A measure of watershed nonlinearity: interpreting a variable instantaneous unit hydrograph model on two vastly different sized watersheds

Abstract. The linear unit hydrograph used in hydrologic design analysis and flood forecasting is known as the transfer function and the kernel function in time series analysis and systems theory, respectively. This paper reviews the use of an input-dependent or variable kernel in a linear convolution integral as a quasi-nonlinear approach to unify nonlinear overland flow, channel routing and catchment runoff processes. The conceptual model of a variable instantaneous unit hydrograph (IUH) is characterized by a nonlinear storage-discharge relation, q = c^Ns^N, where the storage exponent N is an index or degree of watershed nonlinearity, and the scale parameter c is a discharge coefficient. When the causative rainfall excess intensity of a unit hydrograph is known, parameters N and c can be determined directly from its shape factor, which is the product of the unit peak ordinate and the time to peak, an application of the statistical method of moments in its simplest form. The 2-parameter variable IUH model is calibrated by the shape factor method and verified by convolution integral using both the direct and inverse Bakhmeteff varied-flow functions on two watersheds of vastly different sizes, each having a family of four or five unit hydrographs as reported by the well-known Minshall (1960) paper and the seldom-quoted Childs (1958) one, both located in the US. For an 11-hectare catchment near Edwardsville in southern Illinois, calibration for four moderate storms shows an average N value of 1.79, which is 7% higher than the theoretical value of 1.67 by Manning friction law, while the heaviest storm, which is three to six times larger than the next two events in terms of the peak discharge and runoff volume, follows the Chezy law of 1.5. At the other end of scale, for the Naugatuck River at Thomaston in Connecticut having a drainage area of 186.2 km², the average calibrated N value of 2.28 varies from 1.92 for a minor flood to 2.68 for a hurricane-induced flood, all of which lie between the theoretical value of 1.67 for turbulent overland flow and that of 3.0 for laminar overland flow. Based on analytical results from the small Edwardsville catchment, the 2-parameter variable IUH model is found to be defined by a quadruplet of parameters N, c, the storm duration or computational time step Δt, and the rainfall excess intensity i(0), and that it may be reduced to an 1-parameter one by defaulting the degree of nonlinearity N to 1.67 by Manning friction. For short, intense storms, the essence of the Childs – Minshall nonlinear unit hydrograph phenomenon is encapsulated in a peak flow equation having a single (scale) parameter c, and in which the impact of the rainfall excess intensity increases from the linear assumption by a power of 0.4. To illustrate key steps in generating the direct runoff hydrograph by convolution integral, short examples are given.