel, Generalized Extreme Value (GEV), Two Component Extreme Value (TCEV) [35], and others.

_{T}parameter of the DDF curve in the presence of a rainfall trend. To this end, the proposed procedure estimates the a

_{T}parameter linked to several continuous sub-datasets with different ending years and lengths. Starting from a minimum of 15 years, the length of each sub-dataset is increased by one, up to a maximum of 35 years. This assumption is supported by the evidence of an intensification of the hydrological cycle in the last 30–35 years, probably due to the upward trend of temperature occurring in Europe [36,37,38]. In Italy, the change point of the annual average temperature has been identified at the beginning of the 1980s [39]. The evidence of a statistically significant increase in mean annual rainfall over the last 30 years in Sicily [40] further supports this assumption. Finally, the purpose of this study is the assessment of implications of climate change on urban drainage system design; thus, referring to long-term trends (more than 35–40 years) seems to be inadequate because the design return periods of drainage systems typically range between 5 and 10 years.

_{T}parameter are obtained, one for each ending year of the processed sub-datasets. To evaluate the 95

^{th}, 50

^{th}, and 5

^{th}percentiles of each a

_{T}series, the likelihood function value L

_{i}is assessed for each ith element (ending year of the sub-dataset), according to the following equation:

_{T}series. Then, the 95

^{th}, 50

^{th}, and 5

^{th}percentiles of each a

_{T}series are obtained from the cumulative distribution function (CDF) of L

_{i}, as shown in Figure 2.

**Figure 2.**Example of CDF of L

_{i}linked to an a

_{T}series and of the related 95

^{th}, 50

^{th}, and 5

^{th}percentiles.

_{T}in time, the linear regressions of its percentiles series are performed. These regressions are analyzed considering the last 30 years of the dataset because of the return periods commonly considered in drainage system design (5–10 years); therefore, referring to short-term trends is more appropriate.

^{th}and 5

^{th}percentile regression laws represent the upper and lower uncertainty bands linked to the appraisal of the a

_{T}parameter in time. Therefore, the width of these bands provides a quantification of the uncertainty inherent to the a

_{T}appraisal.

_{T}parameter values for future projections. This procedure can be used to generate short-term projections that can be helpful in the design of urban drainage networks (up to the next 30–40 years). The availability of uncertainty bands, connected with the trend projection, makes the design process robust giving to the designer a measure of trend estimation reliability. Indeed, long-term projections require the availability of more recent data.

#### 2.2. The Case Study: Dataset

^{2}in Southern Italy, characterized by a Mediterranean climate with mild winters and hot and generally dry summers, mainly influenced by hot winds blowing from the North African coast (Sirocco). Figure 3a shows the rain gauges in the analyzed area together with the Paceco location, a small town with a population of approximately 12,000 inhabitants, for which the urban drainage system has been selected as the case study. The mean annual precipitation over this area ranges between 500 and 600 mm (Figure 3b). Annual maxima rainfall series were not available for this site. Therefore, four rain gauges were selected in the surrounding area to detect statistically significant trends: Lentina (Contrada), S. Andrea Bonagia, Trapani and Birgi Nuovo. For these rain gauges, the available historical annual maxima rainfall series of durations 1, 3, 6, 12 and 24 h for the period from 1950 to 2008 were elaborated and provided by Osservatorio delle Acque-Regione Siciliana (OA-RS).

**Figure 3.**(

**a**) Location of Paceco and the surrounding rain gauges; (

**b**) mean annual precipitation in north-western Sicily; and (

**c**) orthophotograph of Paceco.

## 3. Results and Discussion

#### 3.1. Pre-Analysis Results

#### 3.2. Bayesian Procedure Results

**Table 1.**Level of significance, magnitude and p-value of trends for each duration and rain gauge. The symbol “+” indicates a positive trend.

Duration | Rain Gauge | ||||
---|---|---|---|---|---|

Lentina (Contrada) | S. Andrea Bonagia | Trapani | Birgi Nuovo | ||

1 h | trend | +/α = 0.01 | +/α = 0.05 | +/α = 0.01 | no trend (α < 0.1) |

β (mm/year) | 0.153 | 0.206 | 0.198 | 0.160 | |

p-value | 0.002 | 0.010 | 0.00002 | 0.571 | |

3 h | α | no trend (α < 0.1) | +/α = 0.05 | +/α = 0.05 | no trend (α < 0.1) |

β (mm/year) | −0.030 | 0.117 | 0.140 | −0.114 | |

p-value | 0.214 | 0.032 | 0.011 | 0.151 | |

6 h | α | no trend | +/α = 0.05 | +/α = 0.05 | no trend (α < 0.1) |

β (mm/year) | −0.054 | 0.092 | 0.164 | −0.126 | |

p-value | 0.267 | 0.025 | 0.037 | 0.368 | |

12 h | α | no trend (α < 0.1) | +/α = 0.05 | no trend (α < 0.1) | no trend (α < 0.1) |

β (mm/year) | 0.064 | 0.183 | 0.165 | 0.006 | |

p-value | 0.155 | 0.019 | 0.166 | 0.690 | |

24 h | α | no trend (α < 0.1) | +/α = 0.05 | no trend (α < 0.1) | no trend (α < 0.1) |

β (mm/year) | 0.177 | 0.308 | 0.209 | 0.184 | |

p-value | 0.481 | 0.101 | 0.845 | 0.760 |

_{T}(Equation (2)). The GEV is a continuous probability distribution that combines the Gumbel, Frechet, and Weibull distributions. It is based on extreme value theory [41] and is frequently used to model extreme rainfall, providing a good fitting of the measured data in the Sicilian basin [42,43]. In this study, GEV parameters were estimated using the l-moments [44].

_{T}parameter of the related DDF curve was estimated. Starting from a minimum of 15 years, the length of each sub-dataset was increased by one, up to a maximum of 35 years.

_{T}parameter were obtained, one for each ending year of the processed sub-datasets (ranging from 1964 to 2008). The a

_{T}series showed an increasing length: for the first four series (with an ending year ranging from 1964 to 1967), the a

_{T}parameter assumed the same value, due to the low number and the related reduced data variability of the sub-datasets linked to these series (Figure 4).

_{i}was assessed for each ith element of each a

_{T}series, and the related 95

^{th}, 50

^{th}, and 5

^{th}percentiles were obtained from the cumulative distribution function of L

_{i}. Three a

_{T}values were obtained as the 95

^{th}, 50

^{th}and 5

^{th}percentiles of the a

_{T}series for each year, from 1964 to 2008. Figure 5 shows the 95

^{th}, 50

^{th}and 5

^{th}percentiles of the a

_{T}parameter and the related linear regressions (obtained considering the last 30 years).

**Figure 5.**The 95

^{th}, 50

^{th}and 5

^{th}percentiles of the a

_{T}parameter and linear regressions (obtained considering the last 30 years).

_{T}in time. The coefficient of determination (R

^{2}) and the standard error of regression (SER) values show strong agreement between the regression laws and percentile values. All the percentile series are clearly affected by a positive trend, similar to that observed for the annual maximum rainfall for d = 1 h. In regards to the 50

^{th}percentile, the a

_{T}value showed a 44.5% increase, increasing from 27.1 mm (1964) to 39.1 mm (2008).

**Table 2.**Linear regression equations, and coefficient of determination (R

^{2}) and standard error of regression (SER) values for the 95

^{th}, 50

^{th}and 5

^{th}of a

_{T}percentiles.

Percentile | Equation | R^{2} | SER |
---|---|---|---|

95^{th} | a_{T} = 0.441·t − 844.4 | 0.92 | 0.961 |

50^{th} | a_{T} = 0.402·t − 768.4 | 0.95 | 0.899 |

5^{th} | a_{T} = 0.393·t − 752.9 | 0.93 | 1.193 |

_{T}value related to 2008 is higher (13.4%) than the corresponding value obtained using the whole historical dataset for the definition of the DDF curve (a

_{T}= 34.4 mm). This result highlights the need to verify the hydraulic performance of the urban drainage system under future climate scenarios.

^{th}and 5

^{th}percentiles linear regressions identify the upper and lower limits of the uncertainty band linked to the a

_{T}assessment, respectively. The width of the uncertainty bands provides information about the reliability of the future projection. The results show that the 50

^{th}and 5

^{th}percentile linear regressions tend to get closer in time, whereas the 95

^{th}percentile linear regression tends to diverge from that of the 50

^{th}percentile, probably due to the presence of some outliers of the annual maximum series in recent years. This result indicates that the width of the upper uncertainty band will increase for future a

_{T}projections. However, due to the gradual overlapping of the 50

^{th}and 5

^{th}lines, the width of the lower uncertainty band will decrease.

#### 3.3. DDF-Curves of Climate Change Scenarios

_{T}parameter increase of 44.5%. Therefore, the DDF parameters need to be updated for future climate scenarios.

_{T}parameters of the DDF curves were estimated for the Trapani rain gauge for two climate change scenarios, 2025 and 2050. Following the temporal analogues approach, climate scenarios were generated by supposing that the changes in precipitation detected by the trend analysis will proceed in the future with the same pattern, assuming a linear trend.

_{T}parameter values for the 2025 and 2050 projections, the linear regression equations previously described (Table 2) have been used. Table 3 reported the a

_{T}values for each scenario. It has to be underlined that this procedure can be used to define short-term projections. Indeed, whenever the percentile linear regressions tend to converge, at a certain time, they will intersect. For this reason, the linear regressions require updating by including the recent data. Using the available dataset, the procedure applied here is able to provide projections up to 2050, but projections more distant in time are less reliable.

Percentile | a_{T} | ||
---|---|---|---|

2008 * | Scenario 2025 | Scenario 2050 | |

95^{th} | 41.54 | 48.60 | 59.62 |

50^{th} | 39.13 | 45.64 | 55.69 |

5^{th} | 37.22 | 43.54 | 53.38 |

_{T}values of Table 3 for each scenario (2025 and 2050), three different DDF curves have been defined, one per percentile (95

^{th}, 50

^{th}and 5

^{th}). In every scenario, n

_{T}has been set equal to 0.25 (the value obtained from the statistical analysis of the whole historical dataset up to 2008). Figure 6 shows the DDF curve used for the design conditions of the Paceco drainage system, the DDF curve evaluated starting from the series recorded over the 1950–2008 period, and the DDF curves for the 2025 and 2050 scenarios.

**Figure 6.**DDF curves for the design conditions for the historical series (1950–2008) and for the 2025 and 2050 scenarios.

^{th}, 50

^{th}and 5

^{th}percentiles of a

_{T}indicate a rainfall depth increase of 41%, 33% and 27%, respectively, if compared with the DDF curve obtained for the 1950–2008 series (blue solid line). In the 2050 scenario, the increases in rainfall depths are 55%, 62% and 73%, respectively, for the curves obtained from the 95

^{th}, 50

^{th}and 5

^{th}percentiles of a

_{T}. For both future scenarios, for a given rainfall duration d, the error with respect to the 50

^{th}percentile related to the evaluation of h(d)

_{T}(ratio of the distance between the 95

^{th}and 5

^{th}percentiles divided the 50

^{th}percentile value) is equal to ~11%.

_{T}parameter variations in time was neglected because the analysis was focused on the intensification of rainfall events for d = 1 h, which is usually adopted for the purpose of design and verification of urban drainage systems, due to the low concentration time of urban watersheds. To obtain more reliable predictions for rainfall events of durations greater than 1 h, further developments of this study should involve the evaluation of n

_{T}variations due to climate change.

#### 3.4. Analysis of Trend Implications on Urban Drainage System Design

_{T}= 30.75 mm and n

_{T}= 0.28, respectively. Those parameter values were obtained by means of univariate statistical analysis of the annual maxima series recorded at the Trapani rain gauge during 1953–1991. The design maximum capacity for pipes was set equal to 60%.

_{T}and n

_{T}):

- a
_{T}= 30.75 mm and n_{T}= 0.28 (design conditions); - a
_{T}= 34.40 mm and n_{T}= 0.25 (historical dataset up to 2008);

Max Pipe Capacity | Network Pipe Volume | |||||||
---|---|---|---|---|---|---|---|---|

Design Conditions | 2008 * | 2025 | 2050 | |||||

5^{th} perc** | 50^{th} perc | 95^{th} perc | 5^{th} perc | 50^{th} perc | 95^{th} perc | |||

0%–20% | 25.64% | 24.55% | 5.33% | 5.33% | 5.25% | 5.12% | 5.12% | 5.05% |

20%–40% | 24.87% | 25.53% | 27.63% | 24.21% | 21.40% | 6.88% | 5.50% | 4.38% |

40%–60% | 38.99% | 38.70% | 11.28% | 12.47% | 10.80% | 18.11% | 17.85% | 16.33% |

60%–80% | 10.50% | 10.15% | 20.63% | 17.16% | 10.62% | 5.14% | 4.15% | 3.66% |

80%–100% | 0.00% | 1.06% | 35.13% | 40.82% | 51.93% | 64.76% | 67.38% | 70.58% |

_{T}obtained by adopting the DDF curve related to the whole historical dataset instead of the design DDF curve (in Figure 6, the blue solid line and green dot line, respectively).

^{th}percentile of the 2025 scenario, it can be observed that the potential future increase of rainfall produces a hydraulic surcharge of 40.82% of the network pipe volume. The system performance worsens when the 2050 scenario is simulated (the surcharged network pipe volume is equal to 67.38% for the 50

^{th}percentile). This behavior is illustrated in Figure 9, where a comparison in terms of the network volume percentage related to maximum pipe capacity ranges are shown for all the simulated conditions.

**Figure 9.**Scenario comparison in terms of network volume percentage related to maximum pipe capacity ranges.

^{th}, 50

^{th}and 5

^{th}percentiles of a

_{T}for the 2025 and 2050 scenarios, providing information about the effect of the uncertainty linked to the evaluation of the a

_{T}on the system hydraulic performance. In regards to the 2025 scenario, the effect of uncertainty in the estimation of the a

_{T}parameter is moderate: for the 5

^{th}percentiles, results of the simulations (Figure 10a) show that 35.13% of the network pipe volume is surcharged (Table 4), and this percentage increases for simulations conducted for the 50

^{th}and 95

^{th}percentile conditions (40.82% and 51.93%, respectively). For the 2050 scenario (Figure 10b), the network volume percentage for the different maximum pipe capacity ranges are quite similar for all simulated conditions: most of the pipes are already surcharged (64.76% - 70.58% for the simulation performed using the 5

^{th}and 95

^{th}percentiles of a

_{T}, respectively. Therefore for the 2050 scenario, the uncertainty related to the a

_{T}evaluation does not transfer to the hydraulic behavior of the system.

**Figure 10.**Network volume percentage related to the maximum pipe capacity ranges: (

**a**) 2025 climate scenario and (

**b**) 2050 climate scenario.

## 4. Conclusions

_{T}parameter of the DDF curves of future climate scenarios and the uncertainty related to its estimation. The proposed procedure was applied to estimate the DDF curves for a case study in the northwestern part of Sicily, in the town of Paceco. First, a preliminary trend analysis performed to determine whether the study area was affected by an increase or decrease in annual maxima rainfall. This analysis highlighted the presence of a statistically significant trend in extreme rainfall with a duration of one hour. Subsequently, a Bayesian procedure was performed to account for the above-mentioned trend of the DDF curve assessment in future climate scenarios. These curves were used as input to investigate the implications of climate change on the hydraulic behavior of the Paceco drainage system.

_{T}parameter increase of 44.5%. Due to this variation, the DDF parameters have been updated to define climate scenarios at 2025 and 2050. In the 2025 scenario, the DDF curves obtained for the 95

^{th}, 50

^{th}and 5

^{th}percentiles of a

_{T}indicate a rainfall depth increase of 41%, 33% and 27%, respectively, if compared with the DDF curve obtained for the historical series (1950–2008). In the 2050 scenario, these increases are 55%, 62% and 73%, The implications of these trends on the performance of the drainage system of Paceco (recently designed and built) have been investigated, comparing the network pipe volume percentage related to several maximum pipe capacity ranges in each scenario. With regard to the 50

^{th}percentile of the 2025 scenario, the potential future increase of rainfall would produce a hydraulic surcharge of 40.82% of the network pipe volume. This percentage is equal to 67.38% in the 2050 scenario. In summary, for future projections at 2025 and 2050, the analyzed drainage system will likely face an increased frequency of drainage system surcharge episodes, due to the positive trend of extreme rainfall intensities in the study area.

## Acknowledgments

## Conflicts of Interest

## References

- Trenberth, K.E.; Dai, A.; Rasmussen, R.M.; Parsons, D.B. The changing character of precipitation. Bull. Am. Meteorol. Soc.
**2003**, 84, 1205–1217. [Google Scholar] [CrossRef] - Alexander, L.V.; Zhang, X.; Peterson, T.C.; Caesar, J.; Gleason, B.; Klein Tank, A.M.G.; Haylock, M.; Collins, D.; Trewin, R.; Rahimzadeh, F.; et al. Global observed changes in daily climate extremes of temperature and precipitation. J. Geophys. Res.
**2006**, 111. [Google Scholar] [CrossRef] - Middelkoop, H.; Daamen, K.; Gellens, D.; Grabs, W.; Kwadijk, J.C.; Lang, H.; Parmet, B.W.; Schädler, B.; Schulla, J.; Wilke, K. Impact of climate change on hydrological regimes and water resources management in the Rhine basin. Clim. Chang.
**2001**, 49, 105–128. [Google Scholar] [CrossRef] - Intergovernmental Panel on Climate Change (IPCC). Summary for Policymakers. Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Rep. of the Intergovernmental Panel on Climate Change; Stocker, TF., Qin, D., Plattner, G.K., Tignor, M., Allen, S.K., Boschung, J., Nauels, A., Xia, Y., Bex, V., Midgley, P.M., Eds.; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Freni, G.; la Loggia, G.; Notaro, V. Uncertainty in urban flood damage assessment due to urban drainage modelling and depth—Damage curve estimation. Water Sci. Technol.
**2010**, 61, 2979–2993. [Google Scholar] [CrossRef] [PubMed] - Fu, G.; Kapelan, Z. Flood analysis of urban drainage systems: Probabilistic dependence structure of rainfall characteristics and fuzzy model parameters. J. Hydroinform.
**2013**, 15, 687–699. [Google Scholar] [CrossRef] - Ahn, J.; Cho, W.; Kim, T.; Shin, H.; Heo, J.H. Flood frequency analysis for the annual peak flows simulated by an event-based rainfall-runoff model in an urban drainage basin. Water
**2014**, 6, 3841–3863. [Google Scholar] [CrossRef] - Notaro, V.; Fontanazza, C.M.; la Loggia, G.; Freni, G. Identification of the best flood retrofitting scenario in an urban watershed by means of a Bayesian Decision Network. In Urban Water II; Mambretti, S., Brebbia, C.A., Eds.; WIT Press: Ashurst, New Forest, UK, 2014; pp. 341–352. [Google Scholar]
- Butler, D.; McEntee, B.; Onof, C.; Hagger, A. Sewer storage tank performance under climate change. Water Sci. Technol.
**2007**, 56, 29–35. [Google Scholar] [CrossRef] [PubMed] - Kleidorfer, M.; Möderl, M.; Sitzenfrei, R.; Urich, C.; Rauch, W. A case independent approach on the impact of climate change effects on combined sewer system performance. Water Sci. Technol.
**2009**, 60, 1555–1564. [Google Scholar] [CrossRef] [PubMed] - Papa, F.; Guo, Y.; Thoman, G.W. Urban drainage infrastructure planning and management with a changing climate. In Proceedings of the 57th Canadian Water Resources Association Annual Congress—Water and Climate Change: Knowledge for Better Adaptation, Montréal, QC, Canada, 16–18 June 2004; p. 6.
- Grum, M.; Jørgensen, A.T.; Johansen, R.M.; Linde, J.J. The effect of climate change on urban drainage: An evaluation based on regional climate model simulations. Water Sci. Technol.
**2006**, 54, 9–15. [Google Scholar] [PubMed] - Fontanazza, C.M.; Freni, G.; la Loggia, G.; Notaro, V. Uncertainty evaluation of design rainfall for urban flood risk analysis. Water Sci. Technol.
**2011**, 63, 2641–2650. [Google Scholar] [CrossRef] [PubMed] - Fontanazza, C.M.; Freni, G.; Notaro, V. Bayesian inference analysis of the uncertainty linked to the evaluation of potential flood damage in urban areas. Water Sci. Technol.
**2012**, 66, 1669–1677. [Google Scholar] [PubMed] - Notaro, V.; Fontanazza, C.M.; Freni, G.; Puleo, V. Impact of rainfall resolution in time and space on the urban flooding evaluation. Water Sci. Technol.
**2013**, 68, 1984–1993. [Google Scholar] [CrossRef] [PubMed] - Wang, Y.; McBean, E.A. Uncertainty characterization of rainfall inputs used in the design of storm sewer infrastructure. J. Urban Rural Water Syst. Model.
**2013**, 22. [Google Scholar] [CrossRef] - Arnbjerg-Nielsen, K. Significant climate change of extreme rainfall in Denmark. Water Sci. Technol.
**2006**, 54, 1–8. [Google Scholar] [CrossRef] [PubMed] - Burlando, P.; Rosso, R. Extreme storm rainfall and climatic change. Atmos. Res.
**1991**, 27, 169–189. [Google Scholar] [CrossRef] - Mailhot, A.; Duchesne, S. Design Criteria of Urban Drainage Infrastructures under Climate Change. J. Water Resour. Plan. Manag.
**2009**, 136, 201–208. [Google Scholar] [CrossRef] - Larsen, A.N.; Gregersen, I.B.; Christensen, O.B.; Linde, J.J.; Mikkelsen, P.S. Potential future increase in extreme one-hour precipitation events over Europe due to climate change. Water Sci. Technol.
**2009**, 60, 2205–2216. [Google Scholar] [CrossRef] [PubMed] - Nguyen, V.; Nguyen, T.; Cung, A.A. Statistical Approach to Downscaling of Sub-daily Extreme Rainfall Processes for Climate-Related Impacts Studies in Urban Areas. Water Sci. Technol. Water Suppl.
**2007**, 7, 183–192. [Google Scholar] [CrossRef] - Olsson, J.; Berggren, K.; Olofsson, M.; Viklander, M. Applying climate model precipitation scenarios for urban hydrological assessment: A case study in Kalmar City, Sweden. Atmos. Res.
**2009**, 92, 364–375. [Google Scholar] [CrossRef] - Willems, P.; Vrac, M. Statistical precipitation downscaling for small-scale hydrological impact investigations of climate change. J. Hydrol.
**2011**, 402, 193–205. [Google Scholar] [CrossRef] - Mirhosseini, G.; Srivastava, P.; Stefanova, L. The impact of climate change on rainfall Intensity-Duration-Frequency (IDF) curves in Alabama. Reg. Environ. Chang.
**2013**, 13, 25–33. [Google Scholar] [CrossRef] - Palutikof, J.P. Some possible impacts of greenhouse gas induced climatic change on water resources in England and Wales. In The Influence of Climate Change and Climate Variability on the Hydrologic Regime and Water Resources; IAHS: Vancouver, Canada, 1987; pp. 585–596. [Google Scholar]
- Krasovskaia, I.; Gottschalk, L. Stability of river flow regimes. Nord. Hydrol.
**1992**, 23, 137–154. [Google Scholar] - Liuzzo, L.; Freni, G. Analysis of Extreme Rainfall Trends in Sicily for the Evaluation of Depth-Duration-Frequency Curves in Climate Change Scenarios. J. Hydrol. Eng.
**2015**, 20. [Google Scholar] [CrossRef] - Mann, H.B. Nonparametric tests against trend. Econ. J. Econ. Soc.
**1945**, 13, 245–259. [Google Scholar] [CrossRef] - Kendall, M.G. Rank Correlation Methods, 3rd ed.; Hafner Publishing Company: New York, NY, USA, 1962. [Google Scholar]
- Helsel, D.R.; Hirsch, R.M. Statistical Methods in Water Resources; Elsevier: Reston, VA, USA, 1992; Volume 49. [Google Scholar]
- Hirsch, R.M.; Slack, J.R.; Smith, R.A. Techniques of trend analysis for monthly water quality data. Water Resour. Res.
**1982**, 18, 107–121. [Google Scholar] [CrossRef] - Burlando, P.; Rosso, R. Scaling and multiscaling models of depth-duration-frequency curves for storm precipitation. J. Hydrol.
**1996**, 187, 45–64. [Google Scholar] [CrossRef] - Ranzi, R.; Mariani, M.; Rossini, E.; Armanelli, B.; Bacchi, B. Analisi e Sintesi Delle Piogge Intense del Territorio Bresciano; Technical Report N. 12; University of Brescia: Brescia, Italy, 1999; p. 94. [Google Scholar]
- Gupta, V.K.; Waymire, E. Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res. Atmos
**1990**, 95, 1999–2009. [Google Scholar] [CrossRef] - Rossi, F.; Fiorentino, M.; Versace, P. Two-component extreme value distribution for flood-frequency analysis. Wat. Resour. Res.
**1984**, 20, 847–856. [Google Scholar] [CrossRef] - Klein Tank, A.M.G.; Wijngaard, J.B.; Konnen, G.P.; Bohm, R.; Demaree, G.; Gocheva, A.; Mileta, M.; Pashiardis, S.; Hejkrlik, L.; Kern-Hansen, C.; et al. Daily surface air temperature and precipitation dataset 1901–1999 for European Climate Assessment (ECA). Int. J. Climatol.
**2002**, 22, 1441–1453. [Google Scholar] [CrossRef] - Jones, P.D.; Moberg, A. Hemispheric and large-scale surface air temperature variations: An extensive revision and an update to 2001. J. Clim.
**2003**, 16, 206–223. [Google Scholar] [CrossRef] - Vose, R.S.; Easterling, D.R.; Gleason, B. Maximum and minimum temperature trends for the globe: An update through 2004. Geophysical. Res. Lett.
**2005**, 32. [Google Scholar] [CrossRef] - Toreti, A.; Desiato, F. Temperature trend over Italy from 1961 to 2004. Theor. Appl. Climatol.
**2008**, 91, 51–58. [Google Scholar] [CrossRef] - Liuzzo, L.; Bono, E.; Sammartano, V.; Freni, G. Analysis of spatial and temporal rainfall trends in Sicily during the 1921–2012 period. Theor. Appl. Climatol.
**2015**. [Google Scholar] [CrossRef] - Coles, S.; Bawa, J.; Trenner, L.; Dorazio, P. An Introduction to Statistical Modeling of Extreme Values; Springer: London, UK, 2001; Volume 208. [Google Scholar]
- Bordi, I.; Fraedrich, K.; Petitta, M.; Sutera, A. Extreme value analysis of wet and dry periods in Sicily. Theor. Appl. Climatol.
**2007**, 87, 61–71. [Google Scholar] [CrossRef] - Noto, L.V.; La Loggia, G. Use of L-moments approach for regional flood frequency analysis in Sicily, Italy. Water Res. Manag.
**2009**, 23, 2207–2229. [Google Scholar] [CrossRef] - Hosking, J.R.M.; Wallis, J.R. Some statistics useful in regional frequency analysis. Water Resour. Res.
**1993**, 29, 271–281. [Google Scholar] [CrossRef] - Rossman, L.A. Storm Water Management Model: User’s Manual 5.1; U.S. Environmental Protection Agency: Cincinnati, Ohio, USA, 2009.
- OFWAT. Preparing for the Future—Ofwat’s Climate Change Policy Statement; OFWAT: UK, 2008. Available online: https://www.ofwat.gov.uk (accessed on 22 September 2015).

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**MDPI and ACS Style**

Notaro, V.; Liuzzo, L.; Freni, G.; La Loggia, G.
Uncertainty Analysis in the Evaluation of Extreme Rainfall Trends and Its Implications on Urban Drainage System Design. *Water* **2015**, *7*, 6931-6945.
https://doi.org/10.3390/w7126667

**AMA Style**

Notaro V, Liuzzo L, Freni G, La Loggia G.
Uncertainty Analysis in the Evaluation of Extreme Rainfall Trends and Its Implications on Urban Drainage System Design. *Water*. 2015; 7(12):6931-6945.
https://doi.org/10.3390/w7126667

**Chicago/Turabian Style**

Notaro, Vincenza, Lorena Liuzzo, Gabriele Freni, and Goffredo La Loggia.
2015. "Uncertainty Analysis in the Evaluation of Extreme Rainfall Trends and Its Implications on Urban Drainage System Design" *Water* 7, no. 12: 6931-6945.
https://doi.org/10.3390/w7126667