Constructing Design Rainfall Hyetographs Using Trivariate Plackett Family of Copulas

Multivariate stochastic analyses via copulas are receiving increasing attention in the hydrologic literatures due to the flexibility they offer in construction of joint distributions with various combinations of marginals and dependence structures. Among the many choices of dependence models, the Frank family of Archimedean copulas has been popular for many bivariate problems. However, there are limitations to extending the application of copulas to trivariate and higher dimensions, namely difficulties in preserving all lower-level mutual dependencies and the compatibility problem in multivariate statistics. In this study, we examine a non- Archimedean copula from the Plackett family that is founded on the theory of constant cross product ratio. It is found that the Plackett family not only performs well at the bivariate level, but also allows a hierarchical multivariate stochastic analysis where the lower-level dependencies between variables can be fully preserved. The feasible range of Plackett parameters that would result in valid (compatible) 3-copulas is determined numerically. This trivariate Plackett family of copulas is then applied to construct the design rainfall hyetograph for several stations in Indiana where the estimated parameters lie in the feasible region. Based on a given design rainfall depth and duration, conditional expectations of rainfall features such as expected peak intensity, time to peak, and percentage cumulative rainfall at 10% cumulative time increments are estimated. The results of this study suggest that the constant cross product ratio theory can be extended to continuous random variables, and that it provides further flexibility for multivariate stochastic analyses of rainfall.