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Hydrological events should be described through several correlated variables, so multivariate HFA has gained popularity and become an active research field during recent years. However, at present multivariate HFA mainly focuses directly on fitting the frequency distribution without confirming whether the assumptions are satisfied. Neglecting testing these assumptions could get severely wrong frequency distribution. This paper uses multivariate Mann-Kendal testing to detect the multivariate trends of annual flood peak and annual maximum 15 day volume for four control hydrological stations in the Upper Yangtze River Basin. Results indicate that multivariate test could detect the trends of joint variables, whereas univariate tests can only detect the univariate trends. Therefore, it is recommended to jointly apply univariate and multivariate trend tests to capture all the existing trends.

Hydrological frequency analysis (HFA), foundation of the hydraulic engineering design and water resources management, is mainly used for purposes of improving hydrological extreme events prediction. Researchers can estimate possible hydrological variable value of certain frequency through analyzing frequency distribution of the hydrological variable. [

For univariate HFA, extensively studies have been devoted to treat the above steps [

In fact, existing research has shown that, due to the impact of climate change and human activities, the physical mechanism of rainfall runoff process has changed a lot, the hydrological stationarity assumption has been challenged. Therefore, before step c it is necessary to apply trend analysis to the hydrological variables of interest. If the hydrological variables have significant tendency, researchers should use nonstationary HFA method, otherwise can go to step c directly.

For hydrological multivariate frequency analysis, if one only uses trend testing for each variable separately, he/she can only test whether there exists trend for the single variable and can’t check out whether there exists changing trend of the correlation between joint variables. Ignoring the testing step in multivariate HFA may lead to very inaccurate or wrong results and hence to inappropriate decisions. Therefore, this paper introduces the widely used multivariate trend testing method in water quality data to the multivariate hydrological events in order to test the stationarity of joint variables prior to selecting a probability distribution model and estimating the parameters.

For the univariate trend analysis, Mann-Kendal test [

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.

[

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} = [S_{1}, S_{2},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).

The first multivariate extension of the MK test was proposed by [_{M}^{−1} is the inverse matrix of C_{M} when it has full rank, otherwise a generalized inverse of C_{M}. The test statistic D is given as:

D is asymptotically c^{2}(q)-distributed in which q is the rank of C_{M}. if the value of D exceeds the significance level according to the c^{2}(q) distribution quantile, the joint variables are considered have a significant trend.

[

The statistic H is approximately normally distributed, with mean zero and variance given by:

in which c_{u}_{,v} is calculated according to Equation (7). For a given significant level, the covariance sum test is the same as the univariate MK test, constructing standard normal variable and then comparing the value with the threshold value.

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.

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

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

It can be seen from

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

It can be seen from

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

Station | Variable | Z value |
---|---|---|

Pingshan | Q | 0.93 |

V | 1.13 | |

Lijiawan | Q | −3.14 |

V | −3.60 | |

Beibei | Q | −1.09 |

V | −2.28 | |

Yichang | Q | −1.01 |

V | −0.35 |

Stations | CIT | Threshold | CST | Threshold |
---|---|---|---|---|

Pingshan | 1.35 | 5.99 | 1.05 | 1.96 |

Lijiawan | 13.03 | 5.99 | −3.52 | 1.96 |

Beibei | 7.73 | 5.99 | −1.75 | 1.96 |

Yichang | 1.35 | 5.99 | −0.73 | 1.96 |

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).

This paper highlights the importance of testing for trend in multivariate hydrological event and briefly introduces two multivariate Mann-Kendal test methods. Four control hydrological stations in the Upper Yangtze River basin are selected for univariate and multivariate MK test for flood event. Results indicate that significant changing trends exist in some stations including univariate or multivariate trend. For the stations with only univariate trend, we could apply nonstational hydrological frequency analysis method, while still using the traditional copula function. But for the stations with both univariate and multivariate trends, we should consider the nonsationarity for the univariate and the correlation between the variables. These tests insure the validity of the HFA results and can help to guide the selection of the appropriate multivariate distribution (margins and copula).

This work was supported by the State Key Program of National Natural Science of China (No. 51239004) and the National Natural Science Foundation of China (No. 51309105).

Lei Ye,Jianzhong Zhou,Xiaofan Zeng,Muhammad Tayyab, (2015) Hydrological Mann-Kendal Multivariate Trends Analysis in the Upper Yangtze River Basin. Journal of Geoscience and Environment Protection,03,34-39. doi: 10.4236/gep.2015.310006