GUIDE WAVE ANALYSIS AND FORECASTING - WMO

The following distributions are briefly described in the order shown (see Johnson and Kotz, 1970, inter alia, for
further details):
• Normal or Gaussian
• Log-normal
• Gamma
• Weibull
• Exponential
• Rayleigh
• Generalized extreme value (GEV)
• Fisher-Tippett Type I or Gumbel
• Fisher-Tippett Type III
• Generalized Pareto
The symbols – x and s are used for the mean and standard deviation of the series x i (i = 1, n), i.e.:
x
where Σ indicates summation over i (i = 1, n).
The symbol Γ has its usual meaning as the gamma function:
Values are widely tabulated.
1. Normal or Gaussian distribution
Probability paper
Moments estimators
α˜
= x
˜ β = s
s
2
x
=
n
∑
x x
=
n
∑(
i – )
– 1
∞ z– 1 – t
dΓ
Γ() z = ∫ t e dt;
Γ′
() z =
t = 0
dz
f (x) =
1
2π β
mean = α
variance = β 2
x =Φ –1 ⎡
( p) ⎢Φ(x)=
∫
⎣⎢
y = (h)
Maximum likelihood estimators
αˆ
= x
ˆ ( – )
β = ∑ xix n
i
2
2
⎡
⎣
(x – α )2
exp –
2β 2 ⎢
x
–∞
1
2π
ANNEX III
PROBABILITY DISTRIBUTIONS FOR WAVE HEIGHTS
⎤
⎥ – ∞