First the MK statistic S is calculated as:

where X_j and X_i are the hydrological data values in years j and i, respectively, with j > i, N is the total number of years and sgn() is the sign function:

sgn(X) = 1 if X > 0; =0 if X = 0; =−1 if X < 0 (2)

The statistics S is approximately normally distributed, with mean zero and variance given by:

E[S] = 0 (3)

Var(S) = N(N − 1)(2N + 5)/18 (4)

The standard normal variable Z is then formulated as:

If Z is positive, the hydrological time series has an increase trend, otherwise it has a decrease trend. The null hypothesis is rejected at significance level α if |Z| > Z_1-α/2, indicating the hydrological time series has significant changing trend.

[9] developed an improved the multivariate nonparametric MK test for trend detection in water quality time series, as the extension of univariable MK test. [1] firstly introduced the multivariate MK test to hydrology field and summarized the procedures of multivariate MK test calculation.

First univariate MK test statistic is calculated according to Equation (1) for each variable, then constructing a vector using these univariate MK test statistic as:

S_multi = [S₁, S₂,¼, S_d] (6)

where d is the total number of joint variables. S_multi is asymptotically d-dimensional normal with zero mean and covariance matrix C_M = (c_u,v) _u,v=1,¼,d, with c_u,v = cov(S_u, S_v). The covariance matrix is calculated as:

in which:

when u = v, the variance is calculated according to Equation (4).

This paper analyzes the annual flood peak (Q) and annual maximum 15 d flood volume (V) multivariate trend in the Upper Yangtze River Basin by taking four control hydrological stations (Pingshan, Lijiawan, Beibei and Yichang stations) into consideration. Data series of flood peak and flood volume (V) are available for the period 1955-2005 corresponding to the four stations. Figure 1 presents the locations of the upper Yangtze River basin and the control hydrological stations

Time series of the two hydrological variables Q and V corresponding to the selected four hydrological stations

for the period 1955-2005 altogether 8 time series data are plotted in Figure 2. Figure 2 also presents the linear regression line of the time series data.

According to the univariate MK test described in Section 2.1, we applied the trend analysis for the Q and V corresponding to the four stations. Given the significant level α = 0.05, the threshold value is 1.96. When the absolute Z value for the variable is higher than 1.96, the variable has a significant changing trend. The results are shown in Table 1.

It can be seen from Table 1 that the absolute Z values of both the Q and V for Pingshan and Yichang stations did not reach 1.96, indicating no significant changing trend for the both variables. The absolute Z value of only V for Beibei station passed 1.96 and meanwhile the Z value was negative, which showed significant decreasing trend for V. For Lijiawan station, Z values for both Q and V passed the threshold value, indicating significant decreasing trend for both variables.

According to the multivariate MK test described in Section 2.3, we applied the CIT and CST trend analysis for the joint variables (Q and V) corresponding to the four stations. Given the significant level α = 0.05, when the absolute Z value for the variable is higher than threshold, the joint variables have a significant changing trend. The results are presented in Table 2.

It can be seen from Table 2 that both the CIT and CST values did not pass the threshold values for Pingshan and Yichang stations, indicating no significant changing trend for the joint variables. For Beibei station, the CIT value passed threshold and the CST value did not pass the threshold. The differences generated by two different multivariate MK test methods encourage us to combine univariate test results to judge the change trend of Beibei station. Both multivariate tests detected significant decreasing bivariate trend for Lijiawan station.

From the results of univariate and multivariate MK test we can find that the no significant change trend existed in annual flood peak, annual maximum 15 d flood volume and the joint variables for Pingshan and Yichang stations. Therefore, traditional HFA method could be used to fit the frequency distribution function of annual flood peak, annual maximum 15 d flood volume and the joint variables.

The annual maximum 15 d flood volume for Beibei station has significant decreasing trend while no significant trend was detected for annual flood peak. Meanwhile, only the CIT test detected significant change trend for joint variables, which indicated that 15 d flood volume dominated the changing trend of multivariate, but the correlation between annual flood peak and annual maximum 15 d flood volume has not changed significantly. Therefore, traditional HFA method could be used to fit the frequency distribution function of annual flood peak and the joint variables. As to annual maximum 15 d flood volume, nonstational HFA method should be applied.

The annual flood peak, annual maximum 15 d flood volume and the joint variables for Lijiawan station have

been detected with significant decreasing trend. Therefore, nonstationarity should be taken into consideration when applying HFA to Lijiawan Station including the nonsationarity for univariate and the correlation (described by copula function).