Investigation on Spectrum Estimation Methods for Bimodal Sea State Conditions

Abstract

The reliable monitoring of sea state parameters is a key factor for weather forecasting, as well as for ensuring the safety and navigation of ships. In the current analysis, two spectrum estimation techniques, based on the Welch and Thomson methods, were applied to a set of random wave signals generated from a theoretical wave spectrum obtained by combining wind sea and swell components with the same prevailing direction but different combinations of significant wave heights, peak periods, and peak enhancement factors. A wide benchmark study was performed to systematically apply and compare the two spectrum estimation methods. In this respect, different combinations of wind sea spectra, corresponding to four grades of the Douglas Scale, were combined with three swell spectra corresponding to different swell categories. The main aim of the benchmark study was to systematically investigate the effectiveness of the Welch and Thomson methods in terms of spectrum restitution and the assessment of sea state parameters. The spectrum estimation methods were applied to random wave signals with different durations, namely 600 s (short) and 3600 s (long), to investigate how the record length affected the assembled sea state parameters, which, in turn, were assessed by the nonlinear least square method. Finally, based on the main outcomes of the benchmark study, some suggestions are provided to select the most suitable spectrum reconstruction method and increase the effectiveness of the assembled sea state parameters.

Sea Spectral Estimation Using ARMA Models

“…The trend of the normalized AIC index is shown in Figure 9 for two different 10 min long time series. In this case the ̂ value must be close to the first drop of the AIC value in order to be able to solve in the least square sense the minimisation problem of Equation (14). Therefore, Figure 9 suggests in this case a value of ̂ of about 5, though, as in the previous case, its interpretation is not straightforward.…”

“…Consequently, different approaches were proposed in the literature to estimate the sea features, starting from either onboard measurements (e.g., ship motion [ 5 , 6 , 7 , 8 , 9 ]) or time signals of wave level. Taking into consideration the latter approach, on the one hand, it is possible to successfully use non-parametric methods such as the Welch’s [ 10 ] and Thomson’s [ 11 ] methods for estimating sea spectrum [ 12 , 13 , 14 ]. On the other hand, parametric methods could also be employed, with the consequent advantage of obtaining a model of the sea spectrum, which can be then used for advanced predictions of sea actions on ships and structures.…”

ARMA models to represent bi-modal sea spectra

“…Recalling that the model of the time series assumed by the Prony's method [27] has the same form of the auto-covariance of Equation (13), it is possible to estimate the AR coefficients by deriving the poles λ i applying the Prony's method to the auto-covariance sequence r x [22] and solving for β, with a least square approach, the following problem: (14) where:…”

“…Consequently, different approaches were proposed in the literature to estimate the sea features, starting from either onboard measurements (e.g., ship motion [5][6][7][8][9]) or time signals of wave level. Taking into consideration the latter approach, on the one hand, it is possible to successfully use non-parametric methods such as the Welch's [10] and Thomson's [11] methods for estimating sea spectrum [12][13][14]. On the other hand, parametric methods could also be employed, with the consequent advantage of obtaining a model of the sea spectrum, which can be then used for advanced predictions of sea actions on ships and structures.…”

Measurement of Sea Waves

“…Wave spectra can be estimated using spectral estimation methods—either non parametrical, such as the fast Fourier transform (FFT) following a proper estimation procedure [ 2 ], or parametrical, based, for example, on autoregressive moving average (ARMA) modeling of the time history [ 3 ].…”