Estimations of on-site directional wave spectra from measured ship responses

doi:10.1016/j.marstruc.2006.06.001

Marine Structures

Volume 19, Issue 1, January 2006, Pages 33-69

https://doi.org/10.1016/j.marstruc.2006.06.001 Get rights and content

Abstract

In general, two main concepts can be applied to estimate the on-site directional wave spectrum on the basis of ship response measurements: (1) a parametric method which assumes the wave spectrum to be composed by parameterised wave spectra; or (2) a non-parametric method where the directional wave spectrum is found directly as the values in a completely discretised frequency-directional domain without a priori assumptions on the spectrum. The paper outlines the theory of these two concepts, and it is shown how to deal with the speed-of-advance problem for operating ships. In addition, the methods include an equivalence of energy in the governing equations and, as regards the parametric concept, a frequency-dependent spreading of the waves is introduced.

The paper includes an extensive analysis of full-scale measurements for which the directional wave spectra are estimated by the two ship response-based methods. Hence, comparisons are made between these estimates and, moreover, the agreement with the corresponding directional wave spectra produced by the wave radar system WAVEX is studied. The agreement between the two methods is reasonable, as well is the agreement between the results of these methods and those of WAVEX. It is difficult to propose one of the ship response-based methods in favour of the other, since they perform equally well.

Introduction

The operational safety of ships can be increased by use of in-service monitoring systems. For such systems to be used efficiently, the on-site directional wave spectrum needs to be estimated and updated continuously. Today, means of obtaining estimations of the directional wave spectrum exist. Such means include moored wave rider buoys and current meters, satellite measurements and wave radar systems. The latter two of these means do not suffer from the problems related to the fixed position of a moored buoy or current meters, but do, on the other hand, require complex computational hardware and have a high initial cost, cf. Tannuri et al. [1], not to mention calibration and maintenance. For this reason it is of interest to be able to estimate the wave spectrum from measured ship responses, which are accessible from the sensor measurements done in in-service monitoring system.

In the literature there exist several papers which deal with the estimation of (directional) wave spectra on the basis of measured ship responses. In general, these works can be split in two groups or as being part of one of two concepts: (1) a concept where the wave spectrum is estimated on the basis of parametric modelling, e.g. Hua and Palmquist [2], Tannuri et al. [1] and the EC project HullMon $+$ [3]; or (2) a concept where the estimation of the wave spectrum is based on non-parametric modelling, e.g. Iseki and Ohtsu [4], Iseki and Terada [5], Waals et al. [6], Isobe et al. [7] and Nielsen [8]. Fundamentally, the two concepts are similar in the sense that both methods in their foundations use linear spectral analysis to set up conditions/equations which relate the measured ship responses—the one hand side—with the wave energy spectrum through complex-valued transfer functions—the other hand side. Thus, from this relation, the principle is to minimise, in the least squares sense, the difference between the two sides, see Fig. 1.

Although Tannuri et al. [1], including Pascoal et al. [9], to some degree compare the two estimation concepts with each other, the comparisons apply for numerical simulations and for a ship not being underway, since these works do not consider the speed-of-advance problem in the derived theory. Similarly, Benoit and Goasguen [10] compares different directional wave analysis methods for a so-called “single-point” measuring system, which record data from a three-displacements buoy, that is located at a fixed position.

The speed-of-advance problem, governed by the deep-water relationship between the encounter and the wave frequency, leads in certain cases to the so-called triple-valued function problem in beam, quartering and following seas, cf. Fig. 3(b). This problem needs to be strictly incorporated in an estimation procedure based on measured ship responses. In Iseki and Ohtsu [4] the elementary problem is dealt with and in that paper it is suggested how to incorporate it.

In the present paper, the theory of a parametric modelling procedure and a non-parametric modelling procedure is derived, and it is outlined in detail how to take into account a vessel being underway, independently if the estimation is performed by parametric or non-parametric modelling. Furthermore, it is shown how to include additional equations, based on Iseki and Terada [5], in both of the estimation concepts, so that it is sought to secure the conservation of energy in the responses. The main part of the present paper is devoted to the analysis of numerical and full-scale data. Thus, on the basis of simulated and measured ship responses, the parametric modelling and the non-parametric modelling procedure are compared. As regards the analysis of full-scale data, comparisons with data from the wave radar system WAVEX have also been conducted.

In principle, any type of response signal, obtained as time series, may be utilised in the wave spectrum estimation as long as a linear complex-valued transfer function exists, that is, can be calculated. However, it is believed that the highest rate of success is achieved when global ship responses, such as pitch, roll, vertically wave-induced bending moment, etc., are used. This means that response signals of e.g. pressure transducers in the hull should not be considered in the estimation, as local effects may affect such a response. It is important to emphasise that at least one of the ship responses must have port/starboard asymmetry, otherwise the modelling procedures cannot differentiate between port and starboard entering waves. This explains also the need of complex-valued transfer functions, so that the amplitudes as well as the phase angles of the responses are obtained.

Specifics on response signals to be used for the estimation of wave spectra can be found in, among others, HullMon+ [3] and Nielsen [8]. Tannuri et al. [1], however, brings an interesting issue to question and, thus, argues correctly that the roll motion, in general, has a non-linear and resonant behaviour and, furthermore, that the roll motion is extremely sensitive to load variations. Tannuri et al. [1] therefore proposes to use the sway response instead of the roll, as the frequency response function of sway is less sensitive to loading conditions. The present work deals primarily with results based on the three responses {heave, roll, pitch}. Though, estimations have also been carried out with the roll response replaced by the sway. Later, in Section 7, more comments are given on this topic.

The paper is organised as follows: In Sections 2–5 theoretical aspects of the modelling procedures are treated. Section 6 deals with a numerical example which verifies and compares the modelling procedures. Then in Section 7, full-scale data is analysed and comparisons between the estimations of the two modelling procedures and those of WAVEX are carried out. Finally, in Section 8, the major conclusions are formulated.

Section snippets

General theory

In this section the general theory of the parametric and the non-parametric modelling procedure are described. Hence, the section deals with facts and conditions which are shared by the two concepts and, subsequently, two sections yield the specifics for the individual concepts.

Bayesian modelling

Without explicit assumptions on the directional wave spectrum, (2.16) expresses $(N^{2} \cdot L + N)$ equations from which $K \cdot M$ unknowns, $E (ω_{m}, β_{k})$ , are to be solved. For a reasonable discretisation, say, $K = 18$ , $M = 30$ and taking $N = 3$ ship responses to form the basis for the estimation, (2.16) is in general underdetermined. The number, L, of encounter frequencies plays a role in the sense that more equations are established by increasing L. However, new information put into the system in this way is of limited

Parametric modelling

Parameterised wave spectra, e.g. Goda [19], are typically considered reliable for describing the variation with frequency of ocean wave spectra. Moreover, the angular spread of wave spectra can be described by certain parameters, e.g. Longuet-Higgins et al. [20] and Goda [19]. On this assumption, the wave spectrum to be estimated from Eq. (2.16) is based on the following 10-parameter bimodal spectrum, e.g. Tannuri et al. [1] and Hogben and Cobb [21], $E (ω, θ) = \frac{1}{4} \sum_{i = 1}^{2} \frac{(((4 λ_{i} + 1) / 4) ω_{p, i}^{4})^{λ_{i}}}{Γ (λ_{i})} \frac{H_{s, i}^{2}}{ω^{4}}$

Solution procedures

From the preceding it appears that both methods, the Bayesian method and the Parametric method, in their fundamentals are conceptually similar and deviate only in the sense of different (prior) assumptions about the wave spectrum to be estimated. In practical computations, though, the solution procedures of the two methods are quite different, since the Bayesian modelling in a number of iterations, based on QR factorisation, solves an overdetermined linear equation system, whereas the

Numerical example

As a means to verify the two estimation concepts, numerical simulations are conducted. Thus, on the assumption of a perfect relationship between ship responses and a wave spectrum through (given) complex-valued transfer functions, generated ship responses can be used to estimate the underlying wave spectrum.

The numerical simulations dealt with are based on the ship responses ${heave, roll, pitch}$ and the complex-valued transfer functions are calculated by a three-dimensional time domain code.

Analysis of full-scale data

In this section full-scale motion measurements of a container ship are studied. The main dimensions of the ship are listed in Table 1, and the ship has been in route on the Pacific Ocean where all the data has been recorded. The data consists of nine sets: $A, B, \dots, I$ , each of a duration of 15 min. The duration is taken in the middle of a 30 min period used for the WAVEX estimation and during this time, the operational conditions were (nearly) constant. The estimations are based on the three

Conclusions

The present work has described two methods to estimate the directional wave energy spectrum on the basis of measured ship responses. The two concepts dealt with are a parametric and a non-parametric method. The Parametric method is based on a summation of a number of parameterised wave spectra, whereas the non-parametric method, denoted the Bayesian method, estimates the directional wave spectrum in a number of discretised points of the wave field, which is divided into a set of frequencies

Acknowledgements

The author heartily thanks Associate Professor Toshio Iseki, Tokyo University of Marine Science and Technology, for his great willingness to share knowledge of related research subjects. Moreover, the assistance and inspiration provided by Professor Jørgen Juncher Jensen, Technical University of Denmark, is highly appreciated.

The analysed full-scale data has been provided by Det Norske Veritas (DNV) and the author would like to thank Dr. Bo Cerup Simonsen and Mr. Øyvind Lund-Johansen for making

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