Abstract

Research on the periodic characteristics of the runoff evolution in the Lower Yellow River is of great importance for flood control, beach regulation and water resources utilization in the Lower Yellow River. By using wavelets to conduct scale analysis of runoff series, the periodic change rule of runoff series on different scales can be obtained. By using the maximum entropy spectrum to analyze the spectrum of runoff, the main period of runoff sequence can be obtained. In this paper, these two methods are applied to the annual runoff of the Lower Yellow River. The results show that: the annual runoff in the Lower Yellow River has multi-scale change law; the four stations have the same main period; there are differences in periodicity between stations, as the catchment area increases, the quasi-periodic value decreases, and the periodic fluctuation becomes more obvious; after 2018, the annual runoff of the Lower Yellow River will be in the dry season. Furthermore, the study can reveal the change law of runoff sequence in the Lower Yellow River to a certain extent, and provide a theoretical basis for river management.

INTRODUCTION

In recent years, influenced by climate change and human activity, the annual runoff of the Lower Yellow River has undergone great changes. Research on the periodicity of runoff in the Lower Yellow River can provide an important basis and theoretical value for flood control and disaster reduction, river management and so on. On the evolution of runoff, domestic and foreign scholars have carried out much research, and obtained some valuable results. Venugopal & Foufoula (1996) used wavelet packet theory to decompose the precipitation series. In 2014, Farajzadeh et al. (2014) used a neural network and autoregressive integrated moving average (ARIMA) model to predict the runoff in the future of Ulmiye basin. Costa-Cabral et al. (2013) studied the snowpack and runoff response to climate change in the Owens Valley and Mono Lake watersheds. Bologov (2002) used stochastic models to study the periodic correlation of seasonal river runoff changes. Stojković et al. (2014) analyzed the periodicity of mean annual and seasonal stream flow in the Danube River basin. Pisarenko et al. (2005) used statistical methods to predict river runoff. Lin et al. (in press) used a neural network model to study the response of surface runoff to climate change. Li et al. (2005) used spectral analysis to analyze and predict groundwater in western Jilin in 2005. Bing et al. (2012) used wavelet analysis to study runoff characteristics in the source regions of the Yangtze and Yellow rivers. Li & Yang (2004) studied the driving factors of runoff change in Yellow River basin. Liu et al. (2015) used wavelet multiscale analysis to study the periodical changes of runoff and sediment discharge in the Weihe River. It can be seen that the above research has been mainly focused on the prediction of runoff, driving mechanism, runoff response to climate change and so on. However, to the best of our knowledge (Zelda et al. 2018), there is limited literature on the quantitative research on the evolution period of runoff; in particular the application of different methods of scientific analysis is still lacking. The maximum entropy spectrum estimation is a nonlinear spectral estimation method with high resolution (Nie & Wang 2005). It can overcome the subjective of autocorrelation function selection in the traditional spectral analysis method. Wavelet analysis has a good ability to solve nonlinear series, especially in the time and frequency domain. In this paper, the maximum entropy spectrum method and Morlet wavelet method are applied to the annual runoff in the Lower Yellow River. By analyzing the periodic change law and future change trend of runoff, this paper provides a theoretical basis for river management in the Lower Yellow River.

METHODS

Maximum entropy spectrum analysis

Entropy was first proposed by Rudolf Clausius in 1865 and is an index of the uncertainty of random variables. It describes the degree of confusion or disorder in a system; the greater the entropy, the higher the uncertainty of the random variable. After that, the maximum entropy spectrum analysis method was proposed by Burg (1967). Entropy spectrum is based on entropy: the maximum entropy method is to extract the maximum amount of information from the known signal, however the information is constrained by the autocorrelation function.

The calculation process of maximum entropy spectrum analysis is as follows: find the autocorrelation function value of the original data and construct the matrix : 
formula
(1)
where R(m + 1) is semi-definite matrices, both the determinant and the principal minor are less than 0.
The prediction formula of the autoregressive signal of maximum entropy spectrum is shown as follows (Chatfield 2009): 
formula
(2)
where is the time series data, ; is autoregressive coefficients; e is the prediction error,
By making some transformations to Equation (2), the prediction error can be calculated by Equation (3): 
formula
(3)
The prediction error spectrum can be described by the error filter coefficient and the convolution of known data: 
formula
(4)
where E(ω) is the spectrum of the prediction error ej; A(ω) is the spectrum of the prediction error filter; X(ω) is the spectrum of the known data X(j). The power spectrum of is shown below (Burg 1975): 
formula
(5)
where is the power of the prediction error.

The prediction error of filter coefficient () and the power of the prediction error () are calculated by the Burg algorithm.

Burg recursive method

The Burg recursive method (Burg 1975) is a typical algorithm for maximum entropy spectrum analysis, it can be used to estimate the parameters of autoregressive model in a simple and effective way. The prediction error factor with a predictive step size of one should satisfy the following equations (Anderson 1974): 
formula
(6)
The recurrence relations of different order filter coefficients is constructed as follows: 
formula
(7)
The recurrence relations between and the least phase of the filter coefficient is as follows: 
formula
(8)
After the filter coefficient is calculated, the spectrum of the prediction error filter can be obtained. The initial value of the predicted error power is shown below: 
formula
(9)
Several methods can be used to determine the order m. For the selection criteria of optimal values of order, there are FPE, AIC and CAT criteria. However, some scholars (Ulrych & Bishop 1975) advocated the FPE criterion for geophysical data, therefore, the FPE criterion is used in this study. The FPE can be defined as (Akaike 1970): 
formula
(10)
where m is order; N is number of samples; and is the prediction error power, which decreases with the increase of m. Because N + m + 1/Nm–1 increases with the increase of m, there must be a point that makes the FPE (m) value minimum at that point, so the corresponding m will be the best order.

The quasi-periodic can be obtained by spectrum analysis. In the maximum entropy spectrum figure, find out the corresponding frequency of the crest value, because the frequency is the reciprocal of the period, therefore the period of time series can be obtained by the taking the reciprocal of frequency.

Wavelet analysis

The measured runoff series is often characterized by complex multi-time scale change. The continuous wavelet transform can effectively reveal the periodicity, trend, catastrophe and other characteristics of the series at different time scales (Wang et al. 2005; Schaefli et al. 2007). The unique advantages of wavelet are embodied in: it can be used for multiscale analysis of hydrologic series (Wang et al. 2005).

Wavelets are a kind of function that can rapidly attenuate to zero. The wavelet function expression is as follows: 
formula
(11)
where is the wavelet basis function.

The wavelet transform

For a given wavelet function , the continuous wavelet transform of the hydrological time series is shown below: 
formula
(12)
where is the sub-wavelet; a is the scale parameter which describes the period length of wavelet; b is the time parameter which describes the translation of wavelet in time.
The wavelet transform (Grossmann & Morlet 1984) of time series is as follows: 
formula
(13)
where is the wavelet transform coefficient; is the square integrable function; a is the scale parameter; b is the time parameter:

—complex conjugate function of

The wavelet coefficient contour line represents the real part of wavelet coefficients. If the value of the real part of wavelet coefficients is positive, it represents the abundant water period of runoff. If the value of the real part of wavelet coefficients is negative, it represents the dry period of runoff.

Wavelet variance

The Morlet function, representing a wave modulated by a Gaussian envelope, is given by: 
formula
(14)
where c is the the constant number; i is the imaginary number.
The wavelet variance is defined as the integral of any wavelet coefficient of different time scales in the time domain. The equation is given below: 
formula
(15)
The wavelet variance figure can reflect the fluctuation of various scales (period) contained in the hydrological time series and the distribution characteristics of its energy with the scale change. The larger the wavelet variance value, the more significant the periodic change of the corresponding time scale.
The relationship between extension scale a and period T is as follows (Torrence & Compo 1998): 
formula
(16)
If , then , and the wavelet can be used for periodic analysis.

CASE STUDY

In this section, 55 years of consecutive runoff from 1960 to 2014 are adopted in the Lower Yellow River from four hydrological stations – Huayuankou station, Gaocun station, Aishan station and Lijin station respectively.

The Huayuankou hydrological station is the first hydrological station in the Lower Yellow River. It is the starting point for the Yellow River and its catchment area is 730,036 km2. Gaocun hydrological station is the first control station for the Yellow River flowing from Henan to Shandong and its catchment area is 734,146 km2. Aishan hydrological station, located in Liaocheng City, Shandong Province, is one of the main hydrological control stations in the Lower Yellow River; its catchment area is 749,136 km2. The Lijin station is the last hydrological station in the Lower Yellow River; it controls the runoff and sediment concentration in the estuary of the Yellow River and its catchment area is 751,869 km2. These four stations are chosen for this research. The locations of the four stations are shown in Figure 1; the annual runoff change of the four stations is shown in Figure 2.

Figure 1

Sketch map of the Lower Yellow River and key hydrologic stations.

Figure 1

Sketch map of the Lower Yellow River and key hydrologic stations.

Figure 2

Annual runoff change trend in the Lower Yellow River.

Figure 2

Annual runoff change trend in the Lower Yellow River.

As can be seen from Figure 2, there is a certain fluctuation of annual runoff change trends in the two stations. Before 1975, the fluctuation was dramatic and the range of fluctuation was large. In 1975–1987, the fluctuation decreased. After 1987, the fluctuation remained steady.

PERIODIC CHARACTERISTICS OF RUNOFF

Maximum entropy spectrum

To determine the implicit period of the measured hydrological series, different series lengths need to be selected (Zhang & Li 2001) which are used to compare the consistency and stability of these test periods in different series lengths. In this section three different time scales are chosen; 45, 50, and 55 years respectively. The analysis results of the four stations are shown in Figures 36.

Figure 3

The maximum entropy spectrum in Huayuankou station.

Figure 3

The maximum entropy spectrum in Huayuankou station.

Figure 4

The maximum entropy spectrum in Gaocun station.

Figure 4

The maximum entropy spectrum in Gaocun station.

Figure 5

The maximum entropy spectrum in Aishan station.

Figure 5

The maximum entropy spectrum in Aishan station.

Figure 6

The maximum entropy spectrum in Lijin station.

Figure 6

The maximum entropy spectrum in Lijin station.

As can be seen from Figures 36, the best coincidence degree of frequency in Huayuankou station is 0.11, the quasi-periodic is about 9 years; the best coincidence degree of frequency in Gaocun station is 0.17, the quasi-periodic is about 6 years; the best coincidence degree of frequency in Aishan station is 0.18, the quasi-periodic is about 5.5 years; and for Lijin station, the best coincidence degree of frequency is 0.2, the quasi-periodic is about 5 years.

It can be seen that the periods of the four hydrological stations decrease successively, which may be related to the catchment area of the four stations. Through the study of runoff series of the four stations, the following hypothesis can be drawn: the greater the catchment area in the same basin, the more frequent the fluctuation of runoff cycle.

From the Huayuankou station to the Lijin station, the catchment area is increasing. Runoff is based on the precipitation, under the same precipitation conditions, without considering the loss along the path and other factors' interference. The Lijin station has the largest water collection area and therefore its runoff change is also the largest. It is likely that the periodic fluctuations are also the largest and the runoff periodic value of Lijin is also minimal.

Wavelet

In this section, the previous calculation steps are used to analyze the annual runoff period in the Lower Yellow River; the results are shown in Figures 710.

Figure 7

Wavelet transform coefficient and wavelet variance of annual runoff in Huayuankou station. (a) Wavelet coefficient figure. (b) Wavelet variance figure.

Figure 7

Wavelet transform coefficient and wavelet variance of annual runoff in Huayuankou station. (a) Wavelet coefficient figure. (b) Wavelet variance figure.

Figure 8

Wavelet transform coefficient and wavelet variance of annual runoff in Gaocun station. (a) Wavelet coefficient figure. (b) Wavelet variance figure.

Figure 8

Wavelet transform coefficient and wavelet variance of annual runoff in Gaocun station. (a) Wavelet coefficient figure. (b) Wavelet variance figure.

Figure 9

Wavelet transform coefficient and wavelet variance of annual runoff in Aishan station. (a) Wavelet coefficient figure. (b) Wavelet variance figure.

Figure 9

Wavelet transform coefficient and wavelet variance of annual runoff in Aishan station. (a) Wavelet coefficient figure. (b) Wavelet variance figure.

Figure 10

Wavelet transform coefficient and wavelet variance of annual runoff in Lijin station. (a) Wavelet coefficient figure. (b) Wavelet variance figure.

Figure 10

Wavelet transform coefficient and wavelet variance of annual runoff in Lijin station. (a) Wavelet coefficient figure. (b) Wavelet variance figure.

It can be seen from Figure 7 that there are obvious inter-annual changes and chronological changes of runoff in the Huayuankou station. There are three time scales which are respectively 20–32 years, 10–18 years, and 3–10 years. The performance characteristics of each scale are shown in Table 1.

Table 1

Wavelet analysis of Huayuankou station

Time scale Change characteristics Shock center Characteristic 
20–32 More less; more less; more  Locality 
10–18 Less more; less more; less more; less more 14 years Global 
3–10 Not obvious  Global 
Time scale Change characteristics Shock center Characteristic 
20–32 More less; more less; more  Locality 
10–18 Less more; less more; less more; less more 14 years Global 
3–10 Not obvious  Global 

It can be seen from Figure 7 that there are three crest values in the wavelet variance figure, the corresponding time scales are 14, 10, and 5 years respectively, but the crest value of the wavelet variance in the 14 years was the largest and is the primary main period of runoff change (Wang & Sun 2016); it plays a major role in the evolution of runoff series. The second and third main periods are 10 and 5 years respectively and the change characteristics of the Huayuankou station in the whole time-domain are determined by these three periods. Due to the limited series, if the time series is extended, it is likely that the next crest value will occur.

The wavelet analysis results of the other three stations are shown in Figures 810.

It can be seen from Figures 710 that the change law of runoff in Gaocun, Aishan, Lijin station is similar to that of Huayuankou station, with the regularity of periodic change on the scale of 20–32, 10–18 and 3–10 years; the four stations have the same main periods, which are 14, 10 and 5 years respectively. By 2014, the contour lines of wavelet coefficients corresponding to large scale did not form a complete closed circle. It is estimated that there will be an increasing trend of runoff in the next period of time.

The crest values of the wavelet variance between the four stations show the following characteristics: as the catchment area increases, the crest value of the wavelet variance also increases gradually, and the periodic fluctuation of runoff is also more frequent. It shows that the first, second and third main periods of the runoff change of Lijin station are more obvious than those of other three stations. The following explanation may be provided: the catchment area of Lijin station is the largest. Under the same precipitation conditions, without considering other conditions, the runoff change is the largest and the cycle change is the most significant; on the other hand, the Lijin station is near the estuary of the Bohai sea, and is greatly influenced by monsoon climate, there is also the potential factor that the runoff change of Lijin station is more obvious than other stations.

PREDICTION OF FUTURE CHANGE TREND OF ANNUAL RUNOFF

According to the primary main period of the wavelet variance figure, the corresponding wavelet coefficients can be analyzed, as shown in Figure 11.

Figure 11

The 14-year scale change trend of annual runoff series in the Lower Yellow River.

Figure 11

The 14-year scale change trend of annual runoff series in the Lower Yellow River.

As can be seen from Figure 11, on the 14-year characteristic time scale, the average period of runoff change in the basin is about 5 years, which has experienced about five cycles of wet and dry seasons. It can be seen that after 2014, the wavelet coefficient of the 14-year scale began to change from a negative value to a positive value. Therefore, in the 1–3 years after 2014, the runoff is in the period of abundant water, and in the 3–4 years after 2018, it will enter the dry season.

DISCUSSION

Through the study of the annual runoff of four stations in the Lower Yellow River, the change characteristics of annual runoff on various scales were obtained. If monthly series are analyzed, the unit of time scale is in months, the change characteristics of runoff at various scales are also more obvious. However, it is possible that the shock centre, change cycle and quasi-period of the wavelet will be different from the annual runoff series, but the details may be more pronounced. In addition, if seasonal series are analyzed, it can be concluded that the periodic fluctuation of Lijin station is still the most significant among the four stations.

Because runoff change is a complex process influenced by many factors, this makes it difficult to accurately analyze the periodic characteristics of runoff. The period detected by a single periodic analysis method is not always reliable. Therefore, this paper adopts two methods of periodic analysis, and from the analysis results, both methods had some similar periodic values. This shows that the detected period is in accordance with certain actual conditions, and the accuracy and reliability are feasible. In addition, the results of the two methods are somewhat different; this is because runoff is based on the precipitation, and is also influenced by many factors in the process of formation. The complexity of these factors leads to the complexity of the annual runoff, which leads to the possibility that the two methods will result in different periodic values.

CONCLUSIONS

In this study, the maximum entropy spectrum method and wavelet analysis method were used to study the periodic regularity of annual runoff in the Lower Yellow River. Through the study of the series of the four stations, the following conclusions were drawn: the annual runoff in the Lower Yellow River is characterized by multi-scale change, of 20–32, 10–18 and 3–10 years respectively; the main period of runoff change is 14, 10 and 5 years, and the periodic shock around 14 years is the strongest. The quasi-cycles of the four stations are 9, 6, 5.5 and 5 years respectively. In the same basin, the larger the catchment area, the more frequent the periodic fluctuation and the smaller the periodic value.

The wavelet method can analyze the change characteristics of runoff at different scales. However, runoff change also has the characteristic of localization, the change characteristics of runoff may be different at different scales of the same time series, and the characteristics of periodic change may also be different. Therefore, the periodic analysis of runoff only makes sense within a certain scale. In addition, the length of time series will also have some influence on the results of periodic analysis. The time series of that year will be transformed into the month series, and the results will also be different.

By analyzing the future cycle trend of runoff in the Lower Yellow River, the runoff stage in the future can be determined, and then river management, river engineering, and ration allocation of water resources can be guided. Through analyzing the change trend of future runoff in the Lower Yellow River, the runoff will enter a dry stage after 2018; therefore, it is suggested that water conservancy engineering departments take appropriate measures to control the industrial and agricultural water, and to build some inter-basin water transfer project. In this way, rivers can run healthily and water for agriculture and industry can be guaranteed.

This research only analyzes the annual runoff series; the monthly and seasonal runoff series were not analyzed. There are many factors influencing the change of annual runoff, which bring uncertainty and randomness. It results in the periodic analysis of time series being disturbed by these factors; the periodic change regularity of runoff is different in natural conditions and under human interference, which will be an interesting topic for further study.

ACKNOWLEDGEMENTS

This work was financially supported by Collaborative Innovation Center of Water Resources Efficient Utilization and Protection Engineering, Henan Province, Water Environment Governance and Ecological Restoration Academician Workstation of Henan Province, Program for Science & Technology Innovation Talents in Universities of Henan Province (No.15HASTIT049). Our gratitude is also extended to reviewers for their efforts in reviewing the manuscript and their very encouraging, insightful and constructive comments.

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