Nonparametric and parametric estimation of wave statistics and spectra

Abstract

A nonparametric bivariate density estimation technique is developed employing tensor product B-splines to provide a concise wave data summary. Most of the existing nonparametric techniques involve a certain level of subjectivity in the choice of smoothing parameters. A criterion based on the least squares concept is proposed to remove the subjective choice of smoothing parameters. Numerical experiments, in which random variables are generated from a known bivariate independent normal distribution and the modified Longuet-Higgins distribution, show that the technique reproduces the population density functions well. However, due to lack of the shape preserving property of B-splines, the positivity of the density function cannot be guaranteed. An alternative spectral estimation procedure is proposed, extending the idea of Bretschneider (1959). The alternative spectrum is the second moment of the wave height of the joint probability density function (pdf) in terms of the frequency domain, and is named the PDF spectra. Comparison of the latter with other spectral estimators such as the FFT spectral window estimator and the autoregressive spectral estimator shows good agreement. The nonparametric joint pdf provides a concise representation of long-term wave data from which one can obtain not only the usual wave statistics, but the wave spectra as well. That is, the wave spectrum is simply a subset statistical function contained in the bivariate pdf for wave height and period.