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to be simply proportional to precipitation intensity. Over the past few decades various
attempts have been made to incorporate nonlinearities in response formulations at the
catchment scale. These can be subdivided into two broad categories, which are briefly
considered in this section.
12.3.1
Functional analysis with nonlinear convolution
As outlined in the Appendix, a logical way to generalize the convolution operation to
nonlinear systems is to make use of a Volterra integral series. In the case of a stationary,
non-anticipatory system with no zero input response and with a finite memory m , this is
Equation (A31), or
m
m
m
y ( t )
=
u 1 (
τ
) x ( t
τ
) d
τ +
u 2 (
τ 1 2 ) x ( t
τ 1 ) x ( t
τ 2 ) d
τ 1 d
τ 2
0
0
0
m
m
m
+
u 3 (
τ 1 2 3 ) x ( t
τ 1 ) x ( t
τ 2 ) x ( t
τ 3 ) d
τ 1 d
τ 2 d
τ 3 +···
0
0
0
(12.46)
As before, the discrete analog of Equation (12.46) can be formulated by assuming that
both rainfall and streamflow consist of piecewise constant values x i and y i , respectively,
within the i th interval of time, where ( i
1)
t
t
i
t . For the purpose of numerical
analysis, Equation (12.46) can therefore be rewritten
m / t
m / t
m / t
y i
=
u 1 , j x i j + 1 +
u 2 , jk x i j + 1 x i k + 1
j = 1
j = 1
k = 1
m / t
m / t
m / t
+
u 3 , jkl x i j + 1 x i k + 1 x i l + 1 +···
(12.47)
j = 1
k = 1
l = 1
Different methods have been used in the past to apply the Volterra series formulation in the
rainfall-runoff context. The main difficulty has invariably consisted of the identification
of the response functions, u 1 ,
u 2 ,
u 3 ,
...,etc., in the case of Equation (12.46) or
u 1 , i ,
u 2 , ij ,
u 3 , ijk ,
...,etc., in the case of Equation (12.47). Among the more practical
examples of its application have been the studies by Amorocho and Brandstetter (1971),
Bidwell (1971), Hino et al . (1971), Diskin and Boneh (1973), Liu and Brutsaert (1978)
and Hino and Nadaoka (1979). Figure 12.23 shows an example of the results that were
obtained with a two-term approximation of Equation (12.47) in the study by Diskin and
Boneh (1973). In most of these studies one of the conclusions was that the nonlinear
formulation is better able to simulate rainfall-runoff behavior of catchment areas than
linear methods. This should not be surprising, as a representation with more adjustable
parameters normally tends to produce a better fit. However, the numerical complexities
of the computations are increased considerably.
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