Linear and nonlinear characteristics of the runoff response to regional climate factors in the Qira River basin, Xinjiang, Northwest China

Mann–Kendall trend test

The nonparametric Mann–Kendall trend test (Kendall, 1975; Mann, 1945) has been widely used to detect change trends in meteorological and hydrological time series (Burn, 2008; Zhang et al., 2010; Xu et al., 2013). The rank-based Mann–Kendall method was used in this study to test for significant tendencies in the temperature, precipitation and runoff time series in the Qira River basin.

In the Mann–Kendall trend test, the null hypothesis (H₀) is that the parameter time series (X_i, i = 1, 2, …, n) are independent and identically distributed random variables, while the alternative hypothesis (H₁) is that there is a tendency in the variable time series (Partal & Kahya, 2006; Li et al., 2010). The standardized statistic of the Mann–Kendall trend test, Z, is defined by: (1)${\displaystyle Z=\left\{\begin{array}{ll}\frac{S-1}{\sqrt{var\left(S\right)}},\hfill & S>0\hfill \\ 0,\hfill & S=0\hfill \\ \frac{S+1}{\sqrt{var\left(S\right)}},\hfill & S<0\hfill \end{array}\right.}$ where (2)${\displaystyle S=\sum _{i=1}^{n-1}\sum _{k=i+1}^{n}sgn\left(x_k-x_i\right),}$ in which x_k and x_i are the series of temperature, precipitation and runoff; n is the length of the time series; k = 1, 2, …, n − 1, j = 2, 3, …, n and (3)${\displaystyle sgn\left(x_k-x_i\right)=\left\{\begin{array}{ll}1,\hfill & x_k-x_i>0\hfill \\ 0,\hfill & x_k-x_i=0\hfill \\ -1,\hfill & x_k-x_i<0.\hfill \end{array}\right.}$ Moreover, S approximately follows the normal distribution, with 0 mean, and the variance is given as: (4)${\displaystyle var\left(S\right)=\frac{n\left(n-1\right)\left(2n+5\right)}{18}.}$ A positive Z statistic means an increasing trend and a negative Z statistic denotes a decreasing trend. If |Z| > 1.96, then the null hypothesis is rejected, i.e., the trend is significant at the 95% confidence level, and vice versa.

Path analysis

Path analysis is a multivariate statistical technique, proposed by Wright (1921), for analyzing the causal relationship between variables. It is an extension of regression analysis, and can partition direct and indirect correlations to differentiate between correlation and causation (Wright, 1934; Li et al., 2011a; Li et al., 2011b; Zhang et al., 2014). The causal relationships for the impact of climatic factors on runoff in the Qira River basin were analyzed using a path diagram, from which the direct and indirect impacts of climatic factors on the river runoff could be readily identified.

Assuming there to be a dependent variable y and multiple independent variables x₁, x₂, …, x_n, the correlation coefficient between any two independent variables, x_i and x_j, is r_ij. The direct path coefficient from the independent variable x_i to the dependent variable y is P_yi, while the indirect path coefficient of x_i, through x_j to y, is r_ijP_yj. The sum of the direct and indirect path coefficients (r_iy) of x_i to y means the correlation coefficient between x_i and y. The expression is given by: (5)${\displaystyle r_{iy}=p_{yi}+\sum _{j=i+1}^n{r}_{ij}{p}_{yj}.}$

Figure 2 shows a schematic diagram of path analysis, describing the independent variables (x₁, x₂, …x_n) and a dependent variable (y) with a residual term (r). Arrows represent the causal relationships. x₁ → y, x₂ → y, …, x_n → y denote direct paths with direct path coefficients of P_y1, P_y2, …, P_yn, respectively. x₁↔x₂, …, x_n−1↔x_n refer to correlations among variables with correlation coefficients for r₁₂, …, r_n−1,n, respectively. x₁↔x₂ → y…, x_n−1↔x_n → y are indirect paths with path coefficients r₁₂P_y2, …, r_n−1,nP_yn, respectively. r → y is the path of the residual term with path coefficient P_ye.

Mann–Whitney test

The Mann–Whitney test is extensively used to determine abrupt changes in climatic and hydrological time series (Ling et al., 2011). The abrupt-change points in meteorological and hydrological series were explored in this research. Given the variable series X = x₁, x₂, …x_n, the time series can be partitioned into two parts: Y = x₁, x₂, …x_n1 and Z = x_n1+1, x_n1+2, …, x_n1+n2. The Mann–Whitney abrupt-change test statistic is expressed as: (6)${\displaystyle U=\frac{\sum _{t=1}^{n1}R\left(x_t\right)-\frac{n_1\left(n_1+n_2+1\right)}{2}}{{\left[n_1{n}_2\left(n_1+n_2+1\right)/12\right]}^{1/2}},}$ where R(x_t) is the rank of the variable series. If |Z| > 1.96, then the null hypothesis H₀ is rejected, i.e., abrupt change is significant at the 95% confidence level, and vice versa.

Wavelet analysis

Wavelet analysis is a time-frequency analysis method that can characterize local features in time and frequency domains, and has been proven as a powerful technique for insight into the periodicity of climate variables and runoff time series without requiring the time series to be stationary (Torrence & Compo, 1998; Labat, 2005; Xu et al., 2013).

Assuming that $f\left(t\right)$ is a discrete signal, such as the time series of climatic factors or runoff, a continuous wavelet function $\phi \left(\tau \right)$ is expressed by: (7)${\displaystyle \phi \left(\tau \right)=\phi \left(a,b\right)=|a{|}^{-1/2}\phi \left(\frac{t-b}{a}\right),}$ where t refers to time, and a and b are the scale and translation parameters, respectively. The continuous wavelet transform of the discrete signal is defined by the convolution of $f\left(t\right)$ and a continuous wavelet function $\phi \left(\tau \right)$: (8)${\displaystyle W_f\left(a,b\right)={\left|a\right|}^{-1/2}{\int }_{-\infty }^{+\infty }f\left(t\right){\phi }^{*}\left(\frac{t-b}{a}\right)dt,}$ where W_f(a, b) indicates the wavelet coefficient, and ${\phi }^{*}\left(\tau \right)$ is the conjugate function of $\phi \left(\tau \right)$. The wavelet variance is used to detect the existing periods in the signal. This can be given by: (9)${\displaystyle W_f\left(a\right)={\int }_{-\infty }^{+\infty }{\left|W_f\left(a,b\right)\right|}^{2}db.}$ The Morlet wavelet function with non-orthogonality was selected in this study as the continuous wavelet function to analyze the characterization of periodicity in climatic factors and runoff time series.

Correlation dimension

The correlation dimension approach has been widely used to determine if climatic and hydrological time series show chaotic dynamic characteristics (Sivakumar, 2007; Xu et al., 2010). This is important for providing information on embedding the attractor and determining the number of parameters that exist in the evolution of the variable.

Suppose that $f\left(t\right)$ is the time series of a variable, such as climatic factors or runoff, it is generated via a nonlinear system with n degrees of freedom. The series of state vectors [$f^{\left(n\right)}\left(t\right)$] is constructed in the n-dimensional phase space: (10)${\displaystyle f^{\left(n\right)}\left(t\right)=\left\{f\left(t\right),f\left(t+\tau \right),\dots ,f\left(t+\left(n-1\right)\right)\tau \right\},}$ where n denotes the embedding dimension and τ is the time delay. If there are deterministic dynamics in the nonlinear system, the trajectory in the phase space converges towards the “attractor.” The correlation dimension of the attractor is calculated by the G-P method (Grassberger & Procaccia, 1983). For a detailed description of the G-P method, readers are referred to Xu et al. (2010). Briefly, however, the expressions are given as: (11)${\displaystyle C\left(r\right)=\frac{1}{N_r^2}\sum _{j=1}^{N_r}\sum _{i=1}^{N_r}\Theta \left(r-|f_i\left(t\right)-f_j\left(t\right)|\right),}$ (12)${\displaystyle \Theta \left(x\right)=\left\{\begin{array}{ll}0,\hfill & x\le 0\hfill \\ 1,\hfill & x>0,\hfill \end{array}\right.}$ (13)${\displaystyle N_r=N-\left(m-1\right)\tau ,}$ where r and N_r are the surveyor’s rod for distance and the number of reference points from N, which is the number of points. When the hypersphere of the radius, r → 0, the correlation exponent (d), i.e., the correlation dimension of the attractor, is expressed by (14)${\displaystyle C\left(r\right)\propto r^d,}$ the correlation exponent (d) is the slope coefficient of $ln\left(C\left(r\right)\right)$ versus lnr, and can be solved by the least-squares method. In this study, the chaotic dynamic characteristics of the temperature, precipitation and runoff time series were analyzed using the correlation dimension approach.

R/S analysis

The R/S analytic method, which is a fractal theory for analyzing long-term correlation characteristics of variable series (Li et al., 2008; Xu et al., 2010), has been widely used in meteorology, hydrology and other fields (Feng et al., 2008). Considering X₁, X₂, …, X_N as the time series of variables, such as annual mean air temperature, accumulated precipitation and runoff, for a period of time N it will be characterized by the mean ($\bar{X}$), the standard deviation (S) and the range (R), which can be briefly calculated via the following expressions:

The mean value series is given as (15)${\displaystyle \overline{X}=\sum _{i=1}^n\left(X_i\right)/N.}$ The accumulative deviation is (16)${\displaystyle X\left(N\right)=\sum _{i=1}^N\left(X_i-\overline{X}\right).}$ The extreme deviation is written by (17)${\displaystyle R\left(N\right)={max}_{1\le i\le n}\left(X\left(i\right)\right)-{min}_{1\le i\le n}\left(X\left(i\right)\right);}$ The standard deviation is (18)${\displaystyle S={\left[\frac{1}{N}\left(\sum _{i=1}^N{\left(X_i-\overline{X}\right)}^2\right)\right]}^{\frac{1}{2}}.}$ The R/S is proposed by the empirical law (Hurst, 1951) (19)${\displaystyle R/S={\left(\alpha N\right)}^H,}$ where H is the so-called Hurst exponent. According to (ln(N), ln(R/S)), H is the slope coefficient, which can be obtained by the least-squares method in a log–log grid.

The Hurst exponent (H) reflects a long range correlation of a time series and its value ranges between 0 and 1. When H > 0.5, the time series has a long-term, enduring characteristic, and the future trend of variable series will be consistent with the past. That is, if the past trend of a series was increasing, the future will also indicate an increasing trend. When H = 0.5, the time series is completely independent, and the change is stochastic or only a short-range correlation (Joan et al., 1991). When H < 0.5, the time series has a long-range anti-persistence and its future tendency will be the opposite change. That is, if the trend of a series was increasing, the future will show a decreasing trend.

Variation trends in average temperature, accumulated precipitation and runoff

The linear trends in the annual average temperature (AN-AT), annual accumulated precipitation (AN-AP) and runoff (AN-AR) time series during the study period (1961–2010) in the Qira River basin were investigated using regression analysis (Fig. 3). The time series of AN-AT and AN-AP showed a total increase, with a rate of increase of 0.28 °C/10a and 17.2 mm/10a during the last 50 years, respectively; whereas, the time series of AN-AR indicated a slight decrease, at a rate of −0.03 × 10⁸ m³/10a.

The statistical significance of the variation trends in the AN-AT, AN-AP and AN-AR time series was assessed using the Mann–Kendall test. The statistic of the nonparametric Mann–Kendall trend test for the AN-AT time series was z = 4.78 > z_0.025 = 1.96, which meant that the null hypothesis was rejected during the study period (1961–2010), i.e., the AN-AT time series in the Qira River basin during that period possessed a significant increasing trend at the 0.05 confidence level. However, the result for the AN-AP time series was z = 1.30 < z_0.025 = 1.96, indicating that the increasing trend of the AN-AP time series in the Qira River basin was not significant at the 0.05 confidence level. Similarly, the AN-AR time series result, z = − 1.05 > z_0.025 = − 1.96, meant that the null hypothesis could not be rejected for the study period (1961–2010). The decrease in the AN-AR time series in the Qira River basin was not a statistically significant trend at the 0.05 confidence level.

The variation trends of seasonal average temperatures during the period 1961–2010 in the Qira River basin, including that of the spring average temperature (SP-AT), summer average temperature (SU-AT), autumn average temperature (AU-AT) and winter average temperature (WI-AT), were identified using regression analysis and the Mann–Kendall trend test (Table 1). All the seasonal average temperature time series increased significantly at the 0.05 confidence level in the last 50 years. The rates of increase in the SP-AT and AU-AT time series were comparable with that of the AN-AT time series. However, among all the seasonal average temperatures, the rates of increase in the SU-AT and WI-AT time series were the smallest and largest, respectively, implying that the temperature in winter featured the most prominent increase, while the temperature in summer increased more slowly during the past 50 years.

Similarly, the seasonal accumulated precipitation time series during the period 1961–2010 in the Qira River basin, including the spring accumulated precipitation (SP-AP), summer accumulated precipitation (SU-AP), autumn accumulated precipitation (AU-AP) and winter accumulated precipitation (WI-AP) time series, were also explored, using the same methods (Table 1). All of the seasonal accumulated precipitation time series showed insignificant increases at the 0.05 confidence level in the last 50 years. Among them, the rate of increase in SU-AP was the largest, meaning the increase in annual accumulated precipitation was concentrated mainly in summer. Meanwhile, the rate of increase in the WI-AP time series was also greater than that of the others, implying an increase in snowfall in winter during the last 50 years.

The linear trends in the spring accumulated runoff (SP-AR), summer accumulated runoff (SU-AR), autumn accumulated runoff (AU-AR) and winter accumulated runoff (WI-AR) time series during the study period (1961–2010) in the Qira River basin were also analyzed, again using the same methods (Table 1). The SP-AR, SU-AR and AU-AR time series showed an insignificant decreasing trend at the 0.05 confidence level during the last 50 years. Only the WI-AR time series showed an insignificant increasing trend at the 0.05 confidence level, with a rate of 0.002 × 10⁸ m³/10a. This is because the increase in snowfall, as well as the increment of glacier-/snowmelt with the rise in temperature in winter, would have led to a simultaneous increase in runoff.

Correlation analysis between average temperature, accumulated precipitation and accumulated runoff

According to the criterion of multiple linear regression analysis, there is a significant linear correlation relationship between every independent variable and the dependent variable. Therefore, path analysis, as a special paradigm of multiple regression analysis, is necessary to verify the linear significance between the independent variables and dependent variable. Moreover, before using path analysis to reveal the impact of the independent variables on the dependent variable, which is required to follow the normal distribution, the normality of the dependent variable needs be tested via the Kolmogorov–Smirnov test (Frank & Massey, 1951).

Correlation analysis was used to verify the linear significance between the temperature, precipitation and runoff. Table 2 shows the Pearson correlation coefficients between the temperature, precipitation and runoff at the annual and seasonal level for the Qira River basin. The correlation coefficient between AN-AT and AN-AR, and between AN-AP and AN-AR, was significant (values of −0.248 and 0.545 at the 0.01 confidence level, respectively). This implies that the closest linear relationship in the Qira River basin existed between runoff and precipitation, while temperature showed a negative effect on runoff due to strong evaporation and/or infiltration.

The correlation between SU-AT and SU-AR at the seasonal level, as well as between SU-AP and SU-AR, was significant (P < 0.01); whereas, between WI-AT and WI-AR, and between SP-AP and SP-AR, the correlation was significant, at the 0.05 confidence level. The other variables in the four seasons showed insignificant correlation at the α ≤ 0.05 significance level. The correlation coefficients between SP-AT, SU-AT, AU-AT and the corresponding seasonal runoff were negative values, implying that strong evaporation and/or infiltration weakened the increase in runoff as temperatures increased. However, the correlation coefficient between WI-AT and WI-AR was 0.335, which was larger than that between WI-AP and WI-AR (−0.046). Precipitation in the form of snowfall, in winter, is accumulated as the snowpack. The process of glacier-/snowmelt in the mountains then begins and directly contributes to runoff when the temperature increases significantly in winter. This is why only the contribution of temperature showed a significant and positive correlation with runoff.

In addition, the correlation coefficient between SP-AP and SP-AR, and that between SU-AP and SU-AR, was also significant and positive (0.291 and 0.403, respectively). This reveals that some of the precipitation in spring was able to occur as rainfall, which may have contributed directly to runoff, and the precipitation in summer was the dominant recharge resource for runoff. These results indicate that the correlation between the precipitation and runoff in spring, summer and autumn was closer than that between temperature and runoff, while the opposite was true in winter.

Impacts of temperature and precipitation on runoff

The requirement for path analysis is that there is a significant linear correlation between independent and dependent variables. Meanwhile, the dependent variables need to follow the normal distribution. In the present study, these two conditions were only satisfied at the annual and summer level, in which the linear correlations between temperature, precipitation and runoff were significant (P < 0.01), and accumulated runoff was found to follow a normal distribution based on the Kolmogorov–Smirnov statistics of 0.087 (P < 0.05) and 0.089 (P < 0.05), respectively. Therefore, path analysis was implemented at the annual and summer level to investigate the impacts of temperature and precipitation on runoff in the Qira River basin (Table 3).

The direct effect of AN-AT on AN-AR was −0.280, and the indirect effect of AN-AP was 0.032l within a year. The total effect of AN-AT was −0.296. The direct effect of AN-AP on AN-AR was 0.561, and the indirect effect of AN-AT via AN-AP on AN-AR was −0.016 during a year. The total effect of AN-AP within a year reached 0.593. The results via path analysis reveal that the impact of precipitation on runoff formation constituted the main driving effect, while that of temperature on runoff showed a negative effect, implying that the occurrence of strong evaporation and/or infiltration, as temperatures increased, indirectly weakened the increase in runoff in the Qira River basin. In addition, the path coefficient of the residual term of AN-AR was 0.791 (P < 0.01), implying that AN-AR was also influenced by other causal factors. Additional factors (e.g., land cover, soil moisture) should therefore be considered in future studies.

Similarly, in summer, the direct effects of SU-AT and SU-AP on AN-AR were −0.366 and 0.423, respectively, and their indirect effects were 0.151 and −0.131, respectively. The total effect of SU-AT and SU-AP on runoff was −0.497 and 0.574, respectively. This also means that, in summer, the main impact upon runoff was from precipitation and, annually, precipitation was the main contributor to runoff. However, the temperature in summer showed a weakening effect on the runoff. Moreover, the path coefficient of the residual term of SU-AR was 0.839 (P < 0.01), implying that other factors also influenced SU-AR. These factors should be investigated in further work.

Abrupt changes in average temperature, accumulated precipitation and runoff

Hydro-climatic processes constitute a highly complicated nonlinear system, and meteorological and hydrological time series generally show characteristics of fluctuation and nonlinear change. Abrupt changes tend to exist in the time series of variables due to changes in the global climate and the influence of human activities. Analysis of the AN-AT, AN-AP and AN-AR time series, for the period 1961–2010 in the Qira River basin, indicated that abrupt changes occurred in 1997, 1987 and 1995, respectively (Fig. 4). The abrupt changes in these time series were verified by applying the Mann–Whitney test. The results showed that the null hypothesis (H₀) was rejected at the 0.05 confidence level, i.e., abrupt changes did indeed occur, at a statistically significantly level, around 1997, 1987 and 1995 for AN-AT, AN-AP and AN-AR, in the Qira River basin, respectively (Table 4). The mean values of AN-AT, AN-AP and AN-AR before the abrupt changes were 0.44 °C, 134.48 mm and 1.29 × 10⁸ m³, respectively, and, after the changes, were 1.37 °C, 162.02 mm and 1.04 × 10⁸ m³ respectively. The increment rates for AN-AT and AN-AP were 207.40% and 20.48%, respectively, while for AN-AR it was −19.81%, after the abrupt changes.

Abrupt changes in temperature, precipitation and runoff at the seasonal level in the Qira River basin were also detected, again by using the Mann–Whitney abrupt change test (Table 4). A common point of abrupt change was verified to have occurred in 1997 with regard to average temperature, in all four seasons. This means that the abrupt change of average temperature in each season remained consistent with that of annual average temperature. The average temperature in the four seasons, to differing degrees, increased after the abrupt change. The increment rate for SP-AT, SU-AT, AU-AT and WI-AT was 64.90% , 4.37%, 176.93% and 10.66%, respectively. Abrupt but statistically insignificant changes (at the 0.05 confidence level) in precipitation and runoff in all seasons except summer were detected. The SU-AP and SU-AR time series featured abrupt changes in 1987 and 1995, respectively, implying that the abrupt changes in annual precipitation and runoff were synchronous with those in summer.

Multiple time scale analysis of average temperature, accumulated precipitation and runoff

To further identify the effects of climatic factors at multiple time scales on the runoff in the Qira River basin, wavelet analysis was used to determine the periodicity, phase change and periodic intensity of temperature, precipitation and runoff at these different time scales in the region. The peak values of the wavelet variance of AN-AT occurred in the fourth year, based on the chi-squared test at the 0.05 confidence level (Fig. 5A). This indicates that the AN-AT time series had a significant 4-year period. Moreover, according to the interdecadal variations, the wavelet variance of AN-AT still featured local maximums in the 11th, 17th and 22nd years. This means that the AN-AT time series also possessed periods of 11, 17 and 22 years, albeit not statistically significant at the 0.05 confidence level. On the basis of the wavelet mean-square variance, the most significant periodic fluctuations occurred during 1971–1973 and 1993–1996, at the 0.05 confidence level (Fig. 5C). Meanwhile, the wavelet coefficients of the 4-year period were in a positive phase during 1971–1972 and 1993–1995, when the AN-AT time series showed its high temperature phase. The other years indicated a lower temperature phase (Fig. 5B).

Similarly, the AN-AP time series in the Qira River basin featured significant periods of 4 and 8 years (Fig. 6A). The periodic changes on the 4- and 8-year time scale, within which the most significant periodic fluctuations occurred during 1980–1997 and 2003–2010, are statistically significant at the 0.05 level (Fig. 6C). The wavelet coefficients revealed alternately high and low precipitation during the significant periodic fluctuations (Fig. 6B). In addition, the AN-AP time series still possessed periods of 10, 15 and 22 years. However, these were not statistically significant at the 0.05 confidence level.

Figure 7 shows that the AN-AR time series in the Qira River basin possessed significant periods of 3 and 8 years (Fig. 7A). Moreover, statistically insignificant periods of 10, 13 and 22 years still existed within the AN-AR time series. In terms of significant periods, there were remarkable periodic changes in AN-AR during 1978–1988 and 2002–2010 (Fig. 7C). From the wavelet coefficients, the AN-AR time series indicated an alternately high and low flow during the significant periodic fluctuations (Fig. 7B). This implies that the period change of the AN-AR time series was caused by the synthetic effect of the AN-AT and AN-AP series at multiple time scales.

In addition, the periodicity and periodic intensity of average temperature, accumulated precipitation and runoff at the seasonal level in the Qira River basin were explored (Table 5). The significant periods of all the variables at the seasonal level occurred during 3–8 years. The significant periods and intensity of the AU-AT and AU-AR time series were consistent with those of AN-AT and AN-AR, respectively, while the significant period and intensity of SP-AP were basically synchronous with AN-AP in the last 50 years. This implies that the multiple time scale characteristics of annual temperature and runoff in the Qira River basin were consistent with the temperature and runoff in autumn, respectively, while that of annual precipitation was consistent with the precipitation in spring.

Chaotic dynamic characteristics of average temperature, accumulated precipitation and runoff

The correlation dimension method is appropriate for identifying chaotic dynamic characteristics of variables. The AN-AT, AN-AP and AN-AR time series in the Qira River basin were reconstructed in phase space, and the values of the correlation integrals, C(r), in the different values of the radius, r, were calculated. The relationship between ln(C(r)) and ln(r) for AN-AT, AN-AP and AN-AR, with different embedding dimensions, m, are shown in Figs. 8A, 8C and 8E, respectively. The correlation exponent (d(m)), which denotes the slope coefficient of ln(C(r)) versus ln(r), increased with embedding dimension and gradually reached a saturated correlation exponent, i.e., the correlation dimension of attractor (d_s).

Figures 8B, 8D and 8F demonstrate the relationship between the correlation exponent (d(m)) and embedding dimensions (m) of AN-AT, AN-AP and AN-AR, respectively. When the embedding dimensions (m) were, respectively, 7, 8 and 7, and the corresponding correlation dimension of attractor (d_s) was 3.46, 2.67 and 2.77, the correlation dimension reached saturation and trended towards stabilization. Because all the values of d_s were greater than 2, and were not equal to integers, this revealed that the AN-AT, AN-AP and AN-AR time series in the Qira River basin featured fractal characteristics, representing complex chaotic systems with a limited degree of freedom. If the dynamic system of variables was expressed, at least the nonlinear differential equations of 4, 3 and 3 independent variables and, at most, those of 7, 8 and 7 independent variables, were needed to describe the dynamic change characteristics of the AN-AT, AN-AP and AN-AR time series, respectively.

Moreover, the embedding dimensions (m_s) and attractor dimensions (d_s) of average temperature, accumulated precipitation and runoff at the seasonal level are shown in Table 6. All the seasonal temperature, precipitation and runoff results remained consistent with their respective annual results, except for the embedding dimension (m_s) and attractor dimension (d_s) of the average temperature in autumn and accumulated precipitation in spring. They required at least the nonlinear differential equations of 3 and 4 independent variables and, at most, those of 6 and 7 independent variables, to describe the dynamic change characteristics of the AU-AT and SP-AP time series, respectively.

Long-term memory characteristics of average temperature, accumulated precipitation and runoff

R/S analysis is able to identify the fractal of variables to reveal the long-term memory very well. To determine the continuity of change tendency of both climatic factors and runoff in the future, Fig. 9 shows scatter diagrams and linear fitting curves of the R/S analyses of the AN-AT, AN-AP and AN-AR time series, based on the abscissa axis for the logarithm of lag time n and the ordinate axis for the logarithm of R/S values, respectively. The values of H for the AN-AT, AN-AP and AN-AR time series were greater than 0.5. This means that they were predicted to continue to persist into the future. The H value of AN-AR, being between the H values of AN-AT and AN-AP, implies that the runoff will be persistently affected by both temperature and precipitation in the future.

Since AN-AT and AN-AP increased gradually over the last 50 years, they will therefore likely continue to do so in the future. AN-AR decreased slightly during the period 1961–2010, implying that annual runoff in the Qira River basin will go on decreasing in the future.

The continuity of change tendency of average temperature, accumulated precipitation and runoff at the seasonal level was investigated based on the results presented in Table 7. All the seasonal variables showed long-term persistent memory, except runoff in autumn, which showed anti-persistency in the future. This implies that the seasonal temperature, precipitation and runoff will maintain their change tendencies in the future, except that runoff in autumn will show a reversed change and gradually increase in time.