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Statistical modeling of extreme with Weibull Distribution

Extreme environmental conditions at sea can have significant impacts on navigation safety and/or successful search and rescue operations. Statistical techniques are crucial for accurately quantifying the likelihood of extreme events and monitoring changes in their frequency and intensity. Extreme events are by definition rare and the SEA service will use the Weibull extreme value distributions in order to evaluate the intensity and frequency of rare events that lie far in the tails of the probability distribution of ocean variables (i.e. events that occur few time in a year in a certain region).

A valid analysis of extremes in the tails of the distribution requires long time series to obtain reasonable estimates of the intensity and frequency of rare events. The service will use time series of daily values of past ten years model outputs to estimate the pdf and evaluate extreme events belonging to the tails of the distribution for significant wave height and period, temperature and intensity of the surface currents.

Weibull Distribution

The most general expression of the Weibull probability density functions (PDF) is given by the three-parameter Weibull distribution expression,

.. math::

f(x) = \frac{\kappa}{\lambda} \left( \frac{x}{\lambda} \right)^{(k-1)} e^{-\left(\frac{x}{\lambda}\right)^k}

where:

  • β is the shape parameter, also known as the Weibull slope
  • η is the scale parameter
  • γ is the location parameter

Frequently, the location parameter is not used, and the value for this parameter can be set to zero. When this is the case, the pdf equation reduces to that of the two-parameter Weibull distribution.

Parameters of the Weibull distribution

The four parameters (mean, standard deviation, skewness, and kurtosis) of the Weibull distribution are calculated by