Introduction

A large amount of precipitation is lost annually. Therefore, studying the precipitation-runoff process and predicting runoff are essential in water resources studies and plans and water engineering discussions. Therefore, choosing a model that can predict runoff from rainfall with acceptable accuracy using effective factors seems essential. The aim of this research is to estimate the components of unit hydrographs resulting from several precipitation-runoff events in the Samian watershed located in Ardabil province using a nonlinear optimization method along with the application of probability distribution functions.



Materials and Methods

In order to determine the optimal values of the distribution function parameters, the objective function is considered as minimizing the sum of the squares of the deviation between the predicted and actual direct runoff hydrographs. The values of the probability distribution functions parameters are optimized using Mathematica software and nonlinear programming method. Finally, the direct runoff hydrographs are calculated using the effective rainfall hydrographs and the corresponding unit hydrographs and compared with the observed direct runoff hydrographs. The performance criteria of mean square error, mean absolute error and correlation coefficient are used to examine the ability of the functions in the calibration and testing stages. The classical least squares method is also used to extract the unit hydrographs and its efficiency is compared with the results of the nonlinear optimization model.



Results and Discussion

In the calibration period, for the first event, the RMSE and MAE errors of the Gumbel (0.0048 and 0.0042) and Nakagami (0.0134 and 0.0095) functions are higher and their correlation coefficients (0.935 and 0.237, respectively) are also lower than those of the other functions. The lowest errors (0.0021 and 0.0016) and the highest correlation coefficient (0.978) are related to the lognormal distribution. The weakest distribution is the Nakagami distribution with RMSE equal to 0.0134, MAE equal to 0.0095, and CC equal to 0.237. In the case of the second event, the RMSE and MAE errors of the Gumbel (0.0110 and 0.0070) and Nakagami (0.0223 and 0.0293) functions are higher and their correlation coefficients (0.682 and -0.087) are also lower than the other functions. The lowest errors (0.0068 and 0.0056) and the highest correlation coefficient (0.874) are related to the lognormal distribution. The weakest distribution is the Nakagami distribution with RMSE of 0.0223, MAE of 0.0293, and CC of -0.087. In the case of the third event, the performance of all functions is satisfactory due to very low RMSE and MAE values and very high correlation coefficient. However, the errors of the gamma and lognormal functions are slightly higher than the others. Also, Weibull and normal distributions show better performance than other distributions based on the values of performance criteria because these distributions showed high ability in predicting the ascending and descending branches of the unit hydrograph.

In the test period, considering the average values of the performance functions of the distributions in each event, it can be said that the lognormal distributions (with average RMSE and MAE errors of 0.0134 and 0.01115 and average correlation coefficient of 0.823) and the normal distribution (with average RMSE and MAE errors of 0.0142 and 0.01275 and average correlation coefficient of 0.739) have higher potential in estimating the unit hydrographs of the basin, respectively. The errors of Pearson and Nakagami distributions in both events were higher than the other distributions. The least squares method also had better performance criteria values than all distributions in both events.



Conclusion

In general, according to the results obtained in the calibration and testing period, it can be concluded that the lognormal distribution is the best distribution for estimating and calculating unit hydrographs (especially in estimating the ascending branch and peak time and discharge) in the Samian basin. Also, the Pearson and Nakagami functions are among the weakest distributions in estimating the hydrographs of that basin. The least squares method, regardless of its shortcomings, is among the appropriate methods in estimating the unit hydrograph.

For future studies, it is suggested to investigate the performance of other statistical distributions such as beta and logistic distributions in extracting the unit hydrograph of the Samian basin. Also, the performance of statistical distributions in determining the unit hydrograph from combined precipitation-runoff events and from precipitation-runoff events related to the warm months of the year in which the basin does not have snow storage should be investigated. Finally, similar research should be conducted in other watersheds of Ardabil province and a suitable model should be proposed for the entire province.



Keywords: Nonlinear optimization, least squares, precipitation-runoff event, Mathematica software, Unit Hydrograph