#### Abstract

Climate change and human activities are two major driving factors for variations in hydrological patterns globally, and it is of significant importance to distinguish their effects on the change of hydrological regime in order to formulate robust water management strategies. Hilbert-Huang transform-based time-frequency analysis is employed in this study to detect abrupt changes and periods of the runoff at five hydrological stations in the Weihe River Basin, China, from 1951 to 2010. The key part of the method is the empirical decomposition mode with which any complicated data set can be decomposed into small number of intrinsic mode functions that admit well adaptive Hilbert transforms. Moreover, an attempt has been made to find out the specific reason for the abrupt point at the five hydrological stations in the Weihe River Basin. The results are presented as follows: (1) annual runoff significantly declined in the basin in intervals of 8~15 years; (2) abrupt changes occurred in 1971, 1982, and 1994 at Huaxian, 1972 and 1982 at Xianyang, 1992 at Zhangjiashan, 1990 at Zhuangtou, and 1984 at Beidao; (3) changes were more frequent and complex in the mainstream and downstream reaches than in tributaries and upstream reaches, respectively.

#### 1. Introduction

Global climate changes and human activities are strongly affecting the spatial and temporal patterns of river runoff and other key hydrological variables [1–3]. Clearly, it is important to characterize these changes and identify the causal factors in order to formulate appropriate water management strategies [4, 5]. Thus, numerous authors have examined these changes in diverse regions using various methods. Notably, Richter et al. [6] studied changes in runoff of the Colorado River employing a method called the Range of Variability Approach (RVA), intended for application in cases where the conservation of native aquatic biodiversity and protection of natural ecosystem functions are primary river management objectives. In a subsequent contribution, Richter et al. [7] demonstrated the utility of the approach for assessing hydrological changes at sites throughout a river basin. Perreault et al. [8] used a Bayesian approach to detect changes in annual energy inflows of eight hydropower systems in Québec and pointed out the (often neglected) need to consider hydrological variations for robust forecasting. Burn and Hag Elnur [9] investigated 18 hydrological variables reflecting various components of hydrological cycles in a network of 248 Canadian catchments selected to reflect natural conditions, using the Mann-Kendall nonparametric test to detect abrupt changes and trends in runoff. They observed more significant trends than expected to occur by chance, with variations in geographic location implying that impacts of the causal factors are not spatially uniform. Similarly, Bao et al. [10] used the Mann-Kendall test to detect abrupt changes in runoff in the Haihe and Kaidu River Basins of China, respectively. They then quantitatively distinguished effects of climate variability and human activities on runoff, which (as mentioned) is essential for formulating appropriate regional water resources assessment and management regimes.

In addition to Mann-Kendall and Bayesian approaches, the Pettitt test [11] and two-stage linear regression [12, 13] have also been used to detect changes in runoff series. However, all of these methods have severe limitations for handling highly complex nonlinear systems; thus, various nonlinear analysis methods have been introduced recently. These methods include rescaled range analysis [14] and approximate entropy [15] techniques, which can solve the problems of nonstationarity and nonlinearity, but the window size affects the results. Wavelet transform analysis can also effectively detect points of change [16–18], but false points may be detected if an inappropriate wavelet function is selected, so other diagnostic tools must also be applied.

In the late 1990s, NASA (the US National Aeronautics and Space Administration) proposed an alternative method of time series analysis [19], the Hilbert-Huang transform (HHT), which combines empirical mode decomposition (EMD) and the Hilbert transform [20]. EMD decomposes complex signals of local characteristics into a series of finitely intrinsic mode functions (IMFs), which are subjected to the Hilbert transform to detect changes in their amplitude and frequency over time, yielding three-dimensional (time-amplitude-frequency) spectra. Since changes over time in the amplitude and frequency of the focal signals are captured, no physically significant harmonic is eliminated which reflects nonlinear and nonstationary processes [21]. HHT incorporates the advantages of wavelet analysis but also provides higher resolution [22]. The EMD method is adaptive to changes in local signals; thus it is highly suitable for processing complex nonlinear and nonstationary signals [23]. Hence, the HHT approach is widely used to process diverse signals in biomedical, chemical engineering, meteorological, seismic, and speech recognition applications [24–26]. This paper serves as an introduction to the HHT runoff time series analysis. Thus, the method is applied to detect abrupt changes in time series of runoff data collected at five hydrological stations (Beidao, Xianyang, Huaxian, Zhangjiashan, and Zhuangtou) in the Weihe River Basin from 1951 to 2010.

#### 2. Study Area and Data

The Weihe River originates in Weiyuan County, Gansu Province, northwest China, and is the largest tributary of the Yellow River. It passes through several major cities, including Tianshui, Xianyang, and Weinan, situated in the southern margins of the Loess Plateau, before discharging into the Yellow River at Tongguan. The Weihe Basin covers an area of 134,800 km^{2}, extending between longitudes 104°00′E and 110°20′E and latitudes 33°50′N and 37°18′N, the stream length is 818 km, and the Qinling Mountains mark its southern borders (Figure 1). The basin is in the continental monsoon climate zone, with cold and dry winters as well as hot and wet summers governed by the Mongolian and West Pacific subtropical high pressure systems, respectively.

This study focuses on detecting changes of annual runoff at the five hydrological stations which are Beidao, Huaxian, Zhuangtou, Zhangjiashan, and Xianyang in Weihe River of Northeastern China. The data analyzed here are annual average runoff data collected at five hydrological stations from 1951 to 2010 archived by the Weihe River Hydrological Bureau. Three of these stations (Beidao, Xianyang, and Huaxian) are located in the mainstream reaches, while the other two (Zhangjiashan and Zhuangtou) are in the largest and second largest tributaries of the Weihe River; the River Jing is 455 km long with basin areas of 454000 km^{2}, accounting for 33.7% of the River Weihe Basin, whereas River Beiluo is 680 km long with basin areas of 26,900 km^{2} accounting for 20% of the River Weihe Basin (Figure 1).

#### 3. Methods

##### 3.1. Empirical Mode Decomposition (EMD)

EMD [19] sequentially decomposes time series of data into a series of IMFs, with smoothing to eliminate riding waves. Each IMF must have the same number of extreme values and zero crossings, with its envelopes being symmetric with respect to zero. The first condition is similar to the traditional narrowband requirements for a stationary Gaussian process, whereas the second condition guarantees meaningful instantaneous frequencies by turning global into local limits. EMD can decompose any complex signal into finite numbers of IMFs, as follows.

Suppose is original time series; is upper envelope of ; is lower envelope of ; is the mean of upper envelope and lower envelope.

First, local maximum and minimum values of the original time series are calculated to acquire upper and lower envelope sets: . If , meets the two conditions of an IMF. If not, local maximum and minimum values are recalculated until meets the two IMF conditions, is the first component of the IMF, and the residual part of the sequence is .

Finally, these steps are applied to the residual component until it forms a monotonic sequence. The standard deviation (SD) of the IMF is

in formula (1) is the component of the IMF. When the SD is ≤0.3, the IMF is deemed to have sufficient stability for robust physical interpretation.

##### 3.2. Hilbert Transformation

is the sequence of IMF. The Hilbert transform of is defined as Here, is the Cauchy principal value and is the Hilbert transform of . According to (2), when and are complex conjugates, the following analytic signal is obtained:

Here, is the instantaneous amplitude and is the phase, where

And the corresponding instantaneous frequency is [26]

The instantaneous frequency is a single-valued function of time, and the time series of the Hilbert transform must be a single component to make the instantaneous frequency physically meaningful. When we need to predict runoff, component is ignored and the runoff sequence can be reconstructed as follows:

Here, is the time and is imaginary unit. Re is the real part of the complex.

The joint Hilbert-Huang spectrum is defined as

The joint spectrum is a full time-frequency distribution that can accurately reflect the amplitude of the variation across the band with the frequency and time.

In order to analyze the runoff series in greater detail and verify the findings, the instantaneous frequencies and Hilbert-Huang spectra were examined.

##### 3.3. Detecting Variant Points

Points of abrupt changes in a focal time series (“mutation points”) are detected as follows. First, for the decomposition of the IMF components, correlation coefficients between extreme values of each IMF component and the original sequence are calculated, and then the differences in amplitude and intervals of adjacent maxima and minima are calculated. Abrupt changes are located at positions of maximum absolute difference between the extrema and the minimum intervals between them. In other words, maxima of the first-order differentials for each IMF component indicate points of significant change (usually changes in the frequency and/or amplitude of runoff in this context). In order to eliminate influence of different station and different IMF, the absolute first-order differentials were standardized. Frequency variations are reflected in changes in the characteristic scales, while amplitude variations are reflected in changes in peak values of the HHT spectra. Thus, changes in peaks of the instantaneous frequency and the HHT spectra can pinpoint locations of mutation points, as outlined in the Results.

#### 4. Results

##### 4.1. Detection of Abrupt Changes

Positions of abrupt changes (mutation points) in the runoff time series are detected by the following: Decomposing the original signal by EMD, seeking IMF components (Figure 2) in accordance with formula (1). Detecting the extrema in the sequence of IMF components. Calculating differences in the amplitudes and intervals of adjacent maxima and minima and standardizing absolute of differences. Identifying mutation points as positions where differences in extrema are maximal and differences in intervals are minimal. That is the maximum value of standardized coefficients (Figure 3).

**(a) EMD coefficients of runoff at Beidao**

**(b) EMD coefficients of runoff at Huaxian**

**(c) EMD coefficients of runoff at Zhuangtou**

**(d) EMD coefficients of runoff at Zhangjiashan**

**(e) EMD coefficients of runoff at Xianyang**

**(a) Standardization absolute of IMF1 in Beidao**

**(b) Standardization absolute of IMF1 in Huaxian**

**(c) Standardization absolute of IMF1 in Zhuangtou**

**(d) Standardization absolute of IMF1 in Zhangjiashan**

**(e) Standardization absolute of IMF1 in Xianyang**

The correlation coefficients (Table 1) for IMF1 and IMF2 are greater than 0.49, which exceeds the 95% significance level. The years when differentials of the amplitude of IMF1 were maximal at five stations are shown in Figure 3.

As shown in Figure 3(b), there were abrupt changes in runoff at Huaxian in 1971, 1982, and 1994. Abrupt changes were also detected in 1984 at Beidao (Figure 3(a)), 1991 at Zhuangtou (Figure 3(c)), 1994 at Zhangjiashan (Figure 3(d)), and 1972 and 1982 at Xianyang (Figure 3(e)).

Hilbert transforms of each IMF component of the runoff at Huaxian, and the average frequencies and amplitudes obtained (from the arithmetic means of the frequencies and amplitudes during the averaging periods, resp.), are shown in Table 2.

It can be seen from Figure 2 and Table 2 that the amplitude and frequency of IMF1 to IMF4 at Huaxian successively declined, while the wavelength successively increased. Cycles in the runoff at this station with 8.21, 15.38, 25.64, and 32.25 periods, significant quasi 8~15 yearly oscillations, and a declining trend were detected. Significant changes in the Hilbert-Huang spectral peaks occurred in 1971, 1982, and 1994, verifying the previous findings. Results of the analyses of runoff changes at Beidao, Xianyang, Zhangjiashan, and Zhuangtou were verified using the same methodology.

##### 4.2. The Amplitude Variation of Average Runoff

In order to further explain the occurrence of the abrupt change points, the amplitude variation of average runoff is calculated, as shown in Figure 4 and Table 3.

The data in Figure 4 and Table 3 show that average runoff has a significant decrease in Beidao, Zhangjiashan, and Zhuangtou, 53.4%, 69.5%, and 41.4%, respectively. Meanwhile, there are periodic changes in Huaxian and Xianyang in the study period. The average runoff of Huaxian was 9.161 billion m^{3} before 1971 but decreased to 5.895 billion m^{3} during the period of 1972 to 1982, increased to 7.262 billion m^{3} from 1983 to 1994, and dropped to 4.015 billion m^{3} in the past several years. The average runoff of Xianyang is 5.509 billion m^{3} before the first abrupt change point but decreased to 3.032 billion m^{3} from 1972 to 1982 and has increased to 3.357 billion m^{3} since 1983. Its corresponding variation amplitude is 53.4%, 69.5%, and 41.4%, respectively.

##### 4.3. Significance Test of the Mutation Points

Since annual mean runoff series spanning periods with significant mutation points should significantly differ from a white noise series (Gaussian distribution with mean 0 and variance 0.01), we compared the series examined here to computer-generated white noise to test the significance of the mutation points. As shown in Figure 5, the results show that confidence levels were ≥99% for one point (in 1982) and ≥95% for the other two points (in 1971 and 1994). The abrupt changes in runoff detected at the other hydrological stations (Beidao, Xianyang, Zhangjiashan, and Zhuangtou) also met 95% confidence criteria.

#### 5. Discussion

Hydrological cycles and water resources are strongly influenced by climatic factors, such as El Niño events and global climate changes, and human activities, such as large-scale water conservation constructions and ecological restoration measures [10, 27, 28]. The Weihe River Basin is in the continental monsoon climate zone of China and thus is affected by the West Pacific subtropical high pressure system. In 1972, weakness of this system in combination with development of an El Niño event caused substantial atmospheric circulation and climate anomalies in China [29], leading to a sharp fall in precipitation in the Weihe River Basin, thus contributing to the mutation point detected in that year. In 1982, another El Niño event occurred (the strongest warming event in the 20th century), which had a substantial climatic impact on China during the summer, including another major turning point in precipitation in the basin. In 1994, a third ENSO event occurred, while the West Pacific subtropical high pressure system was strong, resulting in a further significant drop (14.9%) in precipitation in the basin [30], which substantially contributed to a 42.4% reduction in average annual runoff.

However, human activities have also markedly influenced runoff patterns. Notably, the construction of terraces, reservoirs, and irrigation canals (e.g., the Dongfanghong pumping and Wei irrigation installations constructed in 1970 and the Baojixia canal completed in 1971), and other water conservancy measures, reduced the efficiency and runoff of watershed source area. Completion of the Fengjiashan reservoir contributed to anomalous conditions in 1982, and continuous increases in industrial and agricultural water consumption, combined with climate change, contributed to both the abrupt change of runoff in 1994 and its tendency to decline. Similarly, increases in the area irrigated by the Jinghui and Huiluo canals, fed by the Jinghe and Beiluohe Rivers, respectively, contributed to both abrupt changes in runoff and the significantly declining trend detected at Zhangjiashan and Zhuangtou [27, 28]. In addition, other reference information [29, 30] of Wei River has shown that there is a certain relationship of precipitation, evaporation, and runoff. Since the 1960s, Wei River annual precipitation has been decreasing in Wei River Basin and an abrupt change of precipitation in 1982 and a sharp variation of evaporation occurred in 1993.

#### 6. Conclusions

This study applied HHT method to analyze the periodicity, trends, and abrupt change point of the annual runoff in the Weihe region. The results can be summarized as follows:(1)The instantaneous frequencies and IMFs, calculated as described above, show multitimescale features with significant 8~15-year periods. Significant () decreasing trends in annual runoff throughout the study period were detected at four hydrological stations at Beidao, Huaxian, Zhangjiashan, and Xianyang but at Zhuangtou it increased before 1968 and then decreased. Thus, annual runoff significantly declined throughout the Weihe River Basin.(2)During the period of records, the Weihe River Basin has been becoming drier. Meantime, the local human activities have become more and more extensive. Because of climate changes and a series of water conservancy measures, there are 3 changes of Weihe River runoff in Huaxian station (1971, 1982, and 1994), two in Xianyang station (1972 and 1982), and one in Zhangjiashan station (1992), Zhuangtou station (1991), and Beidao station (1984). That is, one mutation point is detected in the tributary, two mutation points are detected in the upstream reaches, and three mutation points are detected in the central mainstream and downstream station. These findings indicate that changes were more frequent and complex in the mainstream and downstream reaches than in tributaries and upstream reaches, respectively, presumably because the mainstream reaches are affected by changes in all of their tributaries and all points upstream.(3)The reductions of average annual runoff in Beidao (in the upstream reaches), Zhangjiashan, and Zhuangtou (in the tributary) are 53.4%, 69.5%, 41.4%, respectively. The study period can be divided into four stages by three abrupt change points in Huaxian (in the mainstream reaches and downstream reaches), and the amplitude variation of average runoff is 35.7%, 23.2%, and 44.7%, respectively. The period 1951–2010 can be divided into three stages, and the amplitude variation of average runoff is 45.0% and 10.7% in Xianyang (in the midstream reaches).(4)ENSO events, variations in West Pacific subtropical high pressure systems, and various identified human activities appear to have contributed to the changes in runoff. However, sunspot activity patterns, other climatic factors, and other human activities may also have contributed. Thus, further research is required to elucidate the causal factors comprehensively.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research was supported by the Natural Science Foundation of China (51190093, 51179149), the Ministry of Education in the New Century Talents Program (NCET-10-0933), and Key Innovation Group of Science and Technology of Shanxi (2012KCT-10).