Geoscience Reference
In-Depth Information
We assume here that the input sequence has already been suitably transformed to an effective
input that can be related linearly to the outputs.
For simple patterns of the effective input, this equation can be solved analytically. For ex-
ample, for a sudden input of effective rainfall
u
∗
into an initially dry store at time
t
o
u
∗
T
Q
t
=
exp
{−
(
t
−
t
o
)
/T
}
(B4.1.4)
This is the impulse response or transfer function of the linear store expressed in continuous
time. It has the form of an initial step rise followed by an exponential decline in the outflow.
In hydrology and many other modelling applications, it is often usual to have measurements
of inputs and outputs at discrete time increments (e.g. every hour) rather than in continuous
time. Thus, using a simple explicit finite difference form of the mass balance equation of the
linear store over a discrete time step of length
t
Q
t
−
Q
t
−
t
t
u
t
−
Q
t
−
t
T
=
(B4.1.5)
or
1
Q
t
−
t
t
T
t
T
Q
t
=
u
t
+
−
(B4.1.6)
or
Q
t
=
aQ
t
−
t
+
bu
t
(B4.1.7)
t
T
t
T
where
a
=
1
−
;
b
=
; and to ensure mass balance between total inputs and total outputs,
a
1
.
In hydrological systems, there is sometimes a delay between the start of a rainfall and the
discharge starting to rise. Assuming, for the moment, that this delay can be considered a
characteristic of the system under study, it can be introduced as
+
b
=
Q
t
=
aQ
t
−
t
+
bu
t
−
ı
(B4.1.8)
where, in this discrete time equation, the delay
ı
must be expressed as a number of time steps.
In considering more complex linear transfer function models, it is convenient to introduce
the “backward difference operator”,
z
. This is defined as
z
−1
u
t
=
u
t
−1
(B4.1.9)
so that an input delayed by
ı
time steps may be written as
z
−
ı
u
t
u
t
−
ı
=
(B4.1.10)
and the discrete time mass balance equation can be written as
az
−1
Q
t
+
bz
−
ı
u
t
Q
t
=
(B4.1.11)
or rearranging this
bz
−
ı
Q
t
=
az
−1
u
t
(B4.1.12)
1
−
B4.1.2 Higher Order Transfer Function Models
Higher order transfer function models may then be constructed easily from the basic linear
store building block components in either series or parallel structures (feedback structures are
