PE EIE[R-Rg RESEARCH ON - HJ Andrews Experimental Forest

Usually, we are interested in the output variable
at some particular point in space ( a
particular gaging station), in which case equation
23 becomes
An(t) dng(t)
+ An-1(t)
do-1g(t)
+
dtn dt n-1
▪ + A0(t) g(t) = f(t)
(24)
which describes a spatially lumped parameter ,
time-varying linear system. If we assume that
the parameters in equation 24 are time in -
variant, we obtai n
An ddgnt) + An-1 do-lg(t) +
▪
Aog(t) = f(t)
dtn- 1
(25)
which describes a time invariant, lumpe d
parameter linear system . If we assume that
the system is completely at rest at t=0, we can
write equation 25 in the form of the con -
volution equation
t
g(t) = f h(r) f(t-r) dr (26 )
0
which is the basis of the unit hydrograp h
theory and many other hydrologic techniques.
In equation 26, g(t) represents runoff,
f(t) rainfall, and h(t) is the kernel, or in thi s
case, the unit hydrograph .
In summary, application of equation 26
implies that the watershed behaves as a linear
system, it is time invariant, the rainfall is
uniformly distributed over the watershed
area, and the watershed is completely at res t
at the beginning of the rain . The "effective"
precipitation is used as an input to th e
system. This implies that we know some
method for the separation of the runof f
hydrograph into base flow and direct runoff
(fig. 9) . This approach in fact is a combination,
using system synthesis for the estimation
of the effective precipitation, and syste m
analysis for the estimation of runoff. Methods
using this technique have been developed b y
Snyder (1955), Eagleson et al. (1966), Nash
(1957, 1960), O'Donnell (1960), Dooge
(1965), and others .
Deterministic Nonlinear Hydrologic System s
As it was noted in a previous section, a
time invariant analytic system can be expanded
in Volterra series . Such an expansion
can be written in the form
y(t)=ho+ f h 1( T1) x ( t-r1) dr 1
+ f f h 2 (T1 ,T2) X (t-T1) X(t-T 2 )dr 1 dr 2
+f f hn(T 1 Tn) x(t-T 1 ) ...
x(t-Tn) dr 1
. . . drn
+ (27)
EFFECTIVE RAINFALL = DIRECT RUNOF F
xUI
Y(T)
BASE FLOW
m.'UtAut%A%atU
Figure 9 . Estimation of effective rainfall through hydrograph separation .
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