The main goal of this study is to introduce the Archimedean copulas, which overcome the low accuracy and subjective nature of the traditional double mass curve method, to investigate the precipitation–runoff relationship (PRR) and detect change points in the Weihe River Basin (WRB). With the construction of a joint distribution between precipitation and runoff by the Archimedean copulas, a statistical variable considering the distribution parameter was estimated to judge the change point of the PRR. The results show that: (1) annual precipitation and runoff present decreasing trends that are significant and insignificant, respectively, at the 95% significance level, while annual potential evapotranspiration (PET) increases slightly; (2) change points of the PRR occurred in 1971 and 1994; (3) the annual runoff changed more dramatically than precipitation during the periods from 1972 to 1994 and 1995 to 2010 compared with 1960–1971, which indicates that in addition to precipitation, there are some other non-precipitation factors that are responsible for the change in the PRR; and (4) the contributions to runoff from human activities declined from 1972 to 1994 (84.15%) and 1995 to 2010 (57.16%). These results suggest that human activities (e.g., irrigation, reservoirs, water-and-soil conservation) were the primary driving forces leading to changes in the PRR in the WRB.
INTRODUCTION
The precipitation–runoff relationship (PRR) is an important issue in engineering hydrology, water resource planning and management, and watershed system evolution (Areerachakul & Junsawang 2014; Nourani et al. 2015; Wang et al. 2015). It is widely known that the global environment has undergone a drastic change over the past century, increasing the complexity of the hydrological cycle in many regions (Zhang et al. 2011; Rahmani & Zarghami 2015; Rege et al. 2015). The PRR has undergone great changes in many regions, and presents challenges for water regulation and management. Hence, an in-depth understanding of the PRR (i.e., when, how, and why the PRR changes) is an important issue and should be addressed.
The focus of studies involving the PRR should be when it changes (i.e., detecting the change point(s) of the PRR); however, studies on the change point(s) of the PRR are still lacking. The double mass curve method, which is a simple and practical method, is the most widely used method for testing the long-term evolution tendencies between hydrological and meteorological elements. However, when utilizing the double mass curve method, there is considerable subjectivity in estimating the change point by visual assessment and the inherent variability in hydrologic data (Searcy et al. 1960). Some researchers have also applied the cross wavelet spectrum and wavelet transform coherence to study the PRR (Labat et al. 2001; Charlier et al. 2015); however, the cross wavelet spectrum and wavelet transform coherence were better for analyzing the periodic properties of two time series and failed to identify the change point(s) of the relationship between the two series. Wang et al. (2015) applied the correlation coefficient to detect the change point of the PRR. Although this method was able to estimate a linear relationship, it was unable to handle non-linear relationships (such as the PRR) because it ignores the structure of the dependence. To avoid these restrictions of the methods mentioned above, the Archimedean copulas (Huang et al. 2015; Jiang et al. 2015a) were introduced to resolve the structure change of the PRR in this study. This method was chosen primarily because of its ability to accurately catch non-linear and asymmetric correlation characteristics between variables.
There have been a number of studies on changes in precipitation and runoff worldwide. Mwale et al. (2009) found that precipitation variability accounted for some runoff variability; however, on its own, it was insufficient for describing catchment runoff variability in southern Alberta and parts of northwestern Alberta. Yang & Tian (2009) showed that intensive human activities, especially agricultural water use, resulted in significant runoff change points, while they had little impact on precipitation in the Haihe River Basin. Zhang et al. (2011) found that annual precipitation declined more than runoff in the downstream of the Huaihe River. All of these studies indicate that the changes in runoff and precipitation are asynchronous, and that the response of runoff to precipitation varies in response to a changing environment. Partal (2012) showed that variability of both the runoff and the precipitation is generally similar over time, with a noticeable decrease in runoff at all of the stations in the Aegean region of Turkey. Velpuri & Senay (2013) investigated the long-term (1950–2009) trends in precipitation, runoff, and runoff coefficient in major urban watersheds in the United States and showed that most of the stations exhibited a downward trend in precipitation, while nearly half of the stations showed an upward trend in runoff. For the Weihe River Basin (WRB) specifically, many studies have been conducted on this issue (Zhan et al. 2014; Xiong et al. 2014); however, previous studies have mainly focused on the changes in a single hydrological factor (e.g., runoff, precipitation, temperature). In recent years, the response of runoff to climate variability and human activities has also become a focus of research in this region (Du & Shi 2012; Chang et al. 2015; Jiang et al. 2015b). However, there have been few studies regarding the law of water recurrence under the changing environment and the relationships between hydrological variables, such as precipitation and runoff.
The Weihe River is one of the most important industrial and agricultural production zones in China, and has presented a complicated PRR under the changing environment. Furthermore, water issues in this region have had a negative impact on societal and ecosystem development. The main purposes of this paper are to: (1) detect the structure change of the PRR; (2) explore the precipitation and runoff changes over different time scales; and (3) analyze the influence of human activities on the PRR by the climate elasticity method. The methodology used in this study can also provide useful insights for detecting the change points in relationships between other hydrological series. The results of this research provide insights for water resources management and planning departments.
STUDY AREA AND DATASET
Study area
Locations of hydrological and meteorological stations in the WRB.
The WRB is dominated by semi-arid hydrological characteristics and has a continental monsoon climate (Zuo et al. 2012). The climate is dry and chilly in winter and hot and rainy in summer. The mean annual temperature is 7.8 ∼ 13.5 °C, and the maximum and minimum temperatures have been observed in July and January (42.8 °C and −28.1 °C, respectively). The annual precipitation is 634.9 mm, of which approximately 60% falls between June and October. The mean annual natural runoff of the river is 10.4 billion m3 and accounts for 17.3% of the Yellow River's total discharge. The seasonal variations in runoff are similar to precipitation. The runoff in July and October accounts for approximately 65% of the mean annual runoff. The annual potential evapotranspiration (PET) is 800–1,200 mm and decreases from north to south. Maximum evaporation occurs in June or July, and evaporation between May and August accounts for 46 ∼ 58% of the mean annual evaporation.
During the past 50 years, runoff in the Weihe River has decreased dramatically; in particular, in the late 1990s the average runoff was only 3.60 billion m³ compared to 6.20 billion m³ in the 1950s.
Dataset
Monthly and annual runoff series (1960–2010) from gauging stations throughout the basin, including Huaxian and Zhuangtou, were provided by the hydrology bureau of the Yellow River Conservancy Commission (Figure 1). For this study, we assumed that the sum of runoff from the Huaxian and Zhuangtou stations represented the total runoff generated in the WRB.
The meteorological data (1960–2010) from 21 meteorological stations (Figure 1) inside and outside the WRB, including daily, monthly, and annual precipitation, daily mean temperature, daily maximum temperature, daily minimum temperature, air pressure, relative humidity, wind velocity, and sunshine duration, were provided by the China Meteorological Data Sharing Service System (http://cdc.cma.gov.cn). Daily PET was calculated by the FAO56 Penman–Monteith method using meteorological data (Allen et al. 1998). Based on the results from individual stations, the annual precipitation and PET of the whole catchment were calculated by the Thiessen polygon method on the ArcGIS platform. The Thiessen polygon method is a widely used technique and important method to evaluate regional precipitation or PET quantity (Faisal & Gaffar 2012). The first step of this method is to calculate station weights based on the areas of each station. Then the weights are multiplied by the precipitation or PET at the corresponding station to obtain the regional mean precipitation or PET.
METHODOLOGY
FAO56 Penman–Monteith method
is slope of saturated vapour–pressure curve (kPa/°C), Rn is net radiation at the crop surface (MJ/(m2·day)), G is soil heat flux (MJ/(m2·day)), is psychrometric constant (kPa/°C), T is mean daily air temperature at 2 m height (°C), is wind speed at 2 m height (m/s), is saturation vapour pressure (kPa), is actual vapour pressure (kPa) and is saturated vapour pressure deficit (kPa).The Archimedean copulas
, which is expressed as: is the copula function, is the parameter to be estimated, and u and v are the marginal distributions for F(x) and G(y), respectively.
is:The expression of the Kendall correlation coefficient τ and the parameter of the three copula functions are shown in Table 1.
Three common Archimedean copulas in the field of hydrology research
| Archimedean copula | Cθ(u, v) | Range | Kendall's tau |
|---|---|---|---|
| Clayton | |||
| Frank | |||
| Gumbel–Hougaard |
| Archimedean copula | Cθ(u, v) | Range | Kendall's tau |
|---|---|---|---|
| Clayton | |||
| Frank | |||
| Gumbel–Hougaard |
To build a model of joint distribution on runoff and precipitation, the first step is to obtain an appropriate marginal distribution. In this paper, the generalized extreme value distribution (Gev), logarithmic normal distribution (Logn), and gamma distribution (Gam) were applied to fit the marginal distribution. To identify an appropriate marginal distribution, we utilized root mean square error (RMSE) and Akaike information criteria (AIC) to evaluate distributions.
Thus, the appropriate u and v can be obtained through the two evaluation indices.
Based on the appropriate u and v, the second step is to estimate the parameter of the three copula functions by the relationship of the Kendall correlation coefficient τ and the parameter
(Table 1).
Finally, the joint distribution model is built and the RMSE and AIC indices are again applied to identify the appropriate copula function.
Change point detection based on Archimedean copulas
, , and are the maximum likelihood estimations of parameter .
is large, the original hypothesis can be rejected (i.e., there is a change point) (Ye & Miao 2009). According to the likelihood ratio test methods, the asymptotic distribution of obeys the distribution. The threshold of the statistic for rejecting the original hypothesis is based on Dias (2004). The time estimation of k* is shown as:The process of change point detection can be briefly summarized as follows:
Step 1: Select the appropriate marginal distributions for monthly precipitation and runoff.
Step 2: Estimate the parameter
of the joint function and establish the joint function for u and v.
Step 3: Estimate the log likelihood ratio statistic
by Equation (14).
Step 4: Judge the change point by Equation (15).
If multiple change points exist, the binary segmentation method would be applied to detect the change points between multiple variables. This scenario includes the following: (1) the existence of a single point between the original time series that is detected by the method mentioned above; (2) if no such point exists, the original hypothesis should be accepted; otherwise, the original time series would be divided into two subsequences, and each of them is subject to continued detection; and (3) the process is finished when there is no change point for every subsequence.
Attribution analysis by climate elasticity approach
RESULTS
Trends in annual hydro-meteorological variables
Annual runoff, precipitation, and PET changes in 1960–2010 in the WRB.
The MK test results of monthly precipitation, runoff, and PET in 1960–2010 in the WRB. The horizontal dashed lines represent the critical value of the 95% significance level.
The MK test results of monthly precipitation, runoff, and PET in 1960–2010 in the WRB. The horizontal dashed lines represent the critical value of the 95% significance level.
Detection of PRR change points
Selection of the appropriate marginal distribution
Fitted effect of different marginal distributions on monthly precipitation and runoff in the WRB
| Hydrological variables | Model of marginal distribution | RMSE | AIC |
|---|---|---|---|
| Monthly precipitation | Gev | 0.048 | −1611.35 |
| Logn | 0.061 | −1485.58 | |
| Gam | 0.028 | −1906.21 | |
| Monthly runoff | Gev | 0.018 | −2138.10 |
| Logn | 0.012 | −2342.58 | |
| Gam | 0.047 | −1625.78 |
| Hydrological variables | Model of marginal distribution | RMSE | AIC |
|---|---|---|---|
| Monthly precipitation | Gev | 0.048 | −1611.35 |
| Logn | 0.061 | −1485.58 | |
| Gam | 0.028 | −1906.21 | |
| Monthly runoff | Gev | 0.018 | −2138.10 |
| Logn | 0.012 | −2342.58 | |
| Gam | 0.047 | −1625.78 |
Monthly precipitation (a) and runoff (b) frequency curves on different marginal distributions.
Monthly precipitation (a) and runoff (b) frequency curves on different marginal distributions.
Selection of the appropriate copula function
Comparisons of theoretical cumulative frequency obtained by Archimedean copulas with empirical cumulative frequency.
Comparisons of theoretical cumulative frequency obtained by Archimedean copulas with empirical cumulative frequency.
Therefore, the RMSE and AIC were again applied to select the appropriate joint distribution model, and as a result, the Gumbel–Hougaard copula was chosen (Table 3).
The goodness-of-fit test of Archimedean copulas
| Joint function | RMSE | AIC |
|---|---|---|
| Clayton copula | 0.43 | −443.32 |
| Frank copula | 0.44 | −440.03 |
| Gumbel–Hougaard copula | 0.42 | −458.14 |
| Joint function | RMSE | AIC |
|---|---|---|
| Clayton copula | 0.43 | −443.32 |
| Frank copula | 0.44 | −440.03 |
| Gumbel–Hougaard copula | 0.42 | −458.14 |
Change point detection
Changes of statistics Zn estimated by Equation (14) and the dashed lines represent the occurrence time of change point of PRR.
Changes of statistics Zn estimated by Equation (14) and the dashed lines represent the occurrence time of change point of PRR.
DISCUSSION
Comparative analysis of the PRR change point
Double mass curve of annual precipitation and runoff for 1960–2010 in the WRB.
The judgment of PRR change point(s) is clearly subjective and displays uncertainties (Figure 7). In contrast, determination of the change point(s) by the method based on the Archimedean copulas is accurate and reliable.
Change in PRR
PRR for three periods, 1960–1971, 1972–1994, and 1995–2010 in the WRB.
Change in precipitation and runoff before and after the change points of PRR
Annual and intra-annual changes in precipitation and runoff
Several indices show that mean annual runoff in the periods of 1972–1994 and 1995–2010 was 57.25 and 36.03 mm, or 25.78% and 53.29% lower than the 1960–1971 period, respectively (Table 4). By contrast, the precipitation change was smaller, with reductions of 5.32% and 11.23% for the 1972–1994 and 1995–2010 periods, respectively. The variation coefficients of inter-annual (Inter-Cv) runoff and precipitation were not significantly altered, although the inter-annual variability of runoff was higher than that of precipitation. Based on the concentration degree (CD) (Zhang & Qian 2003) and intra-annual variation coefficients (Intra-Cv), there is little change in precipitation but a more obvious change in runoff.
Changes of characteristics of runoff and precipitation in 1960–1971, 1972–1994, and 1995–2010 in the WRB
| Runoff | Precipitation | |||||
|---|---|---|---|---|---|---|
| Index | 1960–1971 | 1972–1994 | 1995–2010 | 1960–1971 | 1972–1994 | 1995–2010 |
| Average (mm) | 77.13 | 57.25 | 36.03 | 585.64 | 554.49 | 519.86 |
| Inter-Cv | 0.41 | 0.40 | 0.43 | 0.18 | 0.16 | 0.18 |
| CD (%) | 28 | 32 | 34 | 35 | 35 | 36 |
| Intra-Cv | 0.78 | 0.90 | 0.83 | 0.94 | 0.94 | 0.96 |
| Runoff | Precipitation | |||||
|---|---|---|---|---|---|---|
| Index | 1960–1971 | 1972–1994 | 1995–2010 | 1960–1971 | 1972–1994 | 1995–2010 |
| Average (mm) | 77.13 | 57.25 | 36.03 | 585.64 | 554.49 | 519.86 |
| Inter-Cv | 0.41 | 0.40 | 0.43 | 0.18 | 0.16 | 0.18 |
| CD (%) | 28 | 32 | 34 | 35 | 35 | 36 |
| Intra-Cv | 0.78 | 0.90 | 0.83 | 0.94 | 0.94 | 0.96 |
Monthly runoff and precipitation changes in different periods in the WRB.
Changes in precipitation and runoff characteristics
Characteristics of monthly runoff and precipitation from Figure 10 in different periods in the WRB
| Exceedance probabilities | ||||
|---|---|---|---|---|
| Elements | Characteristic values | 5% | 50% | 95% |
| Runoff | R(1960–1971) | 18.37 mm | 4.54 mm | 1.19 mm |
| R(1972–1994) | 15.83 mm | 3.15 mm | 0.69 mm | |
| R(1995–2010) | 8.62 mm | 2.03 mm | 0.55 mm | |
| Change ratio (1972–1994 versus 1960–1971) | −13.83% | −30.62% | −42.02% | |
| Change ratio (1995–2010 versus 1960–1971) | −53.08% | −55.29% | −53.78% | |
| Precipitation | P(1960–1971) | 145.39 mm | 40.95 mm | 1.01 mm |
| P(1972–1994) | 132.34 mm | 30.63 mm | 1.30 mm | |
| P(1995–2010) | 125.57 mm | 27.14 mm | 1.68 mm | |
| Change ratio (1972–1994 versus 1960–1971) | −8.98% | −25.20% | 28.71% | |
| Change ratio (1995–2010 versus 1960–1971) | −13.63% | −33.72% | 66.34% | |
| Exceedance probabilities | ||||
|---|---|---|---|---|
| Elements | Characteristic values | 5% | 50% | 95% |
| Runoff | R(1960–1971) | 18.37 mm | 4.54 mm | 1.19 mm |
| R(1972–1994) | 15.83 mm | 3.15 mm | 0.69 mm | |
| R(1995–2010) | 8.62 mm | 2.03 mm | 0.55 mm | |
| Change ratio (1972–1994 versus 1960–1971) | −13.83% | −30.62% | −42.02% | |
| Change ratio (1995–2010 versus 1960–1971) | −53.08% | −55.29% | −53.78% | |
| Precipitation | P(1960–1971) | 145.39 mm | 40.95 mm | 1.01 mm |
| P(1972–1994) | 132.34 mm | 30.63 mm | 1.30 mm | |
| P(1995–2010) | 125.57 mm | 27.14 mm | 1.68 mm | |
| Change ratio (1972–1994 versus 1960–1971) | −8.98% | −25.20% | 28.71% | |
| Change ratio (1995–2010 versus 1960–1971) | −13.63% | −33.72% | 66.34% | |
Duration curves of monthly runoff and precipitation in different periods in the WRB.
Duration curves of monthly runoff and precipitation in different periods in the WRB.
Compared with the duration curve of the 1960–1971 period, the most remarkable change in monthly runoff duration curves occurs at the 95% exceedance probability in 1972–1994. Here, the curve declines by 42.02%, while precipitation increases by 28.71%. Runoff at the 50% and 5% exceedance probabilities declined by 30.62% and 13.83% compared with precipitation, which declined by 25.20% and 8.98%, respectively.
Compared with the duration curve for the 1960–1971 period, runoff in 1995–2010 decreased by 53% and 50% for the 5% and 95% exceedance probabilities, respectively; these changes are more significant than the change in precipitation (−13.63%, −33.72%, and 66.34%, respectively).
What is responsible for the change in PRR
The correlation between precipitation and runoff decreased for the 1972–1994 and 1995–2010 periods compared to the 1960–1971 period, and annual runoff yield significantly decreased at the same annual precipitation. Compared with runoff and precipitation from 1960 to 1971, the mean annual runoff declined by 25.78% and 53.29% in 1972–1994 and 1995–2010, respectively, whereas mean annual precipitation declined by only 5.32% and 11.23%, respectively. Although the change in monthly runoff is notable, there was little change in monthly precipitation. The asynchronous changes in precipitation and runoff may be the driving force behind the changes in PRR. Due to the minimal changes in precipitation, it is essential to determine the reason for the changes in runoff. To this end, the climate elasticity of runoff method (Liu & Cui 2011) was applied to study the impact of climate changes (i.e., precipitation and PET) and human activities on runoff reduction (Table 6).
Impacts of climate variability and human activities on runoff reduction by climate elasticity of runoff method in the WRB
| Climate change | Human activities | ||||||
|---|---|---|---|---|---|---|---|
| Periods | Runoff (mm) | Precipitation (mm) | PET (mm) | ΔQC (mm) | CC | ΔQH (mm) | CH |
| 1960–1971 | 77.13 | 585.64 | 851.68 | ||||
| 1972–1994 | 57.25 | 554.49 | 830.69 | −3.15 | 15.85% | −16.73 | 84.15% |
| 1995–2010 | 36.03 | 519.86 | 870.47 | −17.61 | 42.84% | −23.49 | 57.16% |
| Climate change | Human activities | ||||||
|---|---|---|---|---|---|---|---|
| Periods | Runoff (mm) | Precipitation (mm) | PET (mm) | ΔQC (mm) | CC | ΔQH (mm) | CH |
| 1960–1971 | 77.13 | 585.64 | 851.68 | ||||
| 1972–1994 | 57.25 | 554.49 | 830.69 | −3.15 | 15.85% | −16.73 | 84.15% |
| 1995–2010 | 36.03 | 519.86 | 870.47 | −17.61 | 42.84% | −23.49 | 57.16% |
Human activities are the dominant factors (84.15%) leading to decreases in runoff for the periods of 1972–1994 and 1995–2010 (84.15% and 57.16%, respectively). Du & Shi (2012) used the measured and natural runoff in the WRB to determine that human activities accounted for 51.11% of the runoff reduction in 1988–2008 compared to 1956–1987, while climate change accounted for 48.49%. Zhan et al. (2014) combined an elasticity coefficient approach and hydrological model (SIMHYD model) to conclude that the contributions of climate change and human activities to runoff change in 1990–2008 compared with 1958–1989 were 34.1 ∼ 47.3% and 52.7 ∼ 65.9%, respectively. Chang et al. (2015) used the variable infiltration capacity hydrological model and found that the contribution to runoff changes for the 1970s, 1980s, 1990s, and 2000s from climate change were 36%, 28%, 53%, and 10%, respectively, and 64%, 72%, 47%, and 90%, respectively, from human activities. The results of our study are similar in that human activities were the main factor driving the decrease in runoff in the WRB. Additionally, the contribution of climate changes to the runoff decrease was enhanced in the 1990s. These results show that human activities are the dominant factor influencing the change in PRR. Irrigated areas, water projects, and water and soil conservation are direct causes of the runoff change in the WRB (Du & Shi 2012; Chang et al. 2015).
Changes of reservoir (large- and middle-sized reservoirs) volumes in the WRB of Shaanxi province for 1960–2010.
Changes of reservoir (large- and middle-sized reservoirs) volumes in the WRB of Shaanxi province for 1960–2010.
To control water and soil loss, soil conservation projects, such as afforestation, grass-planting, and construction of level terraces and check dams have been implemented in the study area beginning in the 1970s. Zhao et al. (2001) used the soil hydrology method to calculate the effect of water reduction from soil and water conservation. They showed that the mean annual water reduction was 2.85 billion m³ from 1990 to 1996, and was mainly caused by human activities and precipitation (51.5% and 48.5%, respectively).
Yao et al. (2011) found that the El Nino event that occurred in 1994 also resulted in a significant drop in precipitation. After 1994, there were fewer soil and water conservation projects in the WRB, and climate change was more notable than before. Thus, the role of climate change on runoff is greater than ever.
CONCLUSIONS
Based on monthly runoff and precipitation from 1960 to 2010, the change in PRR was studied using the Archimedean copulas. We identified the change points in the PRR relationship and identified the driving forces behind the change points. The following conclusions were drawn:
Some of the decreasing trends in annual runoff and precipitation were significant at the 95% significance level by the MK trend test, and others were not; however, there were minimal changes in annual PET. In the WRB, the change points in the PRR occurred in 1971 and 1994; these dates were used to divide the whole time series into three periods: 1960–1971, 1971–1994, and 1995–2010.
The runoff yield was the highest in 1960–1971, followed by 1972–1994, and then 1995–2010 under the same annual precipitation. The inter-annual and intra-annual variations in precipitation were less obvious than for runoff. Comparing the duration curves of different periods showed that the change in monthly runoff declined more than precipitation at various exceedance probabilities. Therefore, it was concluded that the change in PRR was primarily caused by a decline in runoff due to human activities in the WRB.
Irrigated areas, water projects, and water and soil conservation efforts directly contributed to changes in runoff in the WRB. Compared with runoff in 1960–1971, human activities account for 84.15% and 57.16% of the runoff reduction in 1972–1994 and 1995–2010, respectively, while climate changes account for 15.85% and 42.84%. It is clear that human activities have played a more dominant role in changing the PRR than climate change.
ACKNOWLEDGEMENTS
This research was supported by the Natural Science Foundation of China (51190093) and Key Innovation Group of Science and Technology of Shaanxi (2012KCT-10). Sincere gratitude is extended to the editor and the anonymous reviewers for their professional comments and corrections.
