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

can be almost linear, but there is no linea r
system which can be almost nonlinear . In
general, linearity is a limiting case of non -
linearity. Therefore, any theory or technique
adequate for a general nonlinear system i s
equally adequate for linear systems .
Deterministic Linear Hydrologic Systems
The theory behind most linear methods ca n
be generalized in the following manner . Suppose
that s and a are continuous variable s
representing position in space, and t and r
define position in time . Consider the linear
P. D. E . of the general form
L[g(s,t)] = f(s,t) (17)
Figure 8 . Demonstration of a functional .
Hydrologic systems are physically realizabl e
where L is linear P . D. operator of arbitrary
since their outputs (runoff) at time t depen d
order, and g(s,t) some function which satisfie s
only on the past values of their inputs (precipitation)
. The "memory" of a system is th e
equation 17 within a certain region R . Given
the appropriate homogeneous boundary conditions
along R, the solution of equation 1 7
time period between some past time and th e
present for which the output depends only
can be written according to Hildebran d
upon the input . If the output depends only
(1958) as
on the present value of the input, the syste m
is said to be a "no-memory" system . If the
g(s,t) =f f G(s,t ; a,r) f (a,r) da dr (18)
output of a time invariant system is analyti c
R
about zero input at some time to, the syste m If
is "analytic". Analyticity is very important ,
f(s,t) = fs (s) ft (t) (19)
since if a system is analytic, its output can b e
equation 18 can be written in the for m
expanded in Volterra series . Hydrologic
systems are assumed to be analytic.
g(s,t) =f f(r) [f G(s,t ; a,r) do] dr (20)
A deterministic system H is said to be
s
"linear," if given the inputs X 1 (t) and X 2 (t)
such that
If fs (s) is spatially invariant, we may write
y 1 (t) = H [AX 1(t)] (1 4)
f
ti ti
g(s,t) = G (s,t ; r) f (r) dr (21)
y 2 (t) = H [BX 2 (t)] (15)
implies that
Y1 (t) + Y2 (t) = Y[ X (t)] ( 16)
= H [AX 1 (t) + BX 2 (t) ]
= AH [X 1 (t)] + BH [X 2 (t)]
that is, in a linear system, each member of a
sequence of input values influences the out -
put independently of every other . This is the
well known principle of superposition. If a
system does not satisfy the above condition i t
is said to be "nonlinear . " A nonlinear system
where
ti ti
f (r) = f(s,t), and G(s,t ; r) (22)
= fG(s,t ; a,r) da
s
We can write equation 21 in differentia l
equation form as
d
An(s,t)
dtn't) + An 1(s,t) dn-lg(s,t) +
dt n- 1
+ A 0(s,t) g(s,t) = f(s,t) (23)
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