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(ii)
An historical event is known to have occurred and is the largest ever, until an even
larger event occurs during the period of record. In this case the largest event during
the period of record is assigned the return period T r =
( N
+
1), and the historical
event T r =
2]. The remainder of the record is treated as usual with the
second largest during the period of record assigned a value of T r =
[( N
+
1)
/
[( n
+
1)
/
2],
and so on for the third, etc.
(iii)
An historical record is available of all events above a certain base, such as for
example “bankful stage,” and it can be assumed that the distribution of the lesser
events during the regular period of record is typical for that of the entire histor-
ical period. When the return periods or the corresponding plotting positions are
obtained as outlined in the previous two cases (i) and (ii), there is a gap between
data points of the regular record and those of the historical events, which causes
some difficulty in deriving a best-fit curve. Such difficulty can be avoided, or at
least alleviated, by means of Benson's (1950) procedure; this consists of weight-
ing the lesser events (i.e. those below base) of the period of record more heavily
by adjusting or “stretching” their order numbers, so that they cover the historical
period. Consider H to denote the length of the historic period (e.g. the number of
years since the first historical information became available until the present), Z
the total number of events above base over that period, N the number of events
below base during the period of record, and L the number of events that cannot be
used (e.g. incomplete or missing records due to faulty equipment, etc.) during the
period of record. Thus the weight assigned to each of the N lesser events is
( H
Z )
=
W
(13.87)
( N
+
L )
and their adjusted order number is
m =
Wm
(13.88)
For example, if the regular record consists of annual observations, by this procedure
each data point below base is made to represent W years instead of 1 y. The plotting
positions and return periods of the lesser events can now be determined as before in
Section 13.2.4 but with the adjusted order number m ; for instance, with the Weibull
plotting position, these are P m =
m ).
The larger events (i.e. those above base) are not weighted, but treated as usual,
and their order numbers are not adjusted; thus they are in increasing magnitude
( H
m /
( H
+
1) and T r =
( H
+
1)
/
( H
+
1
Z
+
1), ( H
Z
+
2)
,...,
( H
1), H .
Estimation of moments
The same weighting method was also recommended in Bulletin 17B (Interagency Advi-
sory Committee on Water Data, 1982) to adjust the moments for the parameter estimation
of the generalized log-gamma distribution. From Equation (13.13) it follows immedi-
ately that, when the lesser observations are weighted in accordance with (13.87), the
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