Karhunen–Loeve representation of stochastic ocean waves

A new stochastic representation of a seastate is developed based on the Karhunen–Loeve spectral decomposition of stochastic signals and the use of Slepian prolate spheroidal wave functions with a tunable bandwidth parameter. The new representation allows the description of stochastic ocean waves in terms of a few independent sources of uncertainty when the traditional representation of a seastate in terms of Fourier series requires an order of magnitude more independent components. The new representation leads to parsimonious stochastic models of the ambient wave kinematics and of the nonlinear loads and responses of ships and offshore platforms. The use of the new representation is discussed for the derivation of critical wave episodes, the derivation of up-crossing rates of nonlinear loads and responses and the joint stochastic representation of correlated wave and wind profiles for use in the design of fixed or floating offshore wind turbines. The forecasting is also discussed of wave elevation records and vessel responses for use in energy yield enhancement of compliant floating wind turbines.

[1]  Alexander Basilevsky,et al.  Statistical Factor Analysis and Related Methods , 1994 .

[2]  H. Pollak,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals , 1962 .

[3]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[4]  M. Ochi.  Ocean Waves: The Stochastic Approach , 1998 .

[5]  George E. P. Box,et al.  Time Series Analysis: Box/Time Series Analysis , 2008 .

[6]  Kari Karhunen,et al.  Über lineare Methoden in der Wahrscheinlichkeitsrechnung , 1947 .

[7]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .

[8]  P. W. Strike,et al.  Forecasting and control , 1991 .

[9]  Francesco Fusco,et al.  Short-Term Wave Forecasting for Real-Time Control of Wave Energy Converters , 2010, IEEE Transactions on Sustainable Energy.

[10]  Ian C. Moore,et al.  Prolate spheroidal wave functions, an introduction to the Slepian series and its properties , 2004 .

[11]  M. Kac,et al.  On the Theory of Noise in Radio Receivers with Square Law Detectors , 1947 .

[12]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1971 .

[13]  P. Sclavounos.  Nonlinear impulse of ocean waves on floating bodies , 2012, Journal of Fluid Mechanics.

[14]  James D. Hamilton.  Time Series Analysis , 1994 .

[15]  J. Schmee.  An Introduction to Multivariate Statistical Analysis , 1986 .

[16]  A. Basilevsky.  Statistical Factor Analysis and Related Methods: Theory and Applications , 1994 .

[17]  S. Winterstein.  Nonlinear Vibration Models for Extremes and Fatigue , 1988 .

[18]  Jean-Paul Chilès,et al.  Wiley Series in Probability and Statistics , 2012 .

[19]  S. Rice.  Mathematical analysis of random noise , 1944 .

[20]  N. Wiener,et al.  Nonlinear Problems in Random Theory , 1964 .

[21]  FLOATING OFFSHORE WIND TURBINES : TENSION LEG PLATFORM AND TAUGHT LEG BUOY CONCEPTS SUPPOTING 3-5 MW WIND TURBINES , 2010 .

[22]  Siem Jan Koopman,et al.  Time Series Analysis by State Space Methods , 2001 .

[23]  A. Kiureghian.  The geometry of random vibrations and solutions by FORM and SORM , 2000 .

[24]  P. Rousseeuw,et al.  Wiley Series in Probability and Mathematical Statistics , 2005 .

[25]  Paul D. Sclavounos,et al.  Floating Offshore Wind Turbines , 2008 .

[26]  V. Rokhlin,et al.  Prolate spheroidal wavefunctions, quadrature and interpolation , 2001 .

[27]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

[28]  Peter Bajorski,et al.  Wiley Series in Probability and Statistics , 2010 .