Wave duration/persistence statistics, recording interval, and fractal dimension

Abstract

The statistics of sea state duration (persistence) have been found to be dependent upon the recording interval \Delta t. Such behavior can be explained as a consequence of the fact that the graph of a time series of an environmental parameter such as the significant wave height has an irregular, "fractal" geometry. The mean duration, \bar\tau, can have a power-law dependence on \Delta t as \Delta t -> 0, with an exponent equal to the fractal dimension of the level sets of the time series graph. This recording interval dependence means that the mean duration is not a well defined quantity to use for marine operational purposes. A more practical quantity may be the "useful mean duration", \bar\tau^u, estimated from the formula (\sum\tau_i^2)/(\sum\tau_i), where each interval [t_i,t_i+\tau_i] satisfying the appropriate criterion is weighted by its duration. These results are illustrated using wave data from the Frigg gas field in the North Sea.