# Introducing Non-Stationarity Into the Development of Intensity-Duration-Frequency Curves under a Changing Climate

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## Abstract

**:**

_{NS}) method is implemented to develop IDFs for future conditions, introducing non-stationarity where justified, based on the Global Climate Models (GCM). The methodology is applied for Moncton and Shearwater gauges in Northeast Canada. From the results, it is observed that EQM

_{NS}is able to capture the trends in the present and to translate them to estimated future rainfall intensities. Comparison of present and future IDFs strongly suggest that return period can be reduced by more than 50 years in the estimates of future rainfall intensities (e.g., historical 100-yr return period extreme rainfall may have frequency smaller than 50-yr under future conditions), raising attention to emerging risks to water infrastructure systems.

## 1. Introduction

_{NS}) is based on statistical analysis which identifies if the non-stationary GEV model is the best GEV model fitted to the data. Time is adopted as a covariate in the location and scale parameters of the GEV distribution. Different time-variant and trend functions lead to eight combinations of non-stationary models which are considered in this study together with one stationary GEV model. The new framework is applied to Moncton City gauging station data in the Province of New Brunswick, Canada, in order to analyze its performance in developing future IDF curves.

## 2. Study Area and Data Used

## 3. Methodology

_{NS}and is schematically shown in Figure 2.

_{NS}methodology includes:

- Statistical analysis: applied to fit stationary and non-stationary probability distributions to both historical and future projected data. An information criteria method is used to identify the best probability distribution model, and a significance test is performed to assess the statistical significance of the non-stationary model in comparison to the stationary one;
- Updating IDF curves for future conditions (EQM
_{NS}): a modified EQM methodology is applied to generate future sub-daily annual maximum precipitation data, and update IDF curves for future period under non-stationary conditions.

#### 3.1. Statistical Analysis

#### 3.1.1. Theoretical Probability Distribution

_{1},x

_{2},…,x

_{n}}, be independent and identically distributed (i.i.d.) with common cumulative distribution function. The annual maximum series can be approximate to a theoretical probability of extremes (e.g., GEV, Gumbel, Fréchet, Weibull). In fact, GEV probability distribution is a family of three distributions combined into one: Gumbel, Fréchet and Weibull. GEV distribution is applied by many studies of extreme precipitation [5,6,32,37,38]. The GEV cumulative distribution function F(x) is given by Equation (1) for ξ ≠ 0 [35]:

_{1},x

_{2},…,x

_{n}} with n years, the log-likelihood derived from Equation (1) is given by Equation (2), for the stationary case.

#### 3.1.2. Identification of the Best Model

#### 3.1.3. Rainfall Depth Estimation

_{p}is given by Equation (8):

#### 3.2. Non-Stationary IDF Curves under Changing Climate

_{NS}). EQM method captures the distribution of changes between the projected time period and the baseline period (temporal downscaling) in addition to spatial downscaling of the annual maximum precipitation derived from the GCM data and the observed sub-daily data [11].

_{NS}introduces the effects of non-stationarity in historic AMPs time series by adopting the 95th percentile values from the varying parameter(s) (Equations (9) and (10)) in Equation (12). This step ensures that the non-stationarity presented even in the historic period is accounted for generating IDFs for future conditions.

_{NS}uses the quantile delta mapping, since it preserves the relative changes in the precipitation mean and quantiles obtained from the climate models [24,50]. The cumulative probability distribution of the GCM generated rainfall in the future period (${F}_{m,f}$) and the GCM generated rainfall in the baseline period (${x}_{m,h}$) are equated to establish a statistical relationship (Equation (14)). In addition, the relative change between the historical and future period is given by Equation (15):

## 4. Results and Discussions

_{NS}); and (iii) differences between the future rainfall intensities, obtained using stationary and non-stationary conditions.

_{NS}methodology implementation is done using the open-access RStudio programming environment [51], based on its available packages and functions [52].

#### 4.1. Trends in GEV Model Parameters in Historical Observed Data

#### 4.2. Historic IDF Relationships

#### 4.3. Performance of the Modified Spatial Downscaling

#### 4.4. Future IDF Curves

## 5. Summary and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Stationary and non-stationary rainfall depths for different return periods for Moncton station.

**Figure 4.**Stationary and non-stationary rainfall depths for different return periods for Shearwater station.

**Figure 5.**Quantile-mapping method for spatial downscaling for different durations. Lines represent the non-linear equation.

**Figure 6.**Rainfall depth estimation for different return periods and GCM models for the Moncton station (Representative Concentration Pathways (RCP) 8.5 emission scenario). Black point represents the rainfall depth estimated with the multimodel median ensemble. Red x represents the models in which results produce outliers.

**Figure 7.**Rainfall depth estimation for different return periods and GCM models for the Shearwater station (RCP 8.5 emission scenario). Black point represents the rainfall depth estimated with the multimodel median ensemble. Red x represents the models in which results produce outliers.

**Figure 8.**Rainfall intensities, in mm/h, estimated for the multimodel ensemble for the Moncton station. X-axis is duration of precipitation in minutes, and Y-axis is the intensity of precipitation in mm/h.

**Figure 9.**Rainfall intensities, in mm/h, estimated for the multimodel ensemble for the Shearwater station. X-axis is duration of precipitation in minutes, and Y-axis is the intensity of precipitation in mm/h.

Study Area | Station Name | Station ID | Latitude x Longitude | Data Availability |
---|---|---|---|---|

Moncton | Moncton INTL A | 8103201 | 46°7′ N 64°41′ W | 1946–2016 (67 years) |

Halifax | Shearwater RCS | 8205092 | 46°7′ N 64°41′ W | 1955–2016 (59 years) |

Model | Country | Centre Name | Spatial Resolution (Longitude vs. Latitude) |
---|---|---|---|

bcc-csm1-1 | China | Beijing Climate Center, China Meteorological Administration | 2.8 × 2.8 |

bcc-csm1-1-m | China | Beijing Climate Center, China Meteorological Administration | 2.8 × 2.8 |

BNU-ESM | China | College of Global Change and Earth System Science | 2.8 × 2.8 |

CanESM2 | Canada | Canadian Centre for Climate Modeling and Analysis | 2.8 × 2.8 |

CCSM4 | USA | National Center of Atmospheric Research | 1.25 × 0.94 |

CESM1-CAM5 | USA | National Center of Atmospheric Research | 1.25 × 0.94 |

CNRM-CM5 | France | Centre National de Recherches Meteorologiques and Centre Europeen de Recherches et de Formation Avancee en Calcul Scientifique | 1.4 × 1.4 |

CSIRO-Mk3-6-0 | Australia | Australian Commonwealth Scientific and Industrial Research Organization in collaboration with the Queensland Climate Change Centre of Excellence | 1.8 × 1.8 |

FGOALS-g2 | China | IAP (Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China) and THU (Tsinghua University) | 2.55 × 2.48 |

GFDL-CM3 | USA | National Oceanic and Atmospheric Administration’s Geophysical Fluid Dynamic Laboratory | 2.5 × 2.0 |

GFDL-ESM2G | USA | National Oceanic and Atmospheric Administration’s Geophysical Fluid Dynamic Laboratory | 2.5 × 2.0 |

HadGEM2-AO | United Kingdom | Met Office Hadley Centre | 1.25 × 1.875 |

HadGEM2-ES | United Kingdom | Met Office Hadley Centre | 1.25 × 1.875 |

IPSL-CM5A-LR | France | Institut Pierre Simon Laplace | 3.75 × 1.8 |

IPSL-CM5A-MR | France | Institut Pierre Simon Laplace | 3.75 × 1.8 |

MIROC5 | Japan | Japan Agency for Marine-Earth Science and Technology | 1.41 × 1.41 |

MIROC-ESM | Japan | Japan Agency for Marine-Earth Science and Technology | 2.8 × 2.8 |

MIROC-ESM-CHEM | Japan | Japan Agency for Marine-Earth Science and Technology | 2.8 × 2.8 |

MPI-ESM-LR | Germany | Max Planck Institute for Meteorology | 1.88 × 1.87 |

MPI-ESM-MR | Germany | Max Planck Institute for Meteorology | 1.88 × 1.87 |

MRI-CGCM3 | Japan | Meteorological Research Institute | 1.1 × 1.1 |

NorESM1-M | Norway | Norwegian Climate Center | 2.5 × 1.9 |

NorESM1-ME | Norway | Norwegian Climate Center | 2.5 × 1.9 |

GFDL-ESM2M | USA | National Oceanic and Atmospheric Administration’s Geophysical Fluid Dynamic Laboratory | 2.5 × 2.0 |

**Table 3.**List of Generalized Extreme Value (GEV) models used in this study and their parameter combinations. $\mu ,\sigma ,$ and $\xi $ are the GEV parameters and ‘t’ is the scoring year for which maximum is taken.

GEV Model | |
---|---|

ID | Specification |

I | F ($\mu ;\sigma ;\xi )$ |

II | F (${\mu}_{0}+{\mu}_{1}\mathrm{t};\sigma ;\xi )$ |

III | F (${\mu}_{0}+{\mu}_{1}\mathrm{t};{\sigma}_{0}+{\sigma}_{1}\mathrm{t};\xi )$ |

IV | F (${\mu}_{0}+{\mu}_{1}\mathrm{t};{e}^{\left(\sigma 0+\sigma 1\mathrm{t}\right)}$;$\text{}\xi )$ |

V | F ($\mu ;{\sigma}_{0}+{\sigma}_{1}\mathrm{t};\xi )$ |

VI | F ($\mu ;{e}^{\left(\sigma 0+\sigma 1\mathrm{t}\right)};\xi )$ |

VII | F (${\mu}_{0}+{\mu}_{1}\mathrm{t}+{\mu}_{2}{\mathrm{t}}^{2};\sigma ;\xi )$ |

VIII | F (${\mu}_{0}+{\mu}_{1}\mathrm{t}+{\mu}_{2}{\mathrm{t}}^{2};{\sigma}_{0}+{\sigma}_{1}\mathrm{t};\xi )$ |

IX | F (${\mu}_{0}+{\mu}_{1}\mathrm{t}+{\mu}_{2}{\mathrm{t}}^{2};{e}^{\left(\sigma 0+\sigma 1\mathrm{t}\right)};\xi )$ |

Duration (Minutes) | Best GEV-Type | Stationary GEV Model | Best GEV Model (95th Percentile) | LR-Test (p-Value) | ||||
---|---|---|---|---|---|---|---|---|

Location | Scale | Shape | Location | Scale | Shape | |||

5 | I | 5.18 | 2.10 | 0.07 | 5.18 | 2.10 | 0.07 | - |

10 | I | 7.32 | 2.92 | 0.07 | 7.32 | 2.92 | 0.07 | - |

15 | VI | 8.92 | 3.35 | 0.09 | 9.01 | 4.27 | 0.14 | 0.043 |

30 | V | 12.03 | 4.12 | 0.08 | 12.33 | 5.51 | 0.06 | 0.030 |

60 | II | 16.22 | 4.60 | 0.22 | 18.06 | 4.10 | 0.31 | 0.011 |

120 | VII | 22.22 | 5.63 | 0.23 | 25.46 | 5.19 | 0.30 | 0.047 |

360 | II | 35.30 | 11.07 | −0.02 | 39.97 | 10.78 | −0.03 | 0.041 |

720 | II | 43.59 | 14.35 | 0.02 | 50.68 | 13.86 | 0.004 | 0.016 |

1440 | II | 51.44 | 17.45 | 0.05 | 61.62 | 15.95 | 0.08 | 0.002 |

Duration (Minutes) | Best GEV-Type | Stationary GEV Model | Best GEV Model (95th Percentile) | LR Test (p-Value) | ||||
---|---|---|---|---|---|---|---|---|

Location | Scale | Shape | Location | Scale | Shape | |||

5 | I | 4.98 | 1.46 | 0.07 | 4.98 | 1.46 | 0.07 | - |

10 | I | 7.67 | 2.16 | 0.001 | 7.67 | 2.16 | 0.001 | - |

15 | I | 9.88 | 2.80 | −0.13 | 9.88 | 2.80 | −0.13 | - |

30 | I | 13.88 | 3.95 | −0.16 | 13.88 | 3.95 | −0.16 | - |

60 | I | 19.17 | 4.94 | −0.06 | 19.17 | 4.94 | −0.06 | - |

120 | I | 26.30 | 6.95 | 0.04 | 26.30 | 6.95 | 0.04 | - |

360 | V | 44.24 | 11.94 | −0.004 | 43.93 | 14.67 | 0.14 | 0.044 |

720 | I | 54.57 | 14.34 | −0.06 | 54.57 | 14.34 | −0.06 | - |

1440 | II | 60.59 | 16.05 | 0.08 | 66.61 | 15.05 | 0.12 | 0.030 |

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**MDPI and ACS Style**

Feitoza Silva, D.; Simonovic, S.P.; Schardong, A.; Avruch Goldenfum, J.
Introducing Non-Stationarity Into the Development of Intensity-Duration-Frequency Curves under a Changing Climate. *Water* **2021**, *13*, 1008.
https://doi.org/10.3390/w13081008

**AMA Style**

Feitoza Silva D, Simonovic SP, Schardong A, Avruch Goldenfum J.
Introducing Non-Stationarity Into the Development of Intensity-Duration-Frequency Curves under a Changing Climate. *Water*. 2021; 13(8):1008.
https://doi.org/10.3390/w13081008

**Chicago/Turabian Style**

Feitoza Silva, Daniele, Slobodan P. Simonovic, Andre Schardong, and Joel Avruch Goldenfum.
2021. "Introducing Non-Stationarity Into the Development of Intensity-Duration-Frequency Curves under a Changing Climate" *Water* 13, no. 8: 1008.
https://doi.org/10.3390/w13081008