Geoscience Reference
In-Depth Information
Estimation with optimization techniques: the method of least squares
For the purpose of parameter estimation, the Muskingum channel can also be treated
as a black box, to which more objective systems techniques (cf. Section 12.2.1) can be
applied. In what follows, the method of least squares is presented as an example.
The simple form of Equation (7.37) immediately suggests that it should be possible
to estimate the constant coefficients by multiple linear regression. There is, however, one
small complication, in that the coefficients must satisfy the constraint
c
0
+
c
1
+
c
2
=
1.
This constraint can be incorporated by defining the variables
y
=
Q
e2
−
Q
e1
,
x
1
=
Q
i2
−
Q
e1
and
x
2
=
Q
i1
−
Q
e1
, and by recasting (7.37) as follows
y
=
c
0
x
1
+
c
1
x
2
(7.43)
which can be considered as a linear regression equation of
y
on
x
1
and
x
2
, forced through
the origin. By applying the method of least squares to Equation (7.43) it can readily be
shown that optimal values of the coefficients can be calculated with the following
yx
1
x
2
−
yx
2
x
1
x
2
x
1
x
2
−
x
1
x
2
2
c
0
=
and
(7.44)
yx
2
x
1
−
yx
1
x
1
x
2
x
1
x
2
−
x
1
x
2
2
and, of course,
c
2
=
c
1
=
−
c
0
−
c
1
. The summations can be performed over the time range
for which the hydrographs are available; this will yield values of the coefficients, which
perform well “on average” (in the least squares sense). However, if certain features of
the hydrograph require greater accuracy, such as for example the flows in the vicinity of
the peak discharge, it may be desirable to perform the summation over a more narrow
time range.
With these three coefficients
c
0
,
c
1
and
c
2
determined, Equation (7.37) can be applied
to solve any routing problem. In case the Muskingum parameters are needed for one or
other reason, they can be calculated from these coefficients by inversion of (7.38), that
is by using
1
=
(
c
1
+
/
(
c
0
+
K
c
2
)
c
1
)
and
(7.45)
X
=
0
.
5(
c
1
−
c
0
)
/
(
c
1
+
c
2
)
Example 7.3. Application of the multiple regression method
The
Q
i
and
Q
e
hydrographs of Example 7.2, as listed in Table 7.2 and shown in Figure
7.10, can be used to illustrate the method. The reader can verify that the sums needed
in (7.44) have the following values:
yx
1
=
2 772 315,
yx
2
=
1 942 872,
x
1
=
8 383 823,
x
2
=
3 838 571, and
x
1
x
2
=
5 454 548 (in the units of Table 7.2 squared).
With Equation (7.44) these yield the following values of the coefficients
c
0
=
0.018,
