Stochastic analysis of ocean wave states with and without rogue waves

Abstract

This work presents an analysis of ocean wave data including rogue waves. A stochastic approach based on the theory of Markov processes is applied. With this analysis we achieve a characterization of the scale-dependent complexity of ocean waves by means of a Fokker-Planck equation, providing stochastic information on multi-scale processes. In particular, we show evidence of Markov properties for increment processes, which means that a three-point closure for the complexity of the wave structures seems to be valid. Furthermore, we estimate the parameters of the Fokker-Planck equation by parameter-free data analysis. The resulting Fokker-Planck equations are verified by numerical reconstruction. This work presents a new approach where the coherent structure of rogue waves seems to be integrated into the fundamental statistics of complex wave states.

Capturing rogue waves by multi-point statistics

“…Furthermore the statistics of the wave height maxima are well grasped by the reconstructed data, see figure 7(b). Both empirical and reconstructed data follow a generalized gamma distribution very well, as expected from [25]. From this verification of the obtained stochastic process we conclude that both the empirical data and the reconstructed data have the same multi-point statistics.…”

“…The scale has been marked by a vertical red dashed line in figure 2. Note that compared to our previous work [25] the Markov properties are fulfilled without applying a Hilbert-Huang transform (HHT) to the original data, which is due to the fact that we have now included the dependencies on h*. Based on the finding that Markov properties are fulfilled for the evolution of water surface height increments j x with decreasing time scale j t we can now proceed to estimate the corresponding stochastic process via the above mentioned Kramers-Moyal coefficients.…”

“…does the (N + 1)-point statistics reduce to N-scale statistics of the increments i x at scales i t . In our previous work [25] we had applied filtering based on Hilbert-Huang transform techniques (HHT) to the wave data. The filtering removed the dependency on h* by effectively separating off the underlying dominant frequency, i.e.…”

Rogue waves and entropy consumption

“…To convert the series of sea-level height measurements in time to its scale or increment counterparts has profound consequences, as the latter now turns out to be Markovian [5,6]. As a consequence the full analysis of the data, based on the computation of the N -point propagator p(h t |h t−τ1 , ..., h t−τ N −1 ) that predicts the time series of heights, can be decomposed into N − 2 increment propagators p(∆h k,t |∆h k+1,t , h t ) for each scale k together with the initial distribution, p(∆h 0 , τ, h t ) [6], see also supplementary material I.…”

Forecasting, hindcasting and feature selection of ocean waves via recurrent and sequence-to-sequence networks

“…Statistical properties of the ocean surface suggest that for a given time and location, ocean waves may be viewed as the summation of a considerable amount of independent regular waves caused by wind sources or wave interactions [4]. Therefore, the ocean surface can be modeled as a zero-mean Gaussian stochastic process, considering all these waves [4,5].…”