Persistence of Storms and Calms

Persistence refers to the time for which a storm of a given severity or a period of calm is likely to persist. In fact it is often the persistence of calms that are of more interest, for example, to offshore engineers who require a continuous period of calm weather in order to perform some difficult task.

In order to derive the statistics of the persistence of storms or calms it is usually necessary to have a reasonably continuous record of wave height, sampled at regular intervals, over a long period (say at least 1 year). The way such a record would be analysed is illustrated in Figure 4. Such data is not common, and the visual observations of wave height and wind speed used by the NMIMET analysis and by Global Wave Statistics Online certainly do not meet this requirement.


Figure 4 - The concept of Persistence.

Fortunately, however, various approximate methods have been developed [6] [34] which can derive reasonable estimates of the statistics of persistence of storms and calms from the wave height probability distribution.

The procedure used in Global Wave Statistics Online follows that of Kuwashima and Hogben [6], and is outlined in the following. The method is based on that of Graham [34] but modified following comparison of results with extensive measured data sequences.

The exceedance probability P(H>Hs) in the Global Wave Statistics Online database is expressed in terms of a 3 - parameter Weibull distribution:

where:

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

The Global Wave Statistics Online database contains the coefficients xo, b and k for each of the valid sea area / season / directional sector records.

The method given in [6] defines the probability distributions of the durations of 'storms' (Hs>Hs') or 'calms' (Hs< Hs' ) and these are expressed in terms of the exceedance probabilities Q(X>X') for normalised durations:

 

where:

  = mean duration.

These may be written:

where , C and a are to be determined.

For a given height threshold Hs' these may be estimated in terms of the exceedance probability P(Hs > Hs') which is in turn given by the Weibull distribution and the appropriate values of b, k and xo which define it.

Using the notation P' = P(Hs>Hs'), and the subscripts g for 'greater than threshold' and l for 'less than threshold' we have:


where:








Thus we can obtain the exceedance probability Q(X > X') which is one of the three possible presentation options for persistence in Global Wave Statistics Online.

The other two options present the persistence information in terms of (a) the Number of Occurrences to be expected in a given period of time and (b) the Proportion of the Time.

The number of occurrences in a given period of time is even by the expression:


where:

Ng = the number of 'storm' occurrences
Nl = the number of 'calm' occurrences
Q = Probability of exceedance of persistence duration
P = Probability of exceedance of threshold wave height
D = Duration of the period (in years)
8760 = Number of hours in a year 365 x 24

The proportion of the time is given by:

where:

R = proportion of time for duration Y
Y = storm or calm duration (hours)
Ny = Number of occurrences of duration Y in period D
D = Period (years)

This is the integration of the area under the occurrences curve. Evaluating the integral analytically it can be shown that:


where:

Rg = Proportion of time for storms of duration greater than specified value.
Rl = Proportion of time for calms of duration greater than specified value.
(the incomplete Gamma Function)

The validation of the Kuwashima and Hogben method for the estimation of storm and calm persistence statistics is given in full in [6], and is outlined in general terms in 'Persistence Analysis Validation'.