Geoscience Reference
In-Depth Information
Fig. 12.8 Example of convolution operation with
discrete time variables according to
Equation (12.13). The runoff values
y
i
areincmh
−
1
.
x
i
5
y
i
0
12 3
2.0
u
i
1.0
1.0
0.5
0
0
1 2
3
4
56
1 2 3 4
in which the subscript
m
denotes the number of discretized streamflow hydrograph
ordinates,
p
the number of input pulses, and
n
the number of unit response function
ordinates. Clearly, the number of ordinates of each of the functions must satisfy
m
=
p
+
n
−
1
(12.12)
which is illustrated in the following example.
Example 12.3. Numerical convolution
Consider a rainfall event consisting of
p
6 measured
output hydrograph ordinates; this implies a discretized response function with
n
=
3 rainfall input pulses, and
m
=
=
4
ordinates. Equations (12.9) and (12.10) become for this case
y
1
=
x
1
u
1
y
2
=
x
1
u
2
+
x
2
u
1
y
3
=
x
1
u
3
+
x
2
u
2
+
x
3
u
1
(12.13)
y
4
=
x
1
u
4
+
x
2
u
3
+
x
3
u
2
y
5
=
x
2
u
4
+
x
3
u
3
y
6
=
x
3
u
4
This is illustrated in Figure 12.8. for a storm with successive hourly input pulses
x
i
=
2.0, 4.0, 1.0 cm h
−
1
, on a catchment whose discretized unit response function
has the successive ordinates
u
i
=
0.3, 0.4, 0.2, 0.1.
In the identification of the response characteristics of the system, as formu-
lated in Equations (12.9)-(12.13), the ordinates
u
1
,
u
2
,...,
u
n
are the unknowns
that need to be determined. Because there are
m
(
n
) equations available, the sys-
tem is over-determined. Nevertheless, one might be tempted to determine these
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