Abstract

In order to reveal the multi-time scale of rainfall, runoff and sediment in the source area of the Yellow River and improve the accuracy of annual runoff forecast, the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) method is introduced to decompose the measured rainfall, runoff and sediment data series of the Tangnahai hydrological station in the source area of the Yellow River of China. With the co-integration theory, two new error correction models (ECMs) for the forecast of annual runoff in the source area of the Yellow River are constructed. The application of these two methods solves the problem of pseudo-regression caused by nonlinearity and non-stationary of hydrological time series. The results show that rainfall, runoff and sediment in the source area of the Yellow River have multi-time scales and the component sequences have co-integration relationships. For two new ECMs, the CEEMDAN component ECM has better forecast accuracy than the original sequence one. The relative error of all forecasted values is less than 15% except 2009, and the accuracy has reached level A.

HIGHLIGHTS

  • The research on the multi-time scale change law of hydrological variables reveals the multi periodic change law of hydrological variables and provides a scientific basis for the rational development of water resources.

  • The non-stationary and nonlinear processing of hydrological variables can avoid spurious regression and make the result more accurate.

  • Study on the co-integration relationship of rainfall, runoff and sediment.

  • Study on multi-time scale dynamic relationship among rainfall, runoff and sediment.

  • Multi-time scale prediction of river runoff provides a technical reference for the effective protection and scientific operation of water resources.

Graphical Abstract

Graphical Abstract
Graphical Abstract

INTRODUCTION

Rainfall, runoff and sediment are important hydrological variables with complex relationships in the source area of the Yellow River. Accurately grasping the changes in these hydrological variables plays a vital role in the change of water resources throughout the Yellow River Basin (Chen & Guo 2016; Wang et al. 2017, 2018a). Besides, rainfall, runoff and sediment analysis are hot issues for scholars at home and abroad (Ramana et al. 2013; Ling et al. 2017; Li & Liu 2018; Wang et al. 2018b). At present, many scholars have conducted a lot of research on the relationship between rainfall and runoff (Nastiti et al. 2018; Tarasova et al. 2018; Chu et al. 2019), and many others have also conducted research on the relationship between runoff and sediment (Hou et al. 2013; Zhang et al. 2014; Wang et al. 2015a, 2015b) and achieved great results. However, the results of combining the three together for research are relatively few.

Runoff forecasting has always been a hot issue in the field of hydrology (Zhang et al. 2017a, 2017b; Zhao et al. 2017). The runoff forecasting with the historical data can not only realize the rational development and utilization of runoff resources but also has significance for the planning, construction and scheduling of water conservancy projects (Xie et al. 2019). At present, the forecast of river runoff often assumes that the time series is stationary. However, because of the influence of climate change, underlying surface and human activities, the statistical characteristics of hydrological time series always change with time. Therefore, most hydrological time series are nonlinear and non-stationary (Zhang et al. 2013a, 2013b). With the development of computer technology, many researchers use soft computing techniques to study hydrological variables with highly accuracy (Rezaie-Balf et al. 2017; Mosavi et al. 2018; Guru & Jha 2019). The common runoff forecast models include artificial neural network (ANN) model (Meng et al. 2015; Sezen & Partal 2019), support vector regression (SVR) model (Yaseen et al. 2018; Wu et al. 2019) and autoregressive moving average (ARMA) model (Wang et al. 2015a, 2015b, 2019). These models with non-stationary time series data will lead to pseudo-regression (Lee & Yu 2009; Jin et al. 2017), so their hydrological element simulation and forecasting are unbelievable.

Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) is an effective method for dealing with nonlinear and non-stationary time series (Torres et al. 2011), which is an improvement on the empirical mode decomposition (EMD) and ensemble empirical mode decomposition (EEMD) (Huang et al. 1998; Wu & Huang 2009). The EMD and EEMD methods are widely used in the fields of hydrology and water resources (Zhang et al. 2013a, 2013b, 2019a, 2019b; Ouyang et al. 2016; Adarsh & Reddy 2018). However, the EMD method presents the mode confusion, and the EEMD method remains a noise residual problem. The CEEMDAN method solves both of these problems (Colominas et al. 2014) and is applied in many fields (Antico et al. 2016; El Bouny et al. 2019).

The co-integration theory was proposed by Engle & Granger (1987), which can deal with the non-stationary problem of time series and reveal the long-term equilibrium and short-term fluctuation between variables. Now, it is widely used in econometrics and the field of hydrology (Yoo 2007; Zhang et al. 2015). Meanwhile, this theory can also be combined with other data analysis methods so as to improve the accuracy of calculation (Zhang et al. 2017a, 2017b, 2019a, 2019b).

The innovation of this paper is to combine the CEEMDAN method with the co-integration theory to construct the three-variable CEEMDAN co-integration error correction model (ECM) for rainfall, runoff and sediment in the source area of the Yellow River of China to forecast the river runoff. The first is to use CEEMDAN to decompose rainfall, runoff and sediment in multi-time scales, and know the changing laws and poly-cycle and obtain the corresponding stationary time series. The second is to reveal the long-term equilibrium and short-term fluctuation relationship of the original and component time series of rainfall, runoff and sediment in the source area of the Yellow River according to the co-integration theory, and to clarify their influencing relationships. The last is to construct two new ECM models of rainfall, runoff and sediment including the CEEMDAN component ECM model and the original sequence ECM model to forecast the river runoff.

STUDY METHODS AND STEPS

CEEMDAN method

The CEEMDAN method is a time-frequency domain analysis method. It can further eliminate the mode effect by adding adaptive noise and has the strong adaptability and better convergence. Usually, the CEEMDAN method is used to deal with nonlinear and non-stationary time series.

The CEEMDAN algorithm steps are as follows:

Step 1: Add Gaussian white noise s(t) to the original signal x(t) and perform I test. The signal of the ith test can be expressed as:
formula
(1)
Step 2: Perform EMD decomposition on the signal xi(t) of the ith white noise addition, and perform an average of I test, and obtain the first intrinsic model function (IMF) component as:
formula
(2)

In the formula, is the first IMF component after EMD decomposition of signal xi(t) with white noise for the ith time.

Step 3: After decomposing to obtain the first IMF component, calculate the difference r1(t) between the original signal and component:
formula
(3)
Step 4: Add white noise again to the difference signal r1(t) and perform I test. Then, the difference signal ri1(t) of the white noise added to the ith time can be expressed as:
formula
(4)
Step 5: Perform EMD decomposition on the ith white noise signal ri1(t), the ith is component, and the second-order IMF component obtained by the I test is:
formula
(5)
Step 6: At this time, the difference obtained by the decomposition is , and steps 4 and 5 are repeated until rn(t) satisfies one of the following conditions: (1) cannot be further decomposed by EMD; (2) meet IMF conditions; (3) the number of local extreme points is less than 3. Finally, the original signal x(t) can be decomposed into n IMF components and a trend term rn(t):
formula
(6)

Co-integration theory

Co-integration concept

Co-integration describes the long-term equilibrium relationship between time series. If a time series is non-stationary but becomes stationary after d-difference, it is called d-order simple integer, which is recorded as I(d). If the time series itself is stationary, it is recorded as I(0). The two time series are defined as and . If meeting the following conditions:

  • (1)

    XitI(d) and YitI(d), (i = 1,2,…, n), d is an integer;

  • (2)

    there is a constant β, which makes YtβXtI(0);

Then, Xt and Yt are co-integrated, and β is called the co-integration vector.

Stationary test

Before the co-integration test, it is necessary to conduct the stationary test of time series, and the commonly used method is the Augmented Dickey–Fuller (ADF) unit root test (Dickey & Fuller 1979). The formula is as follows:
formula
(7)

In the formula, is the first-order difference of variable yt; α, β, δ, ζi are all parameters; t is the time; p is the lag order; εt is the white noise process.

Co-integration test

The E.G. two-step method is a common method for testing the co-integration relationship between time series, which was proposed by Engle & Granger (1987).

The first step of this method is to use the ordinary least square method (OLS) to regress multiple variables and get a residual sequence;

The second step is to test the stationarity of time series with the ADF unit root test on the residual sequence obtained in the first step. If the residual sequence is stationary, it is proved that the variables are co-integrated.

Error correction model

If the time series is co-integrated, an ECM can be constructed. This model describes the long-term equilibrium and short-term fluctuations between variables, and the modeling steps are as follows:

The first step is to perform a co-integration regression to the variables:
formula
(8)
to obtain k0, k1, k2 and the residual sequence ut;
formula
(9)
The second step is to make ecm(−1) = ut−1 as an error correction term and substitute the error correction model:
formula
(10)

In the formula, is a constant term, and are the coefficients of the difference terms of each variable, which reflects the short-term dynamic changes of the model; ecm(−1) is an error correction term, which reflects the degree to which the former term deviates from the long-term equilibrium in short-term fluctuations; φ is the correction coefficient, also called the adjustment speed, usually a negative value; εt is a white noise sequence.

Study steps

By using the CEEMDAN method, the time series of rainfall, runoff and sediment in the source area of the Yellow River are decomposed to obtain IMF component sequences at different time scales. Furthermore, the co-integration theory is used to construct the ECM for the original time series (ECM-OTS) and the CEEMDAN component sequences (ECM-CEEMDAN), and then, the runoff is forecasted by ECM-OTS and ECM-CEEMDAN, respectively. Finally, the runoff forecasted value of each IMF component is reconstructed to get the runoff forecasted value of ECM-CEEMDAN, and the fitting value and forecast accuracy of these two ECM models are compared to draw a conclusion. The flow chart of study steps is shown in Figure 1.

Figure 1

Flow chart of study steps.

Figure 1

Flow chart of study steps.

RESULTS AND CONCLUSION

Data source

The source area of the Yellow River refers to the area above the Tangnaihai hydrological station, which is located in the northeast of the Qinghai Tibet Plateau of China. The geographic coordinates are between 95°50′–103°30′ E and 32°10′–36°05′ N (as shown in Figure 2), the basin area is 122,000 km2, and the average annual runoff is 20.37 billion m3. The water source is mainly supplied by rainfall, followed by glacial snow melting water and groundwater. The change of runoff in the source area of the Yellow River has a vital influence on the change of water resources in the whole Yellow River Basin.

Figure 2

The location map of the source area of the Yellow River.

Figure 2

The location map of the source area of the Yellow River.

The measured rainfall, runoff and sediment time series from 1966 to 2013 at Tangnaihai hydrological station are obtained by the Bureau of Meteorology and the Bureau of hydrology and water resources are shown as in Figure 3.

Figure 3

Time series of rainfall, runoff and sediment at the Tangnaihai station in the source area of the Yellow River.

Figure 3

Time series of rainfall, runoff and sediment at the Tangnaihai station in the source area of the Yellow River.

Table 1 shows the statistical characteristics of rainfall and runoff time series. The mean value is 556.516 mm for rainfall time series, and 203.885 billion m3 for runoff time series, and 1,277.245 × 104 t for sediment time series. The standard deviation of runoff is less than rainfall and sediment. For the coefficient of variation and skewness coefficient, the calculated value of sediment is larger than that of rainfall and runoff.

Table 1

The statistical parameters of rainfall and runoff time series

Time seriesMeanStandard deviationCoefficient of variationSkewness coefficient
Rainfall 556.516 58.464 0.105 0.206 
Runoff 203.885 55.006 0.270 0.659 
Sediment 1277.245 812.960 0.636 1.534 
Time seriesMeanStandard deviationCoefficient of variationSkewness coefficient
Rainfall 556.516 58.464 0.105 0.206 
Runoff 203.885 55.006 0.270 0.659 
Sediment 1277.245 812.960 0.636 1.534 
CEEMDAN decomposition

The CEEMDAN method is used to decompose the time series of rainfall, runoff and sediment in the source area of the Yellow River for multi-time scales. The decomposition results are shown in Figures 46.

Figure 4

The decomposed components of rainfall.

Figure 4

The decomposed components of rainfall.

Figure 5

The decomposed components of runoff.

Figure 5

The decomposed components of runoff.

Figure 6

The decomposed components of sediment.

Figure 6

The decomposed components of sediment.

With the CEEMDAN method, the annual runoff, rainfall and sediment data series at Tangnaihai hydrological station from 1966 to 2013 are decomposed into a fifth-order mode, including four IMF components and one residual. It reflects the multi-time scale evolution characteristics of rainfall, runoff and sediment in the source area of the Yellow River. The IMF1 component of each variable has the shortest period and the highest frequency, and the period of other components gradually gets longer and their frequency gradually decreases. The periodic changes of the component time series are shown in Table 2.

Table 2

Periodic changes of rainfall, runoff and sediment component time series

Component time seriesPeriodic changes (year)/Res changes
RainfallRunoffSediment
IMF1 2–5 2–5 2–5 
IMF2 5–8 6–9 5–10 
IMF3 9–11 29–30 11–30 
IMF4 28 32 41 
Res First reduce and then increase Reduce Reduce 
Component time seriesPeriodic changes (year)/Res changes
RainfallRunoffSediment
IMF1 2–5 2–5 2–5 
IMF2 5–8 6–9 5–10 
IMF3 9–11 29–30 11–30 
IMF4 28 32 41 
Res First reduce and then increase Reduce Reduce 

It can be seen from Table 2 that rainfall, runoff and sediment have four periodic changes. Specifically, rainfall, runoff and sediment all have the same short-period change, and the periodic year is 2–5 years; in the medium period, although the changing periodic years of the three are different, there is little difference, among which the rainfall is 5–8 years, runoff is 6–9 years, and sediment is 5–10 years. There are great differences between rainfall, runoff and sediment in the medium-long period, among which the span of sediment change is large with 11–30 years, 29–30 years for runoff and 9–11 years for rainfall. In terms of the long-period scale, rainfall is 28 years, runoff is 32 years, and sediment is 41 years. The residual component shows the overall nonlinear trend of rainfall, runoff and sediment. The residual component of rainfall showed a decreasing trend from 1966 to 1981, and an increasing trend from 1982 to 2013, but both residual components of runoff and sediment showed a decreasing trend. It can be seen that rainfall, runoff and sediment all have complex multi-time scale periodic change laws, but they have a good correlation in the short and the medium periods. Moreover, runoff and sediment present different periodic changes in the medium-long and long periods, while for their residual components, they show the better synchronization.

Co-integration analysis

Stationary test

The OTS and components of rainfall, runoff and sediment in the source area of the Yellow River are tested by the unit root test. It is assumed that xi, zi and yi (i = 0, 1, 2, 3, 4, 5) are used to represent the CEEMDAN component of rainfall, runoff and sediment, and x0, z0 and y0 are their original sequences, respectively. The optimal lag order is determined by the Akaike information criterion (AIC), and the unit root test results are given in Table 3.

Table 3

Unit root test results of the OTS and components

Time seriesVariablesADF valueTest type (c, t, k)Test critical values
Stationary or not
1%5%10%
The original x0 −0.5138 (0, 0, 3) −2.6186 −1.9485 −1.6121 No 
y0 −0.8657 (0, 0, 3) −2.6186 −1.9485 −1.6121 No 
z0 −0.3647 (0, 0, 3) −2.6186 −1.9485 −1.6121 No 
Δx0 −7.4505 (0, 0, 2) −2.6186 −1.9485 −1.6121 Yes 
Δy0 −6.2861 (0, 0, 2) −2.6186 −1.9485 −1.6121 Yes 
Δz0 −6.5489 (0, 0, 2) −2.6186 −1.9485 −1.6121 Yes 
The IMF1 x1 −7.8693 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
y1 −7.6006 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
z1 −8.3002 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
The IMF2 x2 −10.8127 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
y2 −14.1319 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
z2 −14.2495 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
The IMF3 x3 −14.6845 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
y3 −10.0076 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
z3 −8.6862 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
The IMF4 x4 −26.8800 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
y4 −23.9409 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
z4 −26.7954 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
The residual x5 −20.3586 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
y5 −13.4521 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
z5 −25.1841 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
Time seriesVariablesADF valueTest type (c, t, k)Test critical values
Stationary or not
1%5%10%
The original x0 −0.5138 (0, 0, 3) −2.6186 −1.9485 −1.6121 No 
y0 −0.8657 (0, 0, 3) −2.6186 −1.9485 −1.6121 No 
z0 −0.3647 (0, 0, 3) −2.6186 −1.9485 −1.6121 No 
Δx0 −7.4505 (0, 0, 2) −2.6186 −1.9485 −1.6121 Yes 
Δy0 −6.2861 (0, 0, 2) −2.6186 −1.9485 −1.6121 Yes 
Δz0 −6.5489 (0, 0, 2) −2.6186 −1.9485 −1.6121 Yes 
The IMF1 x1 −7.8693 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
y1 −7.6006 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
z1 −8.3002 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
The IMF2 x2 −10.8127 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
y2 −14.1319 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
z2 −14.2495 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
The IMF3 x3 −14.6845 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
y3 −10.0076 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
z3 −8.6862 (c, 0, 1) −3.5812 −2.9266 −2.6014 Yes 
The IMF4 x4 −26.8800 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
y4 −23.9409 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
z4 −26.7954 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
The residual x5 −20.3586 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
y5 −13.4521 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 
z5 −25.1841 (c, t, 1) −4.1706 −3.5107 −3.1855 Yes 

Note: In the test type (c, t, k), c is the intercept item, t is the time trend term (t = 0 means no trend) and k is the optimal lag length.

The ADF test values of the OTS of rainfall, runoff and sediment in the source area of the Yellow River are all larger than the critical value of t-test, so they belong to non-stationary time series, but their first-order difference time series are stationary. Meanwhile, their CEEMDAN components are stationary.

Co-integration test

The co-integration test is conducted on their OTS and CEEMDAN decomposition sequence of rainfall, runoff and sediment with the E.G. two-step method. The first step is to perform OLS regression on the original sequence and the same time scale component sequence of rainfall, runoff and sediment to establish a co-integration equation. The second step is to perform a unit root test on the residuals of each co-integration equation. If the residual sequence is stationary, the co-integration relationship exists; otherwise, this relationship does not exist. It can be seen from Table 4 that the ADF test values of the residual sequences of all co-integration equations are less than the critical values of the significance levels of 1, 5 and 10%, so the co-integration relationship exists.
formula
(11)
formula
(12)
formula
(13)
formula
(14)
formula
(15)
formula
(16)
Table 4

Unit root test results of residual sequences of co-integration equations

Residual sequencesADF valueTest type (c, t, k)Test critical values
Stationary or not
1%5%10%
u0 −4.6011 (c, 0, 1) −3.6105 −2.9390 −2.6079 Yes 
u1 −6.8268 (c, 0, 1) −3.6105 −2.9390 −2.6079 Yes 
u2 −7.7600 (c, 0, 1) −3.6156 −2.9411 −2.6091 Yes 
u3 −6.1780 (c, 0, 1) −3.6156 −2.9411 −2.6091 Yes 
u4 −8.7148 (c, 0, 1) −3.6156 −2.9411 −2.6091 Yes 
u5 −5.5610 (c, 0, 1) −3.6156 −2.9411 −2.6091 Yes 
Residual sequencesADF valueTest type (c, t, k)Test critical values
Stationary or not
1%5%10%
u0 −4.6011 (c, 0, 1) −3.6105 −2.9390 −2.6079 Yes 
u1 −6.8268 (c, 0, 1) −3.6105 −2.9390 −2.6079 Yes 
u2 −7.7600 (c, 0, 1) −3.6156 −2.9411 −2.6091 Yes 
u3 −6.1780 (c, 0, 1) −3.6156 −2.9411 −2.6091 Yes 
u4 −8.7148 (c, 0, 1) −3.6156 −2.9411 −2.6091 Yes 
u5 −5.5610 (c, 0, 1) −3.6156 −2.9411 −2.6091 Yes 

In the formula, ut represents the residual sequence of the equation, and the data in brackets are the standard deviation of the corresponding coefficient of the equation.

Establishing ECM

According to the ECM method, the ECM-OTS for x0, z0 and y0 and the ECM-CEEMDAN model for xi, zi and yi (i = 1, 2, 3, 4, 5) is as follows:
formula
(17)
formula
(18)
formula
(19)
formula
(20)
formula
(21)
formula
(22)

In the formula, ecmt(−1) represents the error correction term, and the coefficient before ecmt(−1) is the short-period adjustment coefficient, and the coefficient before the difference terms of each variable represents the short-period dynamic change of the model.

It can be seen that the rainfall, runoff and sediment in the source area of the Yellow River show a long-term equilibrium relationship. The component time series also has a long-term equilibrium relationship at different time scales, and the error correction term coefficients of all equations are all negative, which is consistent with the reverse correction mechanism. It can be seen from Equation (17) that runoff is not only affected by rainfall and sediment but also by the deviation of runoff from the equilibrium level in the previous year. The coefficients of Δx0 and Δy0 are 0.23665 and 0.03767, respectively, which indicates that the short-term influence of rainfall and sediment on runoff in the source area of the Yellow River is different, and the influence of rainfall is stronger than that of sediment. The coefficient before ecmt(−1) is −0.70057, which indicates that the deviation of runoff from equilibrium in this year will be adjusted by 70.06% in the next year.

Annual runoff forecast

The ECM-OTS and the ECM-CEEMDAN models of the annual runoff are established by using the measured data series of rainfall, runoff and sediment from 1966 to 2005, and the forecast test is conducted with the measured data series from 2006 to 2013. Figure 6 shows the fitting between the measured value and the fitted value of the two models. Figure 7 shows the relative error between the fitting value and the measured value of the two models. Table 6 shows the forecasted values and relative errors of the two models during the forecast period.

Figure 7

Fitting between the fitted value and measured value of two models.

Figure 7

Fitting between the fitted value and measured value of two models.

It can be seen from Figure 7 that both models can well describe the dynamic equilibrium relationship between rainfall, runoff and sediment in the source area of the Yellow River. Moreover, the accuracy of runoff fitting value of the ECM-CEEMDAN model is better than that of ECM-OTS.

It can be seen from Figure 8 that in the year that the relative error is greater than 20% from 1967 to 2005, the ECM-CEEMDAN model has only one 28.11% in 2002, but the ECM-OTS model has two years, 20.83% in 1997 and 32.17% in 2002. The average relative error of the ECM-CEEMDAN model is 6.21%, which is 1.42% lower than the 7.63% of the ECM-OTS model. It can be seen that the ECM-CEEMDAN model has better fitting accuracy.

Figure 8

Relative error between the fitted value and the measured value of the two models.

Figure 8

Relative error between the fitted value and the measured value of the two models.

According to the Standard for Hydrological Information and Hydrological Forecasting (GB/T 22482-2008) of China, 20% of the measured value is taken as the allowable error for runoff forecasting. When the error is less than the allowable error, it is available. The percent of qualified forecast times and total forecast times is the qualified rate of forecast. Meanwhile, the degree of agreement between the runoff forecasting process and the measured process can be evaluated by the deterministic coefficient, which is calculated as follows:
formula
(23)

In the formula, is the deterministic coefficient, is the measured value, is the forecasted value, is the mean of the measured values, and n is the length of the sequence.

The accuracy of runoff forecast is divided into three grades according to the qualification rate or the deterministic coefficient, as shown in Table 5.

Table 5

Runoff forecast accuracy class table

Accuracy classABC
Pass rate/% QR ≥ 85 85 > QR ≥ 70 70 > QR ≥ 60 
Deterministic coefficient DC > 0.9 0.9 ≥ DC > 0.7 0.7 > DC ≥ 0.5 
Accuracy classABC
Pass rate/% QR ≥ 85 85 > QR ≥ 70 70 > QR ≥ 60 
Deterministic coefficient DC > 0.9 0.9 ≥ DC > 0.7 0.7 > DC ≥ 0.5 

It can be seen from Table 6 that for the ECM-OTS model, in the forecasted 8 years of 2006–2013, only the relative error of runoff in 2009 exceeded 20%, and its forecasted qualified rate was 87.5%, reaching the level A. Meanwhile, for all predicted years, the relative error of runoff forecast that is less than 10% is 5 years, accounting for 62.5%. While for the ECM-CEEMDAN model, its forecasted relative error in 2009 only is 18.81% which is close to 20%. The whole forecasted qualified rate was 100%. Although the ECM-CEEMDAN model has the same as the ECM-OTS model, with the runoff forecast relative error of 10% in 5 years, its relative error value tends to be smaller on the whole, which indicates that the overall forecast accuracy of the ECM-CEEMDAN model is better. Moreover, the average relative error of the ECM-CEEMDAN model is 8.59%, which is 2.7% lower than 11.29% of the ECM-OTS model. This shows that the ECM-CEEMDAN model has a higher forecast accuracy than the ECM-OTS model.

Table 6

The forecasted values and relative errors of the two models during the forecast period

YearMeasured value (108 m3)ECM-OTS model
ECM-CEEMDAN model
Predicted value (108 m3)Relative error (%)Predicted value (108 m3)Relative error (%)
2006 141.26 164.72 16.60 157.08 11.19 
2007 189.04 177.12 6.31 181.34 4.07 
2008 174.60 157.93 9.55 165.21 5.37 
2009 263.48 197.06 25.21 213.92 18.81 
2010 197.08 210.72 6.92 209.32 6.21 
2011 211.21 198.11 6.20 193.63 8.32 
2012 284.04 232.62 18.10 249.97 11.99 
2013 194.64 191.92 1.40 189.24 2.77 
YearMeasured value (108 m3)ECM-OTS model
ECM-CEEMDAN model
Predicted value (108 m3)Relative error (%)Predicted value (108 m3)Relative error (%)
2006 141.26 164.72 16.60 157.08 11.19 
2007 189.04 177.12 6.31 181.34 4.07 
2008 174.60 157.93 9.55 165.21 5.37 
2009 263.48 197.06 25.21 213.92 18.81 
2010 197.08 210.72 6.92 209.32 6.21 
2011 211.21 198.11 6.20 193.63 8.32 
2012 284.04 232.62 18.10 249.97 11.99 
2013 194.64 191.92 1.40 189.24 2.77 

Furthermore, from the deterministic coefficient of runoff forecast, the DC value of the ECM-OTS model is 0.842, which is the level B, while the DC value of the ECM-CEEMDAN model is 0.901, reaching the level A. This shows that the ECM-CEEMDAN model has the higher degree of agreement between the runoff forecasting process and the measured process.

CONCLUSION

  • (1)

    The CEEMDAN method can reveal the periodic characteristics of rainfall, runoff and sediment on the multi-time scales in the source area of the Yellow River. These three variables have a good correlation in the short and the medium periods. In addition, runoff and sediment show a better synchronization in the trend item, which reveals the law of periodic fluctuations of rainfall, runoff and sediment.

  • (2)

    With the co-integration theory and ECM, the ECM-OTS model and the ECM-CEEMDAN model are established. They can reveal the long-term equilibrium and short-term fluctuations of the original sequence and component sequence of rainfall, runoff and sediment in the source area of the Yellow River, and can also effectively forecast the runoff in this source area.

  • (3)

    Both the ECM-OTS model and the ECM-CEEMDAN model can well describe the dynamic equilibrium relationship between rainfall, runoff and sediment in the source area of the Yellow River. However, the forecast period error of the ECM-CEEMDAN model is less than 20%, and its forecast qualified rate can reach 100%, and the accuracy reaches the level A. Compared with the ECM-OTS model, it has the better forecasting accuracy, which provides a new and more accurate runoff forecasting method.

Although the ECM-CEEMDAN of rainfall, runoff and sediment has the higher prediction accuracy, rainfall, runoff and sediment in practice they are often showing the nonlinear relations affected by other many factors, such as underlying surface. More efforts are needed to reveal the nonlinear relations between rainfall, runoff and sediment. Anyway, the combination of the CEEMDAN method and co-integration theory in this paper provides a better analysis method to reveal the internal periodic changes of the hydrological variable, so it can better reflect the actual characteristic of the hydrological variable, which is available for the hydrological forecasting and water resources management.

ACKNOWLEDGEMENTS

This research is supported by the National Key R&D Program of China (Grant No. 2018YFC0406501), Program for Innovative Talents (in Science and Technology) at University of Henan Province (Grant No. 18HASTIT014) and Foundation for University Youth Key Teacher of Henan Province (Grant No. 2017GGJS006).

DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

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