Derivation of unit hydrograph using a transfer function approach

[1] The unit hydrograph (UH) concept and model have been widely used in the hydrological field over the past decades. However, the estimation of such a model in practice has always been a challenge for researchers and practitioners because such a model is usually ill formed in mathematical terms. The large number of parameters (or the number of ordinates) in a unit hydrograph model are correlated to a certain degree, and this could cause unstable results. So far, the research has been mainly focused on restricting the negative values and smoothing the oscillation by brute force methods, such as linear programming, and has achieved a certain degree of success. However, the number of parameters involved and the lack of stable model response are still a problem. In this study, a new model structure has been proposed that would inherently remove the negative UH ordinates and guarantee a smooth curve. This model is derived by the unit pulse response of a given discrete transfer function in the time domain by restricting its poles along the positive real axis in its Z domain (that is, there are no imaginary components and no negative real values). The model is termed the physically realizable transfer function. The strengths of its structure are that it is numerically stable, physically realizable, parsimonious in parameters, and easy to implement in real time for its state and parameter updating. Its shortcomings are that it has nonlinear pole positions and a more complicated parameter estimation process. A case study with two events in England has been used to demonstrate the application of such a model.