# Probability distribution functions for unit hydrographs with optimization using genetic algorithm

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## Abstract

A unit hydrograph (UH) of a watershed may be viewed as the unit pulse response function of a linear system. In recent years, the use of probability distribution functions (pdfs) for determining a UH has received much attention. In this study, a nonlinear optimization model is developed to transmute a UH into a pdf. The potential of six popular pdfs, namely two-parameter gamma, two-parameter Gumbel, two-parameter log-normal, two-parameter normal, three-parameter Pearson distribution, and two-parameter Weibull is tested on data from the Lighvan catchment in Iran. The probability distribution parameters are determined using the nonlinear least squares optimization method in two ways: (1) optimization by programming in Mathematica; and (2) optimization by applying genetic algorithm. The results are compared with those obtained by the traditional linear least squares method. The results show comparable capability and performance of two nonlinear methods. The gamma and Pearson distributions are the most successful models in preserving the rising and recession limbs of the unit hydographs. The log-normal distribution has a high ability in predicting both the peak flow and time to peak of the unit hydrograph. The nonlinear optimization method does not outperform the linear least squares method in determining the UH (especially for excess rainfall of one pulse), but is comparable.

## Keywords

Genetic algorithm Least squares method Mathematica Nonlinear optimization Probability distribution function Unit hydrograph## Introduction

*n*, \( P_{m} \) is the effective rainfall pulse at a discrete time step

*m*, and \( U_{n - m + 1} \) is the ordinate of the UH at any discrete time step \( n - m + 1 \). If the number of effective rainfall pulses is

*M*and the number of DRH ordinates is

*N*, then there will be \( N - M + 1 \) ordinates in the UH of the watershed. On the other hand, when effective rainfall pulses (\( P_{m} \)’s) and DRH ordinates (\( Q_{n} \)’s) are known from observations, Eq. (1) can be used to determine the ordinates of UH through a reverse process. This reverse process of determining the UH ordinates is sometimes referred to as the “de-convolution” process.

There are many methods to solve Eq. (1) for determining the UH. These methods include successive substitution method **(**Dooge and Bruen 1989 **)**, Collins method (Collins 1939), successive approximation method **(**Newton and Vinyard 1976 **)**, Delaine method **(**Raghavendran and Reddy 1975 **)**, harmonic analysis (O’Donnell 1960), Fourier method **(**Levi and Valdes 1964 **)**, Meixner method (Dooge and Garvey 1978), least squares method (Bruen and Dooge 1984), linear programming method (Deininger 1969), and nonlinear programming method (Unver and Mays 1984), among others; see also Singh (1988) for further details.

Mays and Coles (1980) presented a linear programming (LP) model for the determination of composite UH. This model uses the *f*-index method for the estimation of infiltration losses. Prasad et al. (1999) applied an LP model to estimate the optimal loss-rate parameters and UH by considering the inherent characteristics of infiltration and UH. Mays and Taur (1982) developed a nonlinear programming (NLP) model to determine the optimal UH. This method does not require losses to be specified a priori. Unver and Mays (1984) extended the method of Mays and Taur (1982) by incorporating an infiltration equation to estimate the optimal loss-rate parameters and UH.

Although these methods have been shown to perform well for certain situations, their main disadvantage is that the number of unknowns is equal to the number of unit hydrograph ordinates. Therefore, for larger time bases, these methods may involve difficulties in estimating the unit hydrograph from the rainfall–runoff data, since the number of unknowns is generally large (Bhattacharjya 2004).

Unit hydrographs have common characteristics with probability distribution functions, such as positive ordinates and unit area. As a result, probability distribution functions have recently gained enormous interest in deriving UH. In this approach, the number of unknowns is less and equal to the number of probability distribution parameters. Bardsley (2003) used the inverse Gaussian distribution as an alternative to the gamma distribution as a two-parameter descriptor of the IUH. The inverse Gaussian distribution was capable of deriving some hydrographs where the gamma would fail. Bhattacharjya (2004) used gamma and log-normal probability distributions to represent the UH for developing two nonlinear optimization models and solved them using binary-coded genetic algorithms. The gamma and log-normal distribution estimated the time to peak correctly. Log-normal distribution predicted peak discharge more or less properly; whereas gamma distribution did not satisfactorily estimate the peak discharge. Moreover, the results showed fairly similar performance of the distributions and the linear optimization model. Bhunya et al. (2007) explored the potential of four popular probability distribution functions (Gamma, Chi square, Weibull, and Beta) to derive synthetic unit hydrograph (SUH) using field data. The results showed that the Beta and Weibull distributions are more flexible in hydrograph prediction. Nadarajah (2007) provided simple Maple programs for determining SUH from eleven of the most flexible probability distributions and derived expressions for the unknown parameters in terms of the time to peak, the peak discharge, and the time base. Rai et al. (2010) derived the UH using the Nakagami-m distribution and compared its results with those of seven other distribution functions over 13 watersheds. The Nakagami-m distribution yielded UHs and direct runoff hydrographs successfully. Singh (2011) employed the entropy theory to derive a general IUH equation on two small agricultural experimental watersheds. This equation was specialized into some distributions, such as the gamma distribution, Lienhard distribution, and Nakagami-m distribution. The results indicated that surface runoff hydrographs computed using the derived IUH equation were in satisfactory agreement with the observed hydrographs.

In the present study, a nonlinear unconstrained optimization model is presented to transmute UHs into probability distribution functions. Six probability distribution functions are considered: two-parameter gamma, two-parameter Gumbel, two-parameter log-normal, two-parameter normal, three-parameter Pearson, and two-parameter Weibull distribution. The nonlinear least squares optimization formulation is solved by (1) programming in Mathematica and (2) by applying genetic algorithm. The potential of these six probability distribution functions is tested on data from the Lighvan catchment in the northwest of Iran. The nonlinear optimization method is compared with the traditional linear least squares method. One particular novelty of this study is the use of Mathematica for solving the nonlinear optimization formulation problem involved in deriving UH. Since Mathematica has extensive symbolic and numerical capabilities, it enables the calculations in a simpler, faster, and more accurate manner. It also has several statistical distributions already built-in.

The rest of this paper is organized as follows the next section presents a brief description of the six probability distribution functions, nonlinear least squares optimization method, and formulation to transmute UH into probability distribution, genetic algorithm, and traditional least squares methods. After describing the case study area, the results of calibration and validation of the methods are discussed. Finally, the conclusions are drawn.

## Materials and methods

### Probability distribution functions

In this study, six popular probability distribution functions are considered: gamma, Gumbel, log-normal, normal, Pearson, and Weibull. A brief description of these functions can be found in Table 9.

### Nonlinear least squares optimization method

*n*th ordinates of the estimated and actual direct runoff hydrographs, given by

*n*th ordinate of the actual direct runoff hydrograph, \( U_{n - m + 1} = f\left( x \right) \), where \( f\left( x \right) \) is a probability distribution function and \( x = \left( {n - m + 1} \right) \times \Delta t \).

In this method, the number of unknowns is equal to the parameters of the probability distribution. In this study, this method is performed by programming in Mathematica and by applying genetic algorithm which is briefly described in next sub-section 2.3.

### Genetic algorithm

The genetic algorithm (GA) is a search technique based on the concept of natural selection inherent in the natural genetics, and combines an artificial survival of the fittest with genetic operators abstracted from nature (Holland 1975). The major difference between GA and the classical optimization search techniques is that the GA works with a population of possible solutions, whereas the classical optimization techniques work with a single solution. An individual solution in a population of solutions is equivalent to a natural chromosome. Like a natural chromosome completely specifies the genetic characteristics of a human being, an artificial chromosome in GA completely specifies the values of various decision variables representing a decision or a solution. For most GAs, the candidate solutions are represented by chromosomes coded with either a binary number system or a real decimal number system. These chromosomes are evaluated based on their performance with respect to the objective function. The GA that employs binary strings as its chromosomes is called the binary-coded GA; whereas the GA that employs real-valued strings as its chromosomes is called the real-coded GA. The real-coded GAs offer certain advantages over the binary-coded GAs as they overcome some of the limitations of the binary-coded GAs (Deb and Agarwal 1995; Deb 2000). Regardless of the coding method used, the GA consists of three basic operations: reproduction, crossover or mating, and mutation. Reproduction is a process in which individual strings are copied according to their fitness (Goldberg 1989). Crossover is considered as the partial exchange of corresponding segments between two parent strings to produce two offspring strings. The genetic algorithm picks up two strings from the population to perform crossover with probability *p* _{ c } at a randomly selected point along the string. Mutation is the occasional introduction of new features into the population pool to maintain diversity in the population (Bhattacharjya 2004). Genetic algorithms start with randomly generating an initial population (*p*) of possible solutions. The population is then operated by the three basic operators in order to produce better offspring for the next generation. This process would repeat till the individual is better enough to suit the objective function.

### Linear least squares method

*T*and −1 indicate the transpose and inverse of the matrices, respectively. Further details about this method can be found in Singh Singh (1988). In this study, all the calculations of this method are performed in Mathematica.

## Study area and data

^{2}. The maximum and minimum elevations of the watershed are about 3500 and 2000 m, respectively. The length of longest stream is 17 km. The average stream slope is 11 %. The Lighvan River drains into Talkheh River and Urmia Lake. For this watershed, data availability is generally scarce. For the present analysis, data of rainfall and runoff corresponding to four different storms (Storm A, Storm B, Storm C, and Storm D) are considered for calibration of the models. Data corresponding to two other storms (Storm E and Storm F) are used for validation of the models. Details of these datasets are presented in Table 1.

Storm data for Lighvan watershed, Iran

Time (hr) | Storm A May 23, 2003 | Storm B June 15, 2003 | Storm C May 15, 2005 | Storm D May 16, 2005 | Storm E (test) May 24, 2003 | Storm F (test) March 6, 2004 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

P (mm) | Q (mm/hr) | P (mm) | Q (mm/hr) | P (mm) | Q (mm/hr) | P (mm) | Q (mm/hr) | P (mm) | Q (mm/hr) | P (mm) | Q (mm/hr) | |

1 | 0.04 | 0.003828 | 0.44 | 0.04742 | 0.17 | 0.029789 | 0.4 | 0.015604 | 0.43 | 0.004512 | 0.43 | 0.011756 |

2 | 0.008759 | 0.061746 | 0.02629 | 0.054092 | 0.063483 | 0.92 | 0.024154 | |||||

3 | 0.011794 | 0.039133 | 0.019434 | 0.050357 | 0.038323 | 0.0879 | ||||||

4 | 0.011794 | 0.041869 | 0.016745 | 0.022845 | 0.03496 | 0.089464 | ||||||

5 | 0.005781 | 0.033 736 | 0.016077 | 0.024186 | 0.026276 | 0.155487 | ||||||

6 | 0.000947 | 0.016908 | 0.010153 | 0.024186 | 0.020032 | 0.120408 | ||||||

7 | 0.013179 | 0.008213 | 0.024186 | 0.020032 | 0.089464 | |||||||

8 | 0.013179 | 0.008213 | 0.021511 | 0.018002 | 0.080174 | |||||||

9 | 0.015659 | 0.006289 | 0.018867 | 0.018002 | 0.068148 | |||||||

10 | 0.019426 | 0.005016 | 0.018211 | 0.019014 | 0.057962 | |||||||

11 | 0.020695 | 0.005016 | 0.018211 | 0.020032 | 0.052283 | |||||||

12 | 0.020695 | 0.004382 | 0.018211 | 0.020032 | 0.049482 | |||||||

13 | 0.016908 | 0.003121 | 0.018211 | 0.019014 | 0.049482 | |||||||

14 | 0.014415 | 0.001867 | 0.018211 | 0.019014 | 0.041233 | |||||||

15 | 0.011949 | 0.001243 | 0.015604 | 0.015998 | 0.038535 | |||||||

16 | 0.010725 | 0.003121 | 0.012389 | 0.015998 | 0.037195 | |||||||

17 | 0.010725 | 0.009221 | 0.014019 | 0.037195 | ||||||||

18 | 0.009508 | 0.007967 | 0.01304 | 0.037195 | ||||||||

19 | 0.008297 | 0.007343 | 0.010139 | 0.034536 | ||||||||

20 | 0.007092 | 0.0061 | 0.005434 | 0.030595 | ||||||||

21 | 0.002338 | 0.004512 | 0.02543 | |||||||||

22 | 0.004512 | 0.024154 | ||||||||||

23 | 0.004512 | 0.020367 | ||||||||||

24 | 0.004512 | 0.015408 | ||||||||||

25 | 0.014184 | |||||||||||

26 | 0.012967 | |||||||||||

27 | 0.011756 | |||||||||||

28 | 0.009353 | |||||||||||

29 | 0.006976 | |||||||||||

30 | 0.005798 | |||||||||||

31 | 0.003459 | |||||||||||

32 | 0.0023 | |||||||||||

33 | 0.001147 |

It is relevant to note that the effective rainfall rates are computed using the *Φ*-index for each rainfall hyetograph, and the direct runoff hydrographs are obtained by separating base flow from flow hydrographs using the constant-discharge method.

## Results and discussion

We use six probability distribution functions for deriving unit hydrographs for the above datasets: two-parameter gamma, two-parameter Gumbel, two-parameter log-normal, two-parameter normal, three-parameter Pearson distribution, and two-parameter Weibull. The probability distribution parameters are determined using the nonlinear least squares optimization method by programming in Mathematica and by applying the genetic algorithm. The results are also compared with those obtained using the traditional linear least squares method.

### Nonlinear optimization by programming in Mathematica

In the present analysis, the storm data are used to derive a 1-hour unit hydrograph. All the models used involve an inverse problem that optimizes the probability distribution function parameters by minimizing the difference between the actual and predicted direct runoff hydrographs. The probability distribution parameters are obtained using least squares optimization method.

#### Calibration of the models

Parameters of probability distribution functions calibrated by the nonlinear mathematical optimization method for Lighvan watershed

Storm | Gamma | Gumbel | Log-normal | Normal | Pearson | Weibull | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

\( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \gamma \) | |

A | 0.6774 | 5.2076 | 3.6075 | 1.2123 | 1.2293 | 0.3992 | 3.2550 | 1.3035 | 6.9597 | 0.5211 | 3.7207 | 2.9095 | −0.1073 |

B | 7.0135 | 1.0402 | 2.4152 | 3.3580 | 1.7216 | 1.1307 | 2.1155 | 4.0066 | 1.0400 | 7.0156 | 7.3400 | 1.0124 | 0 |

C | 3.9788 | 1.1616 | 2.7072 | 2.8155 | 1.3491 | 1.0494 | 2.1211 | 2.9119 | 1.1617 | 3.9788 | 4.6761 | 1.0884 | 0 |

D | 5.6905 | 1.4903 | 6.9192 | 6.0451 | 1.8902 | 0.9611 | 5.0665 | 5.8073 | 1.4902 | 5.6911 | 8.9275 | 1.2677 | 0 |

Objective function values for six distribution functions calibrated by the linear and nonlinear mathematical optimization method for Lighvan watershed

Model | Storm A | Storm B | Storm C | Storm D |
---|---|---|---|---|

Gamma distribution | 0.000016 | 0.000911 | 0.000026 | 0.001224 |

Gumbel distribution | 0.000008 | 0.002922 | 0.000382 | 0.002687 |

Log-normal distribution | 0.000022 | 0.000788 | 0.000030 | 0.001051 |

Normal distribution | 0.000005 | 0.002399 | 0.000234 | 0.002055 |

Pearson distribution | 0.000014 | 0.000911 | 0.000026 | 0.001224 |

Weibull distribution | 0.000005 | 0.000914 | 0.000027 | 0.001270 |

Least squares | 2.416E−19 | 1.9184E−18 | 1.1971E−18 | 2.0000E−18 |

Generally, based on the visual comparison at the calibration stage using the nonlinear optimization method, it was observed that the log-normal distribution estimates the time to peak and peak flow properly for all storms. This distribution along with the gamma, Pearson, and Weibull predicts the rising and recession limbs of the unit hydrographs more or less perfectly. Moreover, the log-normal distribution was recognized as the most successful model based on the average value of the objective function.

#### Validation of the models

- 1.Root mean squared error (RMSE):$$ {\text{RMSE}} = \sqrt {\frac{{\sum\limits_{i = 1}^{n} {\left( {Q_{{e_{i} }} - Q_{{o_{i} }} } \right)}^{2} }}{n}} $$(7)
- 2.Mean absolute error (MAE):$$ {\text{MAE}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left| {Q_{{{\text{e}}_{i} }} - Q_{{{\text{o}}_{i} }} } \right|} $$(8)
- 3.Correlation coefficient (CC):where \( Q_{{{\text{o}}_{i} }} \) and \( Q_{{{\text{e}}_{i} }} \) are the$$ {\text{CC}} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {Q_{{{\text{o}}_{i} }} - \bar{Q}_{\text{o}} } \right)\left( {Q_{{{\text{e}}_{i} }} - \bar{Q}_{\text{e}} } \right)}}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{n} \left( {Q_{{{\text{o}}_{i} }} - \bar{Q}_{\text{o}} } \right)^{2} } \sqrt {\mathop \sum \nolimits_{i = 1}^{n} \left( {Q_{{{\text{e}}_{i} }} - \bar{Q}_{\text{e}} } \right)^{2} } }}, $$(9)
*i*th observed and estimated DRH ordinates, respectively; \( \bar{Q}_{\text{o}} \) and \( \bar{Q}_{\text{e}} \) represents the average discharge of the observed and estimated DRH, respectively, and*n*is the number of ordinates.

Performance criteria values for six distribution functions calibrated by the linear and nonlinear mathematical optimization method for Lighvan watershed

Model | RMSE (mm/hr) | MAE (mm/hr) | CC | |||
---|---|---|---|---|---|---|

Storm E | Storm F | Storm E | Storm F | Storm E | Storm F | |

Gamma distribution | 0.010 | 0.027 | 0.006 | 0.023 | 0.619 | 0.863 |

Gumbel distribution | 0.015 | 0.048 | 0.013 | 0.038 | 0.642 | 0.397 |

Log-normal distribution | 0.012 | 0.043 | 0.009 | 0.034 | 0.776 | 0.495 |

Normal distribution | 0.014 | 0.047 | 0.012 | 0.038 | 0.670 | 0.428 |

Pearson distribution | 0.012 | 0.025 | 0.006 | 0.021 | 0.402 | 0.929 |

Weibull distribution | 0.013 | 0.042 | 0.012 | 0.035 | 0.710 | 0.571 |

Least squares | 2.9E−10 | 0.002 | 2.45E−10 | 0.001 | 1.000 | 0.998 |

In general, the results of the validation stage showed that the lognormal distribution performance is satisfactory in predicting peak flow and time to peak. The gamma and Pearson models showed acceptable performances in simulating both limbs of the unit hydrographs. Hence, according to the values of statistical measures, these distributions outperformed the others.

### Nonlinear optimization by applying genetic algorithm

Genetic algorithm parameters

Population size ( | 15* (number of variables) |

Crossover probability ( | 1.00 |

Mutation probability ( | 0.01 |

Generation ( | 200* (number of variables) |

#### Calibration of the models

Parameters of probability distribution functions calibrated by genetic algorithm for Lighvan watershed

Storm | Gamma | Gumbel | Log-normal | Normal | Pearson | Weibull | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

\( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \alpha \) | \( \beta \) | \( \gamma \) | |

A | 0.8700 | 4.4860 | 3.5430 | 1.4040 | 1.3580 | 0.6170 | 2.9960 | 1.4440 | 6.9700 | 0.5030 | 3.7380 | 2.7820 | −0.0180 |

B | 7.0100 | 1.0000 | 2.0085 | 3.3390 | 1.6695 | 1.1663 | 1.3436 | 3.9870 | 1.0848 | 5.9851 | 7.6640 | 1.0000 | −0.5284 |

C | 3.9986 | 1.1843 | 1.8231 | 2.9389 | 1.2484 | 1.0642 | 2.0452 | 2.6178 | 1.2912 | 3.0973 | 3.9400 | 0.9990 | −0.6781 |

D | 4.3955 | 1.5072 | 2.0252 | 6.7924 | 1.6042 | 0.8630 | 3.9806 | 5.9895 | 1.7393 | 4.0860 | 8.9220 | 1.3910 | −0.3195 |

Objective function values for six distribution functions calibrated by the genetic algorithm for Lighvan watershed

Model | Storm A | Storm B | Storm C | Storm D |
---|---|---|---|---|

Gamma distribution | 0.000028 | 0.000926 | 0.000027 | 0.001624 |

Gumbel distribution | 0.000013 | 0.002987 | 0.000457 | 0.003910 |

Log-normal distribution | 0.000055 | 0.000826 | 0.000047 | 0.001644 |

Normal distribution | 0.000013 | 0.002531 | 0.000251 | 0.002149 |

Pearson distribution | 0.000015 | 0.001150 | 0.000080 | 0.001492 |

Weibull distribution | 0.000005 | 0.000923 | 0.000065 | 0.001342 |

Generally, at the calibration stage using the genetic algorithm method, the lognormal, Pearson, and gamma models predicted the time to peak more or less properly for all storms. These models along with the Weibull distribution were also successful in simulating the rising and falling limbs of the UHs for all storms except A. The log-normal distribution showed high ability in estimating the peak value for storms B, C, and D. However, the Pearson model can compute well the peak discharge for storms A, B, and C. The Weibull distribution was distinguished as the most successful model based on the average value of the objective functions because of the excellent ability in preserving the UH shape of storm A.

#### Validation of the models

Performance criteria values for six distribution functions calibrated by the genetic algorithm for Lighvan watershed

Model | RMSE (mm/hr) | MAE (mm/hr) | CC | |||
---|---|---|---|---|---|---|

Storm E | Storm F | Storm E | Storm F | Storm E | Storm F | |

Gamma distribution | 0.010 | 0.016 | 0.007 | 0.012 | 0.697 | 0.922 |

Gumbel distribution | 0.015 | 0.038 | 0.012 | 0.032 | 0.655 | 0.654 |

Log-normal distribution | 0.013 | 0.039 | 0.009 | 0.024 | 0.738 | 0.702 |

Normal distribution | 0.014 | 0.036 | 0.012 | 0.028 | 0.669 | 0.736 |

Pearson distribution | 0.011 | 0.016 | 0.007 | 0.013 | 0.576 | 0.906 |

Weibull distribution | 0.013 | 0.032 | 0.012 | 0.023 | 0.711 | 0.839 |

Generally, at the validation stage, the log-normal distribution showed good performance in predicting the time to peak and peak flow of the UH for storm E. The gamma and Pearson distributions were able to preserve the UH shape. Hence, the gamma distribution with the lowest value of RMSEand MAE, and the highest value of CC is the best model for both storms. The Pearson model indicated similar results with the gamma distribution.

## Conclusions

In this study, a nonlinear model was developed to transmute a unit hydrograph into a probability distribution function. The gamma, Gumbel, log-normal, normal, Pearson, and Weibull probability distribution functions were used to derive 1-hour unit hydrographs. The main advantage of this model is that the number of parameters to be determined is equal to the number of probability distribution parameters. In this case, six different storm datasets from the Lighvan catchment were provided. Four storm datasets were used for models calibration and two for validation. The calibration of models was performed using the nonlinear least squares optimization methods, by programming in Mathematica and by applying the genetic Algorithm, and using the traditional linear least squares method.

- 1.
The log-normal distribution function has a high potential in predicting the peak flow and the time to peak of the UH.

- 2.
The gamma and Pearson distributions are more able in preserving the rising and recession limbs of the UH.

- 3.
The log-normal, gamma, and Pearson distribution functions can be applied for quick and approximate estimation of unit hydrographs for the Lighvan catchment.

- 4.
The genetic algorithm did not improve the models performance significantly compared with the nonlinear mathematical optimization.

- 5.
The nonlinear optimization methods are not superior to the linear least squares method when there is only one excess rainfall pulse, but are comparable. The main disadvantage of the traditional least squares method is that it may generate negative unit hydrograph ordinates especially when the number of excess rainfall pulses is bigger than one.

## Notes

### Acknowledgments

We thank the anonymous reviewers and editor for their constructive and useful comments that helped us improve the quality of the paper.

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