# Estimation of the G2P Design Storm from a Rainfall Convectivity Index

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rainfall Characterization

#### 2.1. Selection of an Irregularity Index for the Storm Temporal Structure

#### 2.2. Rainfall Data

^{−1}. Figure 2 shows the hyetographs corresponding to some of the events taken from the sample. Table 1 shows some relevant empirical statistics obtained from the sample: the storm duration (D), the maximum intensity for a 5 min interval (I

_{5}) and the total rainfall depth (V).

^{2}= 0.529.

#### 2.3. Return Period Assignment

_{1}and X

_{2}, first and second principal components, respectively. The first one, X

_{1}, is found to explain in this case 92.73% of the total variance. Accordingly, it can be reasonably considered a measurement of the storm magnitude, involving both original variables (I

_{10}and I

_{60}). Introducing this new variable implies in statistical terms to handle in a single variable that contains by itself more information than any of the two original ones. On the other hand, it facilitates the return period assignment to the storm, requiring the use of a single statistical variable. Equation (2) shows the linear relationship between X

_{1}and I

_{10}, I

_{60}, resulting from the PCA.

_{1}. In particular, general extreme value (GEV) [39,40], Gumbel [41], SQRT-ETmax [42] and two-component extreme value (TCEV) [43] distributions were compared. In all cases, parameters have been estimated by the method of maximum likelihood estimation (MLE). Table 3 shows the estimated values for each of the four distribution functions considered.

_{j}is the estimate of the cumulative frequency of the jth term, j is the order of the observation from the smallest, and N is the number of observations. This expression provides an almost unbiased plotting position for the special case of Gumbel distribution. In practice, though, it is commonly used and suitable to assign empirical probabilities when dealing with samples of maximum values, as is the case [45,46,47,48,49].

^{2}test and the Akaike information criteria (AIC) [50]. Table 4 shows the resulting values for each of the functions. According to them, the GEV distribution function was selected, providing best statistics for both tests. Quantiles derived from the GEV distribution function are shown in Table 5.

## 3. G2P Design Storm Estimation from the Rainfall Convectivity n-Index

#### 3.1. G2P Design Storm: Analytical Definition

^{−1}) is the rainfall intensity in each instant t, i

_{0}(mmh

^{−1}) is a scale parameter representative of the instantaneous peak intensity of the rainfall event, and φ (min

^{−1}) is the storm shape parameter.

_{L}and t

_{U}being the lower and upper limits of the central interval in the design storm (Equation (6)).

#### 3.2. Relationship between the G2P Shape Parameter and the n-Index

_{1}= 10 min and Δt

_{2}= 60 min.

_{1}= 10 min and Δt

_{2}= 60 min, the theoretical ratio $\frac{{\mathrm{I}}_{10}}{{\mathrm{I}}_{60}}$ is given by Equation (10).

^{−1}). Figure 6 shows the relationship between the n-index and φ, according to Equation (11). The different categories shown in Table 2 are illustrated in the graph. Higher convectivity is linked to upper values of the G2P shape parameter. For coastal Mediterranean regions in Spain, strongly affected by an extreme hydrological regimen, typical values of φ are in the range 0.08 to 0.31 for design purposes [19].

#### 3.3. Practical Estimation of the G2P Design Storm for Urban Drainage Applications

_{0}(scale parameter) in Equation (4). This estimation can be achieved in different ways, depending on the available rainfall information in a particular geographical location, from the simpler case with only IDF curves information available to a case similar to the one introduced herein (Valencia), where a detailed insight analysis of historical storms extracted from high temporal resolution series is available.

**Shape parameter, φ (min**. It derives directly from the possible convectivity quantified in through the n-index, using Equation (11). Figure 6 shows the distribution of n-index values for the particular case study analysed herein. For each of the classes assumed, a representative central n-index value is adopted, i.e., n = 0.3, 0.5, 0.7 and n = 0.9. Different probabilities of occurrence might be assigned through the empirical histogram shown in Figure 4. In any case, Equation (11) provides a unique φ value for each of the selected n-index values. Table 6 shows the G2P shape parameter assigned to each of the representative n-index values.^{−1})**Scale parameter, i**. It basically depends on the return period. For T = 25 years, an X_{0}(mm h^{−1})_{1}quantile is estimated (Table 5): X_{1}= 179.20. Then, solving Equations (2) and (9), we obtain I_{10}= 157.27 mm h^{−1}and I_{60}= 91.88 mm h^{−1}. Finally, i_{0}is obtained from Equation (7), using either Δt = 10 min or Δt = 60 min. The resultant estimated values for the scale parameters are: i_{0}= 160.90 mm h^{−1}(for n = 0.3); i_{0}= 175.33 mm h^{−1}(for n = 0.5); i_{0}= 195.72 mm h^{−1}(for n = 0.7); i_{0}= 249.55 mm h^{−1}(for n = 0.9).

_{10}and I

_{60}values from the ID curve (for T = 25 years, to continue with the same example) are needed to obtain the n-index value (Equation (9)). Then, φ (shape parameter) is obtained after Equation (11), while scale parameter i

_{0}is again calculated from Equation (7).

_{10}= 133.3 (mm h

^{−1}), I

_{60}= 70.1 (mm h

^{−1}) for the adopted return period. Thus, $\frac{{\mathrm{I}}_{10}}{{\mathrm{I}}_{60}}$= 1.9017, and n= 0.359. From Equation (11), φ = 0.0856 min

^{−1}, and finally, i

_{0}= 137.3 mm h

^{−1}, from Equation (7).

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location of the rain gauge station in Valencia. Images taken from Institut Cartogràfic Valencià, Generalitat Valenciana (https://visor.gva.es/visor/, accessed on 3 July 2021) and Google Earth.

**Figure 2.**Representative hyetographs of some events from the sample. (

**a**) Hyetograph of the 10/23/2000 event; (

**b**) hyetograph of the 10/11/2007 event; (

**c**) hyetograph of the 09/18/2018 event; (

**d**) hyetograph of the 11/15/2018 event.

Storm Duration
D (min) | Maximum Intensity I_{5} (mm h^{−1}) | Rainfall Depth
V (mm) | |
---|---|---|---|

Maximum | 8530.0 | 223.2 | 220.8 |

Minimum | 15.0 | 50.4 | 5.2 |

Mean | 1049.3 | 96.9 | 43.8 |

Median | 530.0 | 82.8 | 26.3 |

Standard deviation | 1484.2 | 42.8 | 46.7 |

Bias | 2.9 | 1.2 | 2.1 |

Kurtosis | 10.8 | 0.8 | 3.9 |

n | Type of Curve | Intensity | Temporal Distribution | Interpretation of Rainfall Type |
---|---|---|---|---|

0.00–0.20 | Very gentle | Practically constant | Very regular | Stationary/highly predominantly advective |

0.20–0.40 | Gentle | Lightly variable | Regular | Predominantly advective |

0.40–0.60 | Normal | Variable | Irregular | Effective |

0.60–0.80 | Pronounced | Moderately variable | Very irregular | Predominantly convective |

0.80–1.00 | Very pronounced | Strongly variable | Nearly instantaneous | Highly predominantly convective |

Cumulative Distribution Function (CDF) | Parameters | ||||
---|---|---|---|---|---|

GEV [39,40] | $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{exp}\left[-{\left(1-\frac{\mathsf{\beta}}{\mathsf{\alpha}}\left({\mathrm{x}-\mathrm{x}}_{0}\right)\right)}^{\frac{1}{\mathsf{\beta}}}\right]$ | α | β | x_{0} | |

26.0882 | −0.0480 | 62.324 | |||

Gumbel [41] | $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{exp}(-\mathsf{\lambda}\mathrm{exp}\left(-\mathsf{\theta}\mathrm{x}\right))$ | θ | λ | ||

0.0376 | 10.6652 | ||||

SQRT-ET max [42] | $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{exp}\left[-\mathsf{\kappa}\left(1+\sqrt{\mathsf{\alpha}\mathrm{x}}\right)\mathrm{exp}(-\sqrt{\mathsf{\alpha}\mathrm{x}})\right]$ | α | κ | ||

0.5219 | 41.8091 | ||||

TCEV [43] | $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{exp}\left[{-\mathsf{\lambda}}_{1}\mathrm{exp}\left({-\mathsf{\theta}}_{1}\mathrm{x}\right){-\mathsf{\lambda}}_{2}{\mathrm{exp}(-\mathsf{\theta}}_{2}\mathrm{x})\right]$ | θ_{1} | θ_{2} | λ_{1} | λ_{2} |

0.0456 | 0.0265 | 11.1031 | 1.8180 |

Criterion | SQRT-ET Max | TCEV | Gumbel | GEV |
---|---|---|---|---|

χ2 | 0.1896 | 0.1709 | 0.1542 | 0.1537 |

AIC | 686.096 | 686.100 | 686.078 | 685.890 |

T (Years) | 2 | 5 | 10 | 15 | 25 | 50 |
---|---|---|---|---|---|---|

X_{1} (-) | 100.73 | 129.05 | 150.38 | 163.02 | 179.20 | 201.68 |

Storm Classification | n | φ (min^{−1}) |
---|---|---|

Mainly advective | 0.3 | 0.0745 |

Effective | 0.5 | 0.1163 |

Mainly convective | 0.7 | 0.1799 |

Strongly convective | 0.9 | 0.3189 |

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**MDPI and ACS Style**

Balbastre-Soldevila, R.; García-Bartual, R.; Andrés-Doménech, I.
Estimation of the G2P Design Storm from a Rainfall Convectivity Index. *Water* **2021**, *13*, 1943.
https://doi.org/10.3390/w13141943

**AMA Style**

Balbastre-Soldevila R, García-Bartual R, Andrés-Doménech I.
Estimation of the G2P Design Storm from a Rainfall Convectivity Index. *Water*. 2021; 13(14):1943.
https://doi.org/10.3390/w13141943

**Chicago/Turabian Style**

Balbastre-Soldevila, Rosario, Rafael García-Bartual, and Ignacio Andrés-Doménech.
2021. "Estimation of the G2P Design Storm from a Rainfall Convectivity Index" *Water* 13, no. 14: 1943.
https://doi.org/10.3390/w13141943