Bounding linear rainfall-runoff models with fractional derivatives applied to a barren catchment of the Jordan Rift Valley

Abstract

A new concept is developed to mathematically understand the dynamics of the rainfall-runoff events in a barren catchment of the Jordan Rift Valley. Time series data of rainfall and runoff have been acquired at an observation point in the catchment. Due to the extreme arid environment, water current as the runoff from the catchment is ephemeral, and the rainfall-runoff events are clearly distinguishable from each other. Firstly, a pair of linear autoregressive models with exogenous input (ARX models) is identified to tightly bound each runoff time series using the simplex method of linear programming. The exogenous input part is compatible with the conventional unit hydrograph method, while the autoregressive part is regarded as a discretized differential operator of fractional orders. Then, a linear fractional differential equation is determined to approximate each linear ARX model, which restricts the perturbation of the actual causal relationship between rainfall intensity and runoff discharge. The resulting lower and upper bounding rainfall-runoff models with fractional derivatives are examined in the system-theoretic framework. Finally, a nominal model from which actual nonlinear and stochastic phenomena perturb is arranged to envelope the all upper bounding rainfall-runoff models in the frequency domain, leading to the formulation of a challenging fractional optimal control problem involving stochastic processes.