Deterministic Prediction of Wave-induced Ship Responses Based on Corrected Autocorrelation Functions

Deterministic time series prediction of wave-induced responses of a ship is of importance for ensuring ship’s safety and for supporting ship operation. This paper focuses on a conditional prediction method presented by Refs. 1)-4), by which real time prediction can be made using the measured autocorrelation function (ACF) of the response in study. When responses are predicted using the conditional prediction approach, smoothing and decay of the sample ACFs or accompanying power spectrum densities (PSDs) influence the prediction accuracy4). This point is attributed to the fact that the sample ACFs may include unphysical parts at large lag time, since the practical measured signal is discretized and the measurement time window is limited, see Box et al.5). Besides, sufficiently long time measurements are required to obtain reliable ACFs. However, as the real sea states is non-stationary,and moreover, the variations in ship speed and heading relative to the waves should further facilitate the non-stationarity in the wave-induced responses, it is ideal to obtain ACF and PSD from the shortest possible measurement period. To achieve a consistent method for smoothing ACFs from short-time measurements, the authors have presented a new approach using Prolate Spheroidal Wave Functions (PSWF)6),7). One central point is that the use of PSWFs facilitates smoothed ACFs and PSDs by selecting the memory time of the response to reflect the dynamics of the system. Then, the PSWF-based prediction is expected to yield better predictions in the non-stationary time sequences. However, one of problems in making predictions using PSWF-based ACFs is discussed by Ref. 7). This ‘numerical concern’ is related to inversion of the autocorrelation (AC) matrix, which can become non-positive definite as the PSWF-based ACF is introduced. Although the sample AC matrix, per se, should be non-negative definite8), it is not necessarily the case for the PSWF-based AC matrices. This problem can be avoided to some degree by adding white noise to PSWF-based ACF7). On the other, it is unclear what level of noise is appropriate, and from a practical application standpoint, a consistent method to modify the AC matrix is vital. This paper presents two approaches for making the PSWF-based AC matrix positive definite. One method is based on eigenvalue decomposition (EVD), and the other is based on a modification of PSWF-based PSD. A series of demonstrations concerning time series prediction of experimentally measured heave motion using these approaches, and then the efficiency of these two approaches will be discussed.