A cluster model for flood analysis

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Abstract
A two‐level and two‐dimensional nonhomogeneous point stochastic process is developed to model the flood peaks of a hydrograph. The model is a cluster process of the Neyman‐Scott type and presents the occurrences of flood generating mechanisms (FGM) at the precipitation level as the triggers for clusters of flood peaks at the runoff level. The FGMs are studied in terms of their times of occurrence τ and their volumes ν. The volume of a FGM corresponds to the volume of precipitation in the precipitation cluster that originates at the occurrence time of the FGM. Thus, each precipitation cluster at the precipitation level is represented by its origin time and by its volume as a point in a time volume plane, τ = (τ, ν). The FGM points in the time volume plane of the precipitation level form a two‐dimensional process,Nc(τ), with a nonhomogeneous rate of occurrence λ(τ). It is assumed thatNc(τ) is a Poisson process. The FGMs, in turn, generate a two‐dimensional subsidiary process,Ns(t |τ), in the runoff level. This process represents the occurrences of flood peaks at timetwith magnitudemas a point in the time magnitude plane t = (t, m), determined by the conditional rate of occurrence μ(t|τ), and is also assumed to be a Poisson process. The statistical properties of the flood cluster processN(t), defined as the total number of flood peaks, are found in terms of the probability generating functional of the process. Using a nonparametric methodology, the two‐dimensional parameters λ(τ) and μ(t|τ) are estimated. The theoretical rate of occurrence, the theoretical covariance density, and the theoretical probability mass function of the process are compared with the respective empirical functions obtained for several stations located in the Ohio River Basin. A good fit is found for the analyzed stations as the first two moments and the two‐dimensional probability mass function of the flood peak process are preserved.