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Annual maximum rainfall intensity for several duration and return periods has been analyzed according to the Gumbel distribution. The Intensity-Duration-Frequency (IDF) curves before and after 1980 have been computed and compared. For the city of Toronto, it is shown that the rainfall intensities after 1980 are lower than those from before this date. This is especially clear for those of short duration. Comparing our results with those of other authors, it appears that, for the moment, no general law on the impact of global warming on the curves intensity duration frequency cannot be made. It appears that the impact of global warming on rainfall varies with geographic location and that it is not possible to draw some general conclusions across the planet.

Regarding civil engineering, the knowledge and understanding of climate change is important because, if there are changes in the variables related to hydrological systems, it could imply changes in design criteria, as these are frequently based upon the assumption of the hydrological series stationary. Not doing so, could mean the under or over design of hydraulic infrastructures, thus creating performance deficiencies or over expensive solutions.

The IDF (Intensity Duration Frequency) relationship constitutes an objective tool to quantify precipitation uncertainty, especially in circumstances when a design rainfall event must be determined for a particular water resources project. To perform the analysis, long-term precipitation data from a recording rain gage must be available. The prediction of uncertain environmental variables is often a hydrologic problem of significance in water resources management and water resources design projects. The Gumbel distribution, named after one of the pioneer scientists in practical applications of the Extreme Value Theory (EVT), the German mathematician Emil Gumbel (1891-1966), has been extensively used in various fields including hydrology for modeling extreme events [

The Gumbel distribution is very suitable for modeling extreme event [

where X is a random variable. In our case, X is the rainfall intensity or the rainfall depth for a given duration.

The Gumbel variable is defined by Equation (2)

The parameters a and b are defined by:

where σ is the standard deviation and μ is the mean of the variable.

The empirical distribution of Hazen is used:

where i is the rank of a given data and n is the total number of data.

The rainfall data of Toronto (Canada) has been used for computing the IDF curves before and after 1980.

The rainfall data are from 1940 to 2007. The rainfall station is located at latitude 43.67˚N and longitude 79.4˚W. Its elevation is 112 m. The duration of rainfall are 5 min, 10 min, 15 min, 30 min, 1 h, 2 h, 6 h, 12 h and 24 h. Equation (2) shows that if the Gumbel distribution is valid, it has to a linear relationship between the empirical intensity x and the Gumbel variable u.

Validity of the Gumbel DistributionStarting from Equation (4), the Gumbel variable is computed by:

where ln is the natural logarithm, i is the rank of a given data and n is the total number of data.

The IDF curves

The IDF curves are computed for five return periods T (2, 5, 10, 20 and 50 years). For these return periods, the probability associated with the not exceedance is computed by:

The Gumbel variable are computed by:

For each rainfall duration, there are a specific standard deviation σ and a specific mean μ. Therefore, there are a specific parameters a and b defined by Equation (3).

Equation (2) enable to compute the rainfall depth x for the different durations and return periods. Finally, the rainfall intensity is calculated by dividing the rainfall depth by the duration.

Examination of

Return periods (year) | 2 | 5 | 10 | 20 | 50 |
---|---|---|---|---|---|

Probability associated with the not exceedance | 0.5 | 0.8 | 0.9 | 0.95 | 0.98 |

Gumbel variable | 0.36651292 | 1.49993999 | 2.25036733 | 2.97019525 | 3.90193866 |

Duration | 5 min | 10 min | 15 min | 30 min | 1 h |
---|---|---|---|---|---|

Mean | 9.76393443 | 13.4672131 | 16.5032787 | 20.9852459 | 25.4163934 |

Standard deviation | 4.02181687 | 4.81233558 | 6.60978989 | 8.77967038 | 10.0157573 |

b | 3.13738827 | 3.75406581 | 5.15624603 | 6.84895304 | 7.81321486 |

a | 7.95303392 | 11.3003663 | 13.5270935 | 17.0320302 | 20.9066058 |

Duration | 2 h | 6 h | 12 h | 24 h |
---|---|---|---|---|

Mean | 29.6983607 | 36.59 | 43.3305085 | 48.3409836 |

Standard deviation | 10.6580563 | 12.7690662 | 13.7554786 | 14.7781976 |

b | 8.31426738 | 9.96104988 | 10.7305426 | 11.5283578 |

a | 24.8993655 | 30.840482 | 37.1368393 | 41.6868155 |

Intensity mm/h | 2 years | 5 years | 10 years | 20 years | 50 years |
---|---|---|---|---|---|

5 min | 113.4117503 | 161.4166204 | 193.2000214 | 223.6874228 | 263.1502366 |

10 min | 77.58690533 | 105.544399 | 124.0546925 | 141.8102105 | 164.7929072 |

15 min | 62.53175359 | 87.23827087 | 103.5961351 | 119.2869912 | 139.5971949 |

30 min | 40.15905929 | 55.73317723 | 66.04459846 | 75.93556127 | 88.73839806 |

1 h | 24.92161479 | 33.28195029 | 38.81721985 | 44.1267833 | 50.99946856 |

2 h | 14.73000025 | 19.2794554 | 22.29159057 | 25.18090295 | 28.92082144 |

6 h | 6.084464197 | 7.9127997 | 9.123316892 | 10.28447405 | 11.78747288 |

12 h | 3.611399512 | 4.608573879 | 5.268790077 | 5.902085304 | 6.721821017 |

24 h | 2.012225697 | 2.541975305 | 2.89271564 | 3.229154191 | 3.664639384 |

Intensity mm/h | 2 years | 5 years | 10 years | 20 years | 50 years |
---|---|---|---|---|---|

5 min | 103.1542708 | 135.6783017 | 157.2120401 | 177.867719 | 204.6043765 |

10 min | 73.74453206 | 95.29262978 | 109.5593454 | 123.2443216 | 140.9581195 |

15 min | 60.28632165 | 81.87858867 | 96.17454824 | 109.8875759 | 127.6376836 |

30 min | 37.38801884 | 53.05351799 | 63.42544152 | 73.37443959 | 86.25239708 |

1 h | 22.0043123 | 31.57485927 | 37.91139413 | 43.98955049 | 51.85710044 |

2 h | 12.8216983 | 17.70357911 | 20.935809 | 24.03624151 | 28.04943337 |

6 h | 5.218241187 | 7.131638583 | 8.39847415 | 9.61365324 | 11.18657793 |

12 h | 3.132853403 | 4.139250597 | 4.805573109 | 5.44472566 | 6.272043077 |

24 h | 1.76292092 | 2.314501059 | 2.679695105 | 3.029998002 | 3.483429171 |

This is especially clear for those of short duration. Their intensity decreased, particularly for the return period of 5, 10, 20 and 50 years.

Examination of

Fallot and Hertig [

The research carried out showed that the samples of intensive rainfalls do not exhibit trends, as to affirm or contradict the effects often attributed to the climate change phenomenon (i.e. heavier rainfalls with smaller duration). The study found out that all kinds of behaviors can occur: some samples denote the trends often considered as resulting from the climate change, while exhibit the exact opposite, not allowing the identification of any of the consequences attributed to such phenomenon.

Comparing our results with those of other authors, it appears that, for the moment, no general law on the impact of global warming on the intensity duration frequency relationships can be made. It appears that the impact of global warming on rainfall varies with geographic location and that it is not possible to draw some general conclusions across the planet

ErickCarlier,Jamal ElKhattabi, (2016) Impact of Global Warming on Intensity-Duration-Frequency (IDF) Relationship of Precipitation: A Case Study of Toronto, Canada. Open Journal of Modern Hydrology,06,1-7. doi: 10.4236/ojmh.2016.61001