#### Abstract

According to abrupt diagnosis of runoff, two methods, that is, moving approximate entropy and moving permutation entropy, are used to analyse the abrupt year of the daily river runoff from 1961 to 2006 at Lintong station of Wei River in Loess Plateau. The runoff series are divided into 4 stages. With the analysis of hydrological characters of different stages, we find that there are abrupt changes at the three years 1972, 1983, and 2002. The result shows that moving approximate entropy and moving permutation entropy methods are useful tools for abrupt diagnosis of runoff. The attribution of abrupt change at the Lintong runoff series is primarily due to the reduced precipitation, increased water conservancy project, increased water consumption of industry and agriculture, significantly decreased groundwater table, and increased evaporation.

#### 1. Introduction

Runoff is an important hydrological variable in the hydrological cycle. Climate change and human activities have the potential to cause significant changes in runoff behavior, and it is likely that one of the most serious consequences of global warming will be an increased frequency of extreme hydrological events due to changes in runoff patterns [1–3]. This trend is illustrated by the ongoing intensification of various hydrological components of the water cycle [4–6]. A detailed understanding of hydrological series and the factors that affect them is essential for the sustainable exploitation and use of water resources, disaster prevention, and the effective management of water conservation projects. Two types of change can be discerned in hydrological time series: continuous change and abrupt change. In a continuous change the parameter of interest varies smoothly, whereas an abrupt change involves a sudden jump from one value to a much greater or lower value. Methods for the identification and analysis of nonlinear (i.e., discontinuous) variation have been developed by researchers examining meteorological variation. These techniques have had a substantial impact on modern climatology [7], resulting in a much greater emphasis being placed on complexity, interactions between different components of the climate system, and nonlinear factors than was once the case. The identification of abrupt changes requires careful statistical analysis and is highly sensitive to both the time scale over which an event takes place and the length of the analyzed data series.

After the introduction of nonlinear methods, many scholars came to realize that climatic variation is driven by both internal and external forcing mechanisms, resulting in the development of dynamic techniques for the analysis of climate change. For example, a recent study on five temperature series gathered from meteorological stations within the Yangtze Delta used the concept of conditional entropy to test the commonality of the series’ underlying dynamics [8]. Similarly, GIS data were used in conjunction with information entropy analysis to analyze the spatial and temporal variation in precipitation around the Chaobai River basin [9]. Pincus and coworkers developed the concept of approximate entropy as a tool for analyzing time series and studying system complexity [10]. This approach has been applied to identify abrupt climate change events and mutations in the series’ structural dynamics [11]. Weiping et al. used approximate entropy analysis to investigate the spatial and temporal variation in climate data. Their studies demonstrated that approximate entropy analysis is an effective tool for highlighting structural changes in a system’s dynamics [12]. Hou Wei and colleagues introduced the concept of permutation entropy and used it to analyze the variation in temperature data series for Northern China. By applying empirical mode decomposition to their results, they were able to demonstrate a close relationship between abrupt temperature changes and sunspot activity [13]. In this work, we used a similar approach to analyze the variation in runoff series for the Wei River basin.

#### 2. Methodology

##### 2.1. The Moving Approximate Entropy Test for Identifying Abrupt Changes

Some significant abrupt diagnosis methods of the basic concepts of probability and correlation analyses that are applicable in hydrologic engineering [14] are based on the linear theory and work with the system characteristics such as mean value and trend analysis. In physical sciences entropy relates macroscopic and microscopic aspects of nature and determines the behaviour of macroscopic systems in equilibrium [15]. Since the development of the entropy theory by Shannon in the late 1940s and the principle of maximum entropy (POME) by Jaynes in the late 1950s, there has been a proliferation of applications of entropy in a wide spectrum of hydrology [16, 17]. Approximate entropy was used as a diagnostic and it showed that *ApEn* had good computational efficiency and high robustness in characterizing the severity of structural defect [18]. Permutation entropy is useful in the presence of dynamical or observational noise [19]. These two methods are based on entropy theory, and they diagnose the abrupt point with the inside structure of the time series by using system control equation.

Approximate entropy (*ApEn*) analysis is a method for evaluating dynamic variation in data series that is based on entropy theory. It is conceptually straightforward and can be calculated quickly. Moreover, it is not highly sensitive to noise in the analyzed data set and can reliably provide an overview of the studied system’s properties. This section describes how the approach was adapted to analyze dynamic structural variation in runoff time series. The moving approximate entropy (M-ApEn) is a nonnegative scalar that is used to represent the complexity of a time series; the greater its value, the greater the complexity of the data set. For any time series of the form together with an initial dimensionality of and a tolerance threshold , the approximate entropy can be computed as follows.(1)Construct a set of *m*-dimensional vectors:
(2)Calculate the Euclidean distance between the vectors and :
(3)Determine the number of values of for which is less than* r*; this number is denoted by ;
(4)Compute the average value of the logarithm of for all values of ; the resulting term is denoted by :
(5)Increase *m* by 1 and repeat steps to .(6)By performing steps to iteratively, one obtains the approximate entropy time series:

Steps to reconstruct the -dimensional sequence based on the degree of similarity between two points in terms of their values and position within the vector. Increases in the value of indicate an increased likelihood that there has been a shift to a new mean approximate entropy; the greater the increase, the greater the likelihood that there is a corresponding abrupt change in the underlying data set, and hence the greater its complexity. Differences in the *ApEn* value at different points in time can thus be used to keep the ability of self-similar state and divide the period covered by a data set into separate phases separated by abrupt transitions. The quantity computed in step is the approximate entropy and is used in conjunction with values of and that can be obtained from the literature to compute the moving *ApEn*. In this work, the following values were selected for the moving approximate entropy analysis parameters: , , days, and days. The value was taken to be the standard deviation of the original sequence, .

##### 2.2. The Moving Permutation Entropy Test for Detecting Abrupt Changes

Permutation entropy is a metric that is based on entropy theory and can be used to characterize dynamic variation. In this work, a moving average of the permutation entropy was computed, to give a quantity that was termed the moving permutation entropy (M-PE) and was used to analyze the dynamic structural variation in the Lintong runoff series. The permutation entropy provides a measure of the complexity of a dataset and is computed using the following algorithm.

Given a one-dimensional hydrological time series , for any , phase space reconstruction will produce a one-dimensional vector:

Here, and represent the embedding dimension and the delay time, respectively.

If the th reconstruction components of , that is, are sorted in order of size, one obtains the following:

In the sorting process, if , any vector can be used to generate a sequence of the following form by means of sorting:

For each group, there are a total of Different permutation of the symbol sequence , and denotes one of these sequences. Probability analysis can be used to associate each sequence with a set of probabilities for the group: .

According to Shannon, information entropy can be calculated for such a set of probabilities, , as follows:

Theoretically, when , reaches a maximum of . In practice, is usually normalized against :

HP is the PE value, and its size reflects the degree of randomness in the studied time series. The lower the value of HP, the more regular (i.e., the less complex) the sequence. Conversely, the greater the HP value, the more random (i.e., complex) the sequence.

At present, there is no established method for selecting an appropriate subsequence length () or embedding dimension (). The influence of the subsequence length and the value on the outcome of the analysis has been discussed at length in the literature [19, 20]. It is known that if an excessively low subsequence length is used, the identification of abrupt changes in the time series becomes unreliable, and the reconfigurable vector containing too little information will make the PE algorithm invalid. Based on the literature results [13], the following values for these crucial parameters were adopted in the analyses presented below: a subsequence of length () of 5 years, a sliding step length of 1 day, and an embedding dimensionality () of 5.

#### 3. Study Area

The Wei River is the largest tributary of the Yellow River. Its origin is in Niaoshu Hill in Weiyuan County of Gansu province. The Wei River basin covers parts of the Gansu, Ningxia, and Shaanxi provinces, and it joins the Yellow River in Tongguan County of Shaanxi province (Figure 1). The basin’s catchment area is 134800 km^{2}, and the river runs for 818 km. Its upper reaches, which account for around 70% of its total length, are primarily located in hilly Loess regions, at elevations of 2400 to 1200 m. The lower 10% of its length lies in river valleys at elevations of 1700 m–900 m. The midreaches of the Wei River lie in the northern Loess Plateau of northern Shaanxi province, at elevations of 2000–900 m, and the river plays a central role in Loess deposition. It joins the Yellow River on the alluvial plains of the Guanzhong basin, whose eastern regions have an elevation of 700 m–800 m that falls to 500–320 m on moving westwards. The Qin mountain range, which has many peaks above 2000 m, is located to the south of the Wei River basin. The river’s basin covers both arid and humid regions, with most of the precipitation falling in the north; the mountainous regions and the valleys receive comparatively little rainfall. The majority of the area’s precipitation occurs in July and August, with December and January being the driest months of the year. The period between July and October typically accounts for around 60% of the area’s total annual precipitation. The basin has a continental monsoon climate, with the total rainfall and temperature gradually decreasing on moving from the southeast to the northwest. The average annual evaporation from wet surfaces in the basin is 660 mm–1600 mm; the rate of evaporation is lowest in December and highest in June or July. The period between July and October accounts for 46%–58% of the region’s total annual evaporation. The Wei River basin has many tributaries, most of which join the river on its south bank. However, the larger tributaries generally join the main river through the north bank, producing a fan-shaped basin. The basin has 14 tributaries and a catchment area of more than 100,000 km^{2}. Most of the tributaries on the north bank originate in the Loess Hills and the Loess Plateau. They are generally longer than the southern tributaries, flow over shallower slopes, and have greater sediment loads. The tributaries on the south bank originate in the Qin mountains. They tend to be short and fast flowing, with steep slopes, high runoff volumes, and comparatively low sediment concentrations. The Lintong station is located in the river’s downstream region and covers a drainage area of 97,300 km^{2}, which corresponds to 88.8% of the river’s total drainage area (excluding the Beiluohe basin).

#### 4. Data

Daily streamflow data acquired at the Lintong hydrological station (34°26′N, 109°12′E) during the period from 1961 to 2006 were obtained from the Bureau of Hydrology and Water Resources of Shaanxi province. The first measurements in the data series were acquired in January 1961. Lintong station is located downstream of the Wei River with drainage area of 97299 km^{2}. The locations of the hydrological stations in the Wei River Basin are shown in Figure 2. As seen from Figure 3, annual runoff stays at an almost constant level after some sudden upward jumps are recorded. The maximum annual runoff corresponding to differences is 158.21 billon m^{3}. The minimum and maximum were recorded as 19.61 billion m^{3} and 1.82 billon m^{3}. During observation period mean runoff is 6.59 billon m^{3}.

The hydrological variables considered in this work were the mean annual flow, the mean monthly flow, the maximum annual flow, the minimum annual flow, the timing of hydrologic events, maximum, and minimum flows on a daily, 3-day, 7-day, 30-day, and 90-day basis. These variables were selected because they are widely used in comparing flows and engineering applications. Variables associated with minimum flow were evaluated over four separate periods, that is, 1961–1971, 1972–1982, 1983–2001, and 2002–2006.

#### 5. Analysis and Result

##### 5.1. Results of the Moving Approximate Entropy Test

The M-ApEn method was used to detect abrupt changes in the Lintong station daily runoff time series. In this analysis, the value of was chosen to be 2, the allowed deviation () was = 0.15, the moving step length *L* was 365 d, and the subsequence length was also 365 d. Time points at which the difference between the *ApEn* value and the moving average *ApEn* was greater than 20% of the moving average were characterized as change points; that is, points at which an abrupt change had occurred. The calculated *ApEn* values for the Lintong Station daily runoff time series are shown in Figure 4. The *ApEn* value changes are clearly shown in Figure 1. The results show the four stages of abrupt change; appeared in 1972, 1983, and 2002.(1)The first change point occurs in 1972, at which the difference between the *ApEn* value and the moving average is 22.9% of the moving average.(2)The second change point occurs in 1983, at which the difference between the *ApEn* value and the moving average is 24.2% of the moving average.(3)The third change point occurs in 2002, at which the difference between the *ApEn* value and the moving average is 34.1% of the moving average.

##### 5.2. Results of the Moving Permutation Entropy Test

The M-PE method was used to analyze the daily runoff time series for the Lintong Station and to identify abrupt changes in the daily runoff volume. The parameter values used in the analysis were as follows: , lag time = 1 d, , and a. The results obtained are shown in Figure 5. Inspection of this figure clearly shows that abrupt changes in the PE value occur in the early nineteen seventies and at the beginning of the twenty-first century.

The points of abrupt change in daily runoff volume identified using the M-ApEn method are 1972, 1983, and 2002. Using the M-PE method, abrupt changes were detected in the early 1970s and at the beginning of the 21st century. Considering the results of two methods, the abrupt changes are 1972, 1983, and 2002.

#### 6. Discussion and Conclusions

##### 6.1. Results of Hydrological Characteristics

Based on the results obtained in the M-ApEn and M-PE analyses, the runoff time series can be divided into four separate periods: 1961–1971, 1972–1982, 1983–2001, and 2002–2006.

For each of these periods, the values of selected hydrological variables measured at the Lintong Station were compared to those in the preceding period in an attempt to determine whether the abrupt changes detected by analyzing the daily runoff volume were associated with changes in other hydrological properties. The following noteworthy changes were identified.(1)The mean annual runoff volume during the second period was 6.071 billion m^{3}, compared to 8.74 billion m^{3} during the first period, representing a decrease of 31%. The mean annual runoff in the third period was 5.515 billion m^{3}, which represents a decrease of 9% relative to the second period. The mean annual runoff in the fourth period was 6.057 billion m^{3}, which corresponds to an increase of 10% relative to the third period. Overall, the most pronounced change in the mean annual runoff volume occurred between 1961–1971 and 1972–1982, corresponding to the abrupt change identified in 1972 in the M-ApEn analysis and in the early 1970s in the M-PE analysis. The results of the annual daily runoff volume measured at the Lintong station in the Wei River basin are shown in Figure 6.(2)During the first period (1961–1971), the total runoff volumes in all months other than August were lower than those in the same months during the second period (1972–1982). For the months of March, April, May, and June, the total runoff volumes during the second period were less than half those measured during the first period. The monthly runoff volumes for the third period were generally greater than those for the second period. This was especially true in June, for which the mean runoff volume during the third period was 162% of that for the second period. The June runoff volume increased again during the fourth period, when it was 111% greater than that for the third period. The results of total monthly runoff volumes measured at the Lintong station in the Wei River basin are shown in Table 1.(3)The maximum and minimum runoff volumes during the second period were generally lower than those during the first period; while max1 increased very slightly, the other maximum and minimum values fell by between 3% and 66%. There were very substantial changes between the second and third periods: the maximum values for the third period were all slightly lower than those for the second, whereas the minimum runoff volumes were all substantially greater. All of the maximum and minimum runoff volumes measured during the fourth period were greater than the corresponding values for the third period; the greatest change occurred for Min3, which increased by 44%. The results of minimum and maximum runoff volumes calculated over periods of 1, 3, 7, 30, and 90 days for the Lintong station in the Wei River basin are shown in Table 2.

##### 6.2. Attribution of Abrupt Changes

The variation of runoff series is primarily due to climate change (reductions in precipitation) and increases in human activity. The present study shows that the precipitation changes are strongly impacted with the climate [15]; the climate strongly controlled the runoff conditions, and human activities (as water conservancy and water and soil conservation) are closely related to the runoff [21].

###### 6.2.1. Reduced Precipitation

Inspection of Table 3 clearly shows that, compared with the 1970s, the average annual precipitation is all reduced in every stages. The mean annual precipitation volume during the 1972~1982 period was 340.75 mm, compared to 359.95 mm during the 1960~1971 period, representing a decrease of 5%. The mean annual precipitation in the 1983~2001 period was 31.79 mm, which represents a decrease of 9% relative to the first period. The mean annual precipitation in the 1983~2001 period was 10.57 mm, which represents a decrease of 3% relative to the first period.

For 1960~1971, 1972~1982, and 1983~2001 periods, The mean annual precipitation volumes were all decreased compared to those in the preceding period. The rate of the decrease from 1983 to 2001 was 3.70% which is smaller than before, and the rate of 2002~2006 represents an increase of 6.47% relative to 1983–2001. Hence, the precipitation variation is one of the most important factors for runoff change.

###### 6.2.2. Increased Water Conservancy

A large number of water conservancy projects have been built in the early 1970s, such as Yangmaowan Reservoir (1970, with a total storage capacity of 120 million m^{3}), Shihmen Reservoir (1970, with a total storage capacity of 120 million m^{3}), Lin Gao Reservoir (1971, with a total capacity of 0.33 million m^{3}) and some irrigation districts as Hengshui River irrigation (effective irrigation area of 9407 hm^{2}), Baojixia irrigation (effective irrigation area of 188,553 hm^{2}), Wool Bay irrigation district (effective irrigation area of 16,000 hm^{2}), of Lin Gao Irrigation (effective irrigation area of 5733) The mean annual runoff in 1972–1982 was 6.07 billion m^{3}, which represents a decrease of 2.67 billion m^{3} relative to 1960–1971 period. The human activities are significantly impacting factor on runoff.

A large number of water conservancy projects were built in the early 1980s, some reservoirs as Shitou River Reservoir (1981, with a total capacity of 147 million m^{3}), Taoqupo reservoir (1980, with a total capacity of 0.57 million m^{3}) and some irrigation districts as Dongfanghong irrigation (effective irrigation area of 55800 hm^{2}) The mean annual runoff in 1983–2001 was 5.51 billion m^{3}, which represents a decrease of 0.56 billion m^{3} relative to 1972–1983 period.

The cumulative effects of human activities after 1990s, such as industrial and agricultural water consumption, soil and water conservation water consumption and evaporation continue to increase, Groundwater level and the water consumption of the national economy were significantly decreased. The mean annual runoff in 2002–2006 was 6.06 billion m^{3}, which represents an increase of 0.54 billion m^{3} relative to 1983–2001 period.

##### 6.3. Conclusion

Two methods for the identification of abrupt changes in time series—moving approximate entropy analysis and moving permutation entropy analysis—were used to study the variation of 46-year daily runoff series at the Lintong station in the Wei River basin. Both analyses indicated that abrupt changes occurred in the early 1970s (1972) and at the start of the 21st century (2002); in addition, the moving approximate entropy analysis revealed a third abrupt change that is estimated to have occurred in 1983. These so-called change points were used to separate the 46-year period covered by the runoff volume data set into four separate periods, and the mean values of selected hydrological variables within these periods were compared to evaluate the impact of these abrupt changes. Notable differences between the periods were identified in terms of the total annual runoff, total monthly runoff volumes, and the maximum and minimum runoff volumes over selected periods of time. The changes associated with the transition between the first and second periods in 1972 were especially pronounced. Overall, the observed changes in the selected hydrological variables are consistent with the abrupt changes identified using the two new analytical methods, which suggests that both of the new methods are useful for analyzing runoff series and identifying points of abrupt change. The attribution of abrupt change at the Lintong runoff series is primarily due to the reduced precipitation, increased human activities, increased water consumption of industry and agriculture, significantly decreased groundwater table, and increased evaporation.

#### Acknowledgments

Natural Science Foundation of China (no. 51190093, 51179148, 51179149), National Key Basic Research 973 of China (no. 2012CB417003), Governmental public industry research special funds for projects (no. 201101043), and Program for New Century Excellent Talents in University. Constructive comments from reviewers are gratefully acknowledged.