1. Distribution
1.1. Purpose
Fit Weibull, Rayleigh or Gumbel distributions to data and estimate responses based on the fitted probability distributions. See here for description of how to set this up in the post processor or workflow.
1.2. Description
1.2.1. Fitting of Rayleigh distribution
The cumulative distribution function for a Rayleigh distribution is
where \(\lambda\) is the location parameter and \(\sigma\) is the scale parameter. A maximum likelihood estimator is used to find the parameters.
1.2.2. Fitting of Weibull distribution
The cumulative distribution function for a 3parameter Weibull distribution is
where \(\alpha\) is location, \(\beta\) is scale and \(\gamma\) is shape. The Weibull parameters are using the method of moments. If the user specifies a threshold value, then the parameters are fitted using an iteration over the method of least squares:

The location parameter is calculated using the method of moments.

For location parameters from 0.5 to 1.5 times the location parameter found using method of moments, estimate the scale and shape using method of least squares.

The combination of location, scale and shape which gives the smallest residual of the least square fit is chosen.
1.2.3. Fitting of Gumbel distribution
The cumulative distribution function for a Gumbel distribution is
where \(\mu\) is location and \(\beta\) is scale. The Gumbel parameters can be fitted using the method of moments.
1.2.4. Estimation of responses
The user may choose to compute the response for a given probability level. For Rayleigh and Weibull distributions the user has the option Use return period, which enables the user to give a return period (for instance 3 hours) and estimate a corresponding characteristic extreme value. The characteristic value \(u_N\) of \(\{x_i\}^N_{i=1}\) is defined as \(P(u_N) = 1 − 1/(N+1)\). If the simulation length is \(T_{sim}\) and the user wants an estimate of the peak value corresponding to a length of \(T_{return period}\), then this can be found as \(P(u_{return period}) = 1 − \frac{T_{sim}}{(N+1) T_{return period}}\).
1.3. Input
Only input from Peaks is allowed for the Rayleigh distribution. Any equidistant signal (see Signal Types ) is allowed in Weibull and Gumbel. Note that the option Use return period requires input from Peaks.