VOL. 15, NO. 4 WATER RESOURCES RESEARCH AUGUST 1979 Mathematical Derivation of Linear and Nonlinear Runoff Kernels MIKIO HINO AND KAZUO NADAOKA Department of Civil Engineering, Tokyo Institute of Technology, Tokyo, 152 Japan From a viewpoint of overall grasp of response characteristics of a basin to rainfall,an integral formof expression such as the unit hydrograph,th/• Volterra series,and the Wiener series is superiorto differential equation type expressions. However, hitherto, response or runoff kernels have beenderived from empiricaldata or at the mostfrom numerical simulation data. In this paperthe runoff is represented by the Wiener-Hermiteorthogonal functionalexpansion. By application of the Galerkin technique a set of ordinary nonlinear differential equations for the expansion coefficients of kernelfunctions are derived from the basic equation of motion.The differential equations are solved by the Runge-Kutta-Gillscheme to obtain the runoff kernels. Various features of the theoretical linear and nonlinear runoff kernels thus derived comparewell with the empirical ones. INTRODUCTION In rainfall runoff systems the nonlinearityand the probabi- listicuncertainty are the mostimportantfeatures. Especially in mountainous countries suchas Japan the nonlinearity of ba- sins is so strong that the applicability of the conventional linear models, e.g., the unit hydrograph,is limited. Recently, various efforts have been made to improve the accuracy of runoff prediction by taking the nonlinearityinto account. Among the alternatives to the unit hydrograph method are the storage function method and the tank model. In these models, the nonlinearity is incorporated in the para- metricmanner,i.e., 'parametric hydrology.' The parameters in the models are to be determined, by trials, from empirical data. Stochastic H ydrology Great progresshas been made in 'stochastichydrology,' aiming at derivingobjectively by stochastic theories the infor- mation of hydrologic systems from observed hydrologic data. Various attempts have been made, with varying degrees of success, to apply Wiener's probabilisticnonlinear theory to analyzing the hydrologicdata (e.g., 6 delay filter correlation method [Hino et al., 1971; Kikkawa et al., 1970], multi- regression analysis and M eixner function expansion method [Amorocho and Brandstetter, 1971], stepwise regression method [Bidwell, 1971], Laguerre function expansion by ana- logue circuit, [Kuchment and Borshcheosky, 1971], Forsythe polynomial expansion [Zand and Harder, 1973], multi- regression analysis [Shiraishi et al., 1976], cross- and triple- correlation method [Shikashoet al., 1974], and extended6 delay filter [Hino and Sunada, 1976]). All these analyseshave been made on the basis of real empirical hydrologicdata. One of the shortcomings of these analyses is that the nonlinearkernels thus derivedare depen- dent on the individual hydrologic data and lack universality. Dynamic Hydrology Generally speaking, in cases wherethe basic equations of a phenomenon concerned are given, the phenomenon may in principle be solved mathematicallyand theoretically.Un- doubtedly, the rainfall runoff is the naturalphenomenon gov- erned by the fundamental equations of fluid motion. This attitude led to 'dynamic hydrology.' It is reasonable that a hydrologicsystem exhibits a high degree of nonlinearity, since the basic equations governing the Copyright ¸ 1979 by theAmerican Geophysical Union. Paper number 9W0309. 0043-13977797009W-0309501.00 system are nonlinear. The nonlinearity of the hydraulic equa- tions makes it difficultto obtain analyticalsolutions in closed form. Consequently, in almost all cases the solutionshave been obtained numerically. However, the numericalsolutions of the basic hydraulic equations cannot lead to a general understanding of the hydrologic systems. In other words, though it is basedon the rigorous laws describing the local dynamicprocesses, the dynamic approachcould not hitherto give an overall grasp of the total response of a nonlinear system to the input. Owing to the nonlinearityof the equa- tions, a closed form analyticalsolutioncannotbe obtained. On the other hand, althoughthe stochastic approaches are called a 'black box' system, they have several advantages. For instance, the effect of the input to the system is given in the form of an integralform of expression which helps us under- standthe overallresponse of the hydrologic system to different rainfalls. The two approaches, stochastic and dynamic, are appar- ently in sharpcontrastwith eachother. The dynamic hydrol- ogy solves analytically or numerically the basic equations, while the stochastic hydrology depends wholly on empirical data. However, we assert that the two approaches should not be exclusive of each other. In thispaperit is intended to combine these two approaches. In other words, the analytical expression of the linear and nonlinear response kernels will be derived from the fundamen- tal dynamic equations of the hydrologic system. Use will be made of Wiener'stheory of 'orthogonal functional' expansion for a nonlinear system, which has beensuccessfully applied to analyzingthe empirical data, as has been mentioned already. THEORY Model of Hydrologic System The following assumptions are made: 1. The fluctuation of rainfall is a stationary white noise of Gaussian probabilitydistribution. 2. Rainfalls are spatiallyuniform. 3. Only rainfalls effective to the surface runoff are consid- ered. 4. Parameters characteristic of the basin (such as the roughness) are considered to be time invariant. 5. An idealized rectangular basin with constant slope is considered. AssumptionsI and 4 are required for application of the Wiener theory on nonlinear random processes, while assump- 918