WATER RESOURCES RESEARCH, VOL. 19, NO. 1, PAGES 209-224, FEBRUARY 1983 A Cluster Model for Flood Analysis J. E. CERVANTES, ! M. L. KAVVAS, 2 AND J. W. DELLEUR School of Civil Engineering, Purdue University, West Lafayette, Indiana 47907 A two-level and two-dimensional nonhomogeneous point stochastic process is developedto model the floodpeaks of a hydrograph. The modelis a clusterprocess of the Neyman-Scott type and presents the occurrences of flood generatingmechanisms (FGM) at the precipitation level as the triggersfor clusters of flood peaksat the runoff level. The FGMs are studied in termsof their times of occurrence • and their volumes v. The volume of a FGM corresponds to the volume of precipitation in the precipitation cluster that originatesat the occurrencetime of the FGM. Thus, each precipitation cluster at the precipitationlevel is represented by its origin time and by its volume as a point in a time volumeplane, •- - (•, v). The FGM points in the time volumeplaneof the precipitation level form a two-dimensional process,Nc(•), with a nonhomogeneous rate of occurrenceh(•-). It is assumed that Nc(•) is a Poisson process. TheFGMs, in turn,generate a two-dimensional subsidiary process, Ns(tl•), in the runoff level. This process represents the occurrences of flood peaksat time t with magnitude m as a point in the time magnitude plane t - (t, rn), determinedby the conditional rate of occurrence /z(tl•'), and is alsoassumed 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-dimensionhl parameters h(•-) and/z(t[•-) areestimated. Thetheoretical rateof occurrence, thetheoretical covariance density, and thetheoretical probability mass function of theprocess arecompared withtherespective empiricalfunctionsobtainedfor severalstations located in the Ohio River Basin. A goodfit is found for theanalyzed stations asthefirst twomoments and thetwo-dimensional probability mass function of the flood peak process are preserved. INTRODUCTION Flood modeling haspreoccupied the hydrologist for a long time, and several approaches have been made to solve the problem. Two main approaches can be named:the one based on the streamflow partial duration series(pds) and the other based on the streamflow annual flood series (afs). In the afs a maximum value, finite and unique, is associat- ed with the sample process of the streamflow duringa water year and its statisticalproperties are studied. In the case of the pds, the number of points in the sampleis increased by selecting a group of floodsrather than only the largestone. In addition, the sequence of flood peaks is considered as a random sequence of random variables. This formulation makes this approach different from the classical extreme value theory of the afs. Several authors have taken the pds approach. Borgman [1963] and Shane and Lynn [1964] took this approach to find the distribution function of a peak occurrence above a certain level under the assumption that the number of peaks in a certain interval of time (0, t) is a time homogeneous Poissonprocessand that the flow magnitudes Yi follow the exponential distribution. Todorovic and Zelenhasic [ 1970] derived the distribution function of occurrence X(t) in the time interval (to, t) under the assumption that the number of peaks is a nonhomoge- neous Poissonprocess and that the sequence Yi is a se- quence of i.i.d. random variables with exponential distribu- tion. Todorovic and Rousselle [1971] extended this work to the case when the sequence Y•are not i.i.d. and considered different distributions for Yi for different seasons. • Now at Universidad Autonoma, Chapingo, Mexico. 2 Now at the Department of Civil Engineering, University of Kentucky, Lexington, Kentucky 40506. Copyright1983by the American Geophysical Union. Paper number 2W1758. 0043-1397/83/002W- 1758505.00 Todorovic and Woolhiser [1972], under the same assump- tions as Todorovic and Zelenhasic [1970], obtained the distribution function of the time of occurrence T(t) of X(t). Gupta et al. [ 1976] found thejoint distributionfunction of the time of occurrence T(t) and the maximum peak X(t) by using the same assumptions as Todorovic andRousselle [1971]. Todorovic [ 1978]generalizedhis resultsfor the case when Y• is a sequence of randomvariables conditionally independent given the outcome of T(t). With this assumption, he found the distribution of X(t) as a function of the conditional distribution of the flow magnitude given the time of occur- rence. North [ 1980] derived the distribution of X(t) under the assumption that both the number of peaks and their magni- tudes Yi are time dependent. It may be noted that,in the papers listed above, the assumptions in the modelsare usually taken as a mathemati- cal convenience and in the majority of the casesthey do not representprecisely the physicalphenomenon to be modeled. Moreover, none of the referred works takes into account the mechanisms that generatethe flow peaks. Recently, Kavvas and Delleur [ 1975, 1981]and Gupta and Waymire [1979] have studied the space-time behavior of precipitationand their results show that the mechanisms that generate precipitation cause the clustering phenomenon in precipitations. Since precipitation is the main source of runoff, this clustering phenomenon is inherited by the flow sequences. Kavvas [1980]has studiedthe phenomenon of clustering in the flow sequences and presented a trigger model of the Neyman-Scott [Neyman and Scott, 1958] type in which the mechanisms that generate the flows are consideredas cen- ters of clusters of flood peaks and are taken at the same plane as the flowsthemselves. He usedthe probability generating functions of the cluster process to find the theoretical statisticalpropertiesof the model and assumed parametric forms for the functionsthat determine the process. It is the objective of this work to generalizethe one-plane 209