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The liequency factor K is a function of the coefficient of
skew and the return period and may be obtained from
tables (e.g., Harter 1969; USWRC 1967). The form of
Equation 4.5, which is based on the use of a frequency factor,
is preferred to the more formal type of LP3 equation
(e.g., Equation A.3) for the method of moments fitting
technique, as it makes the computations very much easier.
The frequency factor idea has heen propounded by Foster
(1924) and Chow (1951), and the derivation of the factor
for the LP3 distribution is explained by NERC (1975, pp.
39-40) and Kite (1976, pp. 198-204,2291.
Method 2 This method is the same as Method l, except
that an adjustment is made to the computed skew coefficient.
An adjustment is warranted because the computed
skew value is likely to be unreliable for a data sample of
typical size. Indeed, it has been suggested (Beard and Frederick
1975) that at least 100 sample items are needed to obtain
a skew value that is representative of the population
statistic. Since most hydrological data samples are much
smaller than this, various efforts have been made to improve
the reliability of the computed skew value through
the use of generalised skew coeflficients (Beard 1977). One
example is the use of a regional skew value taken from isolines
of computed skew values.
ln the early stages of this New Zealand study, computed
skew values lbr flow stations in the top half of the South
Island were plotted on a map to determine if there was any
pattern in the skew coefficient. None was evident and,
hence, the possibility of using generalised skew coefficients
in this study was not pursued. Instead, the following tactor
Fu, recommended by Bobée and Robitaille (1975), was used
to adlust for the bias in the skew value that is due to the
length of the data sample.
Fo=
where CS = the computed skew coefficient, and
n : the number of sample items.
The adjustment is made by multiplying the computed
skew coefficient by Fu, but only when Equation 4.6 is
applicable i.e., for samples with 20 or more items.
Mefhod 3 This method is often referred to as the log-
Normal method and uses the two-parameter log-Normal
distribution, as distinct from the three-parameter one (see
Kite 1976). The method has long been advocated for use in
hydrological frequency analysis (e.g., Hazen l9t4), and appeals
because of its simplicity the fitted frequency distribution
plots -
as a straight line on log-Normal probability
paper.
The application of the method involves the same computations
as for Method l, except that the coefficient of skew
of the logarithms of the series is set to zero.
Method 4 This uses the maximum likelihood (ML)
method to fit the CEV distribution to a data sample. This
method of fitting is generally recognised as the most efficient
for estimating the distribution parameters, and its use is
recommended when the design events must be extracted
from a small or irregular series (WMO 1969). However, the
ML method involves equations that have no explicit solution.
The solution is complex and requires the use of an
iterative numerical scheme, and is only worthwhile attempting
with the aid of a computer.
Method 5 Although the GEV distribution incorporates
EVI as a special case, only rarely will the application of
Method 4 result in the EVI distribution being fitted to a
data sample. To ensure that a fit was obtained with the EVI
distribution, this distribution was fitted separately (by the
ML method) to the sample by setting the shape parameter k
in the CEV distribution to zero.
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Method ó This method is often called the "Gumbel
method" after Gumbel (1941, 1954) and is probably the
one most commonly employed in hydrology. lt has had
wide use in New Zealand and was the method adopted by
the New Zealand Meteorological Service (Robertson l9ó3)
when determining rainfall depth-duration-f'requency relationships
from New Zealand data.
Melhod 7 This method follows the procedure devised by
Jenkinson (1955, 1969) and also described by Samuelsson
(1972). The method emphasises the extreme part of annual
series and as shown by Samuelsson, it can be applied as the
standard one to extreme values which belong to several different
kinds of frequency distribution. A larger series of
S-year maxima is produced from the annual series by considering
all possible combinations of items of five in the
original series. The EVI distribution is then fittcd to rhe
series of 5-year maxima by the ML method.
If an annual series is used in a lrequency analysis instead
of a series of 5-year maxima, it is quite possible that the
series may be non-homogeneous in that, for example, the
smaller items may belong to one distribution (e.g., EV2)
and the larger ones to another (e.g., EV3). Further, it can
be shown mathematically (WMO 1969) that the lower parr
(37V0) of the series may not even belong to the extreme
value distribution as it is defined. The advantage of the Jenkinson
method is that it generally overcornes this problem
of non-homogeneity of data. The use of 5-year maxima can
be thought of an increasing by fivefold the degree ol independence
in the data, so that these maxinra should then
form a homogeneous set of data that confbrms to EV
theory.
4.4 Evaluation of the frequency analys¡s
methods
4.4.1 General
Prior to the development of the regional curves, two
evaluation tests were carried out on 42 flood records altogether,
using the seven frequency analysis methods described
in section 4.3 and contained in the computer program
FRAN. The purpose of the tests was twofold:
(¡) to observe, and to indicate to users of FRAN, the relative
merits of the seven different methods on individual
New Zealand flood records;
(ii) to assist in the selection of a frequency distribution
that would adequately describe the regional curves.
This section describes the tests and discusses the results
obtained.
4.4.2 Evaluation criteria and method
Most studies that have attempted to discriminate between
frequency analysis methods have relied, at least to some extent,
on objective goodness-of-fit indices. Recent examples
of such studies are those carried out by Benson (1968),
Beard (1974), Kite (1976) NERC (t975), Kopiuke ef ø/.
(1976) and Bobée and Robitaille (1977). However, as is generally
acknowledged (e.g., Benson 1968), the classical
goodness-of-fit indices such as Chi-square and Kolmogorov-Smirnov
are not sufficiently sensitive or powerful
enough, because of the small samples found in hydrology,
to distinguish between the worth of different frequency analysis
methods. Moreover, NERC (1975) found that other
goodness-oi-fit indices had major weaknesses and concluded
that, because of the deficiencies ol goodness-of-fìt
indices, a visual inspection must be made of the probability
plots. The judgement on the performance of a method is
then a subjective one, "... but the objective tests that are
available are so ineffective that their objectivity alone is insufficient
to recommend them" (NERC 1975).
In the evaluation tests, much more emphasis was placed
on the probability plots than on the Chi-square value,
which the computer program calculated. F-ollowing an ex-
Water & soil technical publication no. 20 (1982)