Non-Normally Distributed Extreme ValueStatistics in Offshore Design

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Non-Normally Distributed Extreme Value Statistics in Offshore Design

DANIEL GHARANFOLI

Degree project in Naval Architecture Second cycle Stockholm, Sweden 2013

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“Statistics are no substitute for judgment.”

Henry Clay

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ROYAL INSTITUTE OF TECHNOLOGY

Abstract

KTH Centre for Naval Architecture

Department of Aeronautical and Vehicle Engineering

Master Thesis

byDaniel Gharanfoli

Extreme value behavior of a moored semi-submersible vessel is investigated. There is a need for alternative methods other than the Rayleigh peak model when investigating non-Gaussian processes. In this context the Rayleigh peak model will generally under- estimate extreme values. Four methods are investigated in this study with data from 1000 seeds. They are; construction of an empirical cumulative distribution function, mean of maximas, a LF/WF spectral partition and peak distribution tail fitting. In turn six peak distributions are investigated. It was found that global motions are more sensitive than point accelerations to estimation errors, and the more accurate methods should be applied to global motions. A fitted Weibull peak distribution proved to be the most conservative for both MPM value and 90^th percentile estimations. It was also found that a mean of 10 maxima was a good estimation of a MPM value. Longer seeds than three hours are recommended in order to include higher maxima and lower minima.

Further comparison studies are recommended.

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ROYAL INSTITUTE OF TECHNOLOGY

Sammanfattning

KTH Centre for Naval Architecture

Department of Aeronautical and Vehicle Engineering

Examensarbete

av Daniel Gharanfoli

Denna studie undersöker extremvärdesbeteendet av en förankrad flytande plattform.

Det finns ett behov av alternativa metoder till användandet av den Rayleigh topp- fördelade modellen när man undersöker icke normalfördelade processer. I dessa sam- manhang underskattar Rayleigh modellen extremvärden. Fyra alternativa metoder undersöks med hjälp av data fr˚an 1 000 simuleringar. Metoderna är; skapandet av en empirisk fördelningsfunktion, medelvärdet av maxima, en l˚agfrekvens/v˚agfrekvens spektrum-uppdelning och en “svans”-anpassning p˚a toppfördelningen. Sex stycken an- passningar undersöks. Det visade sig att globala rörelser är känsligare än punkt acceler- ationer för felen i uppskattningarna, och de mer exakta metoderna rekommenderas för de globala rörelserna. En Weibull-fördelad anpassning visade sig vara mest konservativ av de undersökta metoderna för b˚ade MPM värden och 90 percentiler. Det visade sig ocks˚a att medelvärdet av 10 maxima var en bra uppskattning av ett MPM värde. Längre simuleringar än tre timmar rekommenderas för att inkludera högre maxima och lägre minima. Vidare jämförelsestudier rekommenderas.

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Acknowledgements

To those that have made this thesis possible, and for guiding me to what I have become today. You have my deepest gratitude, and I thank you for your extended contributions.

For allowing me the chance to graduate, and for their helping hands, Jenny Trumars

Karl Garme.

For their daily advice, and generous contributions, Carl Almrot

Jonas Andersson G¨oran Johansson.

For his words and assistance, Anders Ros´en.

For a lifetime of support, encouragement and guidance, My family.

For their encouragement and support, My friends.

For extending my faith and boundaries, Ragnvald Løkholm Alvestad.

For making me follow through academic studies, My teachers.

For giving me confidence, My students.

For this L^ATEXtemplate, Sunil Patel: www.sunilpatel.co.uk

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List of Figures

2.1 PDF . . . 4

2.2 Process, peaks and maximum . . . 8

2.3 Process, peak and maximum PDF . . . 9

2.4 Narrow and broad banded spectrums . . . 11

3.1 Standardization . . . 14

3.2 ECDF . . . 16

3.3 LF/WF PSD . . . 17

3.4 Tail fitting. . . 20

4.1 Basic system overview . . . 21

4.2 Coordinate system and thruster positions . . . 24

4.3 Panel model . . . 25

4.4 Morison model . . . 25

4.5 Mooring system . . . 27

5.1 Benchmark convergence . . . 33

5.2 Benchmark convergence ECDF . . . 33

5.3 Observed range method 2 global accelerations . . . 36

5.4 Scatter method 2 global accelerations . . . 36

5.5 Observed range method 2 local accelerations. . . 38

5.6 Scatter method 2 local accelerations . . . 38

5.7 Scatter method 2 global motions . . . 39

5.8 Observed range method 2 global motions . . . 40

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List of Figures viii

5.24 Mean of maxima convergence, global accelerations . . . 63

5.25 Mean of maxima convergence, local accelerations . . . 64

5.26 Mean of maxima convergence, global motions . . . 64

5.27 Stationarity . . . 68

A.1 The Normal distribution . . . 74

A.2 The Gumbel distribution. . . 75

A.3 The Fr´echet distribution . . . 75

A.4 The Weibull distribution . . . 76

A.5 The Rayleigh distribution . . . 77

A.6 The Logistic distribution. . . 77

A.7 The Nakagami distribution . . . 78

A.8 The Exponential distribution . . . 78

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List of Tables

3.1 Behaviour of Fⁿ . . . 19

4.1 Point locations . . . 23

4.2 Main particulars . . . 23

4.3 Hydrostatics. . . 24

4.4 Restoring coefficients . . . 24

4.5 Drag coefficients . . . 26

4.6 Wind force coefficients . . . 26

4.7 Current force coefficients . . . 27

4.8 Mooring cable directions . . . 28

4.9 Mooring system stiffness . . . 28

4.10 Mooring line properties . . . 28

5.1 Results of screening . . . 32

5.2 Relative error, accelerations global coordinates mean of maxima. . . 35

5.3 Relative error, accelerations local coordinates mean of maxima . . . 37

5.4 Relative error, global motions mean of maxima . . . 40

5.5 Relative error, global motions mean of maxima . . . 41

5.6 Relative error, accelerations global coordinates LFWF . . . 44

5.7 Relative error, accelerations local coordinates LFWF . . . 46

5.8 Relative error, global motions LFWF . . . 49

5.9 Relative error, global motions LFWF . . . 50

5.10 Relative error, accelerations global coordinates fit . . . 54

5.11 Relative error, accelerations local coordinates fit . . . 56

5.12 Relative error, global motions fit . . . 58

5.13 Relative error, global motions fit . . . 59

5.14 Kurtosis and skewness, accelerations global coordinates fit . . . 60

5.15 Kurtosis and skewness, accelerations local coordinates fit. . . 61

5.16 Kurtosis and skewness, global motions fit . . . 62

5.17 Amplitudes relative to mean. . . 65

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Abbreviations

API American Petroleum Institute CDF Cumulative Distribution Function DNV Det Norske Veritas

DP Dynamic Positioning

ECDF Empirical Cumulative Distribution Function EPDF Empirical Probability Density Function FFT Fast Fourier Transform

FWD ForWarD

GEV Generalized Extreme Value

IMO International Maritime Organization ISO International Standard Organization

LF Low Frequency

MIMOSA -

MPM Most Probable Maximum

PDF Probability Density Function PID Proportional Integral Derivative

PS Port Side

PSD Power Spectral Density

RAO Response Amplitude Operator

SB StarBoard

SIMO SImulation of Marine Operations

WADAM Wave Analysis by Diffracion And Morison Theory WAFO Wave Analysis for Fatigue and Oceanography

WF Wave Frequency

x

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Physical Constants

Gravitational constant g = 9.81m/s² Density of water ρ = 1025kg/m³

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Symbols

F CDF -

f PDF (data⁻¹)¹

G GEV family CDF -

X Random variable data

E(X) Expectation data

V ar(X) Variance data²

s(X) Skewness -

k(X) Kurtosis -

N Normal distribution -

N_samples Number of samples -

N⁺ Number of zero up crossings -

x, y, z Cartesian coordinates data

n Number of peaks -

M_n Set of maxima data

mn Spectral moments of order n a_n, b_n Constants

S(ω) Power spectral density Power/rad

H_s Significant wave height m

Tp Mean peak period s

T_z Mean zero crossing period s

GM0 Metacentric height m

L Length m

B Breadth m

c₃₃, c₄₄, c₅₅ Hydrostatic stiffness coefficients (heave, roll, pitch)

1Data is either acceleration or motion

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Symbols xiii

Cd Drag coefficient -

F_i Wind & current force N

Mj Wind & current moment Nm

v Wind velocity m/s

KF,i Wind & current force coefficient kNs²/m² K_M,j Wind & current moment coefficient kNs²/m

X_X,Y,Z^accL Accelerations in local coordinates m/s²

X_X,Y,Z^accG Accelerations in global coordinates m/s²

Xsurge,sway,heave Global motions m

Xroll,pitch,yaw Global motions ^o

X(t) Parent process data

X(t)ˆ Parent process amplitudes data

X(t) Parent process mean or offset data

ˆ

x_estimated Estimated amplitude from method 2, 3 or 4 data ˆ

x_benchmark Estimated benchmark amplitude from method 1 data

x^tot_ext Total estimated extreme data

x^{LF,W F}_sign Low or wave frequency partion significant value data x^{LF,W F}_ext Low or wave frequency partion extreme value m/s²

r Correlation factor

α Shape parameter -

αp Quantile -

β Wave direction ^o

η Relative error (in amplitude) -

η_mean Ratio between amplitude and offset -

Ω Sample space data

ω Angular frequency rads⁻¹

σ Standard deviation data

µ Location parameter data

ξ Shape parameter -

γ Peak enhancement factor

Γ Gamma function

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Dedicated to my family for allowing me every opportunity to succeed in life, and to my supervisor Jenny Trumars Ph.D. for

allowing me an opportunity to graduate.

xiv

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Chapter 1

Introduction

During recent years, the demand, availability, procurement and means of consumption of energy has become a matter subjected to daily debate and discussion. A majority of the Earth’s surface is covered by water and oceans. We still only obtain a tiny fraction of the energy that can be obtained at sea. Oil procurement, wind, solar and wave energy are some of the most obvious that comes to mind.

In the case of offshore oil procurement, semi-submersible vessels are used to drill and extract oil. Such a vessel can be anchored to the seabed or it has thrusters to stay in place. It has to withstand whatever the oceans might challenge it with. This brings us to the field of offshore design. Offshore design is in short how to design a marine system in the best manner with regards to its constraints.

The requirements come from many parties with different interests. They come e.g. from governmental and international organizations such as the IMO, ISO and classification societies. They set health, safety, structural and environmental impact requirements.

For offshore structures, structural requirements are mainly governed by the sea states in which it is supposed to operate in. The sea state will induce responses such as accelerations and motions on the vessel. Another response is the so called air gap (the relative distance from platform to wave surface).

To design a vessel for these responses, it is necessary to understand or predict these conditions. The wave elevation is modelled as a stochastic process since it occurs at random. Responses will also occur at random. The design is governed by a level of safety in relation to these responses, and how certain the predictions are.

Extreme value theory or extreme value analysis is a fairly new branch of mathematics pioneered by Leonard Tippett (1902-1985) during the mid-20th century [1]. Despite this,

1

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Chapter 1. Introduction 2

it has emerged as one of the most important statistical disciplines for applied science.

Its applications are widely used, in particular in the fields of structural and geological engineering, finance and risk assessment, traffic predictions, earthquake predictions and ocean wave modelling to name just a few [2].

The aim of extreme value analysis is to quantify the stochastic behaviour of a process at unusually large or small levels, over those that have already been observed. In many cases local time data over a relatively short period of time is regarded, which is used to extrapolate extremes in the context of periods of time larger than that already observed.

E.g. with data from a three hour storm it is possible to deduct stochastic information used in the context of one year. In offshore design, the time context is usually around 100-10000 years [3].

Based on frequency domain calculations, and the assumption that a process is Gaussian, it is not too difficult to calculate the extreme values. A special case of extreme value theory is for narrow banded stationary Gaussian processes. They have a Rayleigh peak- distribution, which in turn has a Gumbel max/min distribution [3]. However, to assess influence on extreme value estimations on non-Gaussian responses, it has become more common to calculate responses in the time domain. A non-Gaussian response is e.g.

the air gap, which in addition can have missing data when the water reaches the deck box. It is then reasonable to question the applicability of the approach with an assumed Gaussian response. (Finding the extreme values for non-Gaussian responses is another matter entirely.)

There is no analytical solution for extreme values of non-Gaussian responses [4]. The objective of this study is to investigate methods for estimating the extreme values from non-Gaussian responses. The suitability of the methods is going to be assessed with regard to time consumption, and results. Another objective of this study is to find appropriate parametric distributions for the sampled data.

There are several ways to determine extreme values. In this study, four are considered. They are; construction of an empirical cumulative distribution function, mean of maxima, a LF/WF spectral partition and peak distribution tail fitting.

Chapter 2 of this report covers key concepts and introduction to theory, including classical extreme value theory. Chapter 3 describes the methodology of this study. Chapter 4 describes the simulation setup used to collect data. Chapter 5 presents the results and the methods are compared, and their applicability in practice is discussed. Chapter 6 concludes the work that has been done, and what recommendations are made. Finally Appendix A describes the investigated parametric distributions.

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Chapter 2

Background Theory

2.1 Basic Statistics

2.1.1 Random Variables

Random variables are variables that vary due to chance (denoted for example X), and does not have a fixed value. They can either be discrete or continuous. Continuous variables can take any continuous range of real values in the sample space Ω, while discrete variables can only take a discrete range in another sample space Ω. Discrete random variables are not considered in this report. All following concepts apply to continuous random processes.

2.1.2 Probability Distribution Functions

There are two important types of probability distribution functions, the cumulative distribution function (CDF), and the probability density function (PDF), see Figure 2.1. The CDF is a non-decreasing function

F (x) = Pr{X ≤ x}, (2.1)

such that F (x−) = 0, and F (x₊) = 1. Where x− and x₊ are the lower and upper bounds of Ω [2]. If the CDF is differentiable, the PDF is defined as

f (x) = dF

dx, (2.2)

and by extension

F (x) = Z x

−∞

f (u)du. (2.3)

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Chapter 2. Background Theory 4

This yields that [2]:

Pr{a ≤ X ≤ b} = Z b

a

f (u)du = F (b) − F (a). (2.4)

If the CDF is not differentiable, there cannot be a PDF by equation (2.2). In these cases a histogram (a bar plot) can be used instead to indicate densities. The mode and a percentile is visually explained in Figure 2.1[3].

Figure 2.1: Visual representation of a 90^th percentile and the mode.

2.1.2.1 Key measurements

The CDF and PDF contain a lot of information. It is convenient to express some of it as scalar values. The first one is the expectation, defined as [2]:

E(X) = Z

Ω

xf (x)dx. (2.5)

The symbolic value of the expectation is that it can be thought of as a pivot point that yields moment equilibrium around either sides of the point in the PDF [3]. It gives a sense of location where the distribution lies. Further the variance is defined as [2]:

Var(X) = σ² = Z

Ω

{x − E(X)}²f (x)dx, (2.6)

and is a measure of spread of data. The standard deviation, defined as the square root of the variance yields the same information, but expressed in the same dimension as the expectation and the variable itself.

Skewness (2.7) [5] and kurtosis (2.8) [6] are measures of the shape of a probability distribution.

s(X) = E(x − µ)³

σ³ , (2.7)

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k(X) = E(x − µ)⁴

σ⁴ , (2.8)

Skewness is a measure of how the mode leans (or is weighted) to either the left, right or not at all. That is how the distribution deviates from the normal distribution in this respect.

Kurtosis is a measure how a distribution deviates from the normal distribution. In the respect that the tails are either “thicker” or “thinner”, than those of the normal distribution.

A Normal distribution has a kurtosis value of 3, and a skewness of 0 [7].

2.1.3 Distribution Fitting/Regression

There are numerous different types of distributions, describing different kinds of data.

They are in most cases composed by the use of elementary functions. Such as the exponent or logarithmic function in various combinations.

Suppose a set of sampled data has been collected. It is in many cases of interest to learn the behaviour of the outcome of these data, and the properties they have. In the best of cases, it is possible to find that they follow a pattern, according to a parametric distribution. This gives the possibility to make a claim that a set of data is for instance so called “normally” distributed, and make inferences beyond the data at hand. That is, if the data matches a Normal distribution. The advantage of parametric models is that a large set of data can be summarized by yet another few measures. The number of parameters varies in amount depending on the current model. There exist for instance a 2 parameter Weibull distribution, and a 3 parameter Weibull distribution. But in general terms the parameters may be found as “location”, “scale” or “shape” parameters of a distribution.

At times, some of these may even be the previously mentioned key measurements (such as expectation or standard deviation). For example the standard normal distribution N (0, 1) with 0-mean, is described by µ = 0 as the location parameter, and σ² = 1 as the scale parameter.

The aim of distribution regression is to find the distribution that best fit the sampled data at hand. In turn to fit a distribution, the distribution parameters need to be estimated.

There are many methods of estimating the fit of parameters. Some are more appropriate for some distributions, while not as good for others. They also put different weight to different parts of the data at hand. A full account of which methods should be used

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is beyond this study. But popular choices include the least squares method, maximum likelihood methods, moment estimators’ methods and so on. The least squares method is known for putting more weight on the data of the upper tail, which is desirable for this study [3]. For practical reasons however, the estimation method was chosen by the available resources and was most often the maximum likelihood methods. This means that estimated extremes will be slightly smaller (see section2.2).

The opposite of the parametric model, would be a non-parametric distribution function.

The disadvantages of these are that they are not described by a few parameters, and they are hard to understand. Non-parametric distributions have the advantage that they fit data points in a “better” way than parametric.

2.1.3.1 Distribution Fit Diagnostics

Suppose a distribution fit has been made, and the parameters have been estimated. Two important questions naturally occur. How good is the estimation? Is it relevant? Often the regression can be found as completely irrelevant. There are two main approaches to dealing with this problem. The first one is hypothesis testing algorithms, and graphical judgement using diagnostic graphs.

Hypothesis testing algorithms include among several, the Kolmogorov-Smirnov test, Anderson-Darling test and Chi square tests. They work in different ways, but they share the same main idea. First state a null hypothesis that the sampled data follows the proposed distribution. Then the algorithms produce a number (this number is generated in different ways depending on the testing algorithm). If the number is higher than a critical value of the algorithm (depending on the number of samples, and number of possible outcomes), then the null hypothesis is rejected at a certain significance level [8] (usually around 5%).

A diagnostic graph in its easiest form is the PDF and CDF. These shapes should obviously match the data. The problem one might encounter is that the axes are linear, and the visual judgement is too rough when it comes to making fine-tuned judgements.

But they certainly provide a good starting platform to make judgements on the general fit. Then there are plots with non-linear axes. The Weibull plot is a good example with logarithmic axes. The sampled points in this graph should simply follow a straight line in order to fit a Weibull distribution [3]. Two more general plots are the probability plots and quantile plots. They hold basically the same information but are presented in different ways. The probability plot may vary in different ways. One way is the pp-plot, where the empirical probability is on the x-axis, and the regression probability is on the y-axis. The sampled points should follow a straight line. Another way is to have the

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x-axis linear with the original data, and a logarithmic y-axis of the probability. The sampled data should follow a line of the model-data [8].

2.1.4 Stationary Random Process

In short this is a stochastic process where the statistic properties for example key parameters mean and variance does not change over time, space or sample length [9]. This means that it is possible for many “short” truncated time series to describe the same process. This is the basic assumption made to generate a large set of comparable data for this study. But obviously the extracted truncated time series is not fully independent of sample length. For example it cannot be as short as 10 s for this investigation, it still need to be “long enough” to include “large enough” maxima and minima.

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2.2 Classical Extreme Value Theory

The following sections aim to model the behaviour of the following expression:

M_n= max{X₁, . . . , X_n}, (2.9) where X₁, . . . , X_n are independent random processes with a common CDF F [2].

2.2.1 Parent, Peak and Maximum Distributions

Before moving on to deeper theory, it is necessary to explain some important concepts.

Suppose a simulation has been made of a semisubmersible platform in a three hour storm and motions and accelerations are measured. Depending on the sampling frequency, this will yield a set of around 30-100 thousand points of the process itself. A subset containing only the peaks of the process will be at around 1000 points. Another subset containing the maximum (or minimum), will only contain one value. It is still possible to create a larger set of maxima (or minima) by performing more simulations, assuming it is a stationary process. It is now possible to create CDFs for all of these sets of data. These are the parent distribution, peak distribution and maximum (or minimum) distribution.

This is illustrated in Figure 2.2and in Figure 2.3.

Figure 2.2: Visual representation of a process, peak and a maximum.

This leads us to the important concept of the extreme value. It is a chosen value in the maximum (or minimum) distribution in Figure2.3, [9], (see section2.5).

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Figure 2.3: Process, peak and maximum distributions (PDFs).

2.2.2 Fundamental model

The main objective of this study is to find and understand the distribution of Mn from equation (2.9). Theoretically M_n can be derived exactly as:

Pr{M_n≤ x} = Pr{X₁ ≤ x, . . . , X_n≤ x}

= Pr{X₁ ≤ x} × . . . × Pr{X_n≤ x}

= {F (x)}ⁿ.

Fmax(x) = {Fpeak(x)}ⁿ. (2.10)

The basic principle of equation (2.10), is that each peak shares the same peak distribution [2]. Thus n is the number of encountered peaks for a narrow banded process. However more generally for broad banded processes, n is the number of zero up crossings. For a narrow banded process it can be shown that the number of zero up crossings is

N⁺ = T 2π

r m₂

m₀, (2.11)

where m₂ and m₀ are spectral moments (see section2.3), and T is the total sample time [9]. This is a very good estimation for a broad banded process.

This short and simple relationship is a very powerful tool, and the basis of extreme value theory. A problem is commonly that Fpeakis unknown. Therefore it is most often estimated by observations, see section 3.7. Small deviations in the estimation of F_peak,

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will lead to “large” deviations in Fmax. There are also limitations to the model, e.g. if n goes to infinity.

2.2.3 Extremal Types Theorem

Another approach is not to look at the peaks (Fpeak), but to look at estimations of (F_peak)ⁿ from observations of maxima (or minima).

Theorem 2.1. If there exists a set of constants {a_n> 0} and {b_n} such that

n→∞lim Pr{(M_n− b_n)/a_n≤ x} → G(x),

where G is a non-degenerate distribution function, then G belongs to one of the families of extreme value distributions [2].

The families of extreme value distributions are the Gumbel, Fr´echet and Weibull distributions (type I, II or III), according to:

I : G(x) = exp n

− exph

−x − b a

io

, −∞ < x < ∞; (2.12)

II : G(x) =







0, x ≤ b,

exp n

−

x−b a

−αo

, x > b; (2.13)

III : G(x) =





 expn

−h

−

x−b a

αio

, x < b,

1, x ≥ b.

(2.14)

These have a scale parameter a > 0, and a location parameter b, and then types II and III have a shape parameter α > 0.

Theorem 2.1 is very powerful in the sense that G(x) only converges to one of three families of distributions, regardless of F_peak.

2.2.4 The GEV distribution

Although theorem 2.1 is a convenient first step, there still remains an uncertainty of which family is the correct model to pursue for specific applications. Luckily all three types are similar. This yields the possibility of reformulating all three types to a single type of distribution, called the generalized extreme value (GEV) distribution.

Theorem 2.2. If there exists a set of constants {an> 0} and {bn} such that

n→∞lim Pr{(Mn− b_n)/an≤ x} → G(x),

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where G is a non-degenerate distribution function, then G belongs to the GEV family:

G(x) = exp n

−h 1 + ξ

x − µ σ

i−1/ξo

, {x : 1 + ξ(x − µ)/σ > 0}, (2.15) where {−∞ < µ < ∞, − ∞ < ξ < ∞, σ > 0} [2].

If the shape parameter ξ is estimated to be positive, the GEV fit represents a type II distribution. If it is negative it represents the type III. Finally if ξ is approximately zero, it represents the type I distribution. Theorem 2.2 is yet another powerful tool, and it greatly simplifies work with investigating sets of maximum (or minimum).

2.3 Spectral Analysis

Frequency domain calculations are a part of this study. Therefor a few key concepts needs to be clarified. Spectral moments m_n of order n are defined as:

mn= Z ∞

0

ωⁿS(ω)dω, (2.16)

where S(ω) is the power spectral density (PSD) [10]. The PSD is partially obtained by the use of an FFT applied to the time signal. m₀ is also the variance σ² from the original time signal [9].

The difference between a narrow banded signal and a broad banded signal is illustrated in Figure2.4. In the time domain a narrow banded signal has usually only one peak per zero mean crossing, and a broad banded may have several peaks per zero mean crossing.

Figure 2.4: Narrow and broad banded spectrums. The broad banded spectrum has more components.

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2.4 Extreme Values of Narrow Banded Stationary Gaus- sian Processes

The calculation of extreme value distributions of a Gaussian parent distribution is somewhat simple, from a narrow banded stationary Gaussian process. It can be shown that the peak distribution is Rayleigh distributed, assuming peaks that are large enough and independent. The Rayleigh distribution also has the scale parameter as the standard deviation of the parent process. Finding the extreme value distribution is now not that difficult, when Fpeak is known, and known to be “true” (as Gaussian parent distributions have a Rayleigh peak distribution). The extreme value distribution is also Gumbel distributed [9].

2.5 Extreme Values in Offshore Design

The definition of an extreme value was earlier mentioned. In order to gain understanding of how a particular set of maxima behave, the aim has been so far to finding a maximum distribution. This behaviour will later be implemented as a basis for decision making.

But what properties of the maximum distribution will be used for this purpose?

Classification societies set requirements based on extreme values that are the mode of the distribution. This extreme value is called the most probable maxima (MPM). It is not the largest maxima, but it occurs most frequently of all maxima. Other requirements are on maxima in between the 85-95^th percentile [10]. This extreme value is generally not as common as other maxima, but it is one of the very largest values observed amongst all the maxima.

Once a value has been chosen, a designer must find a safety factor in accordance to what extreme value has been chosen. Obviously the safety factor varies with the knowledge of how large the extreme is, and if larger values are expected or not.

The difference between the MPM and the 90^th percentile, doesn’t necessarily have to be large. For maximum distributions with low variance, these values are quite similar.

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Chapter 3

Methodology

Time series for the platform motion and acceleration responses are generated using SIMO [11]. The response data makes the foundation of investigating four methods of finding extreme values. They are selected from API:s recommended practice [12], and will be compared to the method of assuming a narrow banded stationary Gaussian process..

However, there is one method that has been modified for this study. More specifically the responses are global motions as surge, sway, heave, pitch, roll and yaw. Furthermore, also point accelerations in X, Y and Z directions are calculated (in both earth-fixed and body-fixed coordinate systems, denoted global and local coordinates). Both maxima and minima distributions are investigated, but at times only the maxima is mentioned in the descriptions. The same principles apply for the minima as for the maxima.

3.1 General Methodology

In this investigation time series are “standardized” to have zero mean according to:

X(t) = X(t) − X(t),ˆ (3.1)

also illustrated in Figure 3.1. Thus only amplitudes or deviations from the mean are investigated. This process is naturally reversible according to:

X(t) = ˆX(t) + X(t), (3.2)

if the actual response values are of interest.

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Chapter 3. Methodology 14

Results are presented as relative errors according to:

η = xˆ_estimated− ˆx_benchmark ˆ

x_benchmark . (3.3)

Negative values are underestimations, and positive values are overestimations of the benchmark. Note that this is in relation to the benchmark amplitude. This means that a relative error of +1 is an amplitude that is twice as large as the benchmark amplitude.

This is done because comparing numerical values between point accelerations and global motions is of no significant meaning. In this way it is possible to compare the methods comprehensively when applied to different types of responses.

Figure 3.1: Illustration of the standardization and relevant variables.

Finally since numerical values are not considered, amplitudes are given a context by dividing parent process maximum amplitudes by the parent process mean value according to:

η_mean= X(t)ˆ

X(t). (3.4)

3.2 Simulation Mean of Maxima Convergence

As previously mentioned the data used for this investigation are from several three hour simulations. It is tested to see how the mean of maximas depended on simulation duration. That is, is three hours enough to include some of the largest maximas (or minimas), if it is only possible to have access to e.g. 10 seeds? Alternatively, is 10 seeds enough to make assumptions of MPM values or 90^thpercentiles?

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3.3 Screening Methodology

For this study non-Gaussian responses are of interest. These must be found and therefore a screening of all responses are made. This is also important because when having to simulate a lot of data and subjected to resource constraints, it is also important to only simulate the relevant data. To avoid statistical uncertainties in extrapolation of quantile α_p, the minimum number of required samples for a certain set should be [3]:

N_samples= 1

1 − α_p (3.5)

If α_p= 0.999, N_samples= 1000. In this investigation it is chosen to do 1000 simulations of the same process, to achieve a reliable ECDF. For the setup described in section 4.1, this means that if all points of measurements will be simulated 1000 times, in all currently available combinations, it will result in a total number of samples (with each seed containing 32767 samples per response) of :

sea states × samples × seeds × headings × (point accelerations + motions) = 1 × 32767 × 1000 × 24 × (21 × 3 + 6) ≈ 5.43 × 10¹⁰.

This number is not feasible for this study, and why it is important to neglect some information, and why a screening is relevant.

For the screening process itself, six seeds (simulations) for one sea state are made. Six might seem as a low number, but considering that they are complete records of all combinations, it is a lot of data. A mean value of kurtosis and skewness is calculated for the responses. If it shows a “significant” deviation from the Normal distribution skewness and kurtosis values of 0 and 3, it is selected as a candidate. Kolmogorov- Smirnov and chi square hypothesis testing algorithms are done to test if the response is normally distributed. If it is indicated as normally distributed, it is rejected as a candidate.

3.4 Method 1 - Construction of an ECDF

The first method is to create an empirical cumulative distribution function (ECDF) of the maximum and minimum distributions, see Figure 3.2. This is done because it is based on actual observations of the maxima (and minima). With a high number of observations, this function should reflect the true outcome of extremes, and will be set as a benchmark for the other methods. Its only downside is that it is very time-consuming.

In many practical cases, only a few (v10) simulations/observations can be afforded.

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The empirical maximum distribution function can be obtained by extracting a set of maxima from several time series (or in general, from sets). That is, from each time series, only one maxima is obtained.

Once the ECDF is constructed, it may be desirable to make an “EPDF” by derivation, to finally extract an MPM value. However for a low number of samples, say a few hundred, the ECDF is not differentiable. Only if the number of samples becomes sufficiently large, the ECDF will approximately become differentiable. So there is not really a derivative to be found (that is non-degenerate). Recalling Theorem2.2, the ECDF should converge to the Generalized Extreme Value distribution. Therefore, in order to find the mode, or the MPM, a GEV regression is made to the ECDF. It does require a few samples (v15, sometimes more, sometimes less) in order for the estimation of parameter algorithm to converge. Once the parameters are found, the MPM is easily obtained from the analytical expression of the GEV distribution (see AppendixA).

Figure 3.2: An ECDF compared to a theoretical shape of the distribution.

An alternative to using a GEV regression is to smooth the rough ECDF using a moving average algorithm, and then finding an approximate EPDF. A benefit of the GEV regression is that it guarantees only one mode of the EPDF.

When finding a quantile, or percentile, the GEV approach is not necessary, as the point can be obtained directly from the ECDF.

This method requires only data on maxima (or minima). Peak distribution data is not relevant as the method goes directly from the process to the maxima, without regarding probabilities of other peaks.

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3.5 Method 2 - Mean of Maxima

This method is very simple. All that needs to be done is to take the mean value of a set of maxima values. This is an inexpensive way of finding a quick estimation of the magnitude of the extreme value. It should give a better estimation of an MPM value, rather than a high numbered percentile. However, if the variance of the outcome is low, it might even be close to a high numbered percentile. But for a low number of observations, it is as all the other methods sensitive to the actual randomness and uncertainties of the observations.

This method requires only data of maxima, and peak distribution data is of no relevance.

3.6 Method 3 - Low and Wave Frequency Partition

MARINTEK has developed this heuristic method of finding extreme responses for design purposes [13]. Thus it does not find the actual maxima, just a value “close” to this for design purposes. It is based on extremes from model tests and simulation studies. It makes a partition, or segmentation of the spectrum S(ω) in two parts (if the spectrum looks like that of Figure 3.3).

Figure 3.3: A smoothed PSD illustrating a low frequency contribution, and a wave frequency contribution, separated by a cut-off frequency. (Generated using WAFO

[14, 15]).

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One part is a low frequency (LF) part, and the other a wave frequency (WF) part. It is given as:

x^tot_ext= max

(x^LF_sign+ x^{W F}_ext x^{W F}_sign+ x^LF_ext

)

(3.6)

where x^{LF,W F}_ext is the estimated extreme value based on either the LF or WF contribution of the spectrum. x^{LF,W F}_sign is the significant response defined as:

x_sign = 2σ, (3.7)

for either the LF or WF contribution (recall section2.3). In words, equation (3.6) has separate contributions from the LF segment and WF segment. It combines the significant value from one contribution, plus the estimated extreme from the other contribution.

This can in turn be done in two ways as (3.6) states. The total extreme value used for design, is the maximum of these two ways of combining.

MARINTEK have a way of finding the extremes x_extfor the LF and WF parts. For LF the so called Stansberg method [16] is used, and for WF it is assumed that the response is a narrow banded Gaussian wave process. In this study, the Stansberg method is not considered, as it is somewhat complex. Instead, a simpler replacement is used. This study then investigates assumed Gaussian segments, plus added significant values. The LF motion component of wind contribution is Gaussian, but the LF wave drift forces is actually more similar to the exponential distribution [16]. The total LF response in itself is somewhere in between the exponential and Gaussian distribution. But it is of interest to see what the LF part yields with the use of Gaussian MPM estimation.

The overall use of the narrow banded Gaussian assumption should be more suited for the more narrow segments of the total broad banded spectrum in this way.

The selected adjusted method requires data of the time series to create a PSD, in order to find the two separate variances used to finding significant values and MPM values.

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3.7 Method 4 - Peak Distribution Tail Fitting

The fourth method will be approaching the extreme value problem from the peak distribution point of view. Recalling equation (2.10) as:

Fmax(x) = {Fpeak(x)}ⁿ.

A common problem with practical implementation of this equation, is that F_peak has to be defined. An empirical set can certainly be found, but it is still not always certain what distribution it may really have. A way of dealing with this problem is to simply assume a peak distribution, fitted to the empirical set at hand [2]. But what makes a good fit? Recall from section2.2that small deviations in F_peak, may result in substantial deviations in (Fpeak)ⁿ. First, regard Table3.1, that illustrates what happens to different parts of (F_peak)ⁿwith n = 1000.

Table 3.1: Behaviour of large values of Fpeak for (F_peak)ⁿ with n = 1000.

F_peak (F_peak)¹⁰⁰⁰ 0.8 ∼ 10⁻⁹⁷ 0.9 ∼ 10⁻⁴⁶ 0.95 ∼ 10⁻²² 0.99 ∼ 10⁻⁵ 0.999 ∼ 0.4

An important lesson from Table 3.1, is that only the very largest values of Fpeak, or in the so called “upper tail” of F_peak, will result in an maxima distribution. Another lesson is also that the tail is very sensitive. Therefore it is essential that the assumed fit of F_peak matches in the upper tail, to avoid errors in (F_peak)ⁿ. This is why this method is also known as ”tail fitting”.

A probability plot (see figure3.4) is used to assess whether the upper tail of an assumed peak distribution will fit in a good way to the empirical data at hand. This is done without regarding the fit to the lower region and the body of the distribution, as they would still only converge to 0 with large numbers of n.

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Figure 3.4: The six fits to a set of peak values. The upper tail behaviour is different.

The following distributions are used in this investigation:

• Rayleigh distribution,

• Weibull distribution,

• Logistic distribution,

• Nakagami distribution,

• GEV distribution,

• Non parametric distribution.

They are to be investigated as hypothetical good fits to the tail of F_peak. All of these have a general good fit to the data at hand. Although it is worth mentioning that their tail behaviour is different from one another, see Figure3.4. The GEV distribution might seem contradictory to resemble a peak distribution. For the tail however, it does have either exponentially or polynomially declining tail behaviour, depending on its shape parameter [2]. As the Weibull distribution is at times used as a peak distribution, it is interesting to see how the GEV behaves (as the Weibull is a special case of the GEV distribution). The Logistic and Nakagami distributions are hypothetically good alternatives as they fit the overall distribution in a good way.

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Chapter 4

Simulation Setup

4.1 Computer Simulations

The global motion and point acceleration data used to investigate the different methods are computer generated, from simulations of a semi-submersible drilling unit like in Figure4.1. The analysis is performed in the time domain using the software SIMO [11].

SIMO is fully capable of creating non-Gaussian responses. The exciting wave forces are proportional to the wave height, and wind and current forces are quadratically proportional to the respective relative velocities. 32767 samples are sampled at a sampling frequency of 3.03 Hz.

Figure 4.1: Basic system overview.

21

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Chapter 4. Simulation Setup 22 4.1.1 Conditions

The simulations are performed for so called survival conditions. This is a condition more severe than what the unit usually operates in, and it still has to survive this condition without damage and serves as a design condition. The environmental conditions with a 100 year return period considered for survival are representative by the North Atlantic.

Only one of the North Atlantic conditions are considered, with sea state parameters Hs= 17m, Tp = 16.5s, Tz = 12.8s and γ = 3.2. The water depth is 300m, with currents of 1.2m/s. The wind velocity is as high as 44 m/s (for 10s mean).

Incoming wave directions does not only excite the vessel from one direction. There is also a spread of wave energy. To clarify if e.g. an incoming wave direction of 180^o is considered, there will also be small contributions from 179^o (and so on). Therefore there can be sway and yaw motions due to head seas. Wind and current forces will then help to accentuate this phenomenon once the vessel has an offset.

4.1.2 Lateral accelerations

The lateral accelerations XâccGX and XâccGY from the time domain analysis in SIMO are calculated in an earth-fixed global coordinate system. These accelerations are trans- formed to XâccLX and XâccLY in the local body-fixed coordinate system as follows:

X_XâccL = X_XâccGcos(X_pitch) − (g + X_ZâccG) sin(X_pitch) (4.1) X_YâccL = X_YâccGcos(X_roll) + (g + X_YâccG) sin(X_roll) (4.2)

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Chapter 4. Simulation Setup 23 4.1.3 Investigated Points

Relevant points on the vessel where point accelerations are calculated are in Table 4.1.

Table 4.1: Locations of point measurements.

Pt. No. Point location Position

1 Pontoon ends SB-FWD

2 SB-AFT

3 PS-AFT

4 PS-FWD

5 Main deck corners SB-FWD

6 SB-AFT

7 PS-AFT

8 PS-FWD

9 Main deck center CENTER

10 Life boat platforms SB-FWD

11 SB-AFT

12 PS-AFT

13 PS-FWD

14 Drill floor center CENTER

15 Top of crane foundations PS

16 SB

17 Accommodation corners PS

18 SB

19 Burner boom SB-AFT

20 Top of tower SB-AFT

21 Centrum helideck SB-AFT

4.1.4 Model Description

Here follows a short summary of the different components of the total model. To describe the full setup and methodology is beyond the scope of this report. The coordinate system is defined according to Figure 4.2, with the z-axis positive “upwards”, so it is a right- handed Cartesian coordinate system. A few vessel main particulars can be found in Table4.2to be aware of the its size.

Table 4.2: Main particulars.

Breath outside pontoos 80 [m]

Length of pontoons 110 [m]

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Chapter 4. Simulation Setup 24

Figure 4.2: Definition of the coordinate system and thruster positions.

4.1.4.1 Hydrostatic properties

Tables 4.3and 4.4 holds some hydrostatic data, with stiffness coefficients. This gives a brief indication of the vessel without also going in to added mass, and dynamic damping.

Table 4.3: Hydrostatic data.

Displacement 56 000 [tonnes]

Water depth 300 [m]

Water density 1025 [kg/m³]

Table 4.4: Hydrostatic restoring coefficients, excluding contributions from mooring system.

Degree of freedom Stiffness at survival draught for GM₀ = 1.0m (transversally).

Heave c₃₃= 12.5· 10³ kN/m Roll c44= 5.5· 10⁵ kN/rad Pitch c₅₅= 5.7· 10⁵ kN/rad

4.1.4.2 Mass distribution

In the analyses the mass is represented with the total mass, centre of gravity and radii of gyration, and added mass.

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4.1.4.3 Added mass and RAO model

A panel model is made to calculate added mass damping and response amplitude op- erators (for each element) according to Figure 4.3. A quarter part is sufficient due to symmetry.

Figure 4.3: Panel model.

4.1.4.4 Drag model

Drag forces on the wetted part of the hull are represented by a Morison model in the frequency domain, composed by slender rod elements (see Figure 4.4). Different rods have different C_dused for the Morison equation [10]. Drag coefficients C_dare calculated according to DNV [10] and are listed in Table 4.5.

Figure 4.4: Slender element layout.

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Table 4.5: Drag coefficients of the Morison model.

Region Drag coefficient

Column

Longitudinal 1.1 Transverse 1.2 Pontoon

Vertical 1.9 Horizontal 1.1 Wing pontoon

Vertical 2.1 Horizontal 0.3

4.1.4.5 Wind force coefficients

The wind force, and wind force moments are calculated as:

F_i = K_F,i· v² (4.3)

Mj = K_M,j· v². (4.4)

Where Fi is the wind force in the i:th degree of freedom {i=1,2,3}, and KF,i is the wind force coefficient in the i:th degree of freedom, and v is the wind velocity. And the same is for Mj, {j=4,5,6}. The wind force coefficients can be found in Table4.6.

Table 4.6: Wind force and moment coefficients for survival draught. Values for heading 225^o is obtained by the use of interpolation.

Heading [^o]

K_F,1 [kNs²/m²]

K_F,2 [kNs²/m²]

K_F,3 [kNs²/m²]

K_M,4 [kNs²/m]

K_M,5 [kNs²/m]

K_M,6 [kNs²/m]

180 -3 0 1 8 -109 3

220 -3 -2 1 72 -91 1

230 -3 -3 1 80 -75 -2

270 0 -3 1 75 3 -2

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4.1.4.6 Current force coefficients

The current forces and moments are calculated in the same way as for wind. The current force coefficients can be found in Table 4.7.

Table 4.7: Current force and moment coefficients for survival draught. Values for heading 225^o is obtained by the use of interpolation.

Heading [^o]

K_F,1 [kNs²/m²]

K_F,2 [kNs²/m²]

K_F,3 [kNs²/m²]

K_M,4 [kNs²/m]

K_M,5 [kNs²/m]

K_M,6 [kNs²/m]

180 -360 0 -250 0 15000 0

220 -330 -790 -780 -9200 12300 -11700

230 -250 -970 -710 -12100 11800 -12100

270 0 -840 60 -7100 2200 900

4.1.4.7 Mooring model

The mooring system (Figure 4.5) properties are based on the information available at the time of this analysis. There is only a mooring system designed for 200m available.

However it is also assumed that it is representative for 300m water depth. In the simulations the 200m mooring system is combined with a 300m radiation-diffraction solution from WADAM [17]. Drag forces on the lines due to currents are not considered.

The mooring system is more specified in Tables4.8,4.9and 4.10.

Figure 4.5: Mooring system layout. A 12 point symmetric system.

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Table 4.8: Mooring cable directions.

Line No. 1 2 3 4 5 6 7 8 9 10 11 12

Line direction rel.

positive x-axis [^o]

340 330 320 220 210 200 160 150 140 40 30 20

Table 4.9: Mooring system stiffness.

Surge Sway Heave Roll Pitch Yaw

Surge 140

kN/m

3000 kN/rad

Sway 50

kN/m

2000 kN/rad

Heave 60

kN/m

Roll 2000

kN/m

210000 kN/rad Pitch 3000

kN/m

210000 kN/rad

Yaw 350000

kN/rad

Table 4.10: Mooring line properties.

Diameter [mm] 84

Axial stiffness [MN] 680

Unit weight in air [kN/m] 1.52 Ratio of weight in water to weight in air 0.87

4.1.4.8 Dynamic positioning model

The spread mooring system previously mentioned is complemented by a dynamic positioning system. It consists of four thrusters on each pontoon, see Figure 4.2. The maximum bollard pull is reduced to in survival conditions according to API recommended practice [12], due to current reduction. The DP system is controlled by a PID regulator in SIMO. It strives to minimize the platform offset by adjusting the available thrust forces.

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Chapter 5

Results and Discussion

5.1 Introduction

5.1.1 Responses

This chapter presents the results of this study. First there is a section on the screening process (section5.2) and what responses are considered. Recall that for this study point accelerations and global motions are considered. Point accelerations are considered in two coordinate systems (global (earth-fixed) and local (body-fixed, see section 4.1.2)).

Note that global motions are the same all over the vessel (that is, independent of where it is measured). Thus the specified investigated points from section5.2are only relevant for point accelerations.

5.1.2 Method 1

Section5.3 presents the results from determining the benchmark values.

29

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Chapter 5. Results and Discussion 30 5.1.3 Methods 2 and 3

The relative error η according to equation (3.3) is calculated to assess how good the methods are. It is presented in both tables and bar plots, (they hold the same information). Results are in general presented by how the responses are excited (incoming wave direction). For global motions there is an additional table and bar plot to present the observed span of η where the different motions are separated instead of incoming wave direction.

The observed span of η comes from 1000 seeds.

After the relative errors are presented, observed trends of kurtosis and skewness influence are presented.

Section 5.4 and 5.5 first presents results for point accelerations in global coordinates, followed by results of point accelerations in local coordinates, and finally results of global motions.

5.1.4 Method 4

The relative error η according to equation (3.3) is calculated to assess how good the method is. It is presented in both tables and bar plots, (they hold the same information) for all six peak distribution fits. Results are in general presented by how the responses are excited (incoming wave direction). For global motions there is an additional table and bar plot to present the observed span of η where the different motions are separated instead of incoming wave direction. In this case only the Weibull fit is considered.

The observed span of η comes from 300 seeds as it is time-consuming to find the parameters for each fit, and six distributions are investigated.

After the relative errors are presented, observed trends of kurtosis and skewness influence are presented.

Section5.6first presents results for point accelerations in global coordinates, followed by results of point accelerations in local coordinates, and finally results of global motions.

5.1.5 Simulation mean of maxima convergence

Section5.7presents how simulation duration affects the largest and smallest values of a seed.

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Chapter 5. Results and Discussion 31 5.1.6 Amplitudes relative to the parent process mean values

Section5.8gives the estimated amplitudes a context relative to the parent process mean values.

5.1.7 Discussion

Section5.9 discusses the results.

5.1.8 Suggested further reading and studies

Section5.10 gives recommendations on how this study can be expanded.

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Chapter 5. Results and Discussion 32

5.2 Results of the Screening Process

Point numbers {6, 8, 13, 15, 17, 20, 21} from Table4.1, are interesting candidates for this study when investigating point accelerations. In turn, incoming wave directions of 180ô (head seas), 225ô (quartering seas) and 270ô (beam seas) with respect to the coordinate system in Figure4.2 are interesting. This is from a set of 24 different directions, (360ô with 15ô increments).

Heave motion and accelerations in z-direction are Gaussian, as expected. They are however still used in this investigation for the sake of completeness.

This is all summarized in Table5.1.

Table 5.1: Results of the screening process.

Point accelerations

Incoming wave directions 180ô 225ô 270ô

Directions X, Y, Z

(in two coordinate systems) Investigated points {6, 8, 13, 15, 17, 20, 21}

6 Main deck corners, SB-AFT

8 Main deck corners, PS-FWD

13 Life boat platforms, PS-FWD

15 Top of crane foundation, PS

17 Accomodation corners, PS

20 Top of tower, SB-AFT

21 Centrum helideck, SB-AFT

Global motions

Incoming wave directions 180ô 225ô 270ô

Motions All global motions

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5.3 Results Method 1-Construction of an ECDF

As previously mentioned 1000 seeds are performed primarily to construct a reliable ECDF as a benchmark. 1000 seeds is enough for the MPM and 90^th percentile values to converge to benchmarks. In Figure 5.1 it can be seen that some fluctuations occur under 200 seeds, but ultimately stabilize. Figure5.2 illustrates that the ECDF in fact converges to the GEV distribution, and 1000 seeds is a sufficient number.

Figure 5.1: Benchmark MPM (left) and 90^thpercentile (right) convergence in sway motion. Sway have the largest fluctuations, and since it converges, other responses also converge. The benchmark estimations are normalized by dividing with the last

numerical value (at 1000 seeds).

Figure 5.2: Benchmark ECDF convergence to the GEV distribution in sway with 1000 seeds. Other responses have even better fits.

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5.4 Results Method 2-Mean of Maxima

Here follows the results of relative errors of means of 10 seeds to assess how the method is. Note that since it takes 10 seeds to create one observation, only 100 observations can be generated. After the errors follows a section on correlations between corresponding mean skewness and mean kurtosis values (for the 10 seeds) to the relative error.

To clarify, as one observation of an extreme value is generated from 10 seeds, it is in turn necessary to generate a mean of 10 skewness values and 10 kurtosis values in order to observe correlations to skewness and kurtosis. This span of skewness and kurtosis will then be narrower as it is mean values.

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Chapter 5. Results and Discussion 35 5.4.1 Relative errors for accelerations in global coordinates

Table5.2 and Figure5.3 shows the relative error according to equation (3.3) for point accelerations in a global coordinate system. They are created by observing 9 times 4 combinations of scatter diagrams like in Figure 5.4 (with 7 point locations in each).

That is

Directions(X, Y, Z) × Headings(180ô, 225ô, 270ô) × Benchmarks(4) =

= 3 × 3 × 4 = 9 × 4.

Table 5.2: Relative error accelerations, mean of maxima, global coordinates.

Max/crests MPM Max/crests 90^th percentile Wave

heading

Direction Observed span of η [%]

Wave heading

180 X −4 → +12 180 X −18 → −4

180 Y −4 → +11 180 Y −16 → −4

180 Z −4 → +10 180 Z −16 → −4

225 X −4 → +12 225 X −18 → −5

225 Y −4 → +12 225 Y −16 → −4

225 Z −4 → +10 225 Z −16 → −4

270 X −4 → +10 270 X −16 → −4

270 Y −4 → +12 270 Y −16 → −4

270 Z −4 → +10 270 Z −16 → −4

Min/troughs MPM Min/troughs 90^th percentile Wave

heading

Wave heading

180 X −4 → +14 180 X −20 → −5

180 Y −4 → +12 180 Y −16 → −4

180 Z −4 → +12 180 Z −16 → −4

225 X −3 → +12 225 X −18 → −5

225 Y −4 → +10 225 Y −16 → −4

225 Z −4 → +11 225 Z −16 → −4

270 X −4 → +10 270 X −16 → −4

270 Y −3 → +12 270 Y −16 → −3

270 Z −4 → +12 270 Z −16 → −3

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Figure 5.3: Observed range of relative errors for accelerations in global coordinates, method 2.

Figure 5.4: Scatter diagram of means of 10 maxima, with corresponding mean kurtosis and skewness values. Relative error to the corresponding maximum distributions 90^th

percentile. 100 points per group.

Non-Normally Distributed Extreme ValueStatistics in Offshore Design