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

Abstract

Predicting the wave-induced response in the near-future is of importance to ensure safety of ships. To achieve this target, a possible method for deterministic and conditional prediction of future responses utilizing measured data from the most recent past has been developed. Herein, accurate derivation of the autocorrelation function (ACF) is required. In this study, a new approach for deriving ACFs from measurements is proposed by introducing the Prolate Spheroidal Wave Functions (PSWF). PSWF can be used in two ways: fitting the measured response itself or fitting the sample ACF from the measurements. The paper contains various numerical demonstrations, using a stationary heave motion time series of a containership, and the effectiveness of the present approach is demonstrated by comparing with both a non-parametric and a parametric spectrum estimation method; in this case, Fast Fourier Transformation (FFT) and an Auto-Regressive (AR) model, respectively. The present PSWF-based approach leads to two important properties: (1) a smoothed ACF from the measurements, including an expression of the memory time, (2) a high frequency resolution in power spectrum densities (PSDs). Finally, the paper demonstrates that a fitting of the ACF using PSWF can be applied for deterministic motion predictions ahead of current time.

Appendices

Appendix A: derivation of PSWF

The numerical derivation of PSWF is made using the Legendre polynomials-based approach presented by Xiao et al. [20]. The Legendre polynomials P_n are defined by the three-term recursion,

$$P_{n + 1} (u) = \frac{2n + 1}{{n + 1}}uP_{n} (u) - \frac{n}{n + 1}P_{n - 1} (u)$$

(25)

with the initial conditions P₀(u) = 1, P₁(u) = u. The numerical evaluation of PSWF can be made by:

$$\psi_{j} (u,c) = \sum\limits_{k = 0}^{\infty } {\beta_{k}^{j} \overline{P}_{k} (u)}$$

(26)

where $\overline{P}_{n}$ is the normalized version of the Legendre polynomials, i.e.

$$\overline{P}_{n} (u) = P_{n} (u) \cdot \sqrt {n + 0.5}$$

(27)

For each j = 0, 1, …, by denoting the coefficients β in Eq. 26 as a column vector β^j,

$${{\varvec{\upbeta}}}^{{\mathbf{j}}} = (\beta_{0}^{j} ,\beta_{1}^{j} ,...)$$

(28)

The values in β^j are derived by solving the following algebraic eigenvalue problem.

$$({\mathbf{A}} - \chi_{j} \cdot {\mathbf{I}})({{\varvec{\upbeta}}}^{j} ) = 0$$

(29)

where χ_j denotes the eigenvalues corresponding to matrix A. The components in matrix A is given by

$$\begin{gathered} A_{k,k} = k(k + 1) + \frac{2k(k + 1) - 1}{{(2k + 3)(2k - 1)}}c^{2} \hfill \\ A_{k,k + 2} = A_{k + 2,k} = \frac{(k + 1)(k + 2)}{{(2k + 3)\sqrt {(2k + 1)(2k + 5)} }}c^{2} \hfill \\ {\text{otherwise}},A = 0 \hfill \\ \end{gathered}$$

(30)

To solve Eq. 29 an eigenvalue solver based on the Jacobi’s method can be applied. In practice, the number of k is limited to certain value, say, M. In this study, the cut-off number M is determined according to a suggestion given by Boyd [40],

$$M \ge 2N_{e} + 30$$

(31)

where N_e + 1 denotes the number of PSWF to be evaluated. N_e is determined according to the value of Slepian frequency c so that the following condition is satisfied.

$$N_{e} + 1 \ge 2c/\pi$$

(32)

After that, the scaled eigenvalues of each PSWF λ_j are calculated using following relationships [22].

$$\begin{gathered} \lambda _{{2k}} = ( - 1)^{k} \frac{{\sqrt 2 \beta _{0}^{{2k}} }}{{\psi _{{2k}} (0,c)}},\quad k \ge 0 \hfill \\ \lambda _{{2k + 1}} = ( - 1)^{k} \sqrt {\frac{2}{3}} \frac{{c\beta _{1}^{{2k + 1}} }}{{\partial _{u} \psi _{{2k + 1}} (0,c)}},\quad k \ge 0 \hfill \\ \end{gathered}$$

(33)

here $\partial_{u}$ denotes the partial derivative in terms of u. Note that as the eigenvalues of PSWF µ_j should be less than 1, the absolute values of λ_j should be less than $\sqrt {2\pi /c}$.

Appendix B: PSD histories from different time windows

See Appendix Fig. 23.

Fig. 23

PSD histories of heave motion from different time window length