# BMT UK Ltd

## Extreme Wave Height Estimation

The NMIMET analysis method includes the fitting of a 3 parameter Weibull probability distribution to the wave height observations. The fit is performed on the whole dataset by means of a moment matching technique.

The Weibull fit coefficients are stored in the Global Wave Statistics Online database for each valid record. (That is for every sea area, season and directional sector that has not been rejected due to insufficient wave data or due to other apparent anomalies in the analysis - see Scatter Tables)

The Weibull coefficients can be used to make estimates of the wave height with a given probability of exceedance, or conversely discover the probability of exceedance of a given wave height.

The three parameter Weibull distribution is given by:

where:

P = probability of exceedance
Hs = Significant wave height
x0 =  Fitted Weibull Coefficient
b = Fitted Weibull Coefficient
k = Fitted Weibull Coefficient

Thus in the "Probability of a given wave height" option of Global Wave Statistics Online the required probability of exceedance of a given significant wave height is given immediately by this expression.

For the "Wave height of a given probability" option, the expression is turned around to provide the significant wave height with a given probability of exceedance:

In order to obtain the probability of exceedance of a maximum wave height, we need an expression relating the most probable maximum wave height in a storm to the significant wave height in that storm. This is given by:

where:

Hm = most probable maximum wave height
N = Number of waves in the storm

and:

where:

D = duration of the storm in hours
T= the mean period of the waves in seconds

Thus, provided that assumptions can be made about the duration of a storm and the mean wave period in the storm, we can provide probabilities of exceedance for maximum wave heights, and vice versa.

Global Wave Statistics Online has been written so that the user can select his own values for D and T, but the default values are as follows:

D = 3 hours
T = 15 seconds

Global Wave Statistics Online also works with return period rather than probabilities (e.g. the 100 year return period wave). Thus, it is necessary to connect probability of exceedance with the equivalent return period.

For 'whole year - all directions' data sets this relationship is as follows:

where:

R = return period (in years)
8760 = 365 x 24 i.e. the number of hours in a year

However, if seasonal and/or directional data sets are being used, then the probability has to be adjusted by a factor which depends on (a) the proportion of the year represented by the season, and (b) the proportion of the time the wind blows from that directional sector. Thus the more general expression is:

where:

and:

M = Number of months in the season
Nd = Number of wind observations in a directional sector
Ns = Number of wind observations in a season.

Once the factor F has been included in the calculation the resulting return period is qualified by the same restrictions as the data set. That is, if we are using a data set which is for months Jan-March and the North-West directional sector, and we calculate the 50 year wave, this will be:

The most probable largest wave exceeded on average once in every 50 years during the Jan-March season and from the North-West.

The 'Characteristic Wave Height' calculation is similar in many ways to the expressions outlined above, but with some special additional assumptions and restrictions which have been defined by DnV [4] [5].

The 'Characteristic Wave Height' is used as a design wave condition for short term offshore operations, and is based on the calculation of a maximum wave height or significant wave height with a 10% probability of exceedance during the operational period.

A two parameter Weibull distribution of significant wave height is defined in [4] as:

where P(Hs) is here the probability of non exceedance and Ho and j are parameters fitted to the available wave data.

Reference [4] suggests that the fit of this data should normally be performed on the data points with probabilities of non-exceedance greater than 0.7. However, it should be noted that in Global Wave Statistics Online, the Weibull fit has been made to the data using a 3-parameter Weibull expression, and this fit has been made using all the data by a moment matching technique. In the DnV part of the extreme analysis these three Weibull coefficient are converted to the required 2 coefficients Ho and j.

The characteristic wave height for an operation is given by:

where:

The characteristic significant wave height or characteristic sea state Hks is given by:

where:

H0 and j are Weibull parameters, and is the Gamma function of (d+0.5)
N= 14400 dn where dn is the number of days for the operation

It should be noted that the operational period should not be taken as less than 30 days in offshore operations, and less than 10 days for coastal transportation.