Estimation of autocorrelation function and spectrum density of wave-induced responses using prolate spheroidal wave functions

Abstract

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Real-time deterministic prediction of wave-induced ship responses based on short-time measurements

“…However, in (Takami et al, 2020b), a problem in making predictions using PSWF-based ACFs was found and discussed; a 'numerical concern' related to inversion of the AC matrix, which can become non-positive definite when the PSWF-based ACF is introduced. Although the sample AC matrix, per se, should be nonnegative definite (McLeod and Jimenez, 1984), it is not necessarily the case for the PSWF-based AC matrices.…”

“…This largelag time error influences the prediction accuracy and smoothing of the ACF or the associated PSD is therefore necessary (Nielsen et al, 2018). Recently, the authors presented a new approach for obtaining smoothed ACF and PSD from short-time measurements (Takami et al, 2020b). In (Takami et al, 2020b), the PSWF (Slepian and Pollak, 1961) were used to taper the sample ACF for reducing the large-lag time errors.…”

“…Recently, the authors presented a new approach for obtaining smoothed ACF and PSD from short-time measurements (Takami et al, 2020b). In (Takami et al, 2020b), the PSWF (Slepian and Pollak, 1961) were used to taper the sample ACF for reducing the large-lag time errors. The tapered time length in ACF can be adjusted by changing the number of PSWF.…”

“…The advantage of the PSWF-based approach is that one can adjust the tapered time length to any time, while maintaining high frequency resolution in PSD, unlike the FFT-based approach. Then, it is expected that the PSWF-based smoothing method will be useful for non-stationary wave-induced response predictions, as it allows for using only recent measurements with rather small computational efforts.However, in (Takami et al, 2020b), a problem in making predictions using PSWF-based ACFs was found and discussed; a 'numerical concern' related to inversion of the AC matrix, which can become non-positive definite when the PSWF-based ACF is introduced. Although the sample AC matrix, per se, should be nonnegative definite (McLeod and Jimenez, 1984), it is not necessarily the case for the PSWF-based AC matrices.…”

“…Although the sample AC matrix, per se, should be nonnegative definite (McLeod and Jimenez, 1984), 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 ACF as demonstrated in (Takami et al, 2020b), however, it is unclear what level of noise is appropriate. From this, the authors proposed the EVD based and PSD modification approaches towards the positive definite AC matrix (Takami et al, 2020c).…”