True Form of Instantaneous Unit Hydrograph of Linear Reservoirs

This paper shows that the initial value of any instantaneous unit hydrograph (IUH) should be a positive value, however small. This unique characteristic is clearly demonstrated by the IUH of overland flow derived for a plane of uniform width based on the kinematic wave solution. Not only is a nonzero initial value a distinct and required characteristic of the IUH, it also provides an additional condition for determining the parameters of the IUH. The gamma probability density function has gained wide acceptance as an IUH for a watershed that is represented by a series of identical linear reservoirs. It has an initial zero value. This study shows that the model used in deriving the popular gamma probability density function IUH, with its characteristic zero initial value, is conceptually unsound. This paper presents a true form of IUH which results from a watershed represented by a series of identical linear reservoirs. Unlike the conceptual model used in deriving this gamma probability density function IUH, in this model the unit effective rainfall is distributed uniformly across all of the reservoirs. The resulting IUH contains both a gamma function and an incomplete gamma function, and it possesses the required positive initial value. This IUH may be viewed as a counterpart of the IUH derived from the kinematic wave solution and is named the pseudokinematic-wave IUH (PKW-IUH). By extending the PKW-IUH and using different weights for the series of linear reservoirs to reflect the shape of the watershed, a general form of IUH can be derived. For practical applications a three-parameter IUH, with a storage coefficient and two weights for the linear reservoirs, is proposed. Finally, an example to demonstrate the application of the proposed three-parameter IUH in predicting direct runoff is presented and the results are compared with the runoff predicted by the universally accepted gamma probability density function IUH.