Application of Prolate Spheroidal Wave Functions for Assessment and Prediction of Ship Responses

Prolate Spheroidal Wave Functions (PSWF) proposed by Slepian and Pollak possess several properties, a) Orthogonality, b) Band-limited, and c) Fourier invariance, amongst others. Thanks to these properties, it has been also known that every band-limited time signal and its spectra can be represented by a simple linear superposition of PSWF. In the present paper, three case studies of the application of PSWF to the prediction and assessment of ship responses making use of the above properties are presented. The first case study is concerned with extreme ship response prediction, where PSWF is utilized for ocean wave representation with a small number of stochastic variables. The computational efficiency of the extreme value predictions was demonstrated in combination with the First Order Reliability Method. The second case study is focused on deterministic time series prediction of ship motion. Herein, the autocorrelation functions of the responses are used for the prediction, where PSWF is utilized for smoothing the measured autocorrelation functions and associated power spectrum densities. By taking this means, the improvement of prediction accuracy was demonstrated. The final case is an application to time-domain sea state estimation, i.e. reconstruction of wave profiles, based on response measurements, by exploiting the Fourier invariance of PSWF to detect the phase angles of waves and responses. Use of PSWF offered accurate reconstructions of the incident wave profiles even based on relatively short-time response measurements.