Geoscience Reference
In-Depth Information
in which the coefficient of variation is given by
C
v
=
[exp(
σ
nc
)
−
1]
1
/
2
. The value of
C
s
can be calculated from the data by means of the third of (13.13), and with that value (13.41)
can be solved for
C
v
as follows
C
v
=
0
5
C
s
+
C
s
4
1
/
2
1
/
3
+
0
5
C
s
−
C
s
4
1
/
2
1
/
3
.
+
.
+
(13.42)
The value of
c
can then be obtained from the second of (13.40), by substitution of this
value of
C
v
for [exp(
1]
1
/
2
. It should be pointed out that this technique does not always
produce good results; indeed, the third- and higher-order moments for most hydrologic data
sets tend to be unreliable, so that some other method to estimate the parameters may be
preferable. For instance, Stedinger (1980) has proposed a quantile method using the median
and the smallest and largest observed values.
The idea of the lognormal distribution appears to have been introduced into hydrologic
practice by Horton (1914) after a suggestion by his uncle George W. Rafter in the 1890s.
But Hazen (1914a) was probably the first to state explicitly, “. . . that if the logarithms of
the numbers representing the several floods are used instead of the numbers themselves,
the agreement with the normal law of error is closer.” For many years after that the
lognormal distribution was the main function used in the United States to describe
annual maximal river discharges for design purposes.
σ
2
nc
)
−
Example 13.6. Lognormal distribution applied to annual peak streamflows
The flow rate in the Cayuga Inlet near Ithaca, NY, has been measured since 1935
and the data have been published by the US Geological Survey (see also the web site
http://waterdata.usgs.gov/nwis). The gaging station is located at 42
◦
23
35
N, 76
◦
32
43
W, at 133 m above sea level, and it has a drainage area of 91.2 km
2
; after correction of
the original precipitation data (Korzoun
et al
., 1977) the average annual precipitation
was estimated to be of the order of 1100 mm. After sorting the annual peak flows of the
66 available years of record, each of them was assigned a probability
m
67; the resulting
data points are plotted with lognormal coordinates in Figure 13.9. The sample estimates
of the moments of these flow rates were then calculated by means of (13.13) and found
to be
M
/
16 m
3
s
−
1
,
S
51 m
3
s
−
1
and
g
s
=
=
44
.
=
33
.
1
.
969. For the logarithms the val-
ues of these moments were found to be
M
1281. The
small skew coefficient
g
s
of the logarithms suggests that the data are close to lognormally
distributed and that
c
in (13.39) can probably be neglected. This is confirmed by the fact
that the theoretical line, i.e. (13.35) or (13.36) applied with the logarithms of the data and
with the values of
M
and
S
of the logarithms, and shown as the heavy straight line 1 in
Figure 13.9, provides a good fit to the data. For comparison, some other theoretical
curves are also shown in Figure 13.9. The generalized log gamma distribution (see
Section 13.4.4) was calculated with the first three moments of the logarithms mentioned
a few sentences earlier. The first asymptotic distribution (see Section 13.4.5 below) was
calculated with the parameter values
=
3
.
557,
S
=
0
.
6793 and
g
s
=
0
.
29.08 m
3
s
−
1
, and the gen-
eralized extreme value distribution (see Section 13.4.7 below) with the parameter values
a
α
n
=
0.038 27 and
u
n
=
28.75 m
3
s
−
1
; the power distribution was applied with
=−
0.1057,
b
=
22.24 and
c
=
the parameters
a
=
28
.
23, and
b
=
0
.
4745.
