GUIDE WAVE ANALYSIS AND FORECASTING - WMO

108
where: γ = Euler’s constant ≈ 0.5772, and π 2 =
(3.14159) 2 ≈ 9.8696.
For a data set h i (i = 1, ... n), the estimates of popu-
lation mean and variance, – h and s2 , are given by:
1
h = ∑ hi
n i
(9.10)
2 1
2
s = ∑(
hi– h)
.
n – 1
The moments estimators of A and B, ~ A and ~ B, are
obtained from Equations 9.9 and 9.10, which give:
˜ –
(9.11)
If the individual hi are not available, then the mean and
variance can be estimated from grouped data given in a
histogram or a scatter plot.
The two-parameter log-normal distribution is
readily fitted by this method (see Johnson and Kotz,
1970, for further details). The three-parameter Weibull
distribution can also be fitted by moments, the third
parameter being estimated by comparing the theoretical
and estimated skewness (Johnson and Kotz, l970,
p. 257). In theory, the FT-III could also be fitted
by moments but, in practice, it does not appear to be
done.
The method of maximum likelihood consists of
finding values for the distribution parameters, such as A
and B in the FT-I (Equation 9.5), which maximize the likelihood
that the observed data come from this distribution.
This is probably the most widely used method of estimation
in statistics because it generally gives statistically
optimal estimates for large samples, but the method is
often numerically difficult and time-consuming, even if a
computer is available. For example, for FT-I, the maximum
likelihood estimates, Â and ˆB, are given by:
˜ A = h γB
˜ s
Β ≈ 6 .
π
ˆ – ˆ log exp –
ˆ
exp
ˆ –
–
ˆ
exp –
⎡ ⎛ h ⎞⎤
A = B ⎢ ∑ ⎝ B ⎠
⎥
⎣
⎦
h
h
B
B = ∑ h
.
h
Bˆ
∑
1
i
e
n i
i
i
1 i
i
n i
i ∑
i
(9.12)
These equations have to be solved numerically.
The two-parameter log-normal and the twoparameter
Weibull are readily fitted by maximum likelihood
— also requiring numerical solution of equations
— but problems can arise when used for the threeparameter
Weibull (and the three-parameter log-normal).
(See Johnson and Kotz, 1970, for details.)
Both the method of moments and the maximum
likelihood method assume data are statistically
independent and identically distributed. Neither assumption
applies to wave data but the methods appear to be
robust and to give useful results. To meet more nearly
the requirement for independent data, it might be prefer-
i
GUIDE TO WAVE ANALYSIS AND FORECASTING
able to fit data recorded at, say, 24-hour intervals — thus
obtaining separate estimates from all the midnight
values, another from all the 03.00 values, etc. — but this
is not usually done in practice. The considerable annual
cycle in wave height found in many parts of the world
means that the data are often not identically distributed.
As already mentioned in Section 9.4.1.2, it would be
more satisfactory to remove this cycle before carrying
out any statistical analysis but this is rarely attempted.
9.4.1.4 Estimating the associated value of T – z
An estimate of the zero-downcrossing or zeroupcrossing
wave period, T – z, at the time when the
significant wave height has an extreme value, such as
Hs50, is obtained by assuming a specific value for
significant wave steepness. Often, a value of oneeighteenth
is used, so that T – z is obtained from:
2πHs50 1
= ,
2
gT 18
i.e., using metres and seconds:
z
Tz≈34 . Hs50
.
Measurements indicate that one-eighteenth is
appropriate for open ocean sites, but larger values of
about one-fourteenth apply at sites where waves are
fetch-limited. In practice, a value is often obtained by a
visual inspection and extrapolation of the H s – T – z scatter
plot. These rather crude estimates of T – z are often sufficient,
but if the exact value of T – z is important — for
instance if a structure being designed is sensitive to
small changes in T – z — then a range of values should be
specified. For example, ISSC (1979) recommends a
range from approximately 2.6√H s50 to 3.9√H s50, where
H s50 is in metres. (The precise recommendation is in
terms not of T – z but of the period corresponding to the
spectral peak frequency, which is given as approximately
1.41 T – z.)
9.4.2 Return value of individual wave height
A value for the highest individual wave which might
occur is often obtained from the probability distribution
of the maximum individual wave height during the time
(about 3–6 hours) for which the return value of significant
wave height prevails. The mode — the most likely
value — of this distribution is generally quoted but other
values such as its mean may be used (see Sections 1.3.6
and 9.4.1.4 for the required T – z). Alternatively, a data set
may be obtained, consisting of the most likely highest
wave corresponding to each measurement of significant
wave height, and a return value estimated from this
data set.
Both methods are based upon the assumption that
the highest individual wave, Hmax, occurs during the
time of highest significant wave height. This is not
strictly correct. Hmax will most probably occur during
this time but might occur by chance during a time of
lower significant wave height. Values for the maximum