Return Period Calculation For Multivariate Hydrological Events

Return period refers to average length of time interval of the corresponding hydrological events in hydrology. It is an important measure to determine the degree of hydrological events. The traditional method of Engineering design only consider single variable to describe the regularity of the hydrological events. However, this method of return period does not take other variables into account, so it has some certain limitations. According to the current studies, Copula functions have some obvious advantages in multivariate analysis. In this paper, floods data recorded by twelve hydrological stations in northern part of Shaanxi province is used for multivariate analysis. Copula functions are used to connect marginal distributions of flood variables. And then, different definitions of return period are calculated and compared. The main process of this paper is following. Firstly, Pearson-Ⅲ distribution is fitted flood peak and volume, and lognormal distribution is fitted flood duration. The goodness tests are used to determine whether Pearson-Ⅲ distribution and logarithmic normal distribution can fit flood peak, volume and duration. Secondly, return periods of stationary and non-stationary conditions are calculated. And the dependence between flood variables are analyzed to figure out the dependency structure. Thirdly, the maximum likelihood method based on copulas is used to estimate the parameters of copula functions and get joint probability distribution of flood variables. Three types of information standard are used to determine the best model for each station. The goodness test of multivariate is used to judge whether the selected copulas are suitable or not. Finally, according to the selected joint probability distribution of stations, three kinds of return period are calculated. These return period are the joint return period, the Kendall return period and the corresponding return period. To make a comparison analysis, the flood design values are calculated by maximum likelihood theory. This paper is aim to describe the characteristic of extreme hydrological events, and provide some reference and data support for water conservancy project design and operation. The main conclusions drawn from this study are as follow:(1) Pearson-Ⅲ distribution and lognormal distributions are used to fit the flood variables at 12 stations. And the results of goodness tests indicate that Pearson-Ⅲ distribution and lognormal distribution can be used as theoretical marginal distributions of flood variables.(2) The results of univariate return periods show that the time-varying parameters probability has a good performance to fit the sequence with a trend variability. The Traditional return period calculation method is difficult to satisfy the needs of the non-stationary hydrological sequence. The correlation coefficients and correlation diagrams illustrate that the dependency between flood peak and volume is strong. On the other hand, the dependency structure between flood peak and duration is weak in some degree.(3) Some common Archimedean copulas functions are used to connect the marginal distributions to establish the multivariate joint probability distribution of flood variables. The goodness tests based on copula function show that Archimedean copulas functions can be used as connecting distribution of flood variable. And the theoretical joint probability can also be used for multivariate analysis.(4) The results of return period calculation indicate that the flood design values based on joint return period are largest, and the design values based on Kendall return period are medium among the three return periods, while the design values of corresponding return period are smallest. The main reason is that joint return expand the dangerous region of flood variables. It lead to the risk probability becomes larger under the same design values and the calculating return periods are smaller than others. In other words, the design values would be the largest if given the same return period. On the contrary, corresponding return period shrink the dangerous region of flood variable and get the smallest design values among the three return periods.(5) From the aspect of definition of return period, joint return period and corresponding return period determine the dangerous area according to the values of variables directly. These ways would expand or shrink the dangerous region, but joint return period and corresponding return period has their own return period definition. And they can also satisfy for some certain practical need. For example, if projects consider the all of design values larger than the threshold as dangerous region, corresponding return period can be used as the design standard. Anyway, Kendall return period get the dangerous region by the joint probability distribution. And it avoid some limitation on misestimating the risk probabilities. Kendall return period can tackle some generous multivariate problem. Therefore, if there is no any prerequisite of project, Kendall return period is recommended for multivariate analysis.