# Physical Parameterization of IDF Curves Based on Short-Duration Storms

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sherman Parameterization

- $\mathrm{i}$ rainfall intensity, in mm/h
- T return period, in years
- d storm duration, in minutes

“The Montana curve showed significant deviations at the lower time scales (for durations from 0 to 10 min), although giving a good representation of the decay of expected maximum rainfall intensities for larger durations (see Figure 1). The limitation of this formula is obvious, since it estimates rainfall intensities tending to infinity when time approaches to 0, and therefore a resulting overestimation of rainfall intensities for low t values”[22], p. 676.

#### 2.2. Traditional Methodologies for Estimating IDF Curves

**Method-a.**By applying the logarithmic conversion, it is possible to convert the Equation (2) into a linear equation, and thus to calculate all the parameters related to the equation. $\mathrm{log}\mathrm{i}=\mathrm{log}\mathrm{k}+\text{}\mathrm{m}\mathrm{log}\mathrm{T}-\mathrm{n}\mathrm{log}({\mathrm{d}}^{\mathsf{\theta}}+\mathrm{C})$ [58]. It is the most used procedure in the literature and the most generalized [59]. Non-linear optimization procedures could also be applied [60].

**Method-b.**We proceeded by trial and error, adjusting the IDF curves by changing the value of C, until “visually” curves are transformed into straight lines when the axis of duration, in minutes, is represented in logarithmic scale [42].

#### 2.3. Proposed Methodologies for Estimating IDF Curves

**Method-c**. Value of $\overline{\mathrm{C}}$ weighted with the number of events (${\mathrm{ST}}_{\mathrm{i}}$) for each analyzed storm duration (${\mathrm{d}}_{\mathrm{i}}$), in minutes, divided by the total number of recorded events (TST). An array similar to the one presented in Table 2 is used.

**Method**

**-d.**Then $\hat{\mathrm{C}}$ results from a change in intensity, directly proportional to the change in intensity with respect to duration (in logarithms). To complete the differential, a constant term is included that may or may not exist, depending on the statistics of the sample. This can be expressed as follows:

#### 2.4. Queretaro Hydrometeorological Network

^{2}. Likewise, the typical rainfall intensities (representative) for each station were studied. It is observed that in the entire raining season, the typical duration is 5 and 6 min, except for SJA-09 station (San Juan del Río city Queretaro), for which a more representative storm duration was 9 min. On the other hand, the station with the highest number of storms recorded was M-07 station, with a total of 82 storms for all months, and the one with the least storms recorded was SJA-09 station with 31 storms. Speaking about months, June is the month in which the highest number of storms was registered, with 108 storms analyzed, while in November only 30 storms were recorded. The analysis by duration showed that the duration of 5 min is the one that recorded the most events, with a total of 100 events for that duration, while the duration of 20 min was only recorded 11 times. It is important to note that as the duration increases, the number of events registered decreases. This proves the procedure proposed by [39].

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Traditional representation of an intensity–duration–frequency (IDF) curve at station CH-03. $\mathrm{k}=51.55;\text{}\mathrm{m}=0.53;\text{}\mathrm{n}=0.53;\text{}\mathsf{\theta}=1;\text{}\mathrm{C}=0$.

**Figure 2.**Traditional representation of an IDF curve at station CH-03. $\mathrm{k}=51.55;\text{}\mathrm{m}=0.53;\text{}\mathrm{n}=0.53;\text{}\mathsf{\theta}=1;\text{}\mathrm{C}=50$.

Formulation Known As | k | m | n | $\mathsf{\theta}$ | C |
---|---|---|---|---|---|

Law of Montana [47] | - | - | 0 | - | - |

Sherman [39] | - | - | - | 1 | - |

Bernard [40] | - | 0 | 1 | 1 | 0 |

Talbot/Linsley [56] ^{1} | - | 0 | 1 | 1 | - |

Wenzel/Kimijima [57] | - | 0 | 1 | - | - |

Chow [41] | - | - | 1 | 1 | - |

Koutsoyiannis [42] ^{2} | - | - | - | 1 | - |

Seong [52] | - | 1 | $\left(\mathrm{n}\xb7\mathrm{m}\right)$ | 1 | - |

^{1}For duration (d) between 5 and 20 min and greater than 60 min.

^{2}With ${\mathrm{T}}^{\mathrm{m}}=\mathrm{m}-\mathrm{Ln}\left[-\mathrm{Ln}\left(1-1/\mathrm{T}\right)\right]$.

**Table 2.**Number of storms (${\mathrm{ST}}_{\mathrm{i}}$) for some stations of Queretaro rainfall warning system (RedCIAQ).

Duration of the Storm (minutes) | TST | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ID | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 18 | 20 | |

C-01 | 9 | 5 | 1 | 5 | 8 | 4 | 2 | 3 | 0 | 1 | 3 | 4 | 1 | 2 | 48 |

B-02 | 7 | 14 | 2 | 3 | 4 | 10 | 7 | 5 | 6 | 4 | 3 | 1 | 4 | 3 | 73 |

CH-03 | 14 | 11 | 7 | 6 | 6 | 2 | 3 | 6 | 4 | 2 | 4 | 2 | 2 | 0 | 69 |

CC-04 | 10 | 1 | 0 | 6 | 3 | 5 | 0 | 3 | 1 | 0 | 3 | 0 | 1 | 0 | 33 |

C-05 | 7 | 10 | 7 | 8 | 7 | 6 | 2 | 5 | 2 | 0 | 4 | 7 | 3 | 3 | 71 |

ER-06 | 26 | 11 | 0 | 6 | 9 | 7 | 1 | 4 | 1 | 2 | 5 | 2 | 1 | 2 | 77 |

M-07 | 13 | 16 | 7 | 6 | 10 | 5 | 6 | 2 | 4 | 2 | 7 | 2 | 2 | 0 | 82 |

RP-08 | 10 | 2 | 7 | 2 | 1 | 3 | 4 | 0 | 2 | 3 | 1 | 1 | 2 | 1 | 39 |

SJA-09 | 4 | 4 | 4 | 1 | 6 | 3 | 2 | 2 | 2 | 2 | 0 | 1 | 0 | 0 | 31 |

- | 100 | 74 | 35 | 43 | 54 | 45 | 27 | 30 | 22 | 16 | 30 | 20 | 16 | 11 | 523 |

Station ID | $\mathbf{TST}\text{}\left({\mathbf{d}}_{\mathbf{t}}\right)$ | ${\mathbf{i}}_{\mathbf{t}}\text{}(\mathbf{mm}/\mathbf{h})$ | Method-a (min) | Method-b (min) | Method-c (min) | Method-d (min) |
---|---|---|---|---|---|---|

C-01 | 48(5) | 18.0 | 13.55 | 9.85 | 10.0 | 14.85 |

B-02 | 73(6) | 15.0 | 21.02 | 11.00 | 16.0 | 17.00 |

CH-03 | 69(5) | 15.6 | 12.50 | 8.88 | 10.5 | 13.88 |

CC-04 | 33(5) | 15.0 | 12.00 | 11.18 | 12.5 | 16.18 |

C-05 | 71(6) | 15.0 | 13.03 | 10.98 | 8.0 | 16.98 |

ER-06 | 77(5) | 15.6 | 12.07 | 7.70 | 13.0 | 12.70 |

M-07 | 82(6) | 18.0 | 18.01 | 11.09 | 17.5 | 17.09 |

RP-08 | 39(5) | 15.6 | 15.42 | 11.07 | 12.5 | 16.07 |

SJA-09 | 31(9) | 15.0 | 13.51 | 11.29 | 12.5 | 20.29 |

**Table 4.**Rainfall intensities (mm/h) for different short duration storms, estimated at station CH-03 with different proposed values of parameter C (min).

${\mathbf{d}}_{\mathbf{i}}\text{}\left(\mathbf{min}\right)$ | Typical Storm (In Situ) | C = 0 | (M-a) C= 12.5 | (M-b) C = 10.5 | (M-c) ^{1} C = 8.88 | (M-d) ^{2} C = 13.88 |
---|---|---|---|---|---|---|

20 | 36.09 | 16.50 | 12.75 | 13.19 | 13.58 | 12.48 |

15 | 26.60 | 19.21 | 13.93 | 14.50 | 15.02 | 13.58 |

10 | 23.50 | 23.82 | 15.50 | 16.28 | 17.01 | 15.02 |

5 | 16.96 | 34.39 | 17.71 | 18.88 | 20.02 | 17.01 |

^{1}$\overline{\mathrm{C}}=\frac{1}{69}\left[\left(14\xb75\right)+\left(11\xb76\right)+\cdots +\left(2\xb718\right)+\left(0\xb720\right)\right]=8.88$ min;

^{2}$\hat{\mathrm{C}}=\overline{\mathrm{C}}+{\mathrm{d}}_{\mathrm{t}}\text{}=\text{}8.88\text{}+\text{}5\text{}=\text{}13.88$ min (from Equation (5)).

**Table 5.**Precipitation intensities (mm/h) for different return periods, estimated at station CH-03 for duration of 10 min and different proposed values of parameter C.

Tr | C = 0 | (M-a) = 12.5 | (M-b) = 10.5 | (M-c) = 8.88 | (M-d) = 13.88 |
---|---|---|---|---|---|

20 | 74.4 | 48.4 | 50.9 | 53.1 | 46.9 |

10 | 51.6 | 33.5 | 35.2 | 36.8 | 32.5 |

5 | 35.7 | 23.2 | 24.4 | 25.5 | 22.5 |

**Table 6.**Some evidence that authors calculate parameter C by some traditional procedure; but without recognizing that in their own work (the proof that C could be estimated with a physical meaning measured in situ).

Fact | Values | Reference |
---|---|---|

“the rainfall time series that allows the correct transfer to the model is 30 min” | The storm duration using in this study for IDF curves is 35 min. | [63], p.374 |

“the storm index associated with the precipitation depth for design storms lasts 60 min” | The IDF equation presented is of type [42] where C = 55 | [64], p.16 |

“The parameter C varies between 0 and${C}_{max}=12{d}_{min}$” their data vary between 0.083 h (5 min) and 0.167 h (10 min). The results of his work present two IDF equations. | C for the IDF curves were C = 0.189 (5 min) and C = 0.143 (10 min). The parameters of these equations are obtained by L-moments and an optimization routine. | [42], p.129 |

“IDF curves for precipitation in the Monsoon area of Vietnam, identifies ratios 60-min rainfall intensity and duration for same return period” | The ratios were fitted by Sherman’s equation with C = 76.31 | [65], p.100 |

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**MDPI and ACS Style**

Gutierrez-Lopez, A.; Jimenez Hernandez, S.B.; Escalante Sandoval, C. Physical Parameterization of IDF Curves Based on Short-Duration Storms. *Water* **2019**, *11*, 1813.
https://doi.org/10.3390/w11091813

**AMA Style**

Gutierrez-Lopez A, Jimenez Hernandez SB, Escalante Sandoval C. Physical Parameterization of IDF Curves Based on Short-Duration Storms. *Water*. 2019; 11(9):1813.
https://doi.org/10.3390/w11091813

**Chicago/Turabian Style**

Gutierrez-Lopez, Alfonso, Sergio Bernardo Jimenez Hernandez, and Carlos Escalante Sandoval. 2019. "Physical Parameterization of IDF Curves Based on Short-Duration Storms" *Water* 11, no. 9: 1813.
https://doi.org/10.3390/w11091813