# Long Term Statistics

The Long Term Statistics input data contains statistical descriptions of the weather conditions to calculate extreme values based on selected return period. The data is split into different periods, where you can input different statistical properties for each period.

The long-term statistical distributions are often applied for ultimate and accidental state analyses (ULS and ALS). Combinations of return values are often dictated by rules and regulations, it is for instance common to apply a combination of 100 year return values for wind and waves with a 10 year return value for current on the Norwegian Continental Shelf.

If the directional extreme values are not adjusted to match the omni-directional extreme value they might have to be adjusted according the to procedure given in NORSOK Standard N-006 [norsokn006]

## 1. Wind

The long-term wind speed is often assumed to be Weibull distributed

where \(u \geq max(0,\epsilon)\)

In the equation above, u is the 1-hour mean wind speed, \(\epsilon\) the location parameter, \(\theta\) the scale parameter and \(\gamma\) the shape parameter. The extreme wind speed \(u_R\) for a certain return period \(R\) can be found as

where p is the sector or monthly probability and \(\tau\) is the duration of the event (1 hour for mean wind speed). The relation between probability of exceedance q and the return period is

where T is 1 year.

Usually, the Weibull parameters and the sector probability are given for 12 directions/sectors.

## 2. Wave

The long-term variation of the waves is often assumed to be LogNormal-Weibull distributed:

In the equation above, \(f_{H_sT_p}\) is the joint probability density function for \(H_s\) and \(T_p\), \(f_{H_s}\) is the probability density function for \(H_s\) and \(f_{T_p | H_s}\) is the conditional probability distribution function for \(T_p\) given \(H_s\).

For wind, the extreme value associated with a return period was a single value. Waves are described by a bivariate distribution and hence the extreme values associated with a return period lies on a contour line.

Usually, several combination of Hs and Tp are given for 12 directions and for various probability levels, corresponding to 1, 10, 100 and 10000 year return period.

## 3. Current

The long-term current speed is often assumed to be Weibull distributed

where \(u \geq max(0,\epsilon)\)

In the equation above, u is the 10-minute mean current speed, \(\epsilon\) the location parameter, \(\theta\) the scale parameter and \(\gamma\) the shape parameter. The extreme current speed \(u_R\) for a certain return period \(R\) can be found as

where p is the sector or monthly probability and \(\tau\) is the duration of the event (10 minutes for mean current speed). The relation between probability of exceedance q and the return period is

where T is 1 year.

Usually, the Weibull parameters and the sector probability are given for 12 directions/sectors.