A nonstationary stochastic model for long‐term time series of significant wave height

Abstract

In this paper an attempt is initiated to analyze long‐term time series of wave data and to model them as a nonstationary stochastic process with yearly periodic mean value and standard deviation (periodically correlated or cyclostationary stochastic process). First, an analysis of annual mean values is performed in order to identify overyear trends. It turns out that it is very likely that an increasing trend is present in the examined hindcast data. The detrended time series Y(τ) is then decomposed, using an appropriate seasonal standardization procedure, to a periodic mean value μ(τ) and a residual time series W(τ) multiplied by a periodic standard deviation σ(τ) of Y(τ)=μ(τ)+σ(τ)W(τ). The periodic components μ(τ) and σ(τ) are estimated and represented by means of low‐order Fourier series, and the residual time series W(τ) is examined for stationarity. For this purpose, spectral densities of W(τ), obtained from different‐season segments, are calculated and compared with each other. It is shown that W(τ) can indeed be considered stationary, and thus Y(τ) can be considered periodically correlated. This analysis has been applied to hindcast wave data from five locations in the North Atlantic Ocean. It turns out that the spectrum of W(τ) is very weakly dependent on the site, a fact that might be useful for the geographic parameterization of wave climate. Finally, applications of this modeling to simulation and extreme‐value prediction are discussed.

Long-term time-dependent stochastic modelling of extreme waves

“…A nonstationary stochastic model for long-term time series of significant wave height was presented in Athanassoulis and Stefanakos (1995) which were modelled by decomposing detrended time series to a periodic mean value and a residual time series multiplied with a periodic standard deviation: XðsÞ ¼ X trend ðsÞ þ lðsÞ þ rðsÞWðsÞ: It was then showed that W(s) could be considered stationary. Short-term and long-term wave characteristics of ocean waves were combined in order to develop nested, stochastic models for the distribution of maximum wave heights in Prevesto et al (2000).…”

“…Since then, there are a number of studies reported in the literature which try to identify and assess previous and current trends in extreme wave climate, most with a focus on the North Atlantic, by different hindcast and reanalysis techniques combined with statistical analyses (see e.g. Kalnay et al 1996;Uppala et al 2005;Athanassoulis and Stefanakos 1995). Some of these will be briefly outlined in the following.…”

A Statistical Approach for Wave-Height Forecast Based on Spatiotemporal Variation of Surface Wind

“…where p and K are orders, (α (1) , β (1) , γ (1) , δ (1) , ω (1) ) are unknown constants and ε (1) t is a random variable that follows a zero-mean white noise process. Similarly, we write ∇(ρ (WD),t WS t cos(hWD t )) and ∇(ρ (WD),t WS t sin(hWD t )) (h = 1, .…”

“…Statistical models for dealing with measurements of long-term variations in wave height have been considered mainly from two perspectives: nonstationarity (e.g., Scheffner and Borgman (1992), Athanassoulis and Stefanakos (1995), Guedes Soares and Ferreira (1996)) and nonlinearity (e.g., Scotto and Guedes Soares (2000)). On the other hand, statistical methods for modeling wave height that take into account changes in wind speed and wind direction have also been considered (e.g., Hokimoto and Shimizu (2008), Hokimoto (2012)).…”

Statistical assessment of annual patterns in coastal extreme wave conditions

“…Furthermore, confidence on the existence of a seasonal periodicity in longterm time series of significant wave height, Hs, the most common parameter used to characterize the sea state severity, is very important when using time series models to represent the stochastic evolution of such parameter (e.g., Athanassoulis and Stefanakos [ 1 ]; Guedes Soares et al [2]). Additionally, statistical methods of extreme value analysis used for the derivation of design events, by extrapolating the data outside the range of observations, commonly assume that data are independent and identically distributed.…”