Abstract
Wind-generated waves are often treated as stochastic processes. There is particular interest in their spectral density functions, which are often expressed in some parametric form. Such spectral density functions are used as inputs when modelling structural response or other engineering concerns. Therefore, accurate and precise recovery of the parameters of such a form, from observed wave records, is important. Current techniques are known to struggle with recovering certain parameters, especially the peak enhancement factor and spectral tail decay. We introduce an approach from the statistical literature, known as the de-biased Whittle likelihood, and address some practical concerns regarding its implementation in the context of wind-generated waves. We demonstrate, through numerical simulation, that the de-biased Whittle likelihood outperforms current techniques, such as least squares fitting, both in terms of accuracy and precision of the recovered parameters. We also provide a method for estimating the uncertainty of parameter estimates. We perform an example analysis on a data-set recorded off the coast of New Zealand, to illustrate some of the extra practical concerns that arise when estimating the parameters of spectra from observed data.
Highlights
Due to the random nature of wind-generated gravity waves, it is common to treat them as stochastic processes
We introduce an approach from the statistical literature, known as the de-biased Whittle likelihood, and address some practical concerns regarding its implementation in the context of wind-generated waves
An alternative method to reducing correlations in the periodogram is to use tapered versions of the spectral density estimate in the Whittle likelihood (Dahlhaus, 1988), but in simulations we found omitting frequencies from the fit to be a better solution than tapering in terms of the resulting bias and variance of parameter estimates
Summary
Due to the random nature of wind-generated gravity waves, it is common to treat them as stochastic processes. There is particular interest in the spectral density function of such wave processes. For this reason, it is important that we are able to construct good spectral density estimators. We assume that the spectral density function follows a parametric form, meaning that the inference task becomes estimation of the parameters of this form. Parametric estimators are often preferable because they result in smoother estimates and more concise representations of the spectral density function—and the parameters themselves provide physical interpretation of the nature of the wave process. We can describe the displacement over time by a stochastic process, an indexed family of random variables, which we shall denote = { } ∈R.
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