A dynamic multiple regression approach for quantifying the relative impact of precipitation variations and streamflow generation conditions on runoff

Streamflow and runoff of a watershed are extensively influenced by climate variability (mainly the change of temperature and precipitation) and human activities (water withdrawal from river, groundwater exploitation, hydraulic projects, land use/cover change (LUCC), etc.) (Tomer & Schilling 2009; Zhang et al. 2011; Bao et al. 2012). Many studies discussed the influence of climate change and human activities on runoff process (e.g., Chiew & McMahon 2002; Yang et al. 2004; Brown et al. 2005; Intergovernmental Panel on Climate Change [IPCC] 2007; Zhang & Lu 2009; Du et al. 2012). General consensus has revealed that climate change is likely to enhance the frequency and severity of extreme climate events, causing more severe floods and droughts (e.g., Milliman et al. 2008; Bates et al. 2008; Jung et al. 2012; Thompson 2012), and that human activities tend to reduce the streamflow and runoff, especially in north China (e.g., Li et al. 2007; Wang & Meng 2008; Liu et al. 2009a; Cuo et al. 2013; Zhao et al. 2014). Within the last three decades, hydrologic regime and water shortage have become increasingly serious issues for water resources management at catchment and/or regional scale (Ren et al. 2002; Lakshmi et al. 2012; Kizza et al. 2013; Li et al. 2013).

Therefore, investigating the influence and quantifying the relative effect of climate change and human activities on hydrological circulation and water resources have drawn considerable concerns (e.g., Siriwardena et al. 2006; Ma et al. 2008; St Jacques et al. 2010; Jin et al. 2012; Carless & Whitehead 2013; Zhan et al. 2013). Some progress has been made by means of diverse approaches including the mass balance approach (Milly 1994; Dooge et al. 1999; Jones et al. 2006; Li et al. 2007; Ma et al. 2008; Wang & Hejazi 2011), hydrologic simulation model (Jones et al. 2006; Ma et al. 2010; Wang et al. 2010; Bao et al. 2012; Zhang et al. 2012) and linear regression method (Ye et al. 2003; Huo et al. 2008; Tian et al. 2009; Jiang et al. 2011; Wang et al. 2012). The mass balance approach uses the water-energy balance over a long-term scale, and a typical example of this approach is the Budyko theory (Budyko 1974) in which annual water balance is regarded as a manifestation of the competition between available water and available energy. However, the selection of appropriate governing equations is challenging, and this simple theory may not work in diverse catchments (Gentine et al. 2012). And also, the linear regression method is convenient for utilization but limited in application due to a lower fitting accuracy because this method may not capture the true non-linear nature of the hydrologic system. As for the hydrologic simulation model, although it is technical and sophisticated, it is difficult to use a simulation model over large areas requiring a large amount of data and computational time (Ahn & Merwade 2014). Other than that, in many areas of the world, particularly in China, the historic data of the LUCC before 1980 is difficult to obtain. Combined with the fact that the underlying surface of many watersheds had been largely changed before the 1980s, the application of the hydrological simulation method is limited to some extent.

Aiming at the above questions, the upper Luan River catchment (ULRC) in China was selected as the study area, and the dynamic multiple regression (DMR) method was proposed to quantify the impact of precipitation variations and streamflow generation condition changes on the runoff in the ULRC. In this study, precipitation is selected as an indicator to represent the climate change, and streamflow generation condition is applied to represent the physical characteristics of a basin (such as topography, soil and vegetation) through which human activities affect the streamflow generation process. This approach will not only avoid the collection of underlying surface data, but also improve the accuracy of the linear regression model.

The Luan River is located in the north-east of the North China Plain (115°30′–119°45′E, 39°10′–115°30′N). The total area of the Luan River basin is 44,750 km². In the area, the average air temperature was 7.5 °C, and the average annual precipitation was 520 mm (Feng et al. 2014). The upstream of Luan River is an important water source area of Tianjin and Tangshan. However, in the last 30 years, the streamflow of the upper Luan River basin has had a pronounced tendency to decrease. Especially in 1997–2005, the continuous drought caused a sharp decrease in the runoff of the basin (Feng et al. 2008). It severely threatened the water supply safety of Tianjin and Tangshan city, which has been concerning the Water Administration Department and related experts. In order to estimate the water resources quantity of the Luan River in a period of future time, it is necessary to quantitatively analyze the reason for the runoff decreasing.

Identification of change points of precipitation and runoff is a prerequisite for the division of the rainfall–runoff linear correlationship into different periods. In different periods, the law of precipitation variations and runoff yield conditions may have changed. So, when conducting the rainfall–runoff regression model and quantifying the impacts of precipitation variations and streamflow generation conditions on the runoff, the change points of the linear relationship between precipitation and runoff must be identified.

In order to investigate whether there have been significant changes in the linear relations between annual precipitation and annual runoff during 1960 to 2011, double mass curve analysis was adopted. The double mass curve is drawn on a Cartesian coordinate system for successive accumulated values of one variable against another variable's continuous cumulative values, which is commonly used in the consistency checking and change-point identifying of a linear relation between two hydrological variables. Its theoretical basis is: as long as the sample data of two variables are proportional to each other in the same period, the accumulated values of one variable and that of the other variable can be expressed as a straight line in Cartesian coordinates, and the slope of the line is the proportionality constant for the corresponding points of the two variables. If there is a variation in the slope of the double mass curve at one time point, it means that the linear relation of the two variables has begun to change around the time point, i.e., the streamflow generation capability has changed after the time point (Mu et al. 2010).

After identifying the three changing points of the cumulative curve, a Kolmogorov–Smirnov test (K–S test) was used to check the samples' consistency among rainfall series of the three periods. The two sample K–S test is a non-parameter statistical test that is used to analyze whether the two samples are from the same populations. The null hypothesis of the K–S test is that the discrepancy of two samples is attributed to the sampling error and that the two samples being from the same population cannot be denied; the alternative hypothesis is that the two samples are from different populations leading to their difference. Using 1960–1979 as the reference period, we conducted the K–S test on the annual precipitation in the period of 1980–1991, 1992–1999 and 2000–1011, and the K–S test on annual runoff as well as the four periods. The result is listed in Table 2, in which the Z-stat is the test statistics, and the P-value is the probability of observing an effect given that the null hypothesis is true, whereas α is a significance level that is the probability of rejecting the null hypothesis given that it is true. The significance level (α) is usually set at a threshold value of 0.05.

As is shown in Table 2, compared to the precipitation in 1960–1979, the P-values of the precipitation in 1980–1991, 1992–1999 and 2000–2011 are bigger than 0.05, failing to reject that the precipitation series in 1961–2011 was from the same population. However, comparing the precipitation during 2000–2011 with that of the years 1961–1979, according to the P-value, the probability of the difference being caused by sampling error was merely 0.308 while by coming from different populations was 0.692, which suggested that although no significant change in statistical characteristics occurred, the variation of precipitation during 2000–2011 is worthy of being considered.

As for runoff series, using the period of 1960–1979 as the referred period, P-values of K–S tests of the runoff series during 1980–1991 and 2000–2011 are all less than 0.05, suggesting that a significant change happened in the runoff series in the two periods. Whereas the P-value of the K–S test during 1992–1999 is far larger than 0.05, which indicates that the runoff coefficients in the period of 1960–1979 and 1992–1999 are very close.

Based on the above analysis, the evolution process of the precipitation-runoff relation can be divided into four periods: 1960–1979, 1980–1991, 1992–1997 and 2000–2011. Since 1960–1979 has been chosen as the reference period, the streamflow generation condition of 1960–1979 is correspondingly taken as the referred streamflow generation condition.

According to the water yield mechanisms and confluence process of a river basin, the streamflow can be attributed to either stormflow or baseflow. The stormflow is mainly from overland flow and lateral subsurface flow, while the baseflow is dominantly derived from groundwater discharge. When performing the precipitation-runoff regression analysis, the reliability and accuracy of the precipitation-runoff regression model can be increased by combining the physical mechanisms of runoff generation. This study will establish the precipitation-runoff DMR model with consideration of precipitation distribution characteristics within a year and streamflow generation mechanisms of the catchment.

According to the precipitation distribution characteristics within a year and through a great deal of trial calculations and optimization, a water year of the study area can be defined as from October 1st of the present year to September 30th of the next year in the solar calendar. Moreover, a water year can be divided into two periods of dry season (October 1st–March 31st) and wet season (April 1st–September 31st). The statistical characteristics of wet season and dry season in a water year are summarized in Table 3.

The reason for taking into account precipitation in the flood season of the previous year is that there is usually a spring freshet in March–April, but the correlation analysis showed that the correlation coefficient between the freshet and the precipitation of March–April and the dry season (October–March) was very low and could even be a negative number. As is known from the water yield theory of river basins, the precipitation-runoff should be positively correlated if the runoff is mainly caused by the precipitation. Thus, we can conclude that the spring freshet and the precipitation in November–April has a spurious correlation or no correlation. So, what contributed to the spring freshet? After a great deal of analysis of correlation among the freshet and the previous 12 months’ precipitation, we found out that there was a high correlation between the freshet and the precipitation in the flood season (July–September) of the previous year. Also, the runoff of the wet season had a considerable correlation coefficient with the precipitation in the flood season of the previous year, but a low correlation coefficient with the wet season of the previous year. Thus, it can be inferred that the runoff of the spring freshet in the upstream of the Luan River was mainly caused by the precipitation in the flood season of the previous year. Also, the baseflow in the upstream of the Luan River was also largely affected by the precipitation of the flood season in the previous year, the reason for which is that groundwater discharge has a rather low velocity such that the groundwater from precipitation in the flood season of the previous year needs about 6–9 months to arrive at the basin outlet.

The correlation coefficient matrices of runoff and precipitation are given in Table 4. In Table 4, R_dry (t) and R_wet (t) represents, respectively, runoff series in the dry season and wet season of the t year in a water year; P_dry (t), P_wet (t), P_flood (t − 1) and P_dry (t − 1), respectively, represent the precipitation in the dry season of the t year, in the wet season of the t year, in the flood season of the t − 1 year and in the dry season of the t − 1 year. The reason for selecting P_dry (t), P_wet (t), P_flood (t − 1) and P_dry (t − 1) was based on a large number of trial computed work and analysis.

As is shown in Table 4, the present year runoff in the dry season had a correlation coefficient of 0.058 with the present year precipitation in the dry season, and a correlation coefficient of 0.741 with the previous year precipitation in the flood season, and a correlation coefficient of 0.0002 with the previous year precipitation in the dry season. It has to be pointed out that the correlation coefficients of R_dry (t) with P_dry (t) and P_dry (t − 1) were all small. Thus, the runoff in the dry season of the present year was mainly correlated with the precipitation in the flood season of the previous year. This is consistent with the physical mechanism that the runoff in the dry season is mainly contributed by baseflow, which is derived from the precipitation in the flood season of the previous year. In the meantime, it is also consistent with the fact that the amount of precipitation in the dry season is very small, most of which is eventually evaporated so that both overflow and groundwater are difficult to yield.

As for the present year runoff in the wet season, it had a correlation coefficient of 0.750 with the present year precipitation in the wet season, a correlation coefficient of 0.173 with the present year precipitation in the dry season, and a correlation coefficient of 0.462 with the previous year precipitation in the flood season. The reason for the high correlation with the previous year precipitation in the flood season was because the baseflow of the runoff in the wet season was mainly supplied by groundwater discharge, which has a slow current velocity. The present year baseflow of the Sandaohezi hydrological station dominantly resulted from the infiltration of the precipitation in the flood season of the previous year. Consequently, it can be drawn that the runoff in the wet season of the present year was mainly correlated to the precipitation in the wet season of the present year and the precipitation in the flood season of the previous year.

According to the above result, we conducted the regression analysis of the present year runoff in the dry season and the previous year precipitation in the flood season in 1961–1979, 1980–1991, 1992–1999 and 2000–2011. The regression analysis was also done for the above four stages among the present year runoff in the wet season, the present year precipitation in the wet season and the previous year precipitation in the flood season. The results are presented in Tables 5–8. In Tables 5–8, for the dry season, K₀ is the intercept of the regression equation, and K₁ is the variable slope of P_flood (precipitation in flood season) in the previous year; for the wet season, K₀ is the intercept of the regression equation as well, but K₁ is the variable slope of P_wet (precipitation in wet season) in the present year, K₂ is the variable slope of P_flood (precipitation in flood season) in the previous year.

In Tables 5–8, the variance analysis result showed that seven out of the eight DMR models passed the Fisher test, i.e., the P-value of the two samples’ F-test was less than 0.05, which indicates that the computed runoff value and the measured runoff data were from the same population. Furthermore, R-square (determination coefficients) of all the four wet season DMR models exceeded 0.7, suggesting good linear relations between precipitation and runoff during the wet season. If examined in a water year timescale, the averages of relative residuals of the four annual DMR models were all below 15%, and the maximum relative residuals were all under 40%, with the R-square being between 0.68–0.98, suggesting that the precipitation-runoff DMR method has a good fitting effect overall.

In Table 7, the DMR model in the dry season of 2000–2011 did not pass the significance test, with a P-value of 0.421, suggesting the probability that the variance of the model and the variance of the original series were from different populations was 42.1%. In other words, the probability of the sampling error accounted for 57.9% of the difference between the two variances. Thus, it can be considered that the model could reflect the precipitation-runoff relationship to some extent. However, the R-square of the model was just 13.32%, which means that the independent variable did not explain the dependent variable well. This result is in accordance with the fact that the lower runoff volume in the dry season is susceptible to human activities and other occasional factors. But the low water volume in the dry season has little effect on the accuracy of the DMR model in a year timescale, the R-square of the annual DMR model is still up to 84.79%.

According to Tables 5–8, the regression formulae of precipitation-runoff in the Sandaohezi catchment during 1961–1979, 1980–1991, 1992–1999 and 2000–2011 are given as follows:

The index t is a serial number of the water year sequence ranging from 0 to 50, in which 0 is used to represent the year of 1960, and 50 is used to represent 2011.

The multiple regression model above can predict the runoff in the future in the dry season (from October of the t − 1 year to March of the t year) by input of the precipitation in the flood season (from July to September of the t − 1 year) into Equation (7), on the condition that the runoff yield mechanism will not change in the future.

Similarly, the runoff in the wet season (from April to September of the t year) can be predicted by input of the precipitation in the flood season (from July to September of the t − 1 year) and the precipitation in the wet season (from April to September of the t year) into Equation (8), on the condition that the runoff yield mechanism will not change in the future.

In Equation (9),

• is the relative impact of the precipitation variation between 1961–1979 and 1980–1991 on the streamflow generation conditions of the year 1960–1979;

• is the change of the calculated annual runoff in 1980–1991 compared with the observed annual runoff in 1961–1979;

• is the average of the observed annual runoff during 1961–1979; and

• is the average of the calculated annual runoff on the condition of the precipitation process in 1980–1991 and the streamflow yield capability in 1961–1979.

Similarly to the above, we can calculate the impacts of precipitation variations on the runoff in the periods of 1992–1999 and 2000–2011 compared with 1961–1979. The calculation results are shown in Table 9.

From Table 9, taking 1961–1979 as the reference period and on the streamflow generation condition of Sandaohezi catchment during 1961–1979: (1) the annual runoff decreased by 13.34% caused by the variations of the precipitation process in 1980–1991; (2) the annual runoff increased about 8.46% attributed to the change of precipitation conditions in 1992–1999; (3) and the annual runoff had a significant reduction of 32.67% due to the precipitation change in 2000–2011.

As has been noted, in comparison with the period of 1961–1979, impacts of the rainfall variation on runoff took place mainly in the periods of 1980–1991 and 2000–2011, leading to decrease in runoff similarly. So, it can be induced that there is a climate cycle with a wet period of about 8 years and a dry period of about 12 years. The results indicate that the catchment of Sandaohezi was probably suffering a period of drought weather in the climate change cycle during 2000–2011, and that, for the next 8–10 years, the Sandaohezi catchment will enter a relative wet period.

Similarly, we can quantify the effect of streamflow generation conditions change from 1961–1979 to 1992–1999 and to 2000–2011 under the precipitation conditions of 1961–1979. The calculation results are shown in Table 10.

From Table 10, compared to the observed runoff in 1961–1979 and on the condition of the precipitation process during 1961–1979, the changes of streamflow generation conditions from the period of 1961–1979 to the period of 1980–1991 caused a decrease of runoff in the dry season by 32.23% and in the wet season by 25.62%, and in the meanwhile a decrease of annual runoff by 27.36%. However, the streamflow generation environment transition from 1961–1979 to 1992–1999 led to an increase in the runoff of about 4.63% in the dry season but a decrease of 3.51% in the wet season. The increase in the dry season may come from the ample rainfall in 1992–1999 which will lead to increasing the runoff coefficient, and may also result from the model bias since Equation (7) has a low R-square score. With the economic development in the region and aggravation of the human activities, the streamflow generation conditions of 2000–2011 changed again, and caused a decrease in the runoff in the dry season and wet season and annual runoff, respectively, of 28.57%, 37.50% and 35.15% in the same rainfall condition. In summary, regarding 1961–1979 as the referred period, the streamflow yield condition changing occurred mainly in the periods of 1980–1991 and 2000–2011, but a slight change occurred in 1992–1999. If the current situation persists, the water shortage state will continue in the Sandaohezi catchment, even in the Luan River basin and related regions such as Tianjin city.

From 1980 to 2011, in contrast with the fact that the precipitation variations decreased the annual runoff by 14.42% (Table 9), the changes of streamflow yield conditions led to a decrease of 23.11% in the annual runoff, and therefore we can calculate the relative contribution of precipitation variations and streamflow yield condition changes to runoff. As a result, the relative contribution of precipitation variations is 38.42% and that of streamflow yield condition changes is 61.58%.

By using the established multiple regression models, the impact of precipitation and runoff-generation change on the runoff was quantitatively identified in the upstream Luan River catchment. The result showed that, for the period of 1980–2011 in contrast with the period of 1960–1979, the precipitation variations decreased the annual runoff by 14.42%, while the changes of streamflow yield conditions decreased the annual runoff by 23.11%. As a result, the relative contribution of precipitation variations is 38.42%, while that of the streamflow yield condition changes is 61.58%.

Besides, this study identified the statistical regularity at the Sandaohezi hydrological station that the runoff in the dry season of the present year is mainly related to the precipitation in the flood season of the previous year, and that the runoff in the wet season of the present year is mainly correlated with the precipitation in the wet season of the present year and the precipitation in the flood season of the previous year.

The major innovation of this research is the proposition of the dynamical multiple regression method combined with the consideration of precipitation distribution features within a water year and the streamflow generation characteristics of a catchment. This approach will not only avoid the need for collection of underlying surface data, but will also improve the accuracy of the linear regression model for precipitation and runoff. The research approach and the dynamical multiple regression model can be applied to the quantitative estimation of how much the changing environment affects the runoff in areas lacking the historic data of the underlying surface.

This work was supported by the Opening Fund of the State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, China (Grant No. HESS-1515), the Key Program of Science Foundation for University and Colleges in Henan province of China (Grant No. 15A110037), the Humanities and Social Science Project of the Education Department of Henan Province, China (Grant No. 2015-GH-390), and the National Natural Science Foundation of China (51374100). The authors extend their thanks to two anonymous reviewers for their valuable comments, which greatly improved the quality of this paper.