Geoscience Reference
In-Depth Information
Box 4.1 Linear Transfer Function Models
B4.1.1 The Building Block: The First-Order Linear Store
A
linear store
is a model element for which the predicted output,
Q
[L
3
T
−1
], is directly propor-
tional to the storage,
S
[L
3
] (see Figure B4.1.1). Thus, we assume:
Q
S/T
(B4.1.1)
where
T
[T] is a parameter equivalent to the mean residence time of the store. For water,
the linear store is physically equivalent to a straight-sided bucket with a hole in the bottom
containing a porous material such that the outflow is laminar and proportional to the difference
in head (note that this is not the case for a simple open hole when application of the Bernoulli
equation shows that the velocity and discharge through the hole would be proprtional to S
0
.
5
).
=
(a)
S
Q = S/T
Q
(b)
Q = Q
o
exp(-t/T)
t
Figure B4.1.1
The linear store.
The mass balance equation for the linear store (or bucket) can be written as
dS
dt
=
u
−
Q
(B4.1.2)
dt
is the rate of change of storage with time and
u
[L
3
T
−1
] is an input
rate (here, an effective rainfall). To obtain an equation in the outflow
Q
, since
dS
where the differential
dS
dQ
=
T
,wecan
modify this equation to:
T
dQ
dt
=
u
−
Q
(B4.1.3)
