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Parametric Estimation of the Directional Wave Spectrum

from Ship Motions

Miguel Angel Hinostroza Muñoz

Thesis to obtain the Master of Science Degree in

Naval Architecture and Marine Engineering

Examination Committee

President : Prof. Yordan Ivanov Garbatov

Supervisor : Prof. Carlos António Pancada Guedes Soares

Members of the committee : Prof. Sergey Sutulo

December 2014

II

Acknowledgments

Special thanks are due to Prof. Carlos Guedes Soares for providing the opportunity for me to study

and stay in Portugal, for all the help and guidance in my research and for giving me the opportunity

and confidence to work onboard of Navy ships. I also want to express my gratitude to Prof. Sérgio

Ribeiro, for his valuable advises during full-scale tests.

The experimental component of this work was performed within the project “Experimental and

Numerical Study of Ship Responses in Waves” financed by the Portuguese Foundation for Science

and Technology (Fundação para a Ciência e Tecnologia) under contract PTDC/EMS-ENE/1073/2012.

Fernando Santos has supported me in many ways during this project by providing instrumentation

advices and helping with the material purchase, and by being available to give a hand any time it was

needed.

I am indebted to many of my colleagues whose support and help was crucial throughout the course of

this project. Therefore, my special thanks go to Jose, Guillermo, Danilo, Jesus, Edwin and Jimmy.

Last but not least, my very special gratitude to my parents, Angeles and Julia and to my sisters who

always helped me focus on my work. Despite the long geographical distance between us, they always

have been by my side to help me look at the bright side in hard times and to share happy moments

Lisbon, December, 2014

Miguel A. Hinostroza Muñoz

III

Abstract

A parametric estimation of the directional wave spectrum derived from ship motions is presented. The

estimation of the sea-state parameters is essential for several applications on open-sea such as

offshore platforms installation/operation and safe ship navigation. Traditionally, the sea-state

parameters have been obtained using a waverider buoy. However, the analogy between the ship and

the buoy is clear, even when the ship is moving with forward speed and, in general, is characterized

by a more complex underwater geometry. Thus, it is possible to obtain an estimate of the wave

spectrum by processing its wave-induced responses similar to the estimation of the traditional wave

rider buoy. The wave spectrum can be estimated from procedures based on measured ship

responses: (1) A parametric method which assumes the wave spectrum to be composed by

parameterized wave spectra; or (2) a non-parametric method where the directional wave spectrum is

found directly as the values in a completed discretized frequency-directional domain without a priori

assumptions on the spectrum.

This thesis deals with a parametric procedure, incorporating the speed-of-advance problem for ships.

In order to verify the estimation method numerical simulations and full-scale tests were carried out.

Thus, the estimated procedure was tested in simulations from generated ship responses based on a

theoretically known wave spectrum. Following the numerical simulations, were performed estimations

using full-scale data from trials carried out onboard of Portuguese Navy vessels. This thesis also

presents the developed system for ship motion monitoring that has capabilities to record ship

responses and sea elevations.

Keywords: Sea-state estimation, Ship motions, Decision support systems, Full-scale trials.

IV

Resumo

Nesta tese é apresentada a estimação paramétrica do espectro direccional de ondas baseada nos

movimentos de navios. A estimação dos parâmetros do estado do mar é essencial em aplicações em

mar aberto, tais como, na instalação/operação de plataformas offshore e na navegação segura de

navios. Tradicionalmente, tais parâmetros têm sido obtidos usando bóias oceânicas. Contudo, a

analogia entre o navio e a bóia é evidente, mesmo quando aquele possuí velocidade de avanço, e

tem uma geometria mais complexa. Consequentemente, é possível obter uma estimativa do espectro

da onda por processamento das respostas induzidas pela onda no navio, similarmente à estimação

com bóia oceânica. O espectro de onda pode ser estimado a partir de respostas de navios: (1) um

método paramétrico que assuma o espectro como sendo composto por espectros de onda

parametrizados; ou (2) um método não-paramétrico no qual o espectro direccional de onda é obtido

directamente como os valores num domínio direccional-frequência discretizado e completo sem

suposições à priori sobre a forma do espectro.

Esta tese apresenta o procedimento paramétrico, incorporando o problema da velocidade de avanço

em navios. Para avaliar-se o método de estimação foram conduzidas simulações numéricas e testes

em escala real. Portanto, o procedimento de estimação foi testado com base em simulações de

respostas de navios geradas a partir de ondas conhecidas. Após as simulações numéricas foram

efectuadas estimações usando dados de testes em escala real realizados em navios da Marinha

Portuguesa. Esta tese apresenta também um sistema desenvolvido para monitorização dos

movimentos de navios.

Palavras-chave: Estimativa do estado do mar, Resposta de navios em ondas, Sistemas de apoio à

navegação, Ensaios à escala real.

V

Table of contents

Acknowledgments .......................................................................................................................................... II

Abstract ............................................................................................................................................................. III

Resumo .............................................................................................................................................................. IV

1 Introduction ............................................................................................................................................................... 1

1.1 Principle of functioning of a waverider buoy ..................................................................... 2

1.1.1 Fields of measurement ......................................................................................................................... 2

1.1.2 Method of Operation ............................................................................................................................. 2

1.2 Use of ship motions to estimate wave spectra.................................................................... 3

1.3 Data acquisition of ship motions ............................................................................................. 4

1.3.1 Definition of decision support systems ......................................................................................... 4

1.3.2 Importance of DSS .................................................................................................................................. 5

1.3.3 Typical signals to be monitored in DSS ......................................................................................... 5

1.3.4 Example of onboard DSS ...................................................................................................................... 6

1.4 Objectives of the dissertation and organization of the text........................................... 8

2 Parametric estimation of directional wave spectrum ................................................................................ 8

2.1 Available methods of estimation of spectrum .................................................................... 8

2.1.1 Theoretical formulation of parametric estimation ................................................................... 9

2.1.2 Numerical considerations ................................................................................................................. 15

2.2 Proposed method of parametric estimation .................................................................... 18

2.2.1 Estimated parameters ........................................................................................................................ 19

3 Validation with numerical simulations......................................................................................................... 20

3.1 Time simulation of ship motions in waves ....................................................................... 20

3.1.1 Formulation ............................................................................................................................................ 21

3.2 Estimation of directional wave spectra from motions of a bulk carrier ................ 22

3.2.1 Directional estimations ...................................................................................................................... 23

3.3 Estimation of directional wave spectra from motions of a small vessel ................ 26

3.3.1 Directional estimations ...................................................................................................................... 26

4 Validation with full scale ship trials ............................................................................................................... 29

4.1 Development of the data acquisition system ................................................................... 29

4.1.1 Description of the LabVIEW Program .......................................................................................... 30

4.1.2 Sensor Locations ................................................................................................................................... 31

4.1.3 Hardware Connections ....................................................................................................................... 31

4.2 Description of the full scale ship trials ............................................................................... 32

VI

4.3 Analysis of the results of the full scale ship trials .......................................................... 34

4.3.1 Data A ........................................................................................................................................................ 34

4.3.2 Data B ......................................................................................................................................................... 35

4.3.3 Data C ......................................................................................................................................................... 36

4.3.4 Data D ........................................................................................................................................................ 37

4.3.5 Data E ......................................................................................................................................................... 38

5 Conclusions .............................................................................................................................................................. 39

References ....................................................................................................................................................... 40

VII

List of figures

Figure 1: Potential wave buoy mooring options: single point taut mooring (left) and s-mooring (right). . 3

Figure 2: Analogy between waverider buoy and ships. ........................................................................... 4

Figure 3: Laptop on the bridge with user interface of DSS, Perera et al. (2012). ................................... 4

Figure 4: Onboard sensor locations (Perera et al., 2012) ....................................................................... 7

Figure 5: Diagram of parametric estimation procedure. .......................................................................... 9

Figure 6: Ship course, α, wave direction, θ, and relative motion 𝛽 respect to a fixed direction. ............ 9

Figure 7: Notation of encounter angles and sign convention of β (in degrees)..................................... 10

Figure 8: Relationship between encounter and wave frequencies: a) All-directional relationship (an

arbitrary example); and b) cross section at β = θ with θ ∈ [−90; 90] deg. (Nielsen, 2005). .................. 11

Figure 9: Discretization of wave frequencies and encounter angles. .................................................... 16

Figure 10: Weighting to avoid too many discretized frequencies. ......................................................... 16

Figure 11: Algorithm for simulations of ship motions. ........................................................................... 20

Figure 12: Directional estimations for ship 1 in head waves. ................................................................ 23

Figure 13: Hs and Tp for 20 estimations of ship 1. ............................................................................... 24

Figure 14: Directional estimations for ship 1 in following waves. .......................................................... 25

Figure 15: Directional estimations for ship 2 in head Waves ................................................................ 27

Figure 16: Hs and Tp for 20 estimations of ship 2. ............................................................................... 27

Figure 17: Directional estimation for ship 2 in following waves. ............................................................ 28

Figure 18: Software architecture of the developed system. .................................................................. 30

Figure 19: User interface of the developed system. .............................................................................. 31

Figure 20: On board location of sensors and equipment. ..................................................................... 31

Figure 21: Hardware structure of the developed system. ..................................................................... 32

Figure 22: a) Sets of data collected during the trials; b) Place of the tests, Lisbon. ............................. 34

Figure 23: Estimated wave spectrum for Data A ................................................................................... 34

Figure 24: Comparison between estimated and measured wave spectra for Data A. .......................... 35

Figure 25: Estimated wave spectrum for Data B ................................................................................... 36

Figure 26: Comparison between the estimated and measured wave spectra for Data B. .................... 36

Figure 27: Estimated wave spectrum for Data C. ................................................................................. 37

Figure 28: Comparison between the estimated and measured wave spectra for Data C. ................... 37

Figure 29: Estimated wave spectrum for Data E. .................................................................................. 38

Figure 30: Comparison between the estimated and measured wave spectra for Data E. .................... 38

VIII

List of Tables

Table 1: Typical monitored responses depending on ship type and structure (Slaughter et al., 1997). . 6

Table 2: Bulkcarrier ship characteristics ................................................................................................ 22

Table 3: Considered genetic algorithm parameters. ............................................................................. 23

Table 4: Set of generated and estimated wave parameters for ship 1 ................................................. 25

Table 5: Main characteristics of ship 2 .................................................................................................. 26

Table 6: Set of generated and estimated wave parameters for ship 2 ................................................. 29

Table 7: Main characteristics of the Navy vessel. ................................................................................. 33

Table 8: Characteristics of the collected data. ...................................................................................... 33

1

1 Introduction

Knowledge of environmental conditions is one of the most important aspects for the safety of critical

operations under open-sea. Wave buoys have been a reliable source of environmental data and the

subject addressed herein is whether a vessel can also be used as a directional wave buoy with an

adequate degree of confidence. This has immediate benefits on the following: for the operational

performance of guidance and control systems, for better decision support; for the likelihood of

parametric roll which depends amongst others on the sea-state in which the ship operates; and in the

long-term ocean data can be made available for building climatological data to be used in the design

stage. The evaluation of a vessel’s performance requires input of the sea-state parameters, so

onboard wave estimation is highly relevant for any type of monitoring and/or decision support system

on ships. Thus, if the sea-state is continuously estimated it is possible to raise a warning if, say, the

vessel speed or course is in a region where parametric roll can be triggered (Jensen, 2011).

In the literature there are reports (e.g. Iseki and Ohtsu, 2000; Nielsen, 2005,2006; Pascoal et al., 2007;

Tannuri et al., 2003) about the estimation of sea-state parameters using measured ship responses

(e.g. motion data) in which the ship acts as a “wave rider buoy”, thus the name, “wave buoy analogy”,

for the methodology. The fundamental input to the wave buoy analogy is a set of response

measurements where the individual ones can basically be any one as long as a linear (complex-

valued) transfer function is associated to the response. The wave buoy analogy provides a robust

alternative to wave radars by using of onboard response measurements that are always carried out on

many of today’s navy and commercial vessels. Consequently, the wave buoy analogy is also a

relatively inexpensive estimation concept, since the system development is associated to software

only.

There are two main options for representation of the estimated ocean wave spectra. There is

published work on using parametric and nonparametric formulations. The first formulation consists in

giving a spectral shape with known analytical description but unknown parameters, such as the

JONSWAP, while the second one is a minimization where only a non-negative constraint on the

spectral amplitude is mandatory but the form is otherwise not specified this is sometimes called hyper-

parametric representation.

Nonlinear gradient based minimization procedures have been used in parametric representations by

Tannuri et al. (2003). It was found that the minimization domain is not globally convex, which means

that global search schemes must be used to some point. It is proposed here the use of genetic

algorithms (GA) based in minimization.

For this study the Complex transfer functions are calculated using in-house software developed at

Centre for Marine Technology and Engineering (CENTEC). This code is based on strip theory and

details can be found in Fonseca and Guedes Soares (1998).

The parametric method is used and minimizations are carried out only after adequate conditioning of

equations and with initial values given by a non-negative least square formulation. Further, the convex

minimization for the parametric representation can be initialized by a genetic algorithm. The algorithm

2

has been designed to be completely independent of human intervention to provide initial estimates.

The motions chosen for the estimation were the, heave, roll and pitch as proposed by Nielsen (2005).

The given method is based on the works presented in Nielsen (2005) and Pascoal et al.(2007).

However the main differences respects the previous methods are: 1) Consider the effects of speed of

advance, 2) uses genetic algorithms and 3) performed full-scale estimations.

1.1 Principle of functioning of a waverider buoy

The principle of functioning of the waverider buoy consists on following the surface slope of the waves,

measuring time series of the vertical (heave) acceleration and the two orthogonal components of the

surface slope (pitch and roll). A directional spectrum can be estimated using Data Analysis

Procedures from each triplet of time series wave measurements. Also, the auto- and cross-spectra

between the records and spectral parameters can be computed. Data is typically processed on-board,

and both raw and processed data may be stored or transmitted to shore via a radio or satellite link.

A key aspect of the buoys is their mooring system; this must be designed so that it does not produce

tilt forces acting on the buoy.

1.1.1 Fields of measurement

In their primary role of wave measurement, buoys record vertical acceleration and two orthogonal

components of surface slope. These are processed on-board the buoy to provide the following:

Time series of the sea surface displacement;

Omni-directional and directional spectra;

Integrated spectral parameters, e.g. significant wave height, mean period and energy period.

Additional oceanographic sensors recording parameters such as surface current velocities and

directions, temperature, salinity and meteorological data may also be integrated into the buoy setup.

1.1.2 Method of Operation

Buoys are designed to operate in offshore locations, and are typically deployed from an ocean-going

vessel with heavy lifting capabilities. In normal conditions and in areas with heavy marine traffic, the

buoy mooring comprises a single point taut mooring. However, in hostile and deep water

environments, an s-mooring should be used, as illustrated in Figure 1. Prior to deployment, buoys

should be calibrated on land using a test rig that replicates the motion of the buoy. All operational

buoys should periodically be brought ashore and re-calibrated to maintain measurement accuracy

3

Figure 1: Potential wave buoy mooring options: single point taut mooring (left) and s-mooring (right).

1.2 Use of ship motions to estimate wave spectra

In principle, a ship is itself a wave rider buoy and, hence, from this analogy several studies outline how

wave spectra can be estimated on-site on the basis of measured ship responses, see e.g. Iseki and

Terada (2002), Tannuri et al. (2003), Pascoal et al. (2005) and Nielsen (2005). Basically, the

estimation procedure is based on a set of responses, say three, of which linear complex-valued

transfer functions can be calculated, and the wave spectrum is then extracted from a set of equations

established by linear spectral analysis relating measured response spectra with calculated ones. The

wave buoy analogy is, however, made complicated by the fact that a ship (due to its relatively large

inertia) filters wave excitations to a larger extent than real wave rider buoys, not to mention the more

complex geometry of a ship hull and the speed-of-advance problem. Figure 2 shows the principle of

to consider the ship as a buoy wave.

4

Ship buoy

Recording of

motions

Signals

processing

Wave spectrum

Waves

Buoy responses Ship responses

Figure 2: Analogy between waverider buoy and ships.

1.3 Data acquisition of ship motions

1.3.1 Definition of decision support systems

Decision support systems (DSS) are computer technology solutions that can be used to support

complex decision making and problem solving. A Decision Support System (DSS) is a computer-

based information system that supports business or organizational decision-making activities. DSSs

serve the management, operations, and planning levels of an organization (usually mid and higher

management) and help to make decisions, which may be rapidly changing and not easily specified in

advance (Unstructured and Semi-Structured decision problems).

Figure 3: Laptop on the bridge with user interface of DSS, Perera et al. (2012).

5

1.3.2 Importance of DSS

The DSS are important for two categories of people. These are the officers on the ship and the ship-

owner.

For these users the main application is to avoid:

Seasickness of passengers;

Damage to cargo due to large accelerations;

Local structural damage to forward structure due to slamming, wave impacts and green water

on deck;

Shift of cargo due to a combination of roll and accelerations;

Excitation of large roll motions;

Loss of lateral stability in following waves.

1.3.3 Typical signals to be monitored in DSS

In-service monitoring systems have not yet become standard equipment, although some kind of real-

time operational surveillance to follow/study wave-induced effects is now installed on many ships.

Furthermore, most classification societies have introduced a special class notation for ships with

installed hull surveillance systems. Based on a study carried out by Robinson (1990) of the Lloyd’s

Register of Shipping, identifies four different types of systems, or monitors, which enable ships to be

operated efficiently within their design limits:

Seakeeping monitor : Display of motions;

Loading and structural monitor: Display of still-water and wave-induced stresses,

accelerations and impact loadings;

Machinery and fuel performance monitor : Display of shaft power, rpm and fuel consumption;

Environmental monitor: Display of current values (wind, temperatures, barometric pressure,

water depth, calculated sea state).

The first three are common to all types of ship monitors, whereas the latter is concerned with the

environmental conditions that are not always influent in navigation. However, some of the monitors are

often integrated, so that the integrated in-service monitoring system offers information which deals

with several of the points stated above.

Depending on the type of ship, different kinds of requirements are made for the in-service monitoring

system. A summary of the key responses based on ship type is given In Table 1. Naturally, such a list

is by no means complete, and the key point is that ship characteristics should be reviewed when

requirements for decision support systems are considered.

6

Table 1: Typical monitored responses depending on ship type and structure (Slaughter et al., 1997).

Passenger ship

- Ship motions

- Noise and vibrations

- Bow flare slam

Tanker/products carrier

- Amidship hull girder stress

- Bow/amidships side shell stiffeners

- Explosive environment

Bulk ships

- Still water hull girder stresses

- Cargo hold frame stresses

- Stress concentration at hatch corners

- Forefoot slam

Container ships

- Stress concentration at hatch corners

- Hull girder torsion

- Bow flare slam

- Green water on bow

- Whipping/cargo accelerations

LNG/internal tank

- Bottom slamming

- Temperature/explosive atmosphere

- Sloshing

Barges/platforms

- Towline/mooring tension

- Motions and inertial forces

- Lateral motion

Naval combatant - Bow flare slam

- Local stability due to weapons

1.3.4 Example of onboard DSS

As part of the present thesis the fundamentals of decision support systems are reviewed for illustrative

purposes. This work was proposed by Perera et al. (2012) in the development of an on-board decision

support system for ship navigation under rough weather conditions. Some of this equipment and data

acquisition strategies used in that system have been adopted in the present experimental work.

These DSS monitors several motions related parameters, and, by processing these data, provides the

ship master with the information about the consequences of the different ship handling decisions. The

software developed as well as the processing hardware are also described.

1.3.4.1 Experimental platform

The DSS prototype was installed on a Ro-Ro ship with Lpp = 214.0 m and B =

at initial stage was mounting on the Ro-Ro ship and it has been collecting data from the motions and

calculations results. Once enough data have been collected the same will then be used to calibrate

and validate the system. In this section, a brief presentation of the individual components of the

experimental platform and its operating logic is carried out.

7

The system was divided into three main sub-systems: Motion monitoring sub-system, Stress

monitoring sub-system and Wave condition monitoring sub-system.

The main objective of the Motion monitoring sub-system is to evaluate the vessel motions on the

seaway. It consists of the midship accelerometer, the midship angular rate sensor, and the

inclinometer. The sensor locations are presented in Figure 4. The accelerometer measures the surge,

sway, and heave accelerations. The angular rate sensor measures the yaw angular velocity and the

inclinometer measures the roll and pitch angles.

The purpose of the Stress monitoring sub-system is to evaluate the hull stress condition. It consists of

four strain gauges that are oriented: Two strain gauges located starboard and portside of the midship

and two strain gauges located fore and aft of starboard. The strain gauges locations are also

presented in Figure 4.

The function of the Wave condition monitoring sub-system is to evaluate the wave spectrum. It

consists of the wave height measurement sensor and the bow accelerometer. Their locations are

shown in Figure 4. The bow mounted, down-looking, wave measurement sensor measures the relative

wave height and the bow accelerometer compensates for the vessel motions. In this system the wave

spectrum was estimated by a filtering algorithm (Pascoal and Guedes Soares 2009), different from the

approach used in the present dissertation.

Figure 4: Onboard sensor locations (Perera et al., 2012)

The main conclusions of the Perera et al. (2012) Work were that onboard decision support systems for

ship navigation under rough sea and weather conditions are able to be installed on board of Ro-Ro

ships. But is necessary collect some data as the ship operates in order to calibrate and validate. This

data serve as the basis for the training of a neural network to be implemented, capable of quickly and

accurately giving the expected loads given the present sea state and possible ship’s courses and

speeds. The system was successfully tested in terms of hardware and software integration.

8

1.4 Objectives of the dissertation and organization of the text

In the present thesis, the parametric procedure for estimating wave spectrum, based in ship motions,

is addressed. This method assumes to be composed of one parameterized spectrum or by the

summation of several parameterized spectra, e.g. the generalized JONSWAP spectrum, so that it is

the underlying wave parameters which are sought for. Genetic algorithms (GA) for minimization plants

were applied.

In order to verify the wave estimation method numerical simulations and Full scale tests were carried

out. Hence, in simulations from generated ship responses based on a theoretically know wave

spectrum, the estimated procedure was tested. Following the numerical simulations, were performed

estimations using full scale data from a trials carried out onboard of Portuguese Navy vessels. The

Present thesis also presents a developed system for Ship motion monitoring that has capabilities to

record ship responses and also sea elevations.

This thesis is organized as follows. In Chapter 2, the mathematical formulation for the parametric

method is addressed. In Chapter 3 are presented the numerical simulations. Chapter 4 presents the

estimations based on full-scale data and Chapter 5 is the conclusion.

2 Parametric estimation of directional wave spectrum

2.1 Available methods of estimation of spectrum

Typically there are two methods to estimate the directional wave spectrum from ship motions,

parametric and non-parametric. The first method assumes the wave spectrum to be composed by

parameterized wave spectra. In the second one the directional wave spectrum is found directly as the

values in a completed discretized frequency-directional domain without a priori assumptions on the

spectrum.

Such methods, which are derived by use of linear spectral analysis, are based on either a parametric

or a non-parametric procedure. In the parametric procedure the estimated wave spectrum is assumed

to be composed of one parameterized spectrum or by the summation of several parameterized

spectra, e.g. the generalized JONSWAP spectrum, so that it is the underlying wave parameters which

are sought for. During the estimation process was found that the minimization domain is not convex,

which means that global search schemes must be used to some point. For this reason genetic

algorithms (GA) based minimization were applied herein. The non-parametric method, however, yields

the directional wave spectrum in a number of discretized points of the wave field. The non-parametric

method, called the Bayesian estimation method, is based on Bayesian modeling, which introduces the

use of prior information to make the directional wave spectrum smooth and to avoid an unrealistic

solution.

9

2.1.1 Theoretical formulation of parametric estimation

In this section the detailed formulation of the parametric method is addressed. This is based in the

theory developed by Iseki et al.(2000) and by Nielsen (2005).In the diagram in Figure 5 a general

overview of the estimation process is presented.

Signal Processing

i.e.Filtering

Time Series Recorded

of Ship Responses in

Waves

Cross Spectral

Analysis of the Ship

Responses (MAR)

Matrix Representation of

Measured Spectra

Ship Responses

Initial Assumption of

Wave Spectra

parameters i.e

JONSWAP

Complex Ship

Transfer Functions

Matrix Representation of

Ship Transfer Functions

Inclusion of the Triple-

Valued-Problem for

Encounter Frequency

Transfer Function Matrix

including Encounter

Frequencies

Estimated Wave Spectra

Statement of the

Optimization Problem

Genetic Algorithms for

Solve the Optimization

Problem

Figure 5: Diagram of parametric estimation procedure.

The initial point of the estimation consider a set of time series measured ship responses, and with this

the corresponding response cross spectra, 𝑆𝑖𝑗 (𝜔𝑒), can be obtained. Because the speed of advance

of the ship the measured cross spectra are functions of the encounter frequency 𝜔𝑒.

Figure 6: Ship course, α, wave direction, θ, and relative motion 𝛽 respect to a fixed direction.

10

Figure 7: Notation of encounter angles and sign convention of β (in degrees).

𝛽 = 𝜃 − 𝛼 (1)

Figure 7 shows the typical notation of encounter angles of waves with respect to the ship, and the sign

convention assumed

In the scope of this work some assumptions are adopted in order to simplify the estimation

procedure:1)The ship responses are assumed to be linear with the incident waves, 2) The formulation

of waves consider deep water , 3) The complex-valued transfer functions 𝛷𝑖 (𝜔𝑒 , 𝛽) and 𝛷𝑗 (𝜔𝑒 , 𝛽) for

the 𝑖𝑡ℎ and 𝑗𝑡ℎ responses yield the theoretical relationship between the 𝑖𝑡ℎ and the 𝑗𝑡ℎ components of

the cross spectra and the directional wave spectrum 𝐸 (𝜔𝑒 , 𝜃) through the following integral equation

for details see Bhattacharyya (1978),

𝑆𝑖𝑗(𝜔𝑒) = ∫ 𝛷𝑖(𝜔𝑒 , 𝛽)𝛷𝑗(𝜔𝑒 , 𝛽) 𝐸(𝜔𝑒 , 𝜃)𝑑𝜃𝜋

−𝜋

(2)

where . denotes the complex conjugate of complex numbers.

In order to simplify the formulation, in Figure 6 the fixed position can be taken coincident to the ship

axis, it is wave direction can be given relative to the ship course, that is, 𝛽 = 𝜃. Hence, (2) is rewritten

𝑆𝑖𝑗(𝜔𝑒) = ∫ 𝛷𝑖(𝜔𝑒 , 𝛽)𝛷𝑗(𝜔𝑒 , 𝛽) 𝐸(𝜔𝑒 , 𝛽)𝑑𝛽𝜋

−𝜋

(3)

As a general remark on (3) it should be noted that the directional wave spectrum 𝐸 (𝜔𝑒 , 𝛽) is

convolved with, some known response function 𝛷𝑖 (𝜔𝑒 , 𝛽), to compute a signal𝑆𝑖𝑗 (𝜔𝑒), the cross

spectra. Thus, if the opposite is done in the estimation of the directional wave spectrum, 𝐸 (𝜔𝑒 , 𝛽) is

basically deconvolved from (3), which task is called deconvolution.

As proposed by Iseki (2000), the directional wave spectrum is advantageously estimated in the wave

frequency domain, 𝜔0 , however, when the right-hand side of (3) is transformed into a true wave

frequency, the so-called triple-valued function problem arise. This problem is ruled by the deep-water

relationship between the encounter and the wave frequency,

11

𝜔𝑒 = 𝜔0 − 𝜔02𝐴 , 𝐴 =

𝑉

𝑔𝑐𝑜𝑠𝛽

(4)

where V is the speed of the ship, and g the acceleration of the gravity

When the true wave frequency is represented in function of encounter frequency in following waves,

𝛽 ∈ [−𝜋/2 ; 𝜋/2 ], for certain speeds, three wave frequencies yield correct solution to the triple-valued

function problem, see Figure 8. Therefore, In following sea and with 𝜔𝑒 < 1/4𝐴 , when (3) is

converted into the wave frequency domain, the following three wave frequencies must to be

considered, as it cannot beforehand be calculated which of the frequencies is the real true one

a)

b)

Figure 8: Relationship between encounter and wave frequencies: a) All-directional relationship (an arbitrary example); and b) cross section at β = θ with θ ∈ [−90; 90] deg. (Nielsen, 2005).

𝜔01 =1−√1−4𝐴𝜔𝑒

2𝐴 |

𝑑𝜔01

𝑑𝜔𝑒| =

1

√1−4𝐴𝜔𝑒

𝜔02 =1+√1−4𝐴𝜔𝑒

2𝐴 |

𝑑𝜔02

𝑑𝜔𝑒| =

1

√1−4𝐴𝜔𝑒

(5)

12

𝜔03 =1+√1+4𝐴𝜔𝑒

2𝐴 |

𝑑𝜔03

𝑑𝜔𝑒| =

1

√1+4𝐴𝜔𝑒

Thus, when (5) is inserting in (3), the transformation into the wave frequency domain is

𝑆𝑖𝑗(𝜔𝑒) = ∫ 𝛷𝑖(𝜔01, 𝛽)𝛷𝑗(𝜔01, 𝛽) 𝐸(𝜔01, 𝛽) |𝑑𝜔01

𝑑𝜔𝑒

| 𝑑𝛽𝜋

−𝜋2

+ ∫ 𝛷𝑖(𝜔01, 𝛽)𝛷𝑗(𝜔01, 𝛽) 𝐸(𝜔01, 𝛽) |𝑑𝜔01

𝑑𝜔𝑒

| 𝑑𝛽

𝜋2

−𝜋2

+ ∫ 𝛷𝑖(𝜔02, 𝛽)𝛷𝑗(𝜔02, 𝛽) 𝐸(𝜔02, 𝛽) |𝑑𝜔02

𝑑𝜔𝑒

| 𝑑𝛽

𝜋2

−𝜋2

+ ∫ 𝛷𝑖(𝜔03, 𝛽)𝛷𝑗(𝜔03, 𝛽) 𝐸(𝜔03, 𝛽) |𝑑𝜔03

𝑑𝜔𝑒

| 𝑑𝛽

𝜋2

−𝜋2

+ ∫ 𝛷𝑖(𝜔01, 𝛽)𝛷𝑗(𝜔01, 𝛽) 𝐸(𝜔01, 𝛽) |𝑑𝜔01

𝑑𝜔𝑒

| 𝑑𝛽−

𝜋2

−𝜋

(6)

where is important notice that the second and the third line, which correspond to following seas, is

only consider for 𝜔𝑒 <1

4𝐴.

2.1.1.1 Matrix notation and multivariate expression

As was mentioned, in order to perform fast computational operations it is necessary to organize the

equations in a matrix form. Thus, to perform numerical calculations the integration on the right-hand

side of (6) is discretized as

𝑆𝑖𝑗(𝜔𝑒) = ∆𝛽 ∑ 𝛷𝑖𝑘(𝜔01)𝛷𝑗𝑘(𝜔01) 𝐸𝑘(𝜔01)

𝐾

𝑘=1

+ ∆𝛽 ∑ 𝛷𝑖𝑘(𝜔02)𝛷𝑗𝑘(𝜔02) 𝐸𝑘(𝜔02)

𝐾1

𝑘=1

+ ∆𝛽 ∑ 𝛷𝑖𝑘(𝜔03)𝛷𝑗𝑘(𝜔03) 𝐸𝑘(𝜔03)

𝐾

𝑘=1

(7)

where

13

∆𝛽 =2𝜋

𝐾 𝐸𝑘(𝜔0𝑚) = 𝐸(𝜔0𝑚, 𝛽𝑘) |

𝑑𝜔0𝑚

𝑑𝜔𝑒|𝛽=𝛽𝑘

, 𝑚 = 1,2,3

𝛽𝑘 = −𝜋 + (𝑘 − 1)∆𝛽, 𝛷𝑖𝑘(𝜔0𝑚) = 𝛷𝑖(𝜔0𝑚 , 𝛽𝑘) , 𝑚 = 1,2,3

(8)

Herein 𝐾1 is the number of small “areas” in Figure 8.b which corresponds to following seas (𝐾1 ≈ 𝐾/2 )

and it important to emphasized that only for −𝜋

2< 𝛽 <

𝜋

2 and 𝜔𝑒 <

1

4𝐴 , 𝑆𝑖𝑗(𝜔𝑒) can be calculated

directly from (7). Principally, the triple-valued function problem means that for each 𝜔𝑒 , and the K

different headings considered, there might be up to 𝐾 + 2𝐾1 different combination of frequency

response functions.

Using the discretized format the cross spectrum can be given in matrix notation as,

𝑆(𝜔𝑒) = ∆𝛽𝛷(𝜔01)𝐸(𝜔01)𝛷(𝜔01) 𝑇

+ ∆𝛽𝛷(𝜔02)𝐸(𝜔02)𝛷(𝜔02) 𝑇

+ ∆𝛽𝛷(𝜔03)𝐸(𝜔03)𝛷(𝜔03) 𝑇

(9)

where

𝑆(𝜔𝑒) = [

𝑆𝜂𝜂(𝜔𝑒) 𝑆𝜂𝜑(𝜔𝑒) 𝑆𝜂𝜃(𝜔𝑒)

𝑆𝜑𝜂(𝜔𝑒) 𝑆𝜑𝜑(𝜔𝑒) 𝑆𝜑𝜃(𝜔𝑒)

𝑆𝜃𝜂(𝜔𝑒) 𝑆𝜃𝜑(𝜔𝑒) 𝑆𝜃𝜃(𝜔𝑒)

]

(10)

𝛷(𝜔01) = [

𝛷𝜂1(𝜔01) … 𝛷𝜂𝐾(𝜔01)

𝛷𝜑1(𝜔01) … 𝛷𝜑𝐾(𝜔01)

𝛷𝜃1(𝜔01) … 𝛷𝜃𝐾(𝜔01)

]

(11)

𝐸(𝜔01) = [𝐸1(𝜔01) ⋯ 0

⋮ ⋱ ⋮0 ⋯ 𝐸𝐾(𝜔01)

]

(12)

𝛷(𝜔0𝑖) = [

𝛷𝜂1(𝜔0𝑖) … 𝛷𝜂𝐾1(𝜔0𝑖)

𝛷𝜑1(𝜔0𝑖) … 𝛷𝜑𝐾1(𝜔0𝑖)

𝛷𝜃1(𝜔0𝑖) … 𝛷𝜃𝐾1(𝜔0𝑖)

] , 𝑖 = 2,3

(13)

𝐸(𝜔0𝑖) = [𝐸1(𝜔0𝑖) ⋯ 0

⋮ ⋱ ⋮0 ⋯ 𝐸𝐾1(𝜔0𝑖)

] , 𝑖 = 2,3

(14)

After to calculated 𝑆 (𝜔𝑒) it is noticed that the cross spectrum matrix is Hermitian, i.e. 𝑆𝑖𝑗 = 𝑆��𝑖, with

𝑆𝑖𝑗 = 𝐶𝑖𝑗 + 𝑖 𝑄𝑖𝑗 (15)

15

𝑓 (��) = [𝐸1(𝜔01) 𝐸2(𝜔01) … 𝐸𝐾+2𝐾1−1(𝜔03) 𝐸𝐾+2𝐾1(𝜔03)]𝑇 (19)

The last term, �� is a Gaussian white noise sequence vector with zero mean and variance 𝜎2 ,

introduced only for stochastic reasons.

From the preceding it should be realized that, in general, when N kinds of response signals, i.e. time

series, are considered, then

𝑠𝑖𝑧𝑒(��) = 𝑁2 × 1

𝑠𝑖𝑧𝑒(��) = 𝑁2 × (𝐾 + 2𝐾1)

(20)

And from now on N kinds of time series are assumed.

2.1.2 Numerical considerations

When numerical computations are considered, an elementary however difficult problem arise due to

the triple-valued function problem. Hence, depending on the value of the encounter frequency, the

number +2𝐾1 can be, as pointed previously, variable, which implies a different dimension of �� and

𝑓 (��), for each encounter frequency. One way to overcome this problem and how to take into account

more encounter frequencies at the same time is explained in detail in Nielsen (2005) and is

reproduced in this work.

2.1.2.1 Set of a known wave frequencies

Fundamentally, the idea is that a certain range of wave frequencies is setting before the estimation

process, i.e. the total number 𝑀 of wave frequencies is known/chosen without considerations of the

triple-valued function problem (TVFP). By assigning all of the total 𝐾 heading angles to each of the

chosen wave frequencies, a kind of vector which shows the organization of heading angles and wave

frequencies can be set up, as presented in (21). The setup of this “vector” serves to give a better

visualization and understanding of how the coefficient matrix A is built. Is important the order

considered in the vector composed by the heading and waves frequencies. The ’vector’ is given in the

following way:

(𝛽1, 𝜔1)

(𝛽2, 𝜔1)⋮

(𝛽𝐾 , 𝜔1)

(𝛽1, 𝜔2)⋮

(𝛽𝐾 , 𝜔2)

(𝛽1, 𝜔3)⋮⋮

(𝛽𝐾 , 𝜔𝑀)

(21)

16

and as can see the total number of element pairs is 𝐾 • 𝑀 . Figure 9 presents a polar format

discretization used for the computational domain.

Figure 9: Discretization of wave frequencies and encounter angles.

As stated by the TVFP each encounter frequency has associated one or three wave frequencies for

each heading angle considered. The computed pairs of heading angles and wave frequencies are

distributed in accordance with (21). Clearly, the value of the wave frequency, as found from the TVFP,

is with little chance obtained exactly as one of the specified wave frequencies in (21). Theoretically,

however, the bigger the discretization for the specified wave frequencies then the better the TVFP

value can be approximated from (21). However, in order to avoid too many prespecified wave

frequencies, a weighting based on linear interpolation as proposed by Nielsen (2005) is introduced.

Therefore, by considering the situation depicted in Figure 10, the frequency response function is

calculated as,

𝛷(𝜔01) =𝑏

𝑎 + 𝑏𝛷(𝜔𝑢) +

𝑎

𝑎 + 𝑏𝛷(𝜔𝑢+1) (22)

where is important notice that in interpolation with complex numbers is recommended interpolate the

module and angle ,of each Complex transfer function, separately.

Figure 10: Weighting to avoid too many discretized frequencies.

17

With the pre-specified organization of wave frequencies and heading angles,(21), the coefficient

matrix A in the multivariate model expression increases from size 𝑁2 × (𝐾 + 2𝐾1), to,

𝑠𝑖𝑧𝑒(��) = 𝑁2 × (𝐾 • 𝑀) (23)

and, as result of this transformation, the matrix may now contain a lot of zero elements. The vector of

unknown discretized values of the wave spectrum is to be evaluated according to,

𝑓(𝑥) =

[ (𝑥𝛽1, 𝜔1)

(𝑥𝛽2, 𝜔1)

⋮ (𝑥𝛽𝐾 , 𝜔1)

(𝑥𝛽1, 𝜔2)

⋮(𝑥𝛽𝐾 , 𝜔2)

(𝑥𝛽1, 𝜔3)

⋮⋮

(𝑥𝛽𝐾 , 𝜔𝑀)]

(24)

so that 𝑠𝑖𝑧𝑒(𝑓 (𝑥)) = (𝐾 • 𝑀 ) × 1. Basically, this means that the 𝐾 • 𝑀 unknowns are to be solved

from the 𝑁2 equations, which, indeed, reflects a highly underdetermined equation system.

2.1.2.2 Generalization for more encounter frequencies

Apart from a new dimension of 𝐴 and 𝑓 (𝑥), the multivariate expression (16) still holds. Now, in order

to deal with a set of L encounter frequencies at the same time, and by discretizing the encounter

frequency as,

𝜔𝑒 = 𝑙 • ∆𝜔𝑒 , 𝑙 = 1,2,3, … , 𝐿 (25)

The cross spectrum vector is written as

𝑏 =

[ 𝑏1

𝑏2

⋮𝑏��

]

(26)

where 𝑏𝑙 = 𝑏 (𝑙 • 𝛥𝜔𝑒) and with 𝑏𝑙 determined from (17). It should be noted that 𝑠𝑖𝑧𝑒(𝑏) = (𝑁2 •

𝐿) × 1. Thus, the multivariate model expression in its final version reads

𝑏 = 𝐴𝑓(𝑥) + 𝑤 (27)

where

18

𝐴 =

[ 𝐴1

𝐴2

⋮𝐴��]

, 𝑠𝑖𝑧𝑒(𝐴) = (𝑁2 • 𝐿) × (𝐾 • 𝑀) (28)

𝑤 = [

𝑤1

𝑤2

⋮𝑤��

] , 𝑠𝑖𝑧𝑒(𝑤) = (𝑁2 • 𝐿) × 1 (29)

where w is the Gaussian white noise sequence vector which has zero mean and variance 𝜎2.

It is clear that the inclusion of more encounter frequencies results in more equations to the equation

system from which the wave spectrum is to be found. Usually, this means a less underdetermined

system. Increasing the number of equations in this manner only is, however, not optimal, as new

information put into the system is limited due to column degeneracy.

In order to solve this undetermined system, different authors have presents methods for example.

Akaike (2000) applies the Bayesian theory combined with the inverse theory to solve this

indetermination, using for it a so-called hyper parameter. However the scope of this work dealt with

the solution based on predefined parameterized forms for the wave such as JONSWAP. The

following section presents in detail this formulation.

2.2 Proposed method of parametric estimation

The wave energy spectrum can also be estimated from a procedure which takes the wave spectrum to

be comprised of one or more parameterized - theoretically known - wave spectra. In this chapter such

a procedure for estimation of the wave spectrum is outlined.

Basically, the relationship between the measured response spectra and the directional wave spectrum

is given by expression (27), including information on the equivalence of energy. Without the

introduction of white noise, no assumption is made on the error between the measured and the

calculated response spectra which means that the directional wave spectrum 𝑓 = 𝐸(𝜔, 𝛽) can be

sought in the least squares sense by solving,

min 𝜒2 ≡ min‖𝐴𝑓 − 𝑏‖2 (30)

with ‖. ‖ being the 𝐿2 norm.

Multiple peaked spectra with strong frequency overlap are complicated cases and thus multiple sea or

swell components are out scope of this work, which deals only whit single peaked spectra.

The JONSWAP spectral representation is given by,

19

𝑆𝑊(𝜔|𝐻𝑠, 𝑇𝑝, 𝛾) = 𝑐𝐻𝑠2 exp (−

1.25

𝜔𝑞4

)𝛾𝛼 (31)

with

𝑐 =5

2𝜋{16𝑇𝑝−1[1.15 + 0.168𝛾 − 0.925/ (1.909 + 𝛾) ]}

(32)

and 𝛼 = exp[−(𝜔𝑞−1

2𝛽)] , 𝛽 = {

0.07 , 𝜎𝑞 < 1

0.09 , 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 and 𝜔𝑞 =

𝜔

𝜔𝑝 , 𝜔𝑝 =

2𝜋

𝑇𝑝 Is the peak angular frequency and

𝛾 is the peak intensification factor. The directional spreading function is:

𝐷(𝛽|𝑠, ��) = 22𝑠−1 𝛤(𝑠 + 1)𝛤(𝑠)

𝜋𝛤(2𝑠) 𝑐𝑜𝑠2𝑠(𝛽 − ��) (33)

with |𝛽 − ��| <𝜋

2 and 𝒔 the spreading factor, �� the mean direction of wave propagation, 𝞒(𝒙) the

gamma function ( also known as generalized factorial). 𝞒(𝒏 + 𝟏) = 𝒏! If 𝒏 ∈ 𝑵.

Hence, by application of (32), and taking the wave direction to be coincident to the encounter angle,

the solution of equation (31) implies a non-linear optimization problem from which the best-fit-values

can be determined.

2.2.1 Estimated parameters

The optimal wave spectrum estimated by the parametric method is found from the optimization of the

parameters

{𝐻𝑠 𝑇𝑝 𝛾 �� 𝑠} (34)

The ship motions considered for the estimation procedure are generally a set three motions, i.e.

{heave, roll and pitch}, {sway, heave, pitch}, etc.

The cost function in these procedures may be written in a quite general form as in Pascoal et al.

(2007),

𝑓 = ∑ 𝑒𝑖 𝑒𝑖∗

𝑁2

𝑖=1

(35)

The errors are initially determined on a frequency by frequency basis and will have the following form

for each cross spectrum,

20

𝑒𝑟 = 𝑊𝑟

{∑ 𝐴𝑗,𝑘𝑚∆𝜔 . 𝑓(𝑥𝑘𝑚)𝐿𝑗=1 − ∑ 𝑏𝑟 . ∆𝜔𝑒

𝐿𝑟=1 }

∑ 𝑏𝑟 . ∆𝜔𝑒𝐿𝑟=1

(36)

where, 𝐴𝑗,𝑘𝑚 was defined in eq. (18), 𝑊𝑟 = 𝜆 ∑ |𝐴𝑟𝑀|𝑀𝑖=1

−1 average performed on 𝑟 (38), with

𝜆 = { 5 𝑖𝑓 𝑟 ≡ ℎ𝑒𝑎𝑣𝑒 2.5 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(37)

The value 𝜆 is considered bigger for the heave motion because is well know that to be motion with the

best numerical estimates.

3 Validation with numerical simulations

3.1 Time simulation of ship motions in waves

In this section of the present chapter the formulation for the ship responses in waves is presented. The

method to obtain the ship motions time series has input the complex ship transfer functions and a

known predefined wave spectra. The following diagram of Figure 11 shows the general overview of

the procedure.

Generate Wave Spectrum

i.e. JONSWAP

Start

Interpolating the Complex

Ship Transfer Functions

Complex Ship

Transfer

Functions

Calculate the Ship

Responses i.e. Heave, Roll

and Pitch

End

Figure 11: Algorithm for simulations of ship motions.

21

3.1.1 Formulation

Ship motions can, in principle, be found by a time domain solution.

𝑛 = 𝑛(𝑡) (38)

The generalized equations of motion where the motions, in six degrees of freedom

𝑛 = [𝑛1 , 𝑛2 , 𝑛3 , 𝑛4, 𝑛5, 𝑛6]𝑇, are the surge, sway, heave, pitch, roll and yaw. In addition global ship

responses, e.g. the vertical acceleration and the wave induced bending moment, can be derived from

the ship motion. In the following, a general global ship response 𝑅(𝑡) will be considered, without

differentiation between ship motions and derived responses.

On the ship assumption of a linear relationship between responses and wave excitations, the time

domain solution of the response 𝑅(𝑡) of a ship can be expressed in terms of the complex-valued

frequency response function𝛷𝑅(𝜔𝑒 , 𝛽). In this workm, the time domain solution of the response is,

however, presented with the same format as in Jensen and Capul (2006) although the latter

references consider only unidirectional waves. This, the response is written as a Gaussian process

introduced by a set of uncorrelated, standard normal distributed variables 𝑢𝑚𝑛 and ��𝑚𝑛. Hence,

𝑅(𝑡) = ∑ ∑[𝑢𝑚𝑛𝑐𝑚𝑛(𝑡) + ��𝑚𝑛(𝑡)].

𝑀0

𝑚=1

𝑁0

𝑛=1

(39)

The deterministic coefficients 𝑐𝑚𝑛(𝑡) and 𝑐��𝑛 are given by

𝑐𝑚𝑛(𝑡) = 𝜎𝑚𝑛|𝛷𝑅(𝜔𝑛 , 𝛽𝑚)|cos (𝜔𝑒,𝑛𝑡 + 𝜀𝑚𝑛) (40)

𝑐��𝑛(𝑡) = −𝜎𝑚𝑛|𝛷𝑅(𝜔𝑛, 𝛽𝑚)|𝑠𝑖𝑛 (𝜔𝑒,𝑛𝑡 + 𝜀𝑚𝑛) (41)

𝜎𝑚𝑛2 = 𝐸(𝜔𝑛, 𝛽𝑚)∆𝜔𝑛∆𝛽𝑚 (42)

where it should be noted that the discretized number of wave frequencies 𝑁0 and the discretized

number of headings 𝑀0 not (necessarily) take the same numbers as in the estimation analysis.

Furthermore, it should be realized that the variation over time is expressed in terms of the encounter

frequency.

𝜔𝑒 = 𝜔 − 𝜔2𝐴, 𝐴 =𝑉

𝑔cos (𝛽) (43)

with 𝑉 being the speed of the ship, and 𝑔 the acceleration of the gravity. 𝐸(𝜔𝑛, 𝛽𝑚) is the directional

wave spectrum, under the assumption that the wave direction is measured relative to the ship course,

22

and ∆𝜔 and∆𝛽 are the increments of the discrete wave frequencies and the discrete headings,

respectively. The phase angles are calculated from

𝜀𝑚𝑛 = tan−1𝐼𝑚[𝛷𝑅(𝜔𝑛 , 𝛽𝑚)]

𝑅𝑒[𝛷𝑅(𝜔𝑛 , 𝛽𝑚] (44)

3.2 Estimation of directional wave spectra from motions of a bulk

carrier

In order to test the parametric estimation procedure, numerical simulation based on ship motions, time

series and a known one directional wave spectrum are computed in this section. Thus, it is possible to

check the formulation and the program for many kinds of errors, and to some extent, the applicability

and correctness of the parametric estimation.

It should be mentioned that the generated time series correspond to a specific mean heading direction

between the ship and the waves. However, the estimations carried out for the generated time series

does not contain information about wave direction.

The first model for the simulations is a bulk carrier ship of 161 length between perpendiculars, 27 m of

breadth and 11m of draught. Complete information is shown in the table 2.

Table 2: Bulkcarrier ship characteristics

Main Dimensions, Bulkcarrier Ship1

Length,Lpp 161.7 m

Breadth,B 27.00 m

Draught,T 11 m

Δ 36,586.00 ton

The Ship 1 bulk was found in the library of examples of the MAXSURF software, called ‘ship pro.msd’.

The speed of advance is fixed in 15 knots for this ship. For estimation three responses were

considered, i.e. heave, pitch, and roll, as was suggested by Nielsen (2005) and Tannuri (2003).

Genetic algorithms (GA) are a possible choice to provide an adequate basin for the convex

minimization and can be applied to our problem.

The GA use here was designed by Lothrop (2003) and suffered only minor changes in order to be

applied to this problem. The floating point representation was chosen for all variables. Those

parameters that were problem specific are provided in the Table 3, as presented by Pascoal et al.

(2007).

23

Table 3: Considered genetic algorithm parameters.

Population 1000

Generations 30

Xover 70 % for 30 pairs

Mutate 5%

3.2.1 Directional estimations

In this section, two cases are studied. Each case is based on the JONSWAP spectrum with significant

wave height, Hs = 2 m, and wave period,Tp = 12 sec. However, the mean heading direction between

the waves and ship course was varied. This last parameter is varied in order to test the capabilities of

the algorithm in head and following waves.

3.2.1.1 Head waves

Figure 12 presents the generated and estimated wave spectra in polar form, for head waves for the

bulk ship 1. The generated sea state has a parameter Hs =2 m and Tp=12s and mean direction of the

wave 120 deg. The estimated wave has a Hs= 2.2 m, Tp=12s and mean direction 107 deg. The

estimated wave spectrum has a good agreement for significant wave height and wave period,

however the mean direction presents a divergence of 13 deg. This happens because the discretization

heading considering 30 deg. between two consecutive headings, this problem can be overcome using

better discretization for wave headings.

At least the estimated spectra have good agreement with the generated one, and the errors are

acceptable.

Figure 12: Directional estimations for ship 1 in head waves.

0.05Hz

0.1Hz

0.15HzN

S

W E

Generated Power Spectrum, mean = 120

Generated

Hs = 2m;

Tp = 12s;

Spread = 2;

Peak = 2;

0.05Hz

0.1Hz

0.15HzN

S

W E

Estimated Power Spectrum, estimated mean = 107

Estimated

Hs = 2.2m;

Tp = 12s;

Spread = 2.1;

Peak = 1.9;

24

In order to evaluate the convergence of the estimates for the same generated wave spectra a set of

estimations were computed and are presented in the figure 13. This plot presents the results of 20

estimations for the wave height and the wave period. In the case of the wave height the estimated

values varying between 1.7 and 2.5 m, and a perfect agreement was obtained for the simulation n.8.

The estimations for the wave period shows a estimation values from 11 and 13 sec. Figure 13 reveals

a good convergence of the method for Wave height and wave period in the case of following waves for

the ship 1.

Figure 13: Hs and Tp for 20 estimations of ship 1.

3.2.1.2 Following waves

The Figure 14 presents the generated and estimated wave spectra for following seas on the bulk ship

1, respectively. The generated wave spectrum was constructed using a JONSWAP formulation with

significant wave height of 2m and wave period of 12s and mean theta of 60 deg. The estimated wave

spectrum has Hs=1.6 m and Tp=12s and mean theta =60.5 deg. The estimated wave spectrum has a

good agreement for the Wave period and the Wave mean direction, they are almost the same. But in

the estimation of wave height some error arise, This error is due to that the cost function depends of

the energy content below the power curve, the directional spectra not only consider the three

parameters before discussed also depends on spread function, that compensate the error in wave

height, and also the peak insensitivity of the generated was 2 and this parameters for the estimated

increased at 2.2.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

Simulation no.

Hs[m

]

Ship 1, mean =120

Parametric

true

0 2 4 6 8 10 12 14 16 18 200

5

10

15

Simulation no.

Tp[s

]

Parametric

true

25

Figure 14: Directional estimations for ship 1 in following waves.

From Figure 12 and figure 14 is clear to see good capabilities of the parametric formulation to

estimated directional wave spectra for head and following seas for the bulk ship.

Table 4: Set of generated and estimated wave parameters for ship 1

Case Hs (m) Tp (s) γ β (deg) s

A

Generated 0.5 6 2 150 2

Estimated 0.2 6.9 2.3 130 2.2

Error (%) 60 15 15 13.3 10

B

Generated 0.75 9 2 150 2

Estimated 0.76 8.8 2.1 150 2.3

Error (%) 1.3 2.2 5 0 15

C

Generated 1 8 2 150 2

Estimated 0.75 8.1 1.9 111 2.2

Error (%) 25 1.25 5 26 10

D

Generated 1.5 10 2 150 2

Estimated 2.1 8.7 2.4 161 2

Error (%) 40 13 20 7.3 0

E


Estimated 1.7 12 1.8 150 2.3

Error (%) 15 7.7 10 0 15

F

Generated 2.5 12 2 120 2

Estimated 2.6 13 2.4 110 2.4

Error (%) 4 8.3 20 8.3 20

G


Estimated 2.4 10 1.6 123 2.5

Error (%) 20 0 20 2.5 25

H


Estimated 3.4 18 1.8 149 2.2

Error (%) 15 0 10 0.7 10

0.05Hz

0.1Hz

0.15HzN

S

W E


Generated

Hs = 2m;

Tp = 12s;

Spread = 2;

Peak = 2;

0.05Hz

0.1Hz

0.15HzN

S

W E

Estimated Power Spectrum, estimated mean = 60.5

Estimated

Hs = 1.6m;

Tp = 12s;

Spread = 2.2;

Peak = 2;

26

Table 4 shows the results of several estimations using the genetic algorithm for ship 1. The significant

wave height considered were from 0.5 to 4m and the wave period from 6 to 18 sec. from the table is

seen that the parametric modeling estimate the energy content of the wave spectrum close to the

generated wave fields. Thus, the mean values of the estimated significant wave height Hs are more or

less identical in the individual cases, with the exception of the cases A and D. It is seen that the error

on the significant wave height is, in most of cases, less than 25% (with the exceptions of the cases A

and D). of the generated value. In the case of the wave period Tp good agreement was found in all the

cases with errors less than 15%.

3.3 Estimation of directional wave spectra from motions of a small

vessel

In order to conduct an study of the influence of ship length , One additional ship were considered and

results of the Parametric estimation for directional wave spectrum in head and following seas are

presented in Figures 15 and Figure 17.

The second ship is a small one of 84 m of length between perpendiculars, and 15 m of breadth.

Details of the vessel are presented in table 4; complete information can be found in DELFship,

examples, demo 2.fbm.

Table 5: Main characteristics of ship 2

Main dimensions of ship 2

Length,Lpp 84.1 m

Breadth,B 15.3 m

Draught,T 4.75 m

Δ 3316 ton

3.3.1 Directional estimations

3.3.1.1 Head waves

The figure 15 presents the generated wave spectrum and the estimated wave spectrum, for head seas

on the ship 2. The generated wave spectrum has parameters, Hs=2m, Tp=12s and the mean

direction, 120 deg. The estimated wave spectrum has parameters, Hs=1.9m, Tp=13s and mean

direction, 115 deg. The estimation in this case was pretty good for the significant wave height and the

mean theta, however the wave period present and error compared with the estimation for the bulk ship

1. This error is due to the directly relation between the estimation frequencies and the ship length. For

small ship the transfer function has a resonance frequency at high frequencies and it is the reason

why more significant errors appear in the wave period.

27

Figure 15: Directional estimations for ship 2 in head Waves

In order to verify the convergence of the estimations for ship 2, a set of 20 simulations were computed

and are presented in the Figure 16. The simulations consider the same generated wave as in Figure

15. For the wave height the estimations values are between 1.8 m and 2.5 m, and the best solution

was found in simulation number 12. As expected, the wave period presents bigger errors. In Figure 16

it is clear to see good convergence of the parametric method for ship 2 in head waves.

Figure 16: Hs and Tp for 20 estimations of ship 2.

0.05Hz

0.1Hz

0.15HzN

S

W E


Generated

Hs = 2m;

Tp = 12s;

Spread = 2;

Peak = 2;

0.05Hz

0.1Hz

0.15HzN

S

W E


Estimated

Hs = 1.9m;

Tp = 13s;

Spread = 2.1;

Peak = 2.1;

0 2 4 6 8 10 12 14 160

1

2

3

Simulation no.

Hs[m

]

Ship 2, mean =60

Parametric

true

0 2 4 6 8 10 12 14 160

5

10

15

Simulation no.

Tp[s

]

Parametric

true

28

3.3.1.2 Following waves

In order to evaluate the capabilities of the parametric method in following waves, additional estimation

was carried out. The Figure 17 presents the generated and estimated wave spectrum for ship 2. The

Generated wave spectrum has parameters, Hs=2m, Tp=12s and mean theta direction of 60 deg. The

estimated spectra has Hs=2.2m, Tp=12s and men theta direction, 70 deg. In this case the estimate

spectrum present small errors for all the parameters. These errors were expected because the triple-

valued-problem in encounter frequency.

Figure 17: Directional estimation for ship 2 in following waves.

Table 6 presents the estimated sea state parameters for the same sea states, as have been

presented in table 4 for the parametric estimator of ship 2. As was expected the errors obtained the

second small vessel are less than the first ship. From the table is seen that the parametric modeling

estimate the energy content of the wave spectrum close to the generated wave fields. Thus, the mean

values of the estimated significant wave height Hs are more or less identical in the individual cases. It

is seen that the error on the significant wave height is, in most of cases, less than 20% of the

generated value. In the case of the wave period Tp good agreement was found in all the cases, with

the exception of the case A , It is seen that the error on the wave period is, in most of cases, less than

18% (with the exceptions of the cases A) of the generated value with errors less than 15%.

0.05Hz

0.1Hz

0.15HzN

S

W E


Generated

Hs = 2m;

Tp = 12s;

Spread = 2;

Peak = 2;

0.05Hz

0.1Hz

0.15HzN

S

W E


Estimated

Hs = 2.2m;

Tp = 12s;

Spread = 2;

Peak = 1.9;

29

Table 6: Set of generated and estimated wave parameters for ship 2

Case

Hs (m) Tp (s) γ β (deg) s

A

Generated 0.5 6 2 150 2

Estimated 0.53 7.9 2.4 147 1.6

Error (%) 6 31.7 20 2 20

B

Generated 0.75 9 2 150 2

Estimated 0.62 8.9 1.8 153 2.2

Error (%) 17.3 1.1 10 2 10

C


Estimated 0.79 8 1.7 138 2.1

Error (%) 21 0 15 8 5

D

Generated 1.5 10 2 120 2

Estimated 1.3 9.9 1.5 119 1.7

Error (%) 13.3 1 25 0.8 15

E


Estimated 1.6 14 2.1 146 2.4

Error (%) 20 7.7 5 2.7 20

F

Generated 2.5 12 2 120 2

Estimated 2.6 11 1.5 104 1.9

Error (%) 4 8.3 25 13.3 5

G


Estimated 2.7 11 2.2 115 2

Error (%) 10 10 10 4.17 0

H


Estimated 3.3 19 2.1 132 1.9

Error (%) 17.5 5.6 5 12 5

4 Validation with full scale ship trials

4.1 Development of the data acquisition system

The data acquisition system for monitoring ship motions is presented in this Chapter. The system is

composed by a set of sensors and equipment for navigation such as accelerometers, inclinometers,

gyros, GPS, wave radar and anemometers. The system also has capabilities to monitor and record

additional signals from the vessel system, i.e. from the odometer, shaft rpm, rudder order, etc. The

core of the system is composed by a unit Navigation System which has a set of 3 accelerometers and

3 angular rates. In order to increase the system reliability an additional sensor was added for the

monitoring of the Vessel Motions, i.e. OCTANS that has another 3 accelerometers and 3 angular

rates. This way a couple of measurements for each ship motion was recorded, i.e. surge, sway,

heave, roll, pitch and yaw.

To synchronize the whole signals coming from the sensors, a reconfigurable embedded control and

acquisition system, Compact-Rio, from National Instruments was used. The Compact-Rio is a rugged

30

hardware system that includes I/O modules, a reconfigurable Field-Programmable Gate Array (FPGA)

chassis, and an embedded Real Time controller.

The system presented was already installed onboard and tested with success in a navy vessel of the

Portuguese Navy. The tests were carried out in “Rio Tejo”, Lisbon, Portugal. The Data collected is

analyzed at the end of this Chapter.

4.1.1 Description of the LabVIEW Program

Figure 18 presents the architecture developed.

Crossbow Inertial

System and Ship

Signals

LabVIEW Program 3

User Inteface

LabVIEW program 1

Data Storage

LabVIEW program 2

GPS, Anemometer,

Octans Sensor and

Wave Radar

Laptop Computer

Onboard Equipment

C-RIO 9004 C-Rio 9074

Figure 18: Software architecture of the developed system.

Figure 19 shows the user interface of the main LabVIEW program. In this interface are displayed all

the signal monitored. The picture of the small vessel in dark corresponds to the schematic

representation of the ship behavior in real time. Also, the wind speed and the wind direction are

displayed. Other important chart presents the wave elevation. The ship Speed and Course over

Ground are also displayed in the user interface.

31

Figure 19: User interface of the developed system.

4.1.2 Sensor Locations

Figure 20 presents the location of the sensor onboard for the trial in the Navy vessel. The wave height

measurement sensor was located at the bow of the vessel. The Inertial Measuring unit (IMU) or NAV

system and the IXSEA accelerometers and Inclinometers were located as close as possible to the

center of gravity of the vessel. The GPS antenna was located at the top of the bridge it in order to

avoid some interference between the GPS and Satellites. Finally, the anemometer was located at the

top of the bridge in order to record the true wind speed without interference of bridge structure.

Wave Hight

Measurement

Sensor

OCTANS

sensor

IMU Sensor

GPS

Anemometer

Bow

Accelerometer

Figure 20: On board location of sensors and equipment.

4.1.3 Hardware Connections

Figure 21 shows the hardware used in the tests. This is mainly composed by two Compact-Rio that

are used for data acquisition and signal processing. The C-Rio 9004 was used to record the signals

coming from the Wave Radar, GPS unit, IXSEA and anemometer. The C-Rio 9074 recorded the

32

signals coming from the IMU (NAV440) and can also record several ship signals from the following

equipment: rudder, odometer, Shaft, etc. The Laptop computer was used to display all monitored

signals and to store the data collected. Both Compact-Rios communicated with the Laptop through an

Ethernet switch unit.

C-RIO 9004

TSK

TSKWM-2

Model 4

NI PS 15

OCTANS

NAV 440

Accelerometer

Connection Box

Compact-RIO

Distance Measurement

Sensor

Signal Processor

Power Supply

Gyrocompass and Motion

Sensor

Inertial Measurement

Unit

DG 14

MODEL 85106

Ultrasonic Anemometer

GPS Processor

DG 14

GPS Antenna

C-RIO 9074

Compact-RIO

NI PS 15

Power Supply

Ship Signals

Odometer, Rudder order,

RPM Shaft

NI UES-3880

Ethernet Switch

Laptop Computer

Data Display and

Storage

NI PS 15

Power Supply

Figure 21: Hardware structure of the developed system.

4.2 Description of the full scale ship trials

The on-board trials conducted on the Portuguese Navy vessel from the “Sagitario” class, were carried

out in “Rio Tejo”, with ship speeds between 7-15 knots and low wind conditions. With this test it was

intended to collect full-scale data to verify the parametric estimation method.

The data from GPS and anemometer were compared with the meteorological data available from the

Portuguese coast, which was found to be in reasonable agreement. The results from the two

accelerometers coincide with each other. The wave records (from the wave radar) were found to be

corrupted by low frequency waves, the reason for it is unknown. A clear distinction between this lower

33

frequency waves and the exciting waves (supposedly around a frequency of 1.2 rad/sec) was

possible. Therefore, it is recommended to check the calibration of the wave radar and use a constant

sampling rate between 0.5-1 sec for the sea trials in the future. Additionally, special attention is to be

paid at the sway and heave motions measured by the gyro compass as they appear erratic, even

though the corresponding velocities are measured appropriately.

Ship particulars are presented in table 5,

Table 7: Main characteristics of the Navy vessel.

Portuguese Navy vessel “Sagitario”

Length Overall, LOA 28.4 m

Breadth, B 5.95 m

Depth,D 3.5 m

Draft,T 1.39 m

Displacement 75 ton

The sensors and equipment used for the tests are summarized below. Detailed information on the

hardware can be found in the corresponding manufacturer website.

1. Radar wave height meter (model: WM-2; manufacturer: Tsurumi Seiki): microwave Doppler

with 13° beam angle and linear servo (accelerometer);

2. GPS receiver (model: DG14; manufacturer: Thales\Magellan);

3. 2-axes Ultrasonic anemometer (model: 85106; manufacturer: Young): wind speed and

direction.

4. Inertial Measuring Unit (IMU) – (model:NAV 440; manufacturer: Crossbow): 6-DOF (3

angular rates and 3 linear acceleration) with 3-Axis internal magnetometer;

5. Gyrocompass and motion sensor (model: Octans; manufacturer: IXSEA): 6-DOF, i.e. 3

accelerometers and 3 Fiber Optic Gyroscopes (FOG);

The recorded data is presented in its raw form. Table 6 shows the time of key events and locations

during the trial.

Table 8: Characteristics of the collected data.

Figure 22 shows the set of collected data and the location of the tests.

Item. Hour filename Key locations and events during the trial Status

A 14:54:28 14042014_145428.dat Departure from the Navy dock File provide good Data

B 15:26:21 14042014_152621.dat Crossing under ''Ponte 25 de Abril'' Loss of GPS signal

C 16:28:54 14042014_162854.dat

‘Rescue’ Manoeuvres (Emergency

naval rescue operation of a civilian), not

part of this test Files provide good Data

D 16:45:38 14042014_164538.dat --- Loss of GPS signal

E 16:48:40 14042014_164840.dat Arrived at the Navy Dock

34

a)

b)

Figure 22: a) Sets of data collected during the trials; b) Place of the tests, Lisbon.

4.3 Analysis of the results of the full scale ship trials

The parametric estimations for the collected data during the tests are presented in this sub section of

the Chapter. There are two plots for each data set: 1) The estimated spectra using the measured

motions; 2) The comparison between the estimated spectra and the measured wave spectra recorded

by the radar wave sensor.

4.3.1 Data A

The Data A represents the recordings from the departure from the Navy dock until the cross under the

bridge “25 de Abril”. Figure 23 shows the estimated wave spectra for data A, the significant wave

height, 0.44 m and the wave period, ≈5 s. and the mean theta direction, 175 deg. These parameters

are coherent with the presented by Santos et al.(1999), from which the average wave periods in “Rio

Tejo” were from 5 to 7s. The estimations also are coherent with the presented by Rusu et al.(2009)

from which the significant wave height at Tagus estuary were from 0.2 to 0.6m depending in wind

direction and intensity.

Figure 23: Estimated wave spectrum for Data A

-9.24 -9.22 -9.2 -9.18 -9.16 -9.14 -9.1238.67

38.675

38.68

38.685

38.69

38.695

38.7

38.705Onboard Trials

Longitud (deg)

Latitu

d (

deg)

Data A

Data B

Data C

Data D

Data E

0.1Hz

0.2Hz

0.3Hz

0.4Hz

0.5Hz

N

S

W E


Estimated

Hs = 0.44m;

Tp = 4.9s;

Spread = 2.5;

Peak = 1.6;

35

The comparison between the measured and estimated wave spectra for Data A is given in Figure 24.

The measured spectrum is plotted only in the range of the estimated frequencies. It can be seen

Figure 24 that the measured spectrum has more than one peak. The measured and estimated spectra

present similarities and also differences. These last can come from the fact that estimated spectra

considered deep waters while the measured spectrum shallow water. Also, the presence of wind was

not considered and the accuracy of the equipment were close to the maximum admissible value.

Figure 24: Comparison between estimated and measured wave spectra for Data A.

4.3.2 Data B

The Data B are the corresponding records from the cross under the bridge and straight navigation

along the river. Figure 25 presents the estimated wave spectra for the data B, the significant wave

height, 0.4 m and the wave period, ≈5.3 sec. and the mean theta direction, 183 deg. These

parameters are coherent with the presented by Santos et al. (1999), from which the average periods in

“Rio Tejo” were from 5 to 7 sec. The estimations are also coherent with the presented by Rusu et

al.(2009) from which the significant wave height at Tagus estuary were from 0.2 to 0.6m depending in

wind direction and intensity.

.

0 0.5 1 1.5 2 2.5 3 3.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Freq (rad/s)

Pow

er

(m2.s

)

Estimated Spectra

Measured Spectra

36

Figure 25: Estimated wave spectrum for Data B

The comparison between the measured and estimated wave spectra for the Data B is presented in

Figure 26. The measured spectrum is plotted only in the range of the estimated frequencies, showing

more than one peak. The difference between the two spectra come from the fact there were

considerable changes in the direction of navigation, i.e. turn back. For this case the speed of the

vessel was not always constant.

Figure 26: Comparison between the estimated and measured wave spectra for Data B.

4.3.3 Data C

The Data C are the corresponding records of “Rescue” manoeuvres (Emergency naval rescue

operation of a civilian), not part of the planned test, yet was considered for analyses. The Figure 27

presents the estimated wave spectra for the data C, the significant wave height, 0.3 m, the wave

period, ≈5.2 sec. and the mean theta direction, 215 deg. These parameters are coherent with the

presented by Santos et al. (1999), from which the average wave periods in “Rio Tejo” were from 5 to 7

0.1Hz

0.2Hz

0.3Hz

0.4Hz

0.5HzN

S

W E


Estimated

Hs = 0.4m;

Tp = 5.3s;

Spread = 2;

Peak = 1.2;

0 0.5 1 1.5 2 2.5 3 3.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Freq (rad/s)

Pow

er

(m2.s

)

Estimated Spectra

Measured Spectra

37

sec. The estimations are also coherent with the presented by Rusu et al.(2009) from which the

significant wave height at Tagus estuary were from 0.2 to 0.6m depending in wind direction and

intensity.

Figure 27: Estimated wave spectrum for Data C.

The comparison between the measured and estimated wave spectra for Data C is presented in Figure

28. The measured spectrum is plotted only in the range of the estimated frequencies. For this case, in

particular, the estimated spectrum is in good agreement with measured spectrum. This is because

during the test the direction and the vessel speed were kept constant.

Figure 28: Comparison between the estimated and measured wave spectra for Data C.

4.3.4 Data D

The Data D was not analyzed because it is too short, i.e. ≈8 minutes and as is known the sea-state

can be considered constant every 20 minutes.

0.1Hz

0.2Hz

0.3Hz

0.4Hz

0.5Hz

N

S

W E


Estimated

Hs = 0.3m;

Tp = 5.2s;

Spread = 1.3;

Peak = 0.85;

0 0.5 1 1.5 2 2.5 3 3.50

0.005

0.01

0.015

Freq (m/s)

Pow

er

(m2.s

)

Estimated Spectra

Measured Spectra

38

4.3.5 Data E

The Data E represents the corresponding records of “coming back” to the Navy dock, using the same

sailing path of test A. Figure 29 shows the estimated wave spectrum for data E, the significant wave

height, 0.48 m, the wave period, ≈7.1 sec. and the mean theta direction, 215 deg. These parameters

are coherent with the presented by Santos et al. (1999), in which the average wave periods in “Rio

Tejo” are from 5 to 7 sec. The estimations are also coherent with the presented by Rusu et al.(2009)

from which the significant wave height at Tagus estuary were from 0.2 to 0.6m depending in wind

direction and intensity.

Figure 29: Estimated wave spectrum for Data E.

Figure 30 shows a comparison between the estimated and measured spectra. The errors came from

changing direction and velocities, and by considering shallow instead of deep waters.

Figure 30: Comparison between the estimated and measured wave spectra for Data E.

0.1Hz

0.2Hz

0.3Hz

0.4Hz

0.5Hz

N

S

W E


Estimated

Hs = 0.46m;

Tp = 7.1s;

Spread = 1.7;

Peak = 2;

0 0.5 1 1.5 2 2.5 3 3.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Freqs( rad/s)

Pow

er

(m2.s

)

Estimated Spectra

Measured Spectra

39

5 Conclusions

The feasibility of estimation of directional wave spectrum based on ship responses measurements

was analyzed. Good results were obtained for numerical simulations and full-scale data.

The numerical estimations were computed based on simulated time-series ship responses. These

time-series were calculated using the ship complex transfer functions and known wave spectrum.

The influence of the ship length was addressed and better results were achieved for small vessels as

was presented by Pascoal et al. (2007).

The influence of following waves affects negatively the estimations, due to arise of the triple-valued of

the wave encounter frequency.

A system for monitoring of ship responses was developed and tested with success on a Navy vessel.

The LabVIEW environment was used for programming the system, representing an advantage since

the time spent on programming was reduced. The hardware used has low power consumption, thus it

can be installed in several ships without “disturbing” the vessel normal power consumption.

The use of redundant accelerometers and inclinometers improved the reliability of the vessel motions

recordings.

The estimation of the wave characteristics in “Rio Tejo” was presented and good results were

achieved. Five different recorded data was analyzed and the average significant wave height was and

wave period were 0.4m and 6s, respectively. These parameters are coherent with the presented by

Santos et al. (1999), in which the average wave periods in “Rio Tejo” are from 5 to 7 sec. this are also

coherent with the presented by Rusu et al.(2009) from which the significant wave height at Tagus

estuary were from 0.2 to 0.6m depending in wind direction and intensity.

40

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Fonseca, N. and Guedes Soares, C. (1998). Time-Domain Analysis of Large-Amplitude Vertical Ship Motions and Wave Loads, Journal of Ship Research, Vol. 42, No. 2, pp. 139-153.

Iseki, T., and Ohtsu, K., (2000). “Bayesian Estimation of Wave Spectra Based on Ship Motions,” Control Eng. Pract., 8, pp. 215–219.

Iseki T, Terada D.(2002) Bayesian estimation of directional wave spectra for ship guidance systems. International Journal offshore Polar Engineering.

Lothrop, K., (2003), http://kerry.lothrop.de/multiGA/, January 2005.

Nielsen UD. (2005).Estimation of directional wave spectra from measured ship responses. PhD thesis, Section of Coastal,Maritime and Structural Engineering, Department ofMechanical Engineering, Technical University of Denmark.

Nielsen, U.D., (2006). Estimations of directional wave spectra from measured ship responses. Mar. Struct. 19, 33–69.

Operation and Maintenance Manual (2010), “TSK Wave Height Meter”, Tsurumi Seiki Co. LTD, Japan.

Pascoal, R., Guedes Soares, C., Sørensen, A.J., (2007). Ocean wave spectral estimation using vessel wave frequency motions. J. Offshore Mech. Arct. Eng. 129 (2), 90–96.

Pascoal R., Guedes Soares C., (2008). Non-parametric wave spectral estimation using vessel motions, Applied Ocean Research, vol 30,pag 46-53.

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