# Intensity-Duration-Frequency Curves at Ungauged Sites in a Changing Climate for Sustainable Stormwater Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology of the Research

#### 2.1. The Nonstationary GEV Distribution Function

#### 2.2. Scaling of Rainfall Intensities

_{d}and I

_{λd}, corresponding to durations d and λd, can be related by the following equation [35,44,63], the equality corresponding to similarity of probability distributions:

_{λd}and I

_{d}are characterised by the same distribution function if finite moments of order q exist for both. The qth moments of rainfall intensity are obtained from Equation (5) after performing the following transformations: (i) raising both sides to an exponent q (order of moment) and (ii) taking averages of both parts [44]:

#### 2.3. Rainfall IDF Curves

_{T}, and slope, η, for each return period T. Based on the already assessed pairs of (A

_{T}, logT), the parameters K and c are then assessed through the least squares approach using the linear relationship:

#### 2.4. Stormwater Network Modelling and Management

## 3. Study Area and Available Data

^{2}, and is located 65 km from Heraklion, 7 km from Neapoli and 22 km from Agios Nikolaos. It is a well-known touristic settlement in Crete, characterised by the large number of churches in the wider area, as well as by the production of a large number of quality national products.

## 4. Results and Discussion

#### 4.1. Nonstationary Analysis of Long-Duration Rainfall Maxima

#### 4.2. Rainfall Scaling and Construction of IDF Curves at the Ungauged Site

_{s}= 0.63). When fitting the GEV distribution function to both series of annual monthly maxima and estimating different quantiles of the series, it has been noted that the quantile ratio among the different quantiles is almost constant (ranges in the interval from 1.51 to 1.57). To further examine homogeneity of annual rainfall monthly maxima of the sites of Heraklion and Fourni, the bootstrap Anderson-Darling [78] and the Durbin and Knott [79] tests are also used, resulting in acceptance of the null hypothesis.

#### 4.3. Design of a Stormwater Network in a Changing Climate

^{3}/s. The infiltration model used for subcatchments in the present work was the one based on Curve Number, while the conductivity coefficient used for green spaces was considered equal to 0.035. For the system’s conduits the minimum slope was considered equal to 0.10% and the minimum diameter was set to DN 400 following the Greek standards. Hydraulic analysis of the network was performed using the dynamic wave routing model. The normal flow criterion was based on the flow’s free surface slope and the Froude number of the flow. The Manning’s roughness coefficient of conduits was set at 0.011 [73]. A 10 min simulation time step was selected for both the hydrologic and hydraulic routing computations.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Map of the study area with an overview of the settlement of Fourni in the island of Crete.

**Figure 3.**Time-dependent estimates of GEV (

**a**) location (mm); (

**b**) scale (mm); (

**c**) shape (−) parameters and Gumbel (

**d**) location (mm); (

**e**) scale (mm) parameters fitted to annual rainfall monthly maxima in Fourni, Crete. Red, green, and blue dashed lines represent statistically significant nonlinear trends for the location, scale and shape parameter of the extreme value distributions, respectively. The degree of the best-fitted polynomial trend and its AIC and BIC are included.

**Figure 4.**Time-dependent estimates of 50-year and 100-year extreme monthly rainfall in Fourni, Crete, extracted by fitting: (

**a**) a nonstationary GEV distribution function; (

**b**) a nonstationary Gumbel distribution function. Dashed lines represent maximum likelihood estimates, while light blue and pink areas correspond to 95% confidence intervals for a return period of 50 and 100 years, respectively.

**Figure 5.**(

**a**) IDF and (

**b**) DDF curves at the site of Heraklion, Crete, for return periods 2, 5, 10, 20, 50, 100 and 200 years.

**Figure 6.**Simple scaling of rainfall return level estimates for return periods 2, 5, 10, 20, 50, 100 and 200 years in Heraklion, Crete. Self-similarity indices, −β, for each return period are included.

**Figure 7.**Return level estimates of annual rainfall monthly maxima in Fourni for different return periods and for four different rainfall durations: (

**a**) 10 min; (

**b**) 1 h; (

**c**) 6 h; (

**d**) 1 day in the interval 1979–2012. Extracted rainfall estimates were produced including the adjusted trends in the GEV distribution parameters.

**Figure 8.**(

**a**) IDF and (

**b**) DDF curves at the site of Fourni, Crete, for return periods 2, 5, 10, 20, 50, 100 and 200 years.

**Figure 9.**Overview of the designed stormwater network in Fourni with (

**a**) conduits and manholes and available flood storages; (

**b**) five outlets of the network.

**Figure 10.**Overview of assessed diameters of stormwater network’s conduits in Fourni considering stationarity of annual rainfall monthly maxima.

**Figure 11.**Overview of assessed diameters of stormwater network’s conduits in Fourni considering nonstationarity of annual rainfall monthly maxima.

**Table 1.**Descriptive statistics for annual rainfall and for annual rainfall monthly maxima at the study site of Fourni during the period 1949–2012.

Rainfall Maxima | Descriptive Statistics | ||||||
---|---|---|---|---|---|---|---|

Mean (mm) | Median (mm) | Max (mm) | Min (mm) | Range (mm) | St. Dev. (mm) | Skewness (−) | |

Annual max | 816.3 | 782.0 | 1296.5 | 411.5 | 885.0 | 185.2 | 0.517 |

Monthly max | 245.2 | 239.3 | 498.5 | 112.7 | 385.8 | 69.9 | 0.731 |

**Table 2.**Rainfall depth and intensity maximum likelihood return level estimates in Fourni, for return periods of 20, 50, 100 and 200 years and rainfall durations of 10 min, 1 h and 24 h, extracted by fitting the stationary and a nonstationary (95th percentile of ξ) GEV distribution to annual rainfall monthly maxima.

Duration | Stationary GEV | Nonstationary GEV | |||||||
---|---|---|---|---|---|---|---|---|---|

T (Years) | 20 | 50 | 100 | 200 | 20 | 50 | 100 | 200 | |

10 min | P (mm) | 30.06 | 34.73 | 38.24 | 41.74 | 32.23 | 38.97 | 44.42 | 50.21 |

i (mm/h) | 180.35 | 208.37 | 229.42 | 250.42 | 193.36 | 233.81 | 266.54 | 301.28 | |

1 h | P (mm) | 51.56 | 59.05 | 64.61 | 70.10 | 55.28 | 66.27 | 75.07 | 84.34 |

i (mm/h) | 51.56 | 59.05 | 64.61 | 70.10 | 55.28 | 66.27 | 75.07 | 84.34 | |

24 h | P (mm) | 134.29 | 151.43 | 163.85 | 175.85 | 143.99 | 169.92 | 190.36 | 211.57 |

i (mm/h) | 5.60 | 6.31 | 6.83 | 7.33 | 6.00 | 7.08 | 7.93 | 8.82 |

**Table 3.**Rainfall depth and intensity upper 97.5% return level estimates in Fourni, for return periods of 20, 50, 100 and 200 years and rainfall durations of 10 min, 1 h and 24 h, extracted by fitting the stationary and a nonstationary (95th percentile of ξ) GEV distribution to annual rainfall monthly maxima.

Duration | Stationary GEV | Nonstationary GEV | |||||||
---|---|---|---|---|---|---|---|---|---|

T (Years) | 20 | 50 | 100 | 200 | 20 | 50 | 100 | 200 | |

10 min | P (mm) | 35.13 | 40.74 | 45.03 | 49.37 | 42.02 | 53.39 | 63.38 | 74.64 |

i (mm/h) | 210.76 | 244.42 | 270.18 | 296.23 | 252.11 | 320.35 | 380.29 | 447.82 | |

1 h | P (mm) | 59.53 | 68.75 | 75.82 | 82.97 | 71.21 | 90.11 | 106.72 | 125.42 |

i (mm/h) | 59.53 | 68.75 | 75.82 | 82.97 | 71.21 | 90.11 | 106.72 | 125.42 | |

24 h | P (mm) | 151.73 | 173.96 | 191.05 | 208.33 | 181.50 | 228.00 | 268.91 | 314.94 |

i (mm/h) | 6.32 | 7.25 | 7.96 | 8.68 | 7.56 | 9.50 | 11.20 | 13.12 |

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**MDPI and ACS Style**

Galiatsatou, P.; Iliadis, C.
Intensity-Duration-Frequency Curves at Ungauged Sites in a Changing Climate for Sustainable Stormwater Networks. *Sustainability* **2022**, *14*, 1229.
https://doi.org/10.3390/su14031229

**AMA Style**

Galiatsatou P, Iliadis C.
Intensity-Duration-Frequency Curves at Ungauged Sites in a Changing Climate for Sustainable Stormwater Networks. *Sustainability*. 2022; 14(3):1229.
https://doi.org/10.3390/su14031229

**Chicago/Turabian Style**

Galiatsatou, Panagiota, and Christos Iliadis.
2022. "Intensity-Duration-Frequency Curves at Ungauged Sites in a Changing Climate for Sustainable Stormwater Networks" *Sustainability* 14, no. 3: 1229.
https://doi.org/10.3390/su14031229