# KDE-Based Rainfall Event Separation and Characterization

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Statistical Representation of Rainfall Events

_{t}) values. v, t and b can be regarded as three random variables with units expressed in mm over the catchment for v and in hours for t and b. Using the exponential PDFs to approximate the observed frequency distributions of rainfall event characteristics, these PDFs can be expressed as [9,37]:

#### 2.2. Rainfall Event Characterization Using KDE Approach

#### 2.2.1. KDE for Estimating PDFs of Rainfall Characteristics

_{i}is the ith independent identically distributed sample of the total rainfall sample data x; $K(\cdot )$ is the kernel function, which is the symmetry function and the integral is unity where the upper and lower limit of integration is positive infinity and negative infinity, respectively, i.e., $\underset{-\infty}{\overset{+\infty}{\int}}K\left(x\right)\mathrm{d}x=1$ for $K\left(x\right)>0$ [39,40].

_{t}(x) hereinafter) by a normal density in which the unknown standard deviation is replaced by the estimator $\widehat{\sigma}$. When the Gaussian function is used, the Silverman’s rule-of-thumb formulae [35,42] was used to determine h in this study:

#### 2.2.2. Correction for Boundary Bias of KDE

_{1}, x

_{2}, …, x

_{n}) using KDE. Therefore, the data reflection approach is applicable to correct the boundary bias. The common practice of the data reflection technique is to mirror the data around the boundary. Given a new data set {${x}_{1},-{x}_{1},{x}_{2},-{x}_{2},\dots ,{x}_{n},-{x}_{n}$}, a new reflection KDE incorporating the mirrored data (denoted as $\hat{{f}_{n}\left(x\right)}$) can be expressed as:

_{n}(x) and f

_{t}(x) is defined as $E\left[{\left(\hat{{f}_{n}\left(x\right)}-\hat{{f}_{t}\left(x\right)}\right)}^{2}\right]$ where E[⋅] represents the mathematic expectation function. It is found that $E\left[{\left(\hat{{f}_{n}\left(x\right)}-\hat{{f}_{t}\left(x\right)}\right)}^{2}\right]$ approaches zero when the term of nh approaches positive infinity, which demonstrates that f

_{n}(x) is close to f

_{t}(x) and proves the feasibility of reflection method [43,44].

#### 2.2.3. KDE for Estimating PDFs of Rainfall Characteristics

_{s}intervals with equal width [45,46,47]. Using Simpson’s Rule Formula, the integral of the PDF $\hat{{f}_{n}\left(x\right)}$ with x in the range between a and b obtained is expressed as:

_{s}= (b–a)/2n

_{s}, in which h

_{s}is the width of the interval and n

_{s}is a positive integer. The CDF of the KDE-induced PDF $\hat{{f}_{n}(x)}$ with a specific value of x (denoted as $\hat{{F}_{n}(x)}$) can be calculated using Equation (8).

_{p}= Var(θ)/⟨θ⟩; is defined as the Poisson test statistic. When specifying different MIET values, the numbers of annual rainfall events (denoted as N) change. r

_{p}can be further calculated based on the transformed test statistic (N–1)r

_{p}that follows a chi-square distribution with (N–1) degrees of freedom and a specified level of significance α [48,49].

_{r}) is proposed to evaluate the agreement between the KDE-induced CDF and theoretical exponential distribution for v. For a pair of selected MIET and v

_{t}, R

_{r}can be calculated using Equation (9).

_{t}, their corresponding R

_{r}can be calculated, respectively. The optimal combination of MIET and v

_{t}can be determined when the minimum R

_{r}is achieved.

#### 2.2.4. Procedures of Rainfall Event Separation and Characterization Based on KDE

_{t}: pairs of suitable values for the minimum inter-event time (MIET) (from 6–12 h) and the volume threshold v

_{t}(from 0–5 mm) are selected first;

_{t}, the hourly rainfall time series is divided into several discrete rainfall events following the rule that b is smaller than MIET, as well as the rule that the small rainfall events with a volume less than the threshold v

_{t}should be removed; the three time series of the corresponding rainfall event characteristics, i.e., rainfall event volume v, rainfall event duration t, and inter-event time b, are then obtained;

_{t}, KDE is applied to obtain the PDFs for each variable of v, t, and b based on the Gaussian kernel function and Silverman’s rule-of-thumb formulae;

_{t}combinations;

_{t}are excluded if the corresponding Poisson test results are not acceptable;

_{t}are further excluded if the corresponding K-S statistical test results are not acceptable;

_{t}: for the separated rainfall events with all acceptable pairs of selected MIET and v

_{t}after finishing the step (8), using Equation (9) to calculate R

_{r}for the corresponding CDF of v; the optimal pair of MIET and v

_{t}is determined when the minimum value of the calculated R

_{r}among all pairs is achieved; then the combination of MIET and v

_{t}corresponding to for the minimum R

_{r}is selected as the optimal one;

_{t}determined in step (8).

#### 2.3. Study Area and Data

## 3. Case Study of the Rainfall Event Separation

#### 3.1. Parameters for Rainfall Event Separation

_{t}are 0, 1, 2, 3, 4 and 5 mm as suggested in the reasonable ranges of MIET (6–12 h) and threshold v

_{t}(0–5 mm) for typical urban catchments. The total combination number of different MIET and v

_{t}values is 24 for each study area.

_{t}combination values. More specifically, if the dry time between two adjacent rainfall episodes is less than the selected MIET, these two rainfall episodes are treated as if they belong to the same rainfall event, and they should be further merged into one rainfall episode; otherwise, they are identified as two individual and consecutive rainfall events. Additionally, when the volume of the rainfall episode (v) is no larger than the selected threshold value (v

_{t}), the rainfall volume of this episode should be removed and adjusted to be zero and the duration of this episode will be appended to the inter-event time between the current and the previous rainfall episodes. In such circumstances, the redefined rainfall event is characterized by an event volume and an event duration that are, respectively, equal to those of the previous rainfall episode. However, the inter-event time of the redefined rainfall event should be equal to the original inter-event time between the current and the previous rainfall episodes with the addition to the duration of the current rainfall episode.

_{t}for event separation at Jinan and Hangzhou. It is worth noting from Table 2 that the numbers of rainfall event volume v, rainfall event duration t, and rainfall inter-event time b are equal with the same combination of MIET and v

_{t}, and the decrease in the number of rainfall events is observed with the increase in either MIET or v

_{t}. The rainfall characterization results (i.e., ⟨v⟩,⟨t⟩,⟨b⟩) for all possible pairs of MIET or v

_{t}before performing statistical tests are shown in Table 3. It is found that all the mean values of three event characteristics decrease with the increase in either MIET or v

_{t}from Table 3.

#### 3.2. Results and Discussion

#### 3.2.1. Poisson Test for the Annual Number of Events θ

_{t}(0–5 mm), Poisson tests for the annual number of events were performed for the two study cities. As shown in Table 4, with a level of significance α = 0.1, the Poisson test results for the annual number of events θ for 22 out of 24 cases with pairs of MIET and v

_{t}at Hangzhou indicate the acceptance of the hypothesis that θ follows the Poisson distribution. In the meanwhile, the r

_{p}of Jinan was found to lie within the interval between 0.71 and 1.33 of critical values in all 24 cases with pairs of MIET and v

_{t}, demonstrating that the hypothesis cannot be rejected. It is worth noting that while v

_{t}is too large or too small, the rejection of the Poisson distribution hypothesis may occur.

#### 3.2.2. GOF Test of Exponentiality Using KDE

_{t}for Jinan and Hangzhou. Silverman’s rule of thumb was used for choosing the window width and Gaussian kernel function is used for the kernel function. The algorithms of the convolution of KDE and the data reflection (i.e., mirror the data) for correcting the boundary bias were coded in Python. Each pair of MIET and v

_{t}uses the same window width method, kernel function type, and boundary correction method to estimate the kernel density. As shown in Figure 3, the PDF of the rainfall event sample data obtained by KDE fits theoretically PDF very well overall, although slight downward trends near the origin are observed due to the formula property of kernel density.

_{t}of 2 mm at Jinan. It is found in Figure 4 that both the PDFs obtained from KDE and theoretical values present a decreasing trench with the increase in MIET when v is very small. The maximum values and the PDF curve shape fit favorably between the PDFs obtained from KDE and theoretical PDFs from visual observation. It is noted that the PDFs estimated by KDE have a larger variation compared to the theoretical PDFs for v near the origin.

_{t}for Jinan and Hangzhou were estimated by KDE using Equation (7). The CDFs were further calculated using Equation (8) and K-S test was performed accordingly. It is noted that the pair of MIET and v

_{t}is accepted when the resulting v, t, and b all pass their corresponding K-S tests. The K-S statistical test results for v, t, and b at Hangzhou and Jian are displayed in Table 5 and Table 6, respectively. As shown in Table 5 and Table 6, there are 13 out of 24 pairs of MIET and v

_{t}that are accepted in Jinan, while 5 out of 24 of those are accepted in Hangzhou. It is, generally, found that the pass rates of K-S statistical tests for v is relatively lower than that for t and b. The minimum K-S statistics achieved among the pairs of MIET and v

_{t}that passed the K-S test for v under different MIETs for all cases result in equal v

_{t}values of 3 mm for both Hangzhou and Jinan. This implies that the threshold value v

_{t}may have a greater effect on the GOF test compared to MIET.

#### 3.2.3. Optimal MIET, v_{t} and Rainfall Event Characterization

_{t}for Hangzhou and Jinan are determined, respectively. The results of the corresponding R

_{r}of these acceptable pairs of MIET and v

_{t}calculated using Equation (9) for the two cities are shown in Table 7. It is observed in Table 7 that the minimum values of R

_{r}for Hangzhou and Jinan are 8.84% and 8.97%, respectively, with the corresponding pairs of MIET and v

_{t}are (12 h-3 mm) and (10 h-3 mm), respectively. Therefore, the optimal MIETs determined are 12 h for Hangzhou and 10 h for Jinan whereas the optimal v

_{t}values are 3 mm for both Hangzhou and Jinan. With the selected optimal pair of MIET and v

_{t}, the CDFs obtained from KDE and the theoretical exponential CDFs for v, t and b are plotted as shown in Figure 5. The excellent agreements between the two CDFs for each rainfall characteristic are observed from Figure 5, which further demonstrates the validity of the selected optimal results. The rainfall event characterization can be finally obtained by calculating the mean values of the three event characteristics with the optimal pairs of MIET and v

_{t}for Hangzhou and Jinan. The rainfall event characterization results are: ⟨v⟩ = 25.19 mm, ⟨t⟩ = 23.67 h, ⟨b⟩ = 103.53 h for Hangzhou and ⟨v⟩ = 23.74 mm, ⟨t⟩ = 13.05 h, ⟨b⟩ = 154.12 h for Jinan; the corresponding exponential distribution parameters are ζ = 0.0397 mm

^{−1}, λ = 0.0422 h

^{−1}, and ψ = 0.00966 h

^{−1}for Hangzhou and ζ = 0.0421 mm

^{−1}, λ = 0.0766 h

^{−1}, and ψ = 0.00649 h

^{−1}for Jinan.

## 4. Summary and Conclusions

_{t}), which are two key parameters for rainfall event separation and characterization. A detailed standardized procedure was also provided for rainfall event separation and characterization at any specific site where the exponential distribution is suitable for characterizing the rainfall event statistics. Taking two climatically different cities, Hangzhou and Jinan of China, for a demonstration example, the validation and application of the proposed KDE-based approach were investigated for rainfall event separation and characterization. The results show that the optimal MIETs determined are 12 h for Hangzhou and 10 h for Jinan while the optimal v

_{t}values are 3 mm for both Hangzhou and Jinan. The corresponding distribution parameters of three rainfall event characteristics for two cities are obtained as well.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Restrepo, P.; Eagleson, P. Identification of Independent Rainstorms. J. Hydrol.
**1982**, 55, 303–319. [Google Scholar] [CrossRef] - Dunkerley, D. Identifying Individual Rain Events from Pluviograph Records: A Review with Analysis of Data from an Australian Dryland Site. Hydrol. Process.
**2008**, 22, 5024–5036. [Google Scholar] [CrossRef] - Adams, B.J.; Papa, F. Urban Stormwater Management Planning with Analytical Probabilistic Models, 1st ed.; Wiley: New York, NY, USA, 2000. [Google Scholar]
- Joo, J.; Lee, J.; Kim, J.; Jun, H.; Jo, D.J. Inter-Event Time Definition Setting Procedure for Urban Drainage Systems. Water
**2013**, 6, 45–58. [Google Scholar] [CrossRef] - Lee, E.H.; Kim, J. Development of New Inter-Event Time Definition Technique in Urban Areas. KSCE J. Civ. Eng.
**2018**, 22, 3764–3771. [Google Scholar] [CrossRef] - Guo, Y.; Adams, B. Hydrologic Analysis of Urban Catchments with Event-Based Probabilistic Models: 1. Runoff Volume. Water Resour. Res.
**1998**, 34, 3421–3432. [Google Scholar] [CrossRef] - Carbone, M.; Turco, M.; Brunetti, G.; Piro, P. Minimum Inter-Event Time to Identify Independent Rainfall Events in Urban Catchment Scale. Adv. Mater. Res.
**2014**, 1073, 1630–1633. [Google Scholar] [CrossRef] - Balistrocchi, M.; Grossi, G.; Bacchi, B. An Analytical Probabilistic Model of the Quality Efficiency of a Sewer Tank. Water Resour. Res.
**2009**, 45, W12420. [Google Scholar] [CrossRef] - Wang, J.; Guo, Y. Proper Sizing of Infiltration Trenches Using Closed-Form Analytical Equations. Water Resour. Manag.
**2020**, 34, 3809–3821. [Google Scholar] [CrossRef] - Lucas, W. Design of Integrated Bioinfiltration-Detention Urban Retrofits With Design Storm and Continuous Simulation Methods. J. Hydrol. Eng.
**2010**, 15, 486–498. [Google Scholar] [CrossRef] - Zhang, S.; Guo, Y. SWMM Simulation of the Storm Water Volume Control Performance of Permeable Pavement Systems. J. Hydrol. Eng.
**2014**, 20, 06014010. [Google Scholar] [CrossRef] - Quader, A.; Guo, Y. Peak Discharge Estimation Using Analytical Probabilistic and Design Storm Approaches. J. Hydrol. Eng.
**2006**, 11, 46–54. [Google Scholar] [CrossRef] - Chahar, B.; Graillot, D.; Gaur, S. Storm-Water Management through Infiltration Trenches. J. Irrig. Drain. Eng.
**2012**, 138, 274–281. [Google Scholar] [CrossRef] - Raimondi, A.; Becciu, G. Performance of Green Roofs for Rainwater Control. Water Resour. Manag.
**2021**, 34, 99–111. [Google Scholar] [CrossRef] - Chen, J.; Adams, B. A Framework for Urban Storm Water Modeling and Control Analysis with Analytical Models. Water Resour. Res.
**2006**, 42, W06419. [Google Scholar] [CrossRef] - Wang, J.; Guo, Y. An Analytical Stochastic Approach for Evaluating the Performance of Combined Sewer Overflow Tanks. Water Resour. Res.
**2018**, 54, 3357–3375. [Google Scholar] [CrossRef] - Guo, R.; Guo, Y.; Wang, J. Stormwater Capture and Antecedent Moisture Characteristics of Permeable Pavements. Hydrol. Process.
**2018**, 32, 2708–2720. [Google Scholar] [CrossRef] - Wang, J.; Guo, Y. Stochastic Analysis of Storm Water Quality Control Detention Ponds. J. Hydrol.
**2019**, 571, 573–584. [Google Scholar] [CrossRef] - U.S.EPA. Methodology for Analysis of Detention Basins for Control of Urban Runoff Quality; EPA: Washington, DC, USA, 1986; Volume 440, pp. 5–87. [Google Scholar]
- Shamsudin, S.; Dan’azumi, S.; Aris, A.; Yusop, Z. Optimum Combination of Pond Volume and Outlet Capacity of a Stormwater Detention Pond Using Particle Swarm Optimization. Urban Water J.
**2013**, 11, 127–136. [Google Scholar] [CrossRef] - Zeng, J.; Huang, G.; Mai, Y.; Chen, W. Optimizing the Cost-Effectiveness of Low Impact Development (LID) Practices Using an Analytical Probabilistic Approach. Urban Water J.
**2020**, 17, 136–143. [Google Scholar] [CrossRef] - Hassini, S.; Guo, Y. Exponentiality Test Procedures for Large Samples of Rainfall Event Characteristics. J. Hydrol. Eng.
**2016**, 21, 04016003. [Google Scholar] [CrossRef] - Nojumuddin, N.; Yusof, F.; Yusop, Z. Determination of Minimum Inter-Event Time for Storm Characterization in Johor, Malaysia. J. Flood Risk Manag.
**2016**, 11, S687–S699. [Google Scholar] [CrossRef] - Nix, S.J. Urban Stormwater Modeling and Simulation, 1st ed.; CRC Press: Boca Raton, FL, USA, 1994. [Google Scholar]
- Balistrocchi, M.; Grossi, G.; Bacchi, B. Deriving a Practical Analytical-Probabilistic Method to Size Flood Routing Reservoirs. Adv. Water Resour.
**2013**, 62, 37–46. [Google Scholar] [CrossRef] - Bacchi, B.; Balistrocchi, M.; Grossi, G. Proposal of a Semi-Probabilistic Approach for Storage Facility Design. Urban Water J.
**2008**, 5, 195–208. [Google Scholar] [CrossRef] - Guo, Y.; Baetz, B. Sizing of Rainwater Storage Units for Green Building Applications. J. Hydrol. Eng.
**2007**, 12, 197–205. [Google Scholar] [CrossRef] - Rajagopalan, B.; Lall, U.; Tarboton, D. Evaluation of Kernel Density Estimation Methods for Daily Precipitation Resampling. Stoch. Hydrol. Hydraul.
**1997**, 11, 523–547. [Google Scholar] [CrossRef] - Pavlides, A.; Agou, V.D.; Hristopulos, D.T. Non-parametric Kernel-based Estimation and Simulation of Precipitation Amount. J. Hydrol.
**2022**, 612, 127988. [Google Scholar] [CrossRef] - Cacoullos, T. Estimation of a Multivariate Density. Ann. Inst. Stat. Math.
**1966**, 18, 179–189. [Google Scholar] [CrossRef] - Devroye, L. The Equivalence of Weak, Strong and Complete Convergence in L1 for Kernel Density Estimates. Ann. Stat.
**1983**, 11, 896–904. [Google Scholar] [CrossRef] - Mosthaf, T.; Bárdossy, A. Regionalizing Nonparametric Models of Precipitation Amounts on Different Temporal Scales. Hydrol. Earth Syst. Sci.
**2017**, 21, 2463–2481. [Google Scholar] [CrossRef] - Wang, S.; Wang, S.; Wang, D. Combined Probability Density Model for Medium Term Load Forecasting Based on Quantile Regression and Kernel Density Estimation. Energy Procedia
**2019**, 158, 6446–6451. [Google Scholar] [CrossRef] - Wu, Z.; Bhattacharya, B.; Xie, P.; Zevenbergen, C. Improving Flash Flood Forecasting Using a Frequentist Approach to Identify Rainfall Thresholds for Flash Flood Occurrence. Stoch. Environ. Res. Risk Assess.
**2022**, 37, 429–440. [Google Scholar] [CrossRef] - Jiang, S.; Wang, M.; Ren, J.; Liu, Y.; Zhou, L.; Cui, H.; Xu, C.-Y. An Integrated Approach for Identification and Quantification of Ecological Drought in Rivers from an Ecological Streamflow Perspective. Ecol. Indic.
**2022**, 143, 109410. [Google Scholar] [CrossRef] - Kim, J.S.; Jain, S.; Lee, J.H.; Chen, H.; Park, S.Y. Quantitative Vulnerability Assessment of Water Quality to Extreme Drought in a Changing Climate. Ecol. Indic.
**2019**, 103, 688–697. [Google Scholar] [CrossRef] - Eagleson, P. Dynamics of Flood Frequency. Water Resour. Res.
**1972**, 8, 878–898. [Google Scholar] [CrossRef] - Segarra, R. Reliability-Based Design of Urban Stormwater Detention Facilities with Random Carryover Storage. J. Water Resour. Plan. Manag.
**2020**, 146, 04019076. [Google Scholar] [CrossRef] - Parzen, E. On the Estimation of Probability Density Functions and Mode. Ann. Math. Stat.
**1962**, 33, 1065–1076. [Google Scholar] [CrossRef] - Botev, Z.; Grotowski, J.; Kroese, D. Kernel Density Estimation via Diffusion. Ann. Stat.
**2010**, 38, 2916–2957. [Google Scholar] [CrossRef] - Marron, J.S.; Nolan, D. Canonical Kernels for Density Estimation. Stat. Probab. Lett.
**1988**, 7, 195–199. [Google Scholar] [CrossRef] - Silverman, B.W. Density Estimation for Statistics and Data Analysis, 1st ed.; Routledge: New York, NY, USA, 1998. [Google Scholar]
- Schuster, E. Incorporating Support Constraints into Nonparametric Estimators of Densities. Commun. Stat.-Theory Methods
**1985**, 14, 1123–1136. [Google Scholar] [CrossRef] - Jones, M. Simple Boundary Correction for Kernel Density Estimation. Stat. Comput.
**1993**, 3, 135–146. [Google Scholar] [CrossRef] - Sauer, T. Numerical Analysis, 2nd ed.; Pearson: Boston, MA, USA, 2011. [Google Scholar]
- Darkwah, K.; Nortey, E.; Lotsi, A. Estimation of the Gini Coefficient for the Lognormal Distribution of Income Using the Lorenz Curve. SpringerPlus
**2016**, 5, 1196. [Google Scholar] [CrossRef] - Zhang, S.; Karunamuni, R. Deconvolution Boundary Kernel Method in Nonparametric Density Estimation. J. Stat. Plan. Inference
**2009**, 139, 2269–2283. [Google Scholar] [CrossRef] - Cunnane, C. A Note on the Poisson Assumption in Partial Duration Series Models. Water Resour. Res.
**1979**, 15, 489–494. [Google Scholar] [CrossRef] - Cruise, J.; Arora, K. A Hydroclimatic Application Strategy for the Poisson Partial Duration Model. J. Am. Water Resour. Assoc.
**2007**, 26, 431–442. [Google Scholar] [CrossRef] - Evans, D.; Drew, J.; Leemis, L. The Distribution of the Kolmogorov–Smirnov, Cramer–von Mises, and Anderson–Darling Test Statistics for Exponential Populations with Estimated Parameters. Commun. Stat.-Simulat. Comput.
**2008**, 37, 1396–1421. [Google Scholar] [CrossRef] - Wang, J.; Zhang, S.; Guo, Y. Analyzing the Impact of Impervious Area Disconnection on Urban Runoff Control Using an Analytical Probabilistic Model. Water Resour. Manag.
**2019**, 33, 1753–1768. [Google Scholar] [CrossRef] - Zhou, J.; Gu, B.; Schlesinger, W.; Ju, X. Significant Accumulation of Nitrate in Chinese Semi-Humid Croplands. Sci. Rep.
**2016**, 6, 25088. [Google Scholar] [CrossRef] [PubMed] - Wang, S.; Zhang, Q.; Yue, P.; Wang, J. Effects of Evapotranspiration and Precipitation on Dryness/Wetness Changes in China. Theor. Appl. Climatol.
**2020**, 142, 1027–1038. [Google Scholar] [CrossRef] - Wang, W.; Yin, S.; Xie, Y.; Nearing, M.A. Minimum Inter-event Times for Rainfall in the Eastern Monsoon Region of China. Trans. ASABE
**2019**, 62, 9–18. [Google Scholar] [CrossRef]

**Figure 3.**Comparison of the KDE-based PDFs and theoretical exponential PDFs of rainfall event volume v, rainfall event duration t, rainfall inter-event time b: (

**a**–

**c**). Hangzhou (MIET = 12 h, v

_{t}= 3 mm); (

**d**–

**f**). Jinan (MIET = 10 h, v

_{t}= 3 mm).

**Figure 4.**(

**a**) The KDE-based PDFs for rainfall event volume v with different MIETs and fixed v

_{t}= 2 mm at Jinan; (

**b**) the theoretical exponential PDFs of rainfall event volume v with different MIETs and fixed v

_{t}= 2 mm at Jinan.

**Figure 5.**Comparison of CDF of KDE and theoretical exponential CDF of rainfall event volume v, rainfall event duration t, rainfall inter-event time b for the optimal MIET and rainfall event volume threshold: (

**a**–

**c**) Hangzhou (MIET = 12 h, v

_{t}= 3 mm); (

**d**–

**f**) Jinan (MIET = 10 h, v

_{t}= 3 mm).

Station | Station Number | Latitude | Longitude | Range of Years | Range of Months | Average Annual Precipitation (mm) | Climate Condition |
---|---|---|---|---|---|---|---|

Jinan | 54,823 | N36°60′ | E117°00′ | 1959–2015 | May.–Oct. | 688.5 | Semi-humid |

Hangzhou | 58,457 | N30°23′ | E120°17′ | 1955–2015 | Apr.–Oct. | 1510.0 | Humid |

**Table 2.**Total number of rainfall events with different pairs of MIET and v

_{t}for Jinan and Hangzhou.

Hangzhou | Jinan | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

MIET (h) | 6 | 8 | 10 | 12 | MIET (h) | 6 | 8 | 10 | 12 | ||

v_{t} (mm) | v_{t} (mm) | ||||||||||

0 | 5769 | 5132 | 4628 | 4172 | 0 | 2567 | 2408 | 2290 | 2180 | ||

1 | 3854 | 3525 | 3234 | 2955 | 1 | 1860 | 1797 | 1744 | 1692 | ||

2 | 3276 | 3036 | 2807 | 2590 | 2 | 1606 | 1562 | 1521 | 1490 | ||

3 | 2893 | 2695 | 2517 | 2333 | 3 | 1453 | 1419 | 1390 | 1360 | ||

4 | 2602 | 2444 | 2301 | 2147 | 4 | 1326 | 1301 | 1276 | 1248 | ||

5 | 2364 | 2239 | 2129 | 2000 | 5 | 1227 | 1207 | 1189 | 1164 |

**Table 3.**Rainfall event characteristics with different pairs of MIET and v

_{t}for Jinan and Hangzhou.

Hangzhou | Jinan | |||||
---|---|---|---|---|---|---|

MIET (h)-v_{t} (mm) | ⟨v⟩ (mm) | ⟨t⟩ (h) | ⟨b⟩ (h) | ⟨v⟩ (mm) | ⟨t⟩ (h) | ⟨b⟩ (h) |

6-0 | 10.48 | 8.48 | 44.66 | 13.21 | 7.48 | 87.18 |

6-1 | 15.49 | 11.64 | 66.70 | 18.07 | 9.49 | 117.23 |

6-2 | 17.95 | 12.88 | 78.69 | 20.69 | 10.32 | 135.80 |

6-3 | 20.00 | 13.85 | 88.60 | 22.61 | 10.84 | 148.47 |

6-4 | 21.84 | 14.71 | 98.92 | 24.43 | 11.28 | 161.50 |

6-5 | 23.58 | 15.45 | 109.51 | 26.04 | 11.66 | 174.30 |

8-0 | 11.78 | 10.34 | 49.41 | 14.08 | 8.40 | 92.51 |

8-1 | 16.96 | 14.03 | 71.63 | 18.73 | 10.45 | 120.71 |

8-2 | 19.45 | 15.50 | 83.33 | 21.31 | 11.35 | 138.90 |

8-3 | 21.60 | 16.68 | 93.32 | 23.21 | 11.93 | 151.20 |

8-4 | 23.45 | 17.67 | 103.35 | 24.99 | 12.40 | 163.71 |

8-5 | 25.19 | 18.56 | 113.41 | 26.59 | 12.83 | 176.23 |

10-0 | 13.06 | 12.39 | 53.87 | 14.81 | 9.27 | 96.85 |

10-1 | 18.51 | 16.71 | 76.75 | 19.31 | 11.42 | 123.74 |

10-2 | 21.10 | 18.43 | 88.46 | 21.92 | 12.44 | 141.87 |

10-3 | 23.24 | 19.83 | 98.04 | 23.74 | 13.05 | 154.12 |

10-4 | 25.09 | 20.98 | 107.67 | 25.55 | 13.51 | 166.07 |

10-5 | 26.76 | 21.93 | 116.89 | 27.09 | 13.97 | 177.97 |

12-0 | 14.49 | 14.89 | 58.62 | 15.56 | 10.26 | 101.22 |

12-1 | 20.29 | 20.03 | 82.27 | 19.91 | 12.51 | 126.82 |

12-2 | 22.94 | 22.03 | 93.90 | 22.41 | 13.56 | 144.14 |

12-3 | 25.19 | 23.67 | 103.53 | 24.31 | 14.23 | 156.64 |

12-4 | 27.06 | 25.01 | 112.91 | 26.18 | 14.73 | 169.17 |

12-5 | 28.72 | 26.12 | 121.73 | 27.74 | 15.26 | 180.88 |

Hangzhou | Jinan | |||||
---|---|---|---|---|---|---|

MIET (h)-v_{t} (mm) | Critical Value of r_{p} Ranges (α = 0.10) | Resulting r_{p} | Decision | Critical Value of r_{p} Ranges (α = 0.10) | Resulting r_{p} | Decision |

6-0 | 0.72–1.32 | 1.54 | Reject | 0.71–1.33 | 1.25 | Accept |

6-1 | 0.72–1.32 | 1.25 | Accept | 0.71–1.33 | 0.86 | Accept |

6-2 | 0.72–1.32 | 1.29 | Accept | 0.71–1.33 | 0.97 | Accept |

6-3 | 0.72–1.32 | 1.25 | Accept | 0.71–1.33 | 1.12 | Accept |

6-4 | 0.72–1.32 | 1.22 | Accept | 0.71–1.33 | 1.08 | Accept |

6-5 | 0.72–1.32 | 1.16 | Accept | 0.71–1.33 | 1.21 | Accept |

8-0 | 0.72–1.32 | 1.44 | Reject | 0.71–1.33 | 1.01 | Accept |

8-1 | 0.72–1.32 | 1.15 | Accept | 0.71–1.33 | 0.83 | Accept |

8-2 | 0.72–1.32 | 1.13 | Accept | 0.71–1.33 | 0.95 | Accept |

8-3 | 0.72–1.32 | 1.17 | Accept | 0.71–1.33 | 1.09 | Accept |

8-4 | 0.72–1.32 | 1.11 | Accept | 0.71–1.33 | 1.06 | Accept |

8-5 | 0.72–1.32 | 1.07 | Accept | 0.71–1.33 | 1.16 | Accept |

10-0 | 0.72–1.32 | 1.29 | Accept | 0.71–1.33 | 0.93 | Accept |

10-1 | 0.72–1.32 | 1.01 | Accept | 0.71–1.33 | 0.79 | Accept |

10-2 | 0.72–1.32 | 0.94 | Accept | 0.71–1.33 | 0.93 | Accept |

10-3 | 0.72–1.32 | 0.98 | Accept | 0.71–1.33 | 1.02 | Accept |

10-4 | 0.72–1.32 | 0.98 | Accept | 0.71–1.33 | 0.98 | Accept |

10-5 | 0.72–1.32 | 0.90 | Accept | 0.71–1.33 | 1.10 | Accept |

12-0 | 0.72–1.32 | 1.15 | Accept | 0.71–1.33 | 0.87 | Accept |

12-1 | 0.72–1.32 | 0.82 | Accept | 0.71–1.33 | 0.76 | Accept |

12-2 | 0.72–1.32 | 0.73 | Accept | 0.71–1.33 | 0.84 | Accept |

12-3 | 0.72–1.32 | 0.75 | Accept | 0.71–1.33 | 0.93 | Accept |

12-4 | 0.72–1.32 | 0.74 | Accept | 0.71–1.33 | 0.86 | Accept |

12-5 | 0.72–1.32 | 0.70 | Reject | 0.71–1.33 | 0.95 | Accept |

Rainfall Event Volume v | Rainfall Event Duration t | Rainfall Inter-Event Time b | ||||
---|---|---|---|---|---|---|

MIET (h)- v_{t} (mm) | K-S Statistic | Critical Value (α = 0.10) | K-S Statistic | Critical Value (α = 0.10) | K-S Statistic | Critical Value (α = 0.10) |

6-0 | 0.213 | 0.066 | 0.107 | 0.102 | 0.155 | 0.209 |

6-1 | 0.098 | 0.066 | 0.044 | 0.102 | 0.096 | 0.188 |

6-2 | 0.069 | 0.066 | 0.033 | 0.102 | 0.078 | 0.188 |

6-3 ^{a} | 0.058 | 0.066 | 0.031 | 0.102 | 0.073 | 0.174 |

6-4 | 0.081 | 0.066 | 0.033 | 0.101 | 0.064 | 0.174 |

6-5 | 0.108 | 0.066 | 0.032 | 0.102 | 0.054 | 0.174 |

8-0 | 0.210 | 0.066 | 0.132 | 0.100 | 0.124 | 0.209 |

8-1 | 0.099 | 0.066 | 0.052 | 0.100 | 0.069 | 0.188 |

8-2 | 0.071 | 0.066 | 0.033 | 0.100 | 0.063 | 0.186 |

8-3 ^{a} | 0.058 | 0.066 | 0.019 | 0.099 | 0.056 | 0.176 |

8-4 | 0.069 | 0.066 | 0.016 | 0.099 | 0.048 | 0.174 |

8-5 | 0.093 | 0.066 | 0.017 | 0.098 | 0.040 | 0.174 |

10-0 | 0.202 | 0.059 | 0.143 | 0.086 | 0.087 | 0.209 |

10-1 | 0.097 | 0.059 | 0.055 | 0.086 | 0.055 | 0.188 |

10-2 | 0.068 | 0.059 | 0.033 | 0.085 | 0.047 | 0.186 |

10-3 ^{a} | 0.056 | 0.059 | 0.017 | 0.085 | 0.040 | 0.174 |

10-4 | 0.060 | 0.059 | 0.013 | 0.085 | 0.031 | 0.174 |

10-5 | 0.084 | 0.059 | 0.017 | 0.085 | 0.026 | 0.174 |

12-0 | 0.198 | 0.059 | 0.151 | 0.086 | 0.076 | 0.209 |

12-1 | 0.092 | 0.059 | 0.062 | 0.085 | 0.037 | 0.188 |

12-2 | 0.063 | 0.059 | 0.038 | 0.085 | 0.027 | 0.186 |

12-3 ^{a} | 0.051 | 0.059 | 0.021 | 0.084 | 0.021 | 0.174 |

12-4 ^{a} | 0.055 | 0.059 | 0.020 | 0.084 | 0.020 | 0.174 |

12-5 | 0.079 | 0.059 | 0.026 | 0.084 | 0.017 | 0.174 |

^{a}The pair of MIET and v

_{t}with acceptable K-S test results hypothesizing that the three rainfall event characteristics (v, t, b) follow the exponential distribution.

Rainfall Event Volume v | Rainfall Event Duration t | Rainfall Inter-Event Time b | ||||
---|---|---|---|---|---|---|

MIET (h)- v_{t} (mm) | KS Statistic | Critical Value (α = 0.10) | KS Statistic | Critical Value (α = 0.10) | KS Statistic | Critical Value (α = 0.10) |

6-0 | 0.179 | 0.071 | 0.049 | 0.125 | 0.056 | 0.188 |

6-1 | 0.089 | 0.071 | 0.042 | 0.124 | 0.025 | 0.171 |

6-2 ^{a} | 0.061 | 0.071 | 0.052 | 0.123 | 0.029 | 0.169 |

6-3 ^{a} | 0.051 | 0.071 | 0.055 | 0.123 | 0.026 | 0.169 |

6-4 ^{a} | 0.058 | 0.071 | 0.055 | 0.123 | 0.028 | 0.151 |

6-5 | 0.079 | 0.071 | 0.059 | 0.123 | 0.026 | 0.150 |

8-0 | 0.168 | 0.071 | 0.047 | 0.119 | 0.039 | 0.188 |

8-1 | 0.087 | 0.071 | 0.032 | 0.118 | 0.018 | 0.171 |

8-2 ^{a} | 0.061 | 0.071 | 0.043 | 0.118 | 0.027 | 0.169 |

8-3 ^{a} | 0.051 | 0.071 | 0.049 | 0.117 | 0.025 | 0.169 |

8-4 ^{a} | 0.054 | 0.071 | 0.052 | 0.117 | 0.026 | 0.155 |

8-5 | 0.075 | 0.071 | 0.055 | 0.117 | 0.023 | 0.150 |

10-0 | 0.160 | 0.071 | 0.051 | 0.118 | 0.030 | 0.188 |

10-1 | 0.084 | 0.071 | 0.024 | 0.118 | 0.015 | 0.171 |

10-2 ^{a} | 0.060 | 0.071 | 0.037 | 0.117 | 0.025 | 0.169 |

10-3 ^{a} | 0.049 | 0.071 | 0.044 | 0.117 | 0.023 | 0.169 |

10-4 ^{a} | 0.052 | 0.071 | 0.049 | 0.117 | 0.024 | 0.155 |

10-5 | 0.071 | 0.071 | 0.053 | 0.117 | 0.021 | 0.150 |

12-0 | 0.154 | 0.071 | 0.058 | 0.106 | 0.022 | 0.188 |

12-1 | 0.086 | 0.071 | 0.016 | 0.105 | 0.024 | 0.171 |

12-2 ^{a} | 0.062 | 0.071 | 0.028 | 0.105 | 0.022 | 0.169 |

12-3 ^{a} | 0.050 | 0.071 | 0.035 | 0.105 | 0.022 | 0.169 |

12-4 ^{a} | 0.048 | 0.071 | 0.041 | 0.104 | 0.023 | 0.155 |

12-5 ^{a} | 0.067 | 0.071 | 0.047 | 0.104 | 0.020 | 0.150 |

^{a}The pair of MIET and v

_{t}with acceptable K-S test results hypothesizing that the three rainfall event characteristics (v, t, b) follow the exponential distribution.

Hangzhou | Jinan | ||
---|---|---|---|

MIET (h)-v_{t} (mm) | Rr (%) | MIET (h)-v_{t} (mm) | Rr (%) |

12-3 | 8.84 | 10-3 | 8.97 |

10-3 | 9.67 | 8-3 | 9.08 |

6-3 | 9.77 | 6-3 | 9.20 |

8-3 | 10.19 | 12-3 | 9.29 |

12-4 | 32.54 | 6-2 | 11.88 |

8-2 | 12.09 | ||

10-2 | 12.13 | ||

12-2 | 12.75 | ||

12-4 | 27.35 | ||

10-4 | 29.21 | ||

8-4 | 30.04 | ||

6-4 | 31.24 | ||

12-5 | 34.68 |

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## Share and Cite

**MDPI and ACS Style**

Cao, S.; Diao, Y.; Wang, J.; Liu, Y.; Raimondi, A.; Wang, J.
KDE-Based Rainfall Event Separation and Characterization. *Water* **2023**, *15*, 580.
https://doi.org/10.3390/w15030580

**AMA Style**

Cao S, Diao Y, Wang J, Liu Y, Raimondi A, Wang J.
KDE-Based Rainfall Event Separation and Characterization. *Water*. 2023; 15(3):580.
https://doi.org/10.3390/w15030580

**Chicago/Turabian Style**

Cao, Shengle, Yijiao Diao, Jiachang Wang, Yang Liu, Anita Raimondi, and Jun Wang.
2023. "KDE-Based Rainfall Event Separation and Characterization" *Water* 15, no. 3: 580.
https://doi.org/10.3390/w15030580