Abstract

This article examines data from the Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai hydrological stations in the Nen River basin to understand the hydrological processes occurring in the catchments. Daily precipitation and runoff data from 1955 to 1973 were combined using the smoothed minima trial method to determine the surface runoff concentration time. Then, a genetic algorithm was used to optimize the parameters and obtain an optimal empirical formula. An improved empirical formula was implemented with the genetic algorithm and optimized parameters then incorporated variable average rainfall intensity, correlation between basin area, surface runoff average concentration time, and average rainfall intensity. Finally, an optimized empirical formula (using genetic algorithm to optimize the parameters) and improved empirical formula (incorporating variable average rainfall intensity) were tested by using the daily precipitation and runoff data from the Baishan and Hongshi hydrological stations of the Second Songhua River. The results show that an optimized and improved formula can be used to more accurately estimate hydrologic conditions in the Nen River. Therefore, the improved formula is an efficient method for calculating surface runoff concentration time. Surface runoff concentration time is an important basis for differentiating source waters, which include surface runoff and underground runoff.

INTRODUCTION

The study on baseflow separation has been one of the key and difficult issues in hydrology and ecohydrology. In recent years, it has been of wide concern to scholars at home and abroad, and has made some progress and breakthroughs. There are many methods to separate baseflow (Zhao & Geng 1996; Zhang et al. 2013, 2017; Stewart 2016; Chawanda et al. 2017), but most of the methods are based on the experience of runoff characteristics, and the results are not the same. Since baseflow separation is related to climate, natural geography, hydrogeology, and other sciences, and due especially to the lack of experimental data, as yet, no generally accepted segmentation method has been found. Thus, compared with other analytical techniques of hydrology and water resources, research progress and breakthroughs are more difficult. Among them, the surface runoff concentration time is an important basis for differentiating source waters, which is the sum of slope concentration time and river network concentration time, and the source waters include surface runoff and underground runoff (Shen & Huang 2008).

There is an empirical formula for the surface runoff concentration time. Key to improving the accuracy of the empirical formula includes selection of the essential factors as independent variables and calibration of other relevant parameters. By consulting a large number of studies at home and abroad, it is found that optimal parameters for the empirical formula are still insufficient, and at the same time, relevant works to improve the empirical formula have not been found. Ni et al. (2005) analyzed and discussed segmentation methods of the process flow line, and suggested that N is not only a function of basin characteristics but also is a function of rainfall characteristics. By analyzing the influencing factors (Chen & Li 2011; Zhang 2016; Charters et al. 2016; Shen et al. 2016) of surface runoff concentration time, in semi-arid and semi-humid areas, rainfall intensity is an important indicator of the restriction on the surface runoff concentration time. Of course, there are more catchment physical characteristics that have an effect on the surface runoff concentration time, such as the slope, area, soil texture, geology, and probably the length of channel and channel density. According to our analysis, it is found that rainfall intensity plays a more major role than the others. However, considering the simplicity of an empirical formula, this paper only introduces a main influencing factor, namely, rainfall intensity.

Equation (1) is an empirical formula that is largely affected by the parameters N and F. For a specific watershed, F has a constant value, while N does not have a constant value. Therefore, the formula requires calibration to more reliably estimate surface runoff concentration time. The formula is an important basis for differentiating source waters, which include surface runoff and underground runoff (Shen & Huang 2008). Model parameter optimization methods are generally closely related to parameter ranges, and computational requirements are relatively high because the global optimal solution must be reached (Huang et al. 2010). A genetic algorithm (GA) is a very useful optimization tool, with a major advantage being the wide range of adaptability, and limited restriction on the problem space so that the algorithm can easily handle a variety of constraints and also include a penalty factor. GAs have two operations, namely, genetic operations (crossover and mutation) and evolutionary computation (selection). Genetic operation simulates the process of creating new offspring in each generation and an evolutionary computation that updates the population. The GA is a heuristic algorithm, with a global search feature that models parameters and optimizes for a wide range of applications. If the GA is used to optimize the empirical formula of surface runoff concentration time, then there are important theoretical and practical significances.

Therefore, this article selects five representative hydrological stations, Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai in the Nen River basin, with various catchment areas. Combined with the daily precipitation and daily runoff data from 1955 to 1973, we use smoothed minima to determine the surface runoff average concentration time. Then, the GA is used to optimize the parameters of the surface runoff average concentration time and the empirical formula. Based on the introduction of variable average rainfall intensity, we correlate basin area, surface runoff average concentration time, and average rainfall intensity. Next, the GA is used to optimize the parameters and improve the empirical formula. Finally, using the daily precipitation and daily runoff data in Baishan and Hongshi hydrological stations of the Second Songhua River, the improved empirical formula is tested.

SURFACE RUNOFF CONCENTRATION TIME ESTIMATION METHOD

Recession curve method

Recession curves are drawn on the same graph paper to show overlap, with the line under the envelope being the standard recession curve (Du 2008). The transparent standard recession curve is placed over the paper, and adjusted to cover the segment of the hydrograph to be analyzed. Using this method, the baseline must coincide with the axis so that the two segment tails match, whereby the overlap of the two lines represents the end point of the surface runoff (The Yangtze River Water Resources Commission 1993; Ye & Zhan 2000). The distance, N, represents the interval between the end point of surface runoff and the moment of the measured flow hydrograph peak, namely, surface runoff concentration time (Linsley et al. 1975) (Figure 1).

Figure 1

Recession curve method.

Figure 1

Recession curve method.

Empirical formula method

Theoretically, for a specific basin, when the condition of the underlying surface is relatively stable, then the average concentration time is a constant; for different basins, the concentration time is mainly controlled by the basin area, whereby larger areas result in longer concentration time and smaller areas result in shorter concentration times. The original empirical formula, adopted from classic ‘water principle literature’ textbooks (Xiao 2004; Miao 2007; Shen & Huang 2008), was derived in accordance with the principle of minimum error and based on area being the independent variable and surface runoff concentration time being the dependent variable. Because of the complexity of hydrological phenomena and processes, empirical correlation (Zhao 1984) is most commonly used to solve hydrologic problems and develop quantitative relationships between independent and dependent variables. In segmentation of the hydrograph of the basin outlet section, the method most widely used is the experience segmentation method (Ni et al. 2005). The method is based on the concept of a river basin unit hydrograph that should enable direct runoff during a constant rate rainstorm event, which provides a fixed time of direct runoff to terminate after the flood peak of hydrograph. In the ‘Hydrology Project’, Linsley et al. (1975) suggested that the distance can be expressed by an empirical formula:  
formula
(1)
In the formula, N is days of runoff terminated after peak, d; A is the basin area, km2; F is the basin area, km2.

After N is obtained, we can fix the terminate point B, the area over slash AB represents the total amount of direct runoff in the flood (Figure 1).

The smoothed minima trial method

The smoothed minima baseflow separation (Aksoy et al. 2008, 2009) was performed on a consecutive daily streamflow time series. Given that the data are presented in the form of consecutive average daily flows, Hisdal et al. (2004) presents the method as follows:

  • 1.

    Divide the daily flow data into non-overlapping blocks of 5 days.

  • 2.

    Mark the minima of each of these blocks, and let them be called

    Consider in turn In each case, if  
    formula
    (2)
    is satisfied, then the central value is a turning point for the baseflow line. Continue this procedure until the whole time series is analyzed.
  • 3.

    Let the discharges at the turning points be Q1, Q2Qm. Join the turning points by straight lines to form the baseflow hydrograph. If, on any day, the baseflow estimated by this line exceeds the total flow on that day, the baseflow is set to be equal to the total flow.

It should be noted that Equation (2) is correct for the case of perennial streams, whereas it was modified as:  
formula
(3)
so that it can be used for intermittent streams. Otherwise, it becomes problematic to separate baseflow during zero-flow periods. Indeed, Equation (2) is not applicable during zero-flow periods in intermittent streams and in perennial streams with fixed discharge periods. Therefore, as a generalization, Equation (3) should be used for perennial or intermittent streams. Using the trial method (Zuo et al. 2007; Li et al. 2013) to determine the value of N, where N represents the surface runoff average concentration time (Figure 1), the following steps are taken:
  • 1.

    Adopted from the Wahl & Wahl (1995) written BFI (baseflow index) baseflow calculation program, the smoothed minima method is used to calculate IGF (baseflow index), when N is equal to 1–10, as shown in Figure 2.

  • 2.

    Along with the increasing value of N, IGF decreases (Figure 2) and direct runoff is a smaller proportion of the underground segment. When the value of N increases there will be a corresponding sharp decrease in IGF, whereby direct runoff is reduced to zero for underground segments. Because part of the direct runoff originally belonged to underground runoff, the value of N is requested (Li et al. 2013).

Figure 2

Relationship of N and .

Figure 2

Relationship of N and .

Comparison of estimation methods for the surface runoff concentration time

The recession curve method is the traditional method of using hand-painted lines, which leads to a time-consuming (Xu & Jin 2011), low efficiency, and subjective process. Although the empirical formula method is simple and practical, there are significant deficiencies related to the empirical correlation analysis of the ‘error minimization principle’ as a control condition, which can lead to internal contradictions between the total result and partial results (Zhao 1984). Therefore, in practice, we often use the smoothed minima trial method to obtain surface runoff average concentration time. The smoothed minima trial method is simple, easy to automate with a computer automatically, and can overcome the subjectivity of artificial methods. The result is an efficient method for calculating surface runoff concentration time that can be quickly applied in various countries and regions (Han et al. 2009; Han 2010). Nevertheless, the empirical formula is still an important basis for estimating in engineering practice.

STUDY AREA AND DATA

Background of the study area

The Nen River originates at 1,030 m elevation from the foot of the North Greater Higgnan Mountains near Yilehulishan, and meets the Songhua River in SanCha, Songyuan City, Jilin Province. The Nen River is about 1,370 km in length and has a basin area of 297,000 km2, which is mostly covered by forests, and hilly in the upper reaches of the Nenjiang. Its middle and lower reaches transit from hills to plains, where the terrain becomes flat.

The Nen River basin is located in a temperate continental monsoon climate zone, with higher precipitation rates in the upstream and mountainous parts versus downstream and plain parts of the watershed, while the average annual precipitation is 400–500 mm. Precipitation is extremely unevenly distributed during the year. More than 50% of the annual precipitation occurs from July to August, and over 80% of annual precipitation occurs from June to September. There is very little precipitation during winter, which accounts for less than 5% of the annual precipitation.

This article examines the Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai catchments, with areas of 17,205 km2, 66,382 km2, 108,029 km2, 162,569 km2, and 221,715 km2, respectively. Baishan and Hongshi hydrological stations of the Second Songhua River served as the test data, with catchment areas of 19,000 and 20,300, respectively. The locations of the Nanjing River basin and the second Songhua River basin are shown in Figure 3.

Figure 3

Water system and representative hydrological station distribution in Nenjiang River basin.

Figure 3

Water system and representative hydrological station distribution in Nenjiang River basin.

Data sources

Runoff and rainfall data were measured by the Ministry of Water Resources Songliao Water Resources Commission. Due to the start of farmland irrigation and hydraulic engineering construction in the Nen River basin, river runoff has been significantly affected by human activities. Daily precipitation and daily runoff data from 1955 to 1973 were used as research data for Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai. Daily precipitation and daily runoff data from 1971 to 1980, at Baishan and Hongshi hydrological stations of the Second Songhua River, were used as the test data. According to the daily runoff data, N, results (Table 1) for the seven catchment areas were obtained using the smoothed minima trial method.

Table 1

Average confluence time for surface runoff at seven hydrologic stations

Catchment areaShihuiyaoNierjiTongmengJiangqiaoDalaiBaishanHongshi
Catchment areaShihuiyaoNierjiTongmengJiangqiaoDalaiBaishanHongshi

METHODS FOR OPTIMIZING AND IMPROVING THE EMPIRICAL FORMULA

Optimized empirical formula

The GA (Zhou & Sun 1999; Termansen et al. 2006) is a simulation of Darwin's genetic choice and evolution process of natural selection. Using the calculation model, it is a type of searching optimal solution that simulates the natural evolution process. The method was first suggested by Professor J. Holland in Michigan, USA in 1975. He published an influential book ‘The Adaptation in Natural and Artificial Systems’. The GA published by Holland (1975) used a simple genetic algorithm (SGA). By following the GA, genes of a certain number of individuals (individual) may explain potential population solutions because a population is made of the genes code. Each individual contains a set of chromosomes (chromosome) encoded with the characteristics of the entity. The chromosomes are the main carriers of genetic material, that is, a collection of multiple genes, so their internal performance (genotype) is a combination of genes that determines the shape of the individual external performance. For example, characteristics such as black hair are controlled by the chromosome in a set of certain genes that result in measurable or observable features of the subjects. Therefore, a phenotype is first assigned upon which a genotype map may be encoded.
  • (a)

    Chromosome coding:

For simplicity, Equation (1) is transformed as follows:  
formula
(4)
where and are the constants of the empirical formula, which is the chromosome for the GA. F is the argument of the empirical formula, namely, the basin area, which is input by the GA. By the parameters of the empirical formula of Equation (1), the variables and are in a range of [0, 1]. The chromosome is coded as a floating point value. For example, if one chromosome is , the formula obtained using the constant given above is as follows:  
formula
(5)

Using Equation (5), the results obtained for the formula are: [2.6127, 2.9533, 4.5318, 4.4461, 8.0310], as shown in Figure 4.

  • (b)

    Objective function:

Figure 4

Empirical formula of .

Figure 4

Empirical formula of .

According to the basic principle used to derive the empirical formula, the objective function of the GA can be expressed as follows:  
formula
(6)
where is the surface runoff average concentration time by the empirical formula in a basin, d; and N is the calculated value of the surface runoff average concentration time in the basin, d.
Physical interpretation of the objective function Equation (6): A set of parameters and , are searched based on several conditions. The inverse square of the difference between and N is maximal. To avoid computations, if the objective function value is too small, the expansion coefficient M is introduced. The value of M is determined according to the specific situation. Equation (6) is converted into the following form:  
formula
(7)
  • (c)

    Operation process:

The process used by the GA to optimize the parameters of the empirical formula is shown in Figure 5.
Figure 5

Optimization of the parameters in the text, written as relevant content into a processes map.

Figure 5

Optimization of the parameters in the text, written as relevant content into a processes map.

The empirical formula is calculated based on the best chromosome , as follows:  
formula
(8)
From Table 2, optimizing the empirical formula:  
formula
(9)
Table 2

Genetic algorithm to optimize runoff parameters

ParametersOptimized values
 0.1200 
 0.3055 
ParametersOptimized values
 0.1200 
 0.3055 

The results, after applying the GA to optimize the parameters of the empirical formula are shown in Figure 6. Figure 6 shows the results of the optimized empirical formula are superior to the original empirical formula. At the same time, the results of the original empirical formula and the optimized empirical formula have been compared with the calculated error value; the results of the statistical analysis are shown in Table 3. Obviously, the results of the empirical formula after applying the GA were better than the results achieved with the original empirical formula (Table 4).

Table 3

Comparison of calculation absolute error for original versus optimized empirical models

Catchment areaShihuiyaoNierjiTongmengJiangqiaoDalai
Error of original empirical formula −2.9076 −4.7391 −4.5308 −5.2574 −3.8500 
Error of optimized empirical formula 0.6378 −0.5682 −0.1406 −0.6912 0.8423 
Catchment areaShihuiyaoNierjiTongmengJiangqiaoDalai
Error of original empirical formula −2.9076 −4.7391 −4.5308 −5.2574 −3.8500 
Error of optimized empirical formula 0.6378 −0.5682 −0.1406 −0.6912 0.8423 
Table 4

Parameters used for optimizing empirical model with a genetic algorithm

ParametersValue
The number of population 80,000 
Choose rate 0.05 
Cross rate 0.1 
Mutation rate 0.05 
Algebra future 10 
ParametersValue
The number of population 80,000 
Choose rate 0.05 
Cross rate 0.1 
Mutation rate 0.05 
Algebra future 10 
Figure 6

Comparison of the calculation results N/d; F/km2.

Figure 6

Comparison of the calculation results N/d; F/km2.

IMPROVED EMPIRICAL FORMULA

In humid and semi-humid areas, rainfall intensity is an important indicator of the restriction on concentration time. Spatial and temporal distribution of rainfall and rainfall intensity will influence the spatial distribution and direction of rainstorm movement. Rainfall intensity may reflect the concentration strengths of different basins. Higher rainfall intensity typically results in faster runoff yield, greater peak flow, and a finer hydrograph. If the rainstorm center is located in the downstream portion of the watershed, there will be a shorter concentration time, earlier and higher peak flow, and finer peak shape. If the rainstorm center moves from upstream to downstream portions of the watershed then there will be a faster concentration speed, larger peak flow, and a higher likelihood of flooding in middle and lower reaches relative to a storm that moves from downstream to upstream (Wu & Cen 1995). According to our analysis, it is found that rainfall intensity plays a more major role than the others. However, considering the simplicity of the empirical formula, this paper only introduces a main influencing factor, namely, rainfall intensity. Furthermore, the original empirical formula, if improved, can include the introduction of variable average rainfall intensity, establishing correlation between basin area, surface runoff average concentration time, and average rainfall intensity, and using the GA to optimize the parameters and the empirical formula.

General empirical formula

 
formula
(10)

GA to optimize the model parameters

Optimization model:  
formula
(11)

The results of the model after the GA to optimize the parameters show that the improved empirical formula was better than the optimized empirical formula and the original empirical formula (Figure 7). At the same time, the results of three empirical formulas were compared with calculated error value, and the results of the statistical analysis are shown in Table 5.

Table 5

Statistics for calculation absolute error of various empirical formula

Catchment areaShihuiyaoNierjiTongmengJiangqiaoDalai
Error of original empirical formula −2.9076 −4.7391 −4.5308 −5.2574 −3.8500 
Error of optimized empirical formula 0.6378 −0.5682 −0.1406 −0.6912 0.8423 
Error of improved empirical formula 0.2747 −0.4841 0.4007 −0.2007 −0.1128 
Catchment areaShihuiyaoNierjiTongmengJiangqiaoDalai
Error of original empirical formula −2.9076 −4.7391 −4.5308 −5.2574 −3.8500 
Error of optimized empirical formula 0.6378 −0.5682 −0.1406 −0.6912 0.8423 
Error of improved empirical formula 0.2747 −0.4841 0.4007 −0.2007 −0.1128 
Figure 7

Comparison of the calculation results N/d; F/km2.

Figure 7

Comparison of the calculation results N/d; F/km2.

From Table 1, the measured value of average confluence time for surface runoff in the Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai hydrological stations is 3, 3, 4, 4, and 6. The calculated value of average confluence time for surface runoff in the Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai hydrological stations is 5.9076, 7.7391, 8.5308, 9.2574, and 9.8500 using the original empirical formula. The calculated value of average confluence time for surface runoff in the Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai hydrological stations is 2.3262, 3.5682, 4.1406, 4.6912, and 5.1577 using the optimized empirical formula. The calculated value of average confluence time for surface runoff in the Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai hydrological stations is 2.7253, 3.4841, 3.5993, 4.2007, and 6.1138 using the improved empirical formula. From Table 5 the errors of the original empirical formula are −2.9076, −4.7391, −4.5308, −5.2574, and −3.8500 in the Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai hydrological stations; the errors of the optimized empirical formula are 0.6378, −0.5682, −0.1406, −0.6912, and 0.8423 in the Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai hydrological stations; and the errors of the improved empirical formula are 0.2747, −0.4841, 0.4007, −0.2007, and −0.1128 in the Shihuiyao, Nierji, Tongmeng, Jiangqiao, and Dalai hydrological stations. Table 5 shows that precision is highest for the improved empirical formula after the introduction of rainfall intensity. The second highest precision results from the optimized empirical formula, while the original empirical formula has the lowest precision.

TEST

Daily precipitation and daily runoff data from 1971 to 1980 in Baishan and Hongshi hydrological stations of Second Songhua River were used as the test data.

Testing the optimized empirical formula

Figure 8 shows that the results of the optimized empirical formula were better than the original empirical formula.

Figure 8

Comparison of the calculation results N/d; F/km2.

Figure 8

Comparison of the calculation results N/d; F/km2.

Testing the improved empirical formula

Figure 9 shows that the results of the improved empirical formula were better than the optimized empirical formula and the original empirical formula.

Figure 9

Comparison of the calculation results N/d; F/km2.

Figure 9

Comparison of the calculation results N/d; F/km2.

From Table 1, the measured value of average confluence time for surface runoff in Baishan and Hongshi hydrological stations is 3, 3. The calculated value of average confluence time for surface runoff in Baishan and Hongshi hydrological stations is 6.026, 6.1063 using the original empirical formula. The calculated value of average confluence time for surface runoff in Baishan and Hongshi hydrological stations is 2.4341, 2.4838 using optimized empirical formula. The calculated value of average confluence time for surface runoff in Baishan and Hongshi hydrological stations is 3.0816, 3.1477 using improved empirical formula. From Table 6, the errors of the original empirical formula are 3.026, 3.1063 in Baishan and Hongshi hydrological stations, the errors of the optimized empirical formula are −0.5659, −0.5162 in Baishan and Hongshi hydrological stations, and the errors of the improved empirical formula are 0.0816, 0.1477 in Baishan and Hongshi hydrological stations. Table 6 shows that precision of the improved empirical formula after the introduction of rainfall intensity and optimize was highest, precision of the optimized empirical formula was second highest, and precision of the original empirical formula was lowest.

Table 6

Statistics for calculation absolute error in test watersheds

Catchment areaBaishanHongshi
Error of original empirical formula 3.0260 3.1063 
Error of optimized empirical formula −0.5659 −0.5162 
Error of improved empirical formula 0.0816 0.1477 
Catchment areaBaishanHongshi
Error of original empirical formula 3.0260 3.1063 
Error of optimized empirical formula −0.5659 −0.5162 
Error of improved empirical formula 0.0816 0.1477 

CONCLUSIONS

The following conclusions were drawn:

  1. Classic watershed average concentration time empirical formula was directly and quantitatively connected as a cause, thus without considering the physical processes and mechanisms, they can be rough estimates for surface runoff average concentration time.

  2. Comparison of estimation methods for the surface runoff concentration time, the recession curve method, is the traditional method of using hand-painted lines, which leads to a time-consuming, low efficiency, and subjective process. The smoothed minima trial method is simple, easy to automate with a computer automatically, and can overcome the subjectivity of artificial methods. The result is an efficient method for calculating surface runoff concentration time that can be quickly applied in various countries and regions. Therefore, using smoothed minima to determine the surface runoff average concentration time, as sample data, and then using the GA to optimize the parameters with respect to the original empirical formula, our approach can more accurately estimate surface runoff average concentration time.

  3. Through the introduction of variable average rainfall intensity, establishing correlation between basin areas, the surface runoff average concentration time, and average rainfall intensity, our approach improves the empirical formula and more accurately estimates surface runoff concentration time.

  4. Using the catchment area of Baishan and Hongshi to test, the result can verify the reliability of the above conclusions.

  5. Using the improved formula to calculate surface runoff concentration time is an important basis for differentiating source waters.

ACKNOWLEDGEMENTS

We are grateful to the National Natural Science Foundation of China (No. 51379088), NSFC-NRF Joint Project ‘Study on regional water resources response and sustainable utilization under the changing environment’ (51711540299), ‘13th Five-Year’ Key Research Program ‘River lakes and reservoirs comprehensive control technology and application in SongLiao River basin’ (2017YFC0406005) for providing financial support for this research.

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