Probabilistic flood forecast: bounds and approximations

The probabilistic river stage forecast (PRSF) specifies a sequence of exceedance functions {Ψ̄n:n=1,…,N} such that Ψ̄n(hn)=P(Hn>hn), where Hn is the river stage at time instance tn, and P stands for probability. The probabilistic flood forecast (PFF) should specify a sequence of exceedance functions {F̄n:n=1,…,N} such that F̄n(h)=P(Zn>h), where Zn is the maximum river stage within time interval (t0,tn], practically Zn=max{H1,…,Hn}. In the absence of information about the stochastic dependence structure of the process {H1,…,HN}, the PFF cannot be derived from the PRSF. This article presents simple methods for calculating bounds on F̄n and approximations to F̄n using solely the marginal exceedance functions Ψ̄1,…,Ψ̄n. The methods are illustrated with tutorial examples and a case study for a 1430 km2 headwater basin wherein the PRSF is for a 72-h interval discretized into 6-h steps.