DEVELOPMENT OF A MATLAB-BASED TOOLBOX FOR TIDE GAUGE

RECORDS USING HILBERT-HUANG TRANSFORM

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF

MIDDLE EAST TECHNICAL UNIVERSITY

BY

AYDIN JAMALI

IN PARTIAL FULFILMENT OF THE REQUIREMENTS

FOR

THE DEGREE OF MASTER OF SCIENCE

IN

GEODETIC AND GEOGRAPHIC INFORMATION TECHNOLOGIES

SEPTEMBER 2013

Approval of the thesis:

DEVELOPMENT OF A MATLAB-BASED TOOLBOX FOR TIDE GAUGE

RECORDS USING HILBERT-HUANG TRANSFORM

submitted by AYDIN JAMALI in partial fulfillment of the requirements for the

degree of Master of Science in Geodetic and Geographic Information

Technologies Department, Middle East Technical University by,

Prof. Dr. Canan Özgen _____________________

Dean, Graduate School of Natural and Applied Sciences

Assoc . Prof. Dr. Ahmet Coşar _____________________

Head of Department, Geodetic and Geographic Information Technologies

Prof. Dr Mahmut Onur Karslıoğlu

Supervisor, Civil Engineering Dept., METU _____________________

Examining Committee Members:

Assoc. Prof. Dr. Ali Kılıçoğlu _____________________

General Command of Mapping- Retired

Prof. Dr Mahmut Onur Karslıoğlu _____________________

Civil Engineering Dept., METU

Prof. Dr. Zuhal Akyürek _____________________

Civil Engineering Dept., METU

Assist. Prof. Dr. Uğur Murat Leloğlu _____________________

Geodetic and Geographic Information Technologies Dept., METU

Assoc. Prof. Dr. Utku Kanoğlu _____________________

Engineering Sciences Dept., METU

Date: 16.09.2013

iv

I hereby declare that all information in this document has been obtained and

presented in accordance with academic rules and ethical conduct. I also declare

that, as required by these rules and conduct, I have fully cited and referenced all

material and results that are not original to this work.

Name, Last Name: AYDIN JAMALI

Signature:

v

ABSTRACT

DEVELOPMENT OF A MATLAB-BASED TOOLBOX FOR TIDE GAUGE

RECORDS USING HILBERT-HUANG TRANSFORM

Jamali, Aydin

M.Sc., Department of Geodetic and Geographic Information Technologies

Supervisor: Prof. Dr. Mahmut Onur Karslıoğlu

September 2013, 77 pages

Tidal records are usually processed on the basis of Harmonic Analysis which in fact

works well for linear and stationary data. It is a mathematical procedure which divides

the data in a finite number of harmonic constituents. But the tidal data is in reality

non-linear and non-stationary because of atmospheric and meteorological parameters.

Therefore in this work a MATLAB –based Toolbox has been developed for hourly

tidal data collected from four tide gauge stations at the Turkish costs using Hilbert

Huang Empirical Mode Decomposition (HHEMD) which is an efficient and powerful

method for non-linear and non-stationary time series. Hilbert Huang Empirical Mode

Decomposition (HHEMD) is carried out in two steps. Firstly data are processed by

Empirical Mode decomposition (EMD) which decomposes the signal in a number of

intrinsic mode components (IMFs). The second step is to apply the Hilbert transform

to each IMF in order to find Phase, Amplitude and Instantaneous Frequency of the

signal.

In this study the hourly tidal data are decomposed in tidal and sub-tidal components

using HHEMD and the results of the tidal components are compared with the results

of harmonic analysis. Very long and non-stationary period components are filtered as

sub tidal. Finally the frequency and amplitude of the diurnal and semi-diurnal

component are used to extract the minimum and maximum velocities of the diurnal

and semidiurnal tidal constituents in order to analyze the capability of producing tidal

energies from possible tidal turbines to be located at tidal stations. Also an

exemplarity completeness test has been performed to validate the decomposition of

the time series obtained by the Antalya station utilizing EMD. The result showed that

the signal has been recovered properly.

Keyword: Empirical Mode Decomposition, Hilbert Huang Transform, Intrinsic Mode

Functions, Tidal and sub-tidal phenomena.

vi

ÖZ

MAREOGRAF İSTASYONU KAYITLARI İÇİN HILBERT-HUANG DÖNÜŞÜMÜ

KULLANILARAK MATALAB TABANLI YAZILIM PAKETİ GELİŞTİRİLMESİ

Jamali, Aydin

Yüksek Lisans, Jeodezi ve Coğrafi Bilgi Teknolojileri Bölümü

Tez Yöneticisi: Prof. Dr. Mahmut Onur Karslıoğlu

Eylül 2013, 77 sayfa

Genellikle gelgit kayıtları, doğrusal ve durağan veriler için iyi çalışan harmonik

analize dayalı olarak değerlendirilirler. Harmonik analiz, veriyi sonlu sayıda harmonik

bileşene bölen matematik bir prosedürdür. Fakat gerçekte gelgit verisi atmosferik ve

meteorolojik parametreler nedeni ile doğrusal-olmayan ve durağan-olmayan

karakterdedir. Bu nedenle, bu çalışmada Türkiye sahillerindeki dört adet mareograf

istasyonundan toplanan saatlik gel-git dataları ile doğrusal-olmayan ve durağan-

olmayan zaman serileri için etkili ve güçlü bir metod olan Hilbert Huang Deneysel

Mod Ayrıştırma (HHDMA) (Hilbert Huang Empirical Mode Decomposition) yöntemi

kullanılarak, MATLAB tabanlı yazılım paketi geliştirilmiştir. Hilbert Huang Deneysel

Mod Ayrıştırma yöntemi iki adımda gerçekleştirilir. İlk olarak veri, sinyali bir dizi

yapısal mod bileşenine ( intrinsic mode components) ayıran Deneysel Mod Ayrıştırma

(DMA) (Empirical Mode decomposition) yöntemi ile işlenir. İkinci adım, sinyalin faz,

genlik ve anlık frekanslarının bulunması için ikinci adımda her bir yapısal mod

bileşenine Hilbert Dönüşümü uygulanır.

Bu çalışma kapsamında gelgit verisi, HHDMA kullanılarak gelgitsel (tidal) ve gelgit

dışı (sub-tidal) bileşenlere ayrıştırılmıştır ve gelgit bileşenlerinin sonuçları harmonik

analizle karşılaştırılmıştır. Çok uzun ve durağan-olmayan devir bileşenleri, gelgits dışı

bileşen olarak filtrelenmiştir. Son olarak, gelgit istasyonlarına yerleştirilecek olası

gelgit tribünlerden, gelgit enerjisi üretme yeteneğinin analizinde kullanılabilecek

günlük ve yarı günlük gelgit bileşenlerinin minimum ve maksimum hızlarını elde

etmek için günlük ve yarı günlük bileşenlerin frekans ve genlikleri kullanılmıştır.

Ayrıca DMA kullanan Antalya istasyonundan elde edilen zaman serisi ayrışımlarını

doğrulamak için örnek bir tamlık testi gerçekleştirilmiştir. Sonuç sinyalin uygun

biçimde yeniden elde edildiğini göstermektedir.

Anahtar Kelimeler: Deneysel Mod Ayrıştırma, Hilbert Huang Dönüşümü, Yapısal Mod

Fonksiyonları, Gelgitsel ve gelgitsel dışı etkenler.

vii

To my Shima

viii

ACKNOWLEDGEMENT

I wish to thank my supervisor Prof. Dr Mahmut Onur Karslıoğlu for his guidance,

support and advices during my graduate study.

I also thank to examining committee members Assoc. Prof. Dr. Ali Kılıçoğlu, Prof.

Dr. Zuhal Akyürek, Assist. Prof. Dr. Uğur Murat Leloğlu, Assoc. Prof. Dr. Utku

Kanoğlu for their valuable comments and contributions.

I would like to thank my wife Shima for her encouragement, quiet patience and

unwavering love. I would like to thank my family especially my mother and father for

being such a huge supporter through my study and thank them for everything. I have

to say that I couldn't have completed the thesis without my parent. Also, I thank

Shima’s parents for their unending encouragement.

I would like to convey my deepest thank to my friend Eren Erdoğan who has not left

me lonely in my graduate study.

Last but not least, I would like to express my deepest thank to my friend Matt Piroglu

who has been a source of love and energy.

ix

TABLE OF CONTENTS

ABSTRACT…………..………………………………………………………………. v

ÖZ……………………………………………………………………………………. vi

ACKNOWLEDGEMENT…………………………………………………….......... viii

TABALE OF CONTENTS………………………….................................................. ix

LIST OF FIGURES………………………………………………………………….. xi

LIST OF TABLE………………………………………………………………….... xiv

CHAPTERS

1. INTRODUCTION………………………………………..………………………. 1

1.1 History and Background……………………………………..…………….. 1

1.2 Motivation and purpose of study……………………………………..…..... 2

1.3 Thesis outline………………………………………………………….…... 2

2. TIDES …………………………..……………..………………………………..... 3

2.1 Definition of tide…………………………..…………………………………. 3

2.2 Earth and Sun Geometry………………………………….……..…………… 5

2.3 Mathematical theory of ocean tides ………………………………………..… 5

2.3.1 Tidal acceleration and Tidal potential……………………………………. 6

2.3.1.1 Zonal tidal potential…………………………………………………... 8

2.3.1.2 Tesseral tidal potential………………………………………………... 9

2.3.1.3 Sectorial tidal potential…………………………………..…………… 9

2.4 Doodson expansion…………………………………………………………... 9

2.5 Tidal frequencies……………………………………………………………. 10

2.5.1 Instantaneous frequencies…………………………………..…………… 11

3. HARMONIC ANALYSIS AND TIDAL CONSTITUENT…………..………... 13

3.1 Harmonic analysis ………………………………………………..………… 13

3.2 Harmonic constituent………………………………………………..………. 14

3.3 Doodson number…………………………………………………….……… 14

3.4 Shallow water effects on tidal constituents………………………………..... 15

4. NON-STATIONARY DATA PROCESSING METHODS…………………….. 17

4.1 Methods for processing of non-stationary data and linear system……….…. 17

4.1.1 Spectrogram………..…………………………………………………..... 17

4.1.2 Wavelet analysis ………………………………………………………... 18

4.1.3 Empirical orthogonal function expansion………………………………. 18

4.1.4 Other methods…………………………………………………............... 19

x

4.2 Methods for processing of non-stationary data and non-linear system …...... 19

4.2.1 Empirical mode decomposition………………………………………..... 19

4.2.2 Intrinsic mode functions ………………………………………………... 19

4.2.3 Definition of sifting process……………………………………….…..... 20

4.2.4 Hilbert spectrum………………………………………………………… 25

5. TIDAL STREAM AND TIDAL POWER……………………………………… 27

5.1 Tidal energy………………………………………………………………..... 27

5.2 Tidal power and tidal velocity …………………………………………….... 28

6. MATLAB BASED HHT/HA TOOLBOX……………………………………… 31

6.1 Programing environment…………………………………………………..... 31

6.2 Toolbox definition…………………………………………………………... 31

7. DATA SET, EVALUATION AND RESULTS…………………….………….. 37

7.1 Data set……………………………………………………………………... 37

7.2 Application of Hilbert Huang Transform to tide gauge data records ……… 39

7.2.1 Analysis of tidal data based on the Hilbert Huang Empirical Mode

Decomposition (HHEMD)……………………………………………………… 39

7.2.2 Completeness test………………...…………………………………….. 49

7.2.3 Comparison of the HHT analysis with HA ………...………………….. 51

7.2.4 Analysis of sub-tidal phenomena using HHEMD……………………… 67

7.2.5 Tidal velocity…………………………………………………………… 71

8. CONCLUSION AND FUTURE WORK……………………………………….. 73

8.1 Conclusion……………………………………………………….………….. 73

8.2 Future work…………………………………………………………………. 74

REFERENCES……………………………………………………………………..... 77

xi

LIST OF FIGURES

FIGURES

Figure 1: spring tide………………………………………………………………....... 3

Figure 2: Neap tide……………………………………………………………............. 4

Figure 3: Basic description of the relation between the non-rigid planet and the

celestial body …………………………………………………………………………. 6

Figure 4: Sifting procedure………………………………………………………….. 21

Figure 5: All procedure of HHEMD………………………………………………… 23

Figure 6: IMFs of Antalya (2003-2004)…………………………………………….. 24

Figure 7: A sample of tide gauge data…………………………………………...….. 28

Figure 8: Tidal turbines……………………………………………………………… 30

Figure 9: The graphical user interface for main window of the HHT/HA toolbox … 32

Figure 10: Sifting procedure ………………………………….…………………….. 33

Figure 11: A sample IMFs window generated by EMD for Antalya tide gauge

station………………………………………………………………………………... 33

Figure 12: A sample of the Harmonic analysis result for Antalya tide gauge station. 34

Figure 13: Amplitudes of the selected frequency band of the Antalya tide gauge

station………………………………………………………………………………... 35

Figure 14: The extracted data files of the HHT/HT toolbox ………………………... 35

Figure 15: A sample of tide gauge data format……………………………………… 38

Figure 16: Result of the EMD analysis of Antalya tide gauge data ………………… 40

Figure 17: Result of the EMD analysis of Trabzon tide gauge data ……………....... 41

Figure 18: Result of the EMD analysis of Bozyazi tide gauge data………………… 42

Figure 19: Result of the EMD analysis of Amasra tide gauge data…………...…….. 43

Figure 20: Frequencies for each IMFs of the Antalya tide gauge station data……… 45

Figure 21: Frequencies for each IMFs of the Trabzon tide gauge station data ……... 46

xii

Figure 22: Frequencies for each IMFs of the Bozyazi tide gauge station data……… 47

Figure 23: Frequencies for each IMFs of the Amasra tide gauge station data…...….. 48

Figure 24: Numerical proof of the completeness of the EMD through reconstruction of

the original data from the IMF components…………………………………………. 49

Figure 25: least squares fitting of 4 tide gauge station data. …………………….….. 51

Figure 26: Corresponding frequency and amplitude of the Diurnal tides of the

Trabzon……………………………………………………………………………… 57

Figure 27: Corresponding frequency and amplitude of the semi-diurnal tides of the

Trabzon……………………………………………………………………………… 57

Figure 28: Corresponding frequency and amplitude of the shallow water of the

Trabzon…...................................................................................………….………… 58

Figure 29: Corresponding frequency and amplitude of the sub-tidal of the Trabzon

……………………………………………………………………………………...... 58

Figure 30: Corresponding frequency and amplitude of the diurnal tides of the

Bozyazi......................................................................................................................... 59

Figure 31: Corresponding frequency and amplitude of the semi-diurnal tides of the

Bozyazi………………………………………………………………………...…….. 59

Figure 32: Corresponding frequency and amplitude of the shallow water of the

Bozyazi………………………………………………………………………...…….. 60

Figure 33: Corresponding frequency and amplitude of the sub-tidal of the

Bozyazi……………………………………………………………………...……….. 60

Figure 34: Corresponding frequency and amplitude of the diurnal tides of the

Antalya………………………………………………………………………………. 61

Figure 35: Corresponding frequency and amplitude of the semi-diurnal tides of the

Antalya………………………………………………………………………….…… 61

Figure 36: Corresponding frequency and amplitude of the shallow water tides of the

Antalya……………………………………………………………………………..... 62

Figure 37: Corresponding frequency and amplitude of the sub-tidal of the Antalya.. 62

Figure 38: Corresponding frequency and amplitude of the diurnal tides of the

Amasra………………………………………………………………………………. 63

xiii

Figure 39: Corresponding frequency and amplitude of the semi-diurnal tides of the

Amasra……………………………………………………………………………..... 63

Figure 40: Corresponding frequency and amplitude of the shallow water of the

Amasra………………………………………………………………………………. 64

Figure 41: Corresponding frequency and amplitude of the sub-tidal of the

Amasra………………………………………………………………………………. 64

Figure 42: Sub-tidal and tidal of Antalya…………………………...……………….. 68

Figure 43: Sub-tidal difference between Antalya (blue color line) and Bozyazi (green

color line)……………………………………………………………………………. 69

Figure 44: Sub-tidal difference between Amasra (blue color line) and Trabzon (green

color line)……………………………………………………………………………. 69

Figure 45: Sub-tidal difference between 4tide gauge stations………………………. 70

xiv

LIST OF TABLES

TABLES

Table 1: Long period, Diurnal, Semidiurnal and Shallow water tidal constituents… 14

Table2 : Period of Doodson numbers…………………..…………............................ 15

Table 3: Doodson number of some tidal constituents……………………………….. 15

Table 4: Sifting procedure……………………………………………………….…... 22

Table 5: Tide gauge station names and theirs geodetic coordinates………………… 37

Table 6: Tide gauge station gaps…………………………………………………….. 37

Table 7: Data Flags of Errors …………………….………………………………..... 38

Table 8: harmonic analysis tidal constituents of Antalya…………………………… 52

Table 9: Harmonic analysis tidal constituents of Amasra............................................ 53

Table 10: Harmonic analysis tidal constituents of Bozyazi......................................... 54

Table 11: Harmonic analysis tidal constituents of trabzon.......................................... 55

Table 12: Amplitude and velocity of the diurnal, semidiurnal and shallow water of the

tides in each tide gauges stations……………………………………………………. 65

Table 13: Amplitudes of the Lunar Monthly (Mm), Solar semiannual (Ssa) and Solar

annual (Sa) of the tides in each tide gauges station ………………………………… 66

Table 14: Sea surface difference height for each tide gauge station during

measurement time…………………………………………………………………… 70

Table 15: Minimum and maximum velocity of the diurnal, semidiurnal and shallow

water of the tides in each tide gauges stations………………………………………. 71

1

CHAPTER 1

INTRODUCTION

1.1 History and Background

Tidal analysis has been carried out for more than a century at most coastal countries.

For a long time, tide and tidal analysis was one of the difficult problems of science.

Laplace was the one who first formulated tide mathematically as hydrodynamic

equation. The main interest of the Laplace theory lies in its ability to provide a

practical method for prediction of high and low tides (Simon et al., 2013). Laplace

equation assumes that the tide is linear and can be decomposed into similar period of

oscillations. Because of this linearity assumption, Laplace’s equations couldn’t be

universally applicable. Later in 1867, implementation of tide was promoted based on

the harmonic analysis which has been universally used as a basis of most tidal studies.

But this harmonic model purely harmonic and was not entirely satisfactory and

consequently needed some corrections. As early as 1921, more satisfying purely

harmonic expansions were published as proposed by Doodson (1880-1968) (Simon et

al., 2013).

Tides have normally a harmonic and stationary characteristic. But, some atmospheric

and meteorological parameters such as wind, seasonal temperatures and pressures

superimposed to tides cause non-stationary and non-periodic behaviors. These effects

on tides are known as sub-tidal phenomena and occur over a broad range of longer

time scale and are highly non-stationary. Discussions about these fluctuations can be

found in e.g., Pattullo et al. (1955), Roden (1960), Stub and James (1988).

Donland B.percival and Harold O.Mofjeld also analyzed the sub-tidal sea level

fluctuation at Crescent City, California, during 1980-91 by using the maximal overlap

discrete wavelet transform (MODWT) and they extracted sub-tidal fluctuation from

the hourly data by low pass filtering to remove astronomical tides at diurnal and

higher frequencies (Percival and Mofjeld 1997).

The theoretical foundations of HHT are explained and discussed in detail with its

advantages and possible disadvantages in Huang et al. (1998), Zeiler et al. (2010),

Magrin-Chagnolleau and Baraniuk (1999). HHT has been applied to different

scientific areas in the literature, for instance Rudi et al. (2010) analyzed the hydrologic

time series data from Dedenborn , Erkensruhr and Rollesbroich. The energy spectrum

of the HHT has been compared with the continuous wavelet transform and Fourier

transform. The results of the Rudi et al. (2010)’s studies show that the HHT indicates

much more localized information.

2

Also Ezer and Corlett (2012) used Empirical Mode Decomposition (EMD) and (HHT)

method for sea level data. Zeiler et al. (2010) applied HHT to non-linear and non-

stationary biomedical signal and pointed out to some advantages and disadvantages of

the EMD decomposition. Nevertheless, EMD and HHT have been applying

successfully to solve for many practical problems. A comprehensive treatment of HHT

with its applications from geophysics to image analysis can be found in Norden E.

Huang, Samuel S. Shen (2005).

In addition to sub-tidal effects which are very important for ecosystems, also the tidal

current energy is considered as a renewable energy source and is not yet fully utilized

(Shiono at al., 2002). Reliable energy sources have been prominent issues of the

modern civilization in the new era. Electricity is one of the energy sources that could

provide more than 90 percent of the energy requirement. For that reason analyzing

tidal and sub-tidal phenomena helped to develop such an industry so that new

generation of water and tidal generators are being produced to convert tidal energy

into electricity. European Marine Energy center introduced six types of tidal energy

converters. These are horizontal axis turbines, vertical axis turbines, oscillating

hydrofoils, venture devices, Archimedes screw and tidal kite. Several studies have

been done for the tidal power and mechanism of the tidal data discussing about

advantages and disadvantages of tidal turbines and energies turbines ( Baker (1991),

Gorlov (2001).

1.2 Motivation and purpose of study

Main goal of this study is to investigate the ability of applying HHEMD to tide gauge

records on the basis of a MATLAB Toolbox which has been developed and

implemented within this work. The work is focused particularly on the hourly tide

gauge data of Turkey coastline to filter the sub-tidal phenomena from the astronomical

tides. On the other hand, the minimum and maximum velocities of the tides are

calculated through the amplitudes and frequencies to search for the possibilities of

producing electricity from tidal power.

1.3 Thesis outline

This thesis consists of eight chapters. Background, literature review and motivation of

study are explained in chapter 1 (Introduction). Chapter 2 is dedicated to a brief

definition of theory, models and mathematical backgrounds used in tide analysis.

Definition of the harmonic analysis and its application in tidal analysis are presented

in Chapter 3. Definition of some Non-stationary methods and disadvantage of them on

tidal analysis and also definition of Non-stationary, Non-linear Empirical Mode

Decomposition (EMD) are presented in chapter 4. Chapter 5 is about tidal power and

tidal generator and discussion about the possibilities of these systems in Turkey.

MATLAB-based HHT/HA toolbox is explained in chapter 6. Data and application of

Hilbert Huang transform to tide gauge data records are presented in chapter 7.

Concluding and future works are given in chapter 7 with discussion and summary.

3

CHAPTER 2

TIDES

2.1 Definition of tide

Most of the people know more or less meaning of the tide and see what happens at the

shoreline when tides occur. Tide is a phenomena of the world which is the rise and fall

of ocean and sea levels. Also in a very small scale, tides occur in large lakes and

within the solid crust of the earth. Basically tide mostly results from the gravitational

force of other planets on earth. The earth, sun and moon rotate around a common

center of mass and their wobble is the actual reason of tidal energy.

Indeed, tides are long period waves through the ocean due to the existence of the

forces by the Moon and the Sun. Tides are produced in the ocean and progress toward

the shorelines where the rise and fall of the sea surface occur.

When the tides reach their highest part or crest of the wave, high tide occurs and low

tide becomes the lowest part of the wave or its trough. Height difference between high

tide and low tide is known as tidal range. Some non-astronomical elements such as

form of the coastline, depth of the water, topography of the ocean and some

meteorological effect play an important role in altering the range and interval between

high and low water (NOAA OCEAN SERVICE). During lunar month magnitude of

the high and low tides is changed. When the moon, earth and sun are approximately

positioned on a straight line, highest tides are occurred and called spring tides (Figure

1). The lowest tide in lunar day is called neap tide and occurs when the earth, moon

and sun are at right angle to each other (Figure 2) (Gorlov 2001).

Figure 1: spring tide

4

Horizontal movement of the tides due to vertical motion is called tidal current.

Velocity of the tidal current is important to flood flow (when the tidal current is

coming to the coastline) and Ebb flow (when the tidal current is returning to the sea).

Tide currents can have velocity up to 5m/s in ocean shorelines.

Figure 2: Neap tide

Gravity is one of the main sources that causes the tides. In 1687 Isaac Newton

explained that ocean tides are result of orbital movements of the Sun and the Moon

(Sumich 1996). Newton’s gravitational low depicts that attraction between two bodies

is directly proportional to their masses, and inversely proportional to the square of the

distance between the bodies (Sumich 1996; Thurman 1994). Therefore the greatest of

mass and closest of distance to each other will be equal to the more attraction to each

other. Consequently tidal forces that are related to gravitational attraction are based on

the Newton law and the distance between two perturbation bodies is more critical than

their masses (Ross 1995).

At the center of the earth, the centripetal acceleration of the earth is equal to the

centrifugal acceleration. Centrifugal acceleration due to the rotation of the earth is

constant all over the surface of the earth but centripetal acceleration is produced by the

attraction of the moon and varies due to the distance of the moon. The period of the

rotation of the moon about the common center of mass is about 27.32 days and is

called sidereal month.

At everywhere on the surface of the earth, imbalance exits between the centripetal and

centrifugal acceleration. At the side of the earth that is close to the moon, the

gravitational force is more than centrifugal acceleration and centrifugal force exceeds

at the other side of the earth which is far from the moon (Ross 1995).

5

2.2 Earth-Sun Geometry

Due to the spinning of the earth, the earth surface moves approximately 1675 km per

hour. One rotation takes about 24 hours. Earth has a motion around the Sun which is

called earth revolution (Pidwirny 2006). This motion takes about 365.26 days to

complete one cycle around the Sun. Furthermore, the orbit of this motion is not

circular but elliptical. Earth distance variation with respect to the elliptical orbit is not

responsible for the earth’s seasons. This variation causes the amount of solar radiation

received by earth and magnitude of gravitational effect (Pidwirny 2006).

Ecliptic plane is the flat surface that coincides with the earth’s orbit around the Sun.

There is an angle about 23.5 degree between ecliptic and the spin or rotation axis of

the earth. The relative position of the earth axis changes during the earth revolution

but remains unchanged in four dates (December21, March 21, June 21, and September

21). This declination is responsible for changes in the height of the Sun from horizon

of the earth and it also causes the seasons (Pidwirny 2006).

The earth tidal bulges are related to the position of the moon and the Sun. The position

of these objects changes, relative to the earth’s equator. Over the time, these changes

show a direct effect on daily tidal and tidal current.

As the Moon revolves around the earth and earth revolves around the Sun, their

declination changes during sidereal month and year. These declinations produce two

bulges during the lunar and solar day.

2.3 Mathematical theory of ocean tides

In this section for understanding tide and its behavior, a celestial body is considered to

be rotated around a non-rigid planet like our earth (Figure 3). The attraction of the

Moon and the Sun will give rise to a deformation of the earth. This deformation is

proportional to the force, to the stress itself and this is the main reason to calculate the

force exerted on each point of the earth surface (Simon et al., 2013).

6

Figure 3: Basic description of the relation between the non-rigid planet (Earth)

and the celestial body (Sun or Moon).

Let m be the Mass of the non-rigid planet such as earth and r is its mean radius. O is

the position of the earth center of gravity and chosen as origin of the coordinates

(is considered as geocentric coordinate system). The mass of the celestial body

such as the Moon and the Sun is regarded as M. Q shows position of the center of the

celestial body. Point on the surface of the earth is denoted by a position P. Distance

between Q and P is determined as d and D is the distance between Q and O.

2.3.1 Tidal acceleration and Tidal potential

Tidal acceleration is caused by lunisolar gravitation and to a lesser extent by the

gravitation of other planets. The tidal acceleration at any point on the surface of the

earth can be determined from Newton’s law of gravitation and position of the celestial

bodies (moon, sun and planets). Tidal acceleration occurs at all over the earth which is

defined as the difference between the gravitational force of the earth surface point P,

and the constant part which refers to the earth’s center

. (2.1)

The tidal acceleration deforms the earth’s gravity field symmetrically with respect to

three orthogonal axes with origin at the center of the mass of the earth. This tidal

acceleration field experiences diurnal and semidiurnal variations, which are due to the

rotation of the earth about its own axis.

By applying the law of gravitation to Equation (2.1), the tidal acceleration for the

moon is defined as

(2.2)

7

where is mass of the moon, and and are the distance of the moon from

calculation point and the earth’s center O. The tidal acceleration is given as

(2.3)

and the tidal potential V calculated at the point P on the earth surface due to the

existence of the orbiting body with mass M is given by

(2.4)

with

. (2.5)

Where Is the gravitational constant. The potential of the homogeneous field is

given by

. (2.6)

By inserting Equations (2.4) and (2.6) into Equation (2.3), and by assuming ,

tidal potential will be as

. (2.7)

As result the tidal potential given by

∑

(2.8)

where indicate Legendre polynomials of degree in and are known

as Legendre polynomials.

The main term of the tidal potential series is

(

) (2.9)

Differentiating of equation (2.9) leads to the tidal acceleration. The radial component

is found to be

. (2.10)

According to the Equation (2.10) there is no tidal effect at and . Maximum

values of tidal acceleration occur at and .

8

According to equation (2.9), tidal potential is dependent on the zenith angle and

distance to the celestial body. The variation of tidal potential and acceleration can be

recognized by change to the earth fixed coordinate system ( , ) for the points on the

earth surface and to equatorial system of astronomy ( ) for the perturbing celestial

body like the Moon and the Sun (Equation 2.11 and Equation 2.12).

. (2.11)

The hour angle of the Moon and the Sun is given by

= (2.12)

where LAST (Local Apparent Sidereal Time) is the hour angle of the vernal equinox,

indicate right ascension and GAST (Greenwich Apparent Sidereal Time) is the angle

between the observer’s celestial meridian and the hour circle of the point.

Subsequently, by substituting Equations (2.11) and (2.12) into Equation (2.9),

Laplace’s tidal equation gives in the form

(

)

. (2.13)

The quantities , , vary with time, consequently Equation (2.13) can be

decomposed into three zonal, tesseral and sectorial terms as given below

(

) (2.14)

(2.15)

. (2.16)

2.3.1.1 Zonal tidal potential

The first term of the Laplace’s tidal potential is independent of the earth rotation and

is called zonal tidal potential. Zonal tidal potential shows long period or low frequency

according to zonal tidal potential Equation, it does not contain the hour angle h. It

varies by square of the sine of the declination of the Sun and the Moon, which is really

slow and it has roughly 14 days for the Moon and 6 months period for the Sun. Given

values are evaluated by declination, for the Sun and for the Moon and

latitude of the point on the earth surface. Zonal potential is always positive for latitude

between and S and negative for other locations (Simon et al., 2013).

9

2.3.1.2 Tesseral tidal potential

The second term of the Laplace’s tidal potential is called tesseral tidal potential.

Tesseral tidal potential shows diurnal period. According to the equation (2.15), it

contains the hour angle h which has a period about one day. The period of the hour

angle is roughly 24 hours for the Sun and 24h 50 minutes for the Moon. The

declination effect is very low in comparison with hour angle. The diurnal local

maximum will occur when the Sun or Moon cross the upper or lower meridian of the

observer and tesseral tidal potential effect has its maximum at latitude N S

(Simon et al., 2013) and the potential is zero for the points at the equator and at the

pole.

2.3.1.3 Sectorial tidal potential

The third term of the Laplace’s tidal potential is called sectorial tidal potential and

introduces semidiurnal periods. According to the equation (2.16), it contains 2H with

approximately 12 hours period for the Sun and 12h 25 minutes for the Moon. During

the day, earth divides into four sectors and due to the Cos2H, the components of tide

have two maxima and two minima per day (Simon et al., 2013).

As seen from Equation (2.13) zonal and sectorial tides are symmetric about the

equator, while the tesseral tides are antisymmetric. The diurnal tide has their

maximum at N S and is zero at the poles and equator, while the semidiurnal

tides obtain their maximum at the equator and are zero at the poles. The long periodic

tides have a maximum at the pole.

2.4 Doodson expansion

Laplace equations for tidal potential which is mentioned above are not proper for tidal

phenomena analysis. Because, the term

as well as trigonometric function including

hour angle and declination indicate very complicated time-variation due to the

complexity of the orbital motion of the earth around the Sun and Moon around the

earth.

Laplace expansion is also based on the sinusoidal functions for expanding the Sun and

the Moon potentials whose data should be linear in time. So Laplace expansion will be

valid for an imaginary body with uniform circular motion in the equatorial plane.

Many scientists like Kelvin and Darwin continued applying Laplace method for

improving the harmonic expansion of the tidal potential. In 1922, Doodson solved this

problem and published harmonic expansion containing 389 (100 are long period, 160

are daily, 115 are twice per day, and 14 are thrice per day) components.

10

Doodson expanded tidal potential components in a Fourier series by choosing some

spatial period and frequencies (table 1)

Doodson introduces constant parameters instead of the ones in the tidal potential

formulas. The trigonometric function is multiplied by

. D represents the

distance between the center of the earth and perturbing body and is replaced by the

mean distance “c” that is to show averaged value during a revolution. Also r is the

distance between the center of the earth and point on the surface of the earth and is

replaced by as mean radius of the sphere.

√

(2.17)

where

a: semi-major axis of the earth,

b: semi-minor axis of the earth.

General constant for the Moon and the Sun with respect to the Doodson constant is

produced as

. (2.18)

It is 2.628 for the moon and 1.208 for the Sun (Torge 2001 ). Thus ratio

of these constants depicts that why the effect of the Moon is roughly two times larger

than the Sun.

. (2.19)

With respect to the Doodson constant, we can write components of the tidal potential

as below

(2.20)

(2.21)

(2.22)

2.5 Tidal frequencies

Each constituent of the tide which has a direct relation with the Sun and the Moon,

have known periods.

11

Similarly, lunar day is the time for revolution of the moon around the earth and it

takes 24h 50 minutes. Also moon has direct effect on attraction force on the surface of

the earth. Period of the lunar day is 24h 50 minutes with 0.0403 frequencies.

Moreover other parameters of the Sun and the Moon like temperature and pressure are

effective on tide. But unfortunately we don’t know the period of this effect on the

ocean and the sea, consequently for calculating the frequency of another parameters,

we will need the instantaneous frequency.

2.5.1 Instantaneous frequency

As explained in Fourier analysis, frequency was defined as full wave form of the sine

or cosine function in the whole data length with constant amplitude. Consequently, at

least one full oscillation of the sine or cosine wave will be required. Such a definition

would not make sense for non-stationary data where frequency values’ change from

time to time.

According to the Hilbert transform which will be explained in chapter 4 the

instantaneous frequency is defined as

(2.23)

Based on the Equation (2.23), Hilbert transform is applied to calculate the

instantaneous frequency depending on time. Consequently, it can deduce frequency

and amplitude for each given time (monocomponent) (Huang et al., 1998).

12

13

CHAPTER 3

HARMONIC ANALYSIS AND TIDAL CONSTITUENTS

3.1 Harmonic analysis

As mentioned in the previous chapter, tides occur due to the moon revolving around

the earth and earth revolving around the Sun. Tides are the periodic part of the sea

level variation. Several basic methods of tidal analysis have been developed until now.

A method normally used for prediction is harmonic analysis which helps to figure out

the harmonic constituents of the tides. Tidal harmonic analysis is mathematical

process by which the observation tidal data are separated into a finite number of

harmonic constituents.

Prediction depends on the quality of the tidal analysis. It involves adding up the

separated constituents of the tidal data to produce a signal which is the prediction.

Obviously, number of constituents of the tidal analysis depends on the duration of

tidal observations obtained in tide gauge stations.

Based on the mathematical method of the harmonic analysis, the observed tidal data at

any tide gauge station is decomposed into harmonic constituents, of the form

(3.1)

where is the amplitude, is the angular speed, is period of tidal constants and

indicates phase log on the equilibrium tide at the Greenwich Meridian

For the prediction of tides and filling the gaps of the data, the fitting is adjusted by the

least square fitting model. the sum of the square of the difference between the

observation tidal data and the tidal prediction will be minimum.

By using harmonic analysis method tidal function is modeled as given below

∑ ( ) (3.2)

where, is hourly sea level of time and t depict mean sea level at the

beginning epoch and measurement time respectively, is the trend (mm/year), and

are the coefficients of tidal constituents amplitude, describe frequency of tidal

constituents, is the number of tidal constituents and M is the total number of

significant tidal constituents. Thus, based on the known tidal frequency, the amplitude

of each tidal constituents is calculated

14

√

. (3.3)

Phase angle is given as

. (3.4)

3.2 Harmonic Constituents

Tidal constituents are the periodic changes of the ocean tide according to the

perturbation body position like the Moon and the Sun. Based upon the harmonics

terms and their phases, tides are decomposed to Long term, diurnal, semidiurnal and

shallow water constituents. Table 1 indicates some tidal constituents with their

periods.

Table 1: Long period, Diurnal, Semidiurnal and Shallow water tidal constituents

Tidal

constituent

Period

of tidal constituent

(Solar hours)

Nature Description

MF 327.90 Long period Luni-solar fortnightly

MM 661.30 Long period Lunar monthly

SSA 4383.00 Long period Solar semi annual

Q1 26.87 Diurnal Large lunar elliptic

P1 24.07 Diurnal Solar diurnal

O1 25.82 Diurnal Lunar diurnal

K1 23.93 Diurnal Lunar diurnal

K2 11.97 Semi-Diurnal Luni-solar

N2 12.66 Semi-Diurnal Large lunar elliptic

S2 12.00 Semi-Diurnal Principal solar

M2 12.42 Semi-Diurnal Principal lunar

M4 6.21 Shallow Water Principal lunar

MS4 6.10 Shallow water

3.3 Doodson number

In order to specify the different harmonic components of the tide, Doodson introduces

a particular method that is called Doodson number. Doodson numbers include six

numbers or arguments as basic angular argument.

Most of the large components of the tide are specified as combination of small integer

multiples, positive or negative of six basic Doodson numbers. The Doodson numbers

are given in table 2 (Doodson 1921).

15

Table2 : Period of Doodson numbers

Doodson

nmber

Period Angular

argument

Description

⁄ 14.49205211 Period of mean Lunar Time

⁄ 0.54901652 Period of lunar declination

⁄ 0.04106864 Period of solar declination

⁄ 0.00464184 Period of lunar perigee rotation

⁄ 0.00220641 Period of lunar node rotation

⁄ 0.00000196 Period of perihelion rotation

Where, addition 055555 make the number positive.

(3.5)

Doodson numbers of some tidal Constituents are given in table 3

Table 3: Doodson numbers of some tidal constituents

Name Doodson

number

Period

Semi-diurnal

2 0 0 0 0 0 255555 12.4206

2 2 -2 0 0 0 273555 12.0000

Diurnal

1 1 0 0 0 0 165555 23.9344

1 -1 0 0 0 0 145555 25.8194

Long period

0 2 0 0 0 0 075555 327.85

0 1 0 -1 0 0 065455 661.31

3.4 Shallow water effects on tidal constituents

One of the assumptions in discussion of long wave of sinusoidal form is the amplitude

of the wave which should be much less than the water depth. Therefore, when the tide

deploys into shallow water, this assumption may not be valid, and as might be

expected the wave form would be distorted. In such event crest of the wave propagates

faster than the trough.

Tides in the open ocean usually have smaller amplitude than along the coast. This

event occurs due to the amplification by reflection and resonance and due to the

shoaling. As the waves propagates into shallower water the wave height and the tidal

stream increase.

16

This distortion wave could be represented by the harmonic frequency of the tidal

constituents and is called shallow water constituent (over tides). The calculation of

shallow water constituents is not deduced directly from the tidal force. As shown in

Table 1 the quarter diurnal M4 and MS4 are the most common shallow water

constituents with frequencies twice of the M2 and the sum of the M2 and S2,

respectively (Forrester 1983).

17

CHAPTER 4

NON-STATIONARY DATA PROCESSING METHODS

Historically, Fourier spectral analysis has provided a general method for analyzing

frequency domain distribution of tidal data. Consequently, the part ‘spectrum’ has

become roughly synonymous with the Fourier transform of the data. Because of its

simplicity Fourier analysis has dominated the data analysis since soon after its

introduction, and has been applied to all kinds of data. However the Fourier transform

is valid under some general conditions (see, for example, Titchmarsh 1948). There are

some critical limitations of the Fourier spectral analysis. The system should be linear

and the data should be periodic or stationary.

The stationarity requirement is not special to the Fourier spectral analysis and it is one

of the general restrictions of the available data analysis methods. Other than

stationarity, Fourier spectral analysis also needs linearity. Fourier spectral analysis has

limitations in using such data which is not stationary. The uncritical use of Fourier

spectral analysis and linear assumptions may give misleading results.

In this study, we will present a data analysis method based on the Empirical Mode

Decomposition (EMD) that will produce intrinsic mode functions (IMF) of the signal.

The decomposition is based on the direct extraction of the data which is associated

with various intrinsic time scales. Then Hilbert transform is applied to each IMF to

calculate instantaneous frequency. Consequently any event on the time can be

localized. Adaptivity is the most important aspect of this analysis. As will be

explained later in more detail, locality and adaptivity are the requirements for the

analysis of nonlinear and non-stationary time series.

Combination of the EMD and Hilbert spectrum analysis will have good results for

analyzing various nonlinear and non-stationary data such as tidal records.

4.1 Methods for processing of non-stationary data and linear systems

There are some methods for analyzing the non-stationary data. Most of the methods

depends on Fourier analysis of linear system.

4.1.1 Spectrogram

Spectrogram method is one of the basic methods which successively sliding data as

time window to get time-frequency distribution. Since the spectrogram is based on the

Fourier spectral analysis, the data is assumed to be piecewise stationary. This

18

assumption is not always justified in non-stationary data and cannot guarantee that

each window size adopted always coincides with the stationary time scale.

Furthermore applying of this method has some practical difficulties such as problems

in localization an event in time. This has to do with the window width which should

be narrow. But on the other hand, the frequency resolution will need longer time series

(see, for example, Oppenheim and Schafer 1989).

4.1.2 Wavelet analysis

Fourier transform is a very useful tool for signal analysis. But in this method,

frequency variations in times could not be detected. Short-time Fourier transform or

spectrogram analysis is used for overcoming this problem, however, the length and

scale of the window seems to be a problem too. Wavelet analysis is essentially an

adjustable window Fourier spectral analysis and has the following general equation

| |

∫

(4.1)

where *(.) is the basic wavelet function which satisfies specified very general

condition, is the dilation factor and is the translation of the origin. In

is the energy of in scale at .

Commonly *(.) are not orthogonal for different for continuous wavelets. But

orthogonality is satisfied by selecting a discrete set of . This discrete wavelet analysis

misses the physical signals having scale which is different from the selected discrete

set of . Continuous or discrete wavelet analysis is basically a linear analysis and it is

non-adaptive in nature. In spite of all these problems, wavelet is one of the best non-

stationary data analysis methods (Huang et al., 1998).

4.1.3 Empirical orthogonal function expansion (EOF)

The empirical orthogonal function expansion (EOF) or principal component analysis

(PCA) is the time domain analysis. Empirical orthogonal expansion is computed by

eigenvector and eigenvalue of the covariance matrix of the time series data. The nature

of EOF is briefly given as follows. For any , the EOF will reduce it to

∑ . (4.2)

EOF represents a radical departure from the spectrogram and wavelet, for the

expansion basis is derived from the data. Therefore, EOF is highly efficient method.

Unfortunately, there is no guarantee that EOF components coming from a nonlinear

and non-stationary system such as tidal data will be stationary and linear (see, for

example, Simpson 1991).

19

4.1.4 Other methods for non-stationary and linear data

Other than the above methods, there are various ones such as least square estimation

of the trend. More detail can be found in (e.g. Brockwell and Davis 1991). All the

above methods are designed basing on the Fourier analysis and they have some

defects in one way or more.

4.2 Methods for processing of non-stationary data and non-linear system

In this study, Hilbert Huang transform are applied in two steps to analyze the

nonlinear and non-stationary data. At the first step data is decomposed into a number

of intrinsic mode function components by the empirical mode decomposition method.

The second step is the construction of the instantaneous frequency and amplitude of

the decomposed IMFs by applying the Hilbert transform. The necessary conditions for

the basis to represent a nonlinear and non-stationary time series are completeness,

orthogonality, locality and adaptivity.

Completeness is the standard requirement of the nonlinear system and guarantees the

degree of the precision of the expansion. Orthogonality guarantees positivity of the

energy. These two conditions are the requirements for the linear expansion. But basis

conditions are not satisfied by some of the non-stationary linear methods.

Locality and adaptivity are the particular conditions to the non-linear and non-

stationary data. There is no time scale in non-stationary data. Therefore all epoch of

the data should be identified as effective parameters in analysis. Consequently,

locality is the most prominent condition for the non-stationarity. Adaptivity is also

crucial condition for nonlinear and non-stationary data. By adapting to the local

variations the data can be fully decomposed to its inherent function without any

requirement for the mathematical fitting to the data.

4.2.1 Empirical mode decomposition

Empirical Mode Decomposition is the method that was developed for non-stationary

and nonlinear time series resulting from non-linear systems. The nature of the method

is decomposition of the time series, based on the Intrinsic Mode Functions (IMFs).

Empirical mode decomposition in combination with Hilbert spectral transform is ideal

for extracting essential components of the time series.

4.2.2 Intrinsic mode functions (IMF)

With respect to the instantaneous frequency, the functions should be symmetric and

have the same numbers zero crossings and extrema. For this purpose, a term of

functions as intrinsic mode function (IMF) was proposed with following conditions.

1. The number of extrema and the number of zero crossing should be equal or

their maximum difference should be one.

20

2. In each point the mean value of the envelope in local maxima and in local

minima should be zero.

The first condition is as stationary Gaussian process that requires narrow band. Due to

the second condition, the instantaneous frequency will not have the unwanted

fluctuations and symmetry condition will be satisfied. Indeed, the local mean of the

data should be zero but in the non-stationary data, local mean includes the local time

scale mean that is impossible to define. Local envelope values are produced by cubic

spline interpolation for solving this problem (Huang et al., 1998).

4.2.3 Definition of sifting process

Empirical mode decomposition decomposes the time series data to its intrinsic

oscillation through the sifting process. During this process, most of the riding wave

and oscillations with non-zero crossing between extrema are eliminated and wave

profile will be more symmetric. According the IMF definition, the decomposition

starts by producing an envelope which contains local maxima and minima. In the first

step all local maxima are connected by a cubic-spline interpolation method as the

upper envelope (Figure 4). Similarly the procedure is repeated for local minima. Their

mean value of the envelopes is called m1 and the difference between the data and local

mean value will be the first component.

. (4.3)

21

Figure 4: Sifting procedure. a) Original data. b) Selecting of maximum and minimum

values. c) Envelope by cubic spline all selected vales. d) Calculating the mean value e)

extract the residue by subtracting mean value from original value.

Logically R1 should satisfy all requirements for IMF. Otherwise the previous

procedures will be repeated. The sifting procedure will be repeated until IMF

conditions are being satisfied is Equation (4.4)

(4.4)

and the first IMF component will be c1

. (4.5)

All sifting procedure is shown in table 4 and result of sifting procedure with 10 IMFs

of Antalya (2003-2004) is shown in figure 6

22

Table 4: Sifting procedure

IMF1 IMF2 IMF3 IMFh

0

1

2

3

. . . . . . . . . . . .

n

IM

F

Standard deviation of the neighborhood’s component is deduced from Equation (4.6).

A value for SD is set between 0.2 and 0.3 which is the criteria for sifting process to

stop.

∑| |

. (4.6)

The sifting process will be continued as long as the residue is achieved and will be

stopped as long as not being able to extract more IMF. Consequently, final residue can

be interpreted as the trend related part of the time series. The final equation is given as

∑ . (4.7)

Figure 5 shows the brief process of the Hilbert Huang Empirical Mode

Decomposition.

23

Figure 5: All procedure of HHEMD

24

Figure 6: IMFs of Antalya (2003-2004)

25

4.2.4 Hilbert spectrum

After decomposition of time series and extraction of all IMFs, we can deduce

amplitude, phase and instantaneous frequencies of each IMF by applying Hilbert Hung

transform.

The Hilbert transform is calculated as

∫

(4.8)

where P indicates the Cauchy principal value. According to Hilbert transform,

and forms a complex conjugate pair, therefore analytic signal could be

shown as

(4.9)

where amplitude and phase are given by

(4.10)

. (4.11)

26

27

CHAPTER5

TIDAL STREAM AND TIDAL POWER

Tidal streams are the velocity of the tidal currents. The energy of the tidal stream is

one of renewable energy which is growing in technology of the renewable energy

sector (Haas 2011).

Tidal stream energy generators have same operational details with wind turbines.

However, the potential of the tide in power generation is greater than the wind.

Density of the water is about 800 times greater than the density of the air. It means

that one tidal stream generator can generate significant power at low tidal velocity

compared with wind. Based on the water and tidal characteristics serious studies and

the design of tidal turbines for exploiting tidal energy are growing rapidly.

5.1 Tidal energy

Tide is combined with kinetic and potential energy. Tidal potential energy is produced

when the mass of the water is lifted above the ocean surface due to the attraction of

the Sun and the Moon and can be deduced as

(5.1)

where

E: Potential energy,

g: Acceleration of gravity,

: Sea water density

,

A: Area,

a: Amplitude of tide.

And kinetic energy T of the water with mass m is the energy to do work by its

velocity and can be deduced as

(5.2)

where

: mass,

28

V: velocity of the water motion.

5.2 Tidal power and tidal velocity

Tidal energy is being deduced by two methods. The first one is the conventional

method with construction of the water dam by operation of hydraulic turbines. The

turbines converts the potential energy of the accumulated water, due to the tide in two

side of the dam, into electric energy and are mostly designed as double action (Sea

Figure 7). Traditional tidal construction design is developed for river or canals and has

a poor ecological reputation. Dams block fish migration and destroying their

population also damaging the environment by flooding and swamping adjacent lands

(Gorlov 2001). (see for example, Defne et al., 2011).

Figure 7: Example of tidal dam. When tide is in high position, water flows in to

reservation and in low tide, water flows out of the reservation. Two of them cause turbine

blades rotate.

Due to the environmental problems, the second method or new tidal energy converter

generation is suggested. New tidal energy converters are the kind of turbines that does

not require tidal dams and they can efficiently extract the kinetic energy from a free

unconstrained tidal current without any dams (Gorlov 2001). Figure 8 shows kind of

these tidal turbines. In these methods, tidal current turbines use a generator to produce

energy and change the kinetic energy of current into electricity by setting water

turbine in tidal current. Therefore, it will be considerable advantage by using of water

turbines that can always revolve in a fixed direction without any influence from tidal

current direction (Shiono et al., 2002) and allows utilization of double action tidal

power. This is a very important advantage of them that led to study about them.

Tidal power extracted from all free unconstrained turbines can be calculate as

(5.3)

29

where is the turbine power, is the turbine efficiency, indicate the mass of the

water, shows the total effective frontal area of the turbine in and is the tidal

current velocity .

The principal astronomical ocean tidal constituents are different for various coastal

areas and depending on the type of tide and its amplitude. Tidal velocity of the tide is

one of the most prominent parameter of tide. Also velocity of the tidal current is the

main parameters of the turbine power Equation (5.3). Tidal velocity can be deduced its

variation in three dimensions based on the laws of fluid dynamic.

Navier-Stokes equation describes the motion of the fluid substance and dictates not

position but rather velocity. These equations are based on Newton’s second law of

fluid motion. The velocity of the fluid in a given space and time is described by

solution of Navier-Stokes equation and is called a velocity field or flow field (Temam

1995).

But based on the water turbines that could always revolve in a fixed direction, the

velocity of the tidal current can be deduced based on the frequency and period of the

tidal constituent.

According to the astronomical tidal frequency, tidal angular velocity can be deduced

as

(5.4)

where, is the tidal angular velocity in one direction and is the period of the tidal

constituents.

In this study Hilbert Huang Empirical Mode Decomposition (HHEMD) method has

been applied to the hourly tidal data of Antalya, Amasra, Trabzon and Bozyazi.

Frequency and amplitude of the astronomical tides have been deduced according to

the Hilbert Huang Transform.

According to the frequencies and amplitudes of the tidal constituents the velocity of

the tidal constituents in one direction can be obtained from the Equation (5.5).

. (5.5)

Where is the angular velocity and is the amplitude of the corresponding

tidal constituents.

30

Figure 8: Tidal turbines

a ) shows the horizontal axis tidal turbines, that mostly designed with three or more blade

between 5m and 20m diameter for capture energy from the nature flows of tidal or river

current (www.marineturbines.com).

b ) shows the oscillating hydrofoils tidal turbines. Oscillate up and down like a dolphin’s

tail. The mechanical system is very efficient at taking energy from the flow. In this kind

of turbines energy is taken from a rectangular cross section of water. This allows it to

take full advantage of shallow flows. Changing the amplitude of oscillation of the foils,

means the system can be adjusted for different flow depths (www.engb.com).

c ) shows the vertical axis turbines with blades about 70 cross section area

(www.bluenergy.com).

d ) shows the tidal kite turbine and this kind of turbines are particularly suited to areas

of relatively deep water that experience low velocity currents. In Uk this kind of turbines

are produced from 22 to 40 TWh per year for water flows between 1.2 and 2.2 meters per

second (http://www.theguardian.com).

31

CHAPTER6

MATLAB BASED HHT/HA TOOLBOX

6.1 Programing Environment

Hilbert Huang transform and Harmonic analysis toolbox (HHT/HA) was developed in

MATLAB environment. MATLAB is high performance language for technical and

mathematical computing. It includes computation, visualization and programing in an

easy-to-use environment.

Typically MATLAB uses in math and computation, data Acquisition, data analyzing

and data visualization, scientific and engineering graphics and application

development, including graphical user interface (GUI).

In MATLAB, basic data element is the matrix and simple integer is considered a

matrix of one row and one column that does not require dimensioning. This allows

solving computation problem, especially those works with matrix and vector, in a

fraction of time (The Math Work INC.,2004).

6.2 TOOLBOX definition

In order to run HHT/HA toolbox, MTLAB should be installed in related computer.

HHT/HT toolbox is compatible with MATLAB version 7.11.0(R2010b) and has not

been tested for the previous versions.

HHT/HA consist of 34 MATLAB files with the total size of the 100 KB. Before

running the HHT/HT toolbox, all related files should be pasted into same folder of the

operation system.

HHT/HT can be run within MATLAB simply typing “emd_ha” in the MATLAB’s

command window. This opens the main window of HHT/HT (See Figure 9) with the

pushbuttons to reach following functions of the toolbox.

Open file

Outlier

Empirical Mode decomposition

Harmonic analysis

Hilbert Huang Transform

32

Figure 9: The graphical user interface for main window of the HHT/HA toolbox

With Open file function, observation data file, especially tidal data file for this study

in .txt format import into the HHT/HA toolbox and appear the file pass in “path to

file” section. Note that all gaps of data have been removed when data import and

graphic of the data appear in below of the interface. Figure 6 shows the interface of

the “open file” with related data graphic.

If data have outliers, user can implement Three-sigma outlier test in Outlier section.

After preprocessing of data, the user can process its data by data analysis section. In

empirical mode decomposition, tidal data have been decomposed to its inherent

functions by using empirical decomposition mode. The EMD uses stopping criteria

based on the explanation in previous chapters with maximum 3000 iteration. The

EMD can present all IMFs of the tidal data. Figure 10 shows the sifting procedure of

the Antalya tide gauge station’s data. As can be seen in Figure 10, the first signal

shows the signal in iteration 309 before sifting and includes upper and lower envelope.

Second signal shows the signal after sifting in the same iteration and last one shows

the residue.

33

Figure 10: Sifting procedure

Figure 11 shows the empirical mod decomposition results of Antalya from 2003 to

2010. All IMFs of Antalya have been saved as .mat file (See Figure 14).

Figure 11: A sample IMFs window generated by EMD for Antalya tide gauge

station

In the third section of the data analysis, harmonic analysis can be implemented. This

section decomposed the tidal data to its harmonic constituents by using of the least

34

square fitting. Figure 12 shows the harmonic analysis results with least square fitting

of Antalya2003-2010. All constituents of Antalya that have been extracted by

harmonic analysis have been saved as .out file (See Figure 14).

Figure 12: A sample of the Harmonic analysis result for Antalya tide gauge

station

Frequency and amplitude of the each IMF, are calculated by Hilbert Huang transform

section. Hilbert Huang Transformation section uses EMD and Hilbert Huang

Transform. After data decomposition to its inherent, frequency and amplitude of the

each IMF is calculated by the HHT.

As can be seen in Figure 13 after running the Hilbert Huang Transform, frequency

selection appears in right side of the data analysis panel. In this section can be selected

frequency band for each IMF frequencies and extract amplitudes of the selected

frequency band. Figure 13, shows the amplitude of the Antalya in frequency between

0.0775 and 0.0835.

35

Figure 13: Amplitudes of the selected frequency band of the Antalya tide gauge

station

Figure14: The extracted data files of the HHT/HT toolbox

36

37

CHAPTER7

DATA SET, EVALUATION AND RESULTS

7.1 Data set

Data set used in this study has been collected from tide gauge stations of Turkey.

These tide gauges are Amasra, Antalya, Trabzon, Bozyazi. The detailed information

about the time interval of the gathered data and the geodetic coordinates of the tide

gauge stations are given in Table 5 and the data gaps are shown in Table 6.

Table 5: Tide gauge station names and theirs geodetic coordinates

Station Name latitude longitude Time Span

AMASRA 2005-2010

ANTALYA

2003-2010

TRABZON 2003-2010

BOZYAZI

2008-2010

Table 6: Tide gauge station data gaps

Station From To

AMASRA 2006-08-07 11:00:00 2006-09-25 06:015:00

2009-03-03 07:00:00 2009-06-29 14:15:00

ANTALYA 2006-02-22 19:00:00 2006-12-25 10:15:00

2009-03-30 07:00:00 2009-07-02 12:00:00

TRABZON 2009-03-31 08:30:00 2009-06-01 10:45:00

BOZYAZI 2009-07-05 16:30:00 2009-07-06 08:00:00

In these data outliers have been detected and eliminated. Also all error sources were

classified by data service provider, General Command of Mapping (GCM). Figure 15

illustrate sample of obtained data format.

38

Figure 15: A sample of tide gauge data format

First column shows date and time of observation, second column is its decimal year,

third column indicate sea level height (SLH), and forth columns shows the given error

classification value (data flag). Table 7 shows the description of data flag numbers

which changes from 0 to 9.

Table 7: Data Flags of Errors

0 No quality control

1 Correct value

2 Interpolated value

3 Doubtful value

4 Isolated spike or wrong value

5 Correct but extreme value

6 Reference change detected

7 Constant values for more than a defined time interval

8 Out of range

9 Missing value

39

7.2 Application of Hilbert Huang Transform to tide gauge data records

7.2.1 Analysis of tidal data based on the Hilbert Huang Empirical Mode

Decomposition (HHEMD)

The main purpose of this study is to generate a MATLAB-based toolbox which can

analyze tidal and sub-tidal phenomena using Hilbert Huang Empirical Mode

Decomposition (HHEMD). As mentioned in chapter 4 other tidal analysis methods

such as harmonic analysis (Jay and Flinchem 1999) and wavelet transform cannot

provide accurate results on the necessary time scale and detect non stationary tidal

phenomena properly. Sub-tidal phenomena using wavelet transform has been analyzed

by (Percival and Mofjeld 1997) without giving the instantaneous frequency. HHEMD

method has been applied to the hourly tidal data, collected by the tide gauge stations

from Antalya, Amasra, Trabzon and Bozyazi to understand the characteristics of the

tidal and sub-tidal phenomena. Results have been compared with harmonic analysis

(HA). For this, hourly collected tide gauge data of Antalya, Amasra, Bozyazi and

Trabzon have been decomposed in its components in terms IMFs using Equation (4.9)

(See also Figures 16 to Figure 19).

40

Figure 16: Result of the EMD analysis of Antalya tide gauge data. Vertical axis

shows the sea surface elevation (m) and horizontal axis shows the Time (year).

X(t) is the raw data and other ones are the all IMFs of Antalya tide gauge station.

Time (year)

41

Figure 17: Result of the EMD analysis of Trabzon tide gauge data. Vertical axis

shows the sea surface elevation (m) and horizontal axis shows the Time (year).

X(t) is the raw data and other ones are the all IMFs of Trabzon tide gauge

station.

Time (year)

42

Figure 18: Result of the EMD analysis of Bozyazi tide gauge data. Vertical axis

shows the sea surface elevation (m) and horizontal axis shows the Time (year).

X(t) is the raw data and other ones are the all IMFs of Bozyazi tide gauge station.

Time (year)

43

Figure 19: Result of the EMD analysis of Amasra tide gauge data. Vertical axis

shows the sea surface elevation (m) and horizontal axis shows the Time (year).

X(t) is the raw data and other ones are the all IMFs of Amasra tide gauge station.

Time (year)

44

The maximum number of IMFs is calculated as 13 IMFs for Antalya, Amasra and

Trabzon. Due to the short time interval of the Bozyazi (2 years) EMD provided 12

IMFs. Based on the extracted frequencies the first 4 IMFs have periodic oscillation

modes which contain different frequencies. IMF1 comprises high frequency part of the

signal and containing periods between 2 to 3 hours, IMF2 indicates shallow water

phenomena having 3 to 6 hours periods. IMF3 and IMF4 are associated with

semidiurnal and diurnal variations. Other IMFs namely IMF5 and higher do not have

periodic waves and are perturbed by some other factors like atmospheric pressure or

wind stress than the Moon and the Sun. These can be understood as sub-tidal

phenomena (Ezer and Corlett 2012). The last IMFs (IMF13 for Antalya, Trabzon,

Amasra and IMF12 for Bozyazi) which have zero frequencies can be interpreted as the

trend related part of the time series (Huang et al., 1998). Figure 20 to Figure 23 shows

the instantaneous frequencies of the all IMFs components of the 4 tide gauge stations,

resulting from the Hilbert transform.

45

Figure 20: Frequencies for each IMFs of the Antalya tide gauge station data.

Vertical axis indicates the frequency (Hz) and horizontal axis indicates the Time

(year). Semidiurnal and diurnal tide can be deduced from IMF3 and IMF4.

Time (year)

IMF

s fr

equen

cy (

Hz)

46

Figure 21: Frequencies for each IMFs of the Trabzon tide gauge station data.

Vertical axis indicates the frequency (Hz) and horizontal axis indicates the Time

(year). Semidiurnal and diurnal tide can be deduced from IMF3 and IMF4.

Time (year)

IMF

s fr

equen

cy (

Hz)

47

Figure 22: Frequencies for each IMFs of the Bozyazi tide gauge station data.

Vertical axis indicates the frequency (Hz) and horizontal axis indicates the Time

(year). Semidiurnal and diurnal tide can be deduced from IMF3 and IMF4.

Time (year)

IMF

s fr

equ

ency

(H

z)

48

Figure 23: Frequencies for each IMFs of the Amasra tide gauge station data. Vertical

axis indicates the frequency (Hz) and horizontal axis indicates the Time (year).

Semidiurnal and diurnal tide can be deduced from IMF3 and IMF4.

Time (year)

IMF

s fr

equen

cy (

Hz)

49

7.2.2 Completeness test

To check the completeness of tide gauge data numerically the IMF sequences have

been reconstructed using the data from the Antalya station between years 2003 to

2006. Reconstruction of the IMFs component was started from longest to the shortest

wave in the IMF sequence as seen in Figure 24a to Figure 24j. Vertical axis of the

figures shows the sea surface elevation (m) while horizontal one shows the time

(year). Figure 24a gives the raw tide gauge data and the longest period component as

the blue color line namely IMF11, which can be interpreted as the trend related part of

the time series (Huang et al., 1998). By adding the next longest period component,

IMF10, the trend of the sum, IMF11+IMF10 gives a remarkable fitting to the data as

shown in Figure 24b. By successively adding more components with increasing

frequency, the fitting improved continuously. The gradual change from monotonic

trend to the final reconstruction is illustrated by Figures 24c to Figure 24i. By sum of

IMF components up to IMF1, has been recovered all the signal data.

Figure 24: Numerical proof of the completeness of the EMD through

reconstruction of the original data from the IMF components. a) Raw tidal data

(Brown color line) and the IMF11 component as residue (Blue color line). b) Raw

tidal data (Brown color line) and the sum of IMF11 and IMF10 components

(Blue color line). c) Raw tidal data (Brown color line) and the sum of IMF11 to

IMF9 components (Blue color line). d) Raw tidal data (Brown color line) and the

sum of IMF11 to IMF8 components (Blue color line).

a

c

b

d

Sea

Surf

ace

Ele

vat

ion

(m

)

Time (year)

50

The difference between the reconstructed tidal data from the sum of the IMFs and the

original data is shown in Figure 24j, in which the maximum amplitude is less than

. This difference is related to the limit of the computational precision of the

personal computer (PC) used. Therefore the completeness which is inherent to the

theory of the EMD by Equation (4.7), has been shown also numerically ( See Figure

24j).

Figure 24: Cont. e) raw tidal data (Brown color line) and the sum of IMF11 to

IMF5 components (Blue color line). f) Raw tidal data (Brown color line) and the

sum of IMF11 to IMF4 components (Blue color line). g) Raw tidal data (Brown

color line) and the sum of IMF11 to IMF3 components (Blue color line). h) Raw

tidal data (Brown color line) and the sum of IMF11 to IMF2 components (Blue

color line). i) Raw tidal data (Brown color line) and the sum of IMF11 to IMF1

components (Blue color line). This is the final reconstruction of the data from the

IMFs. It appears no different from the original data. j) The difference between

the original data and the reconstructed one.

e f

g h

i j

Sea

Surf

ace

Ele

vat

ion

(m

)

Time (year)

51

7.2.3 Comparison of the HHT analysis with Harmonic analysis (HA)

To make a comparison of the results generated by HHT with an well-known method

harmonic analysis using least square fitting procedure has been applied to Equation

(3.2) ( Also See Figure 25). Resulting tidal constituents of 4 tide gauge data stations

are given in Table 8 to Table11. The records are limited to 68 constituents for Antalya,

Trabzon, Amasra and to 55 constituents for Bozyazi. The first columns of the tables

contain the abbreviations of tidal harmonic constituents. Second and third columns

show the frequencies and amplitudes of tidal constituents.

Figure 25: least squares fitting of 4 tide gauge station data. a) Amasra b) Antalya

c) Bozyazi d) Trabzon

a b

d c

52

Table 8: Harmonic analysis tidal constituents of Antalya

Tidal harmonic

constituents

Frequency Amplitude Amplitude

error

Phase Phase

error SA 0.0001141 0.0849 0.011 238.50 6.54

SSA 0.0002282 0.0340 0.010 241.71 15.70

MSM 0.0013098 0.0083 0.009 154.77 79.83

MM 0.0015122 0.0054 0.007 207.87 99.68

MSF 0.0028219 0.0088 0.010 199.42 66.54

MF 0.0030501 0.0041 0.008 204.09 128.44

ALP1 0.0343966 0.0001 0.000 53.69 167.08

2Q1 0.0357064 0.0003 0.000 76.82 99.96

SIG1 0.0359087 0.0004 0.000 71.92 58.64

Q1 0.0372185 0.0017 0.000 258.82 16.00

RHO1 0.0374209 0.0004 0.000 101.14 74.16

O1 0.0387307 0.0128 0.000 265.67 1.83

TAU1 0.0389588 0.0018 0.001 156.40 24.02

BET1 0.0400404 0.0005 0.000 264.04 52.87

NO1 0.0402686 0.0009 0.000 271.41 20.12

CHI1 0.0404710 0.0007 0.000 293.15 36.34

PI1 0.0414385 0.0011 0.001 258.59 31.42

P1 0.0415526 0.0060 0.001 298.37 4.62

S1 0.0416667 0.0040 0.001 260.47 12.26

K1 0.0417807 0.0213 0.000 287.64 1.35

PSI1 0.0418948 0.0012 0.001 95.83 24.96

PHI1 0.0420089 0.0008 0.001 34.26 44.48

THE1 0.0430905 0.0003 0.000 194.32 90.36

J1 0.0432929 0.0011 0.000 267.81 23.27

SO1 0.0446027 0.0004 0.000 204.63 60.70

OO1 0.0448308 0.0005 0.000 292.11 28.33

UPS1 0.0463430 0.0001 0.000 347.63 187.54

OQ2 0.0759749 0.0007 0.001 295.71 109.40

EPS2 0.0761773 0.0003 0.001 293.89 167.29

2N2 0.0774871 0.0011 0.001 266.82 78.04

MU2 0.0776895 0.0011 0.001 263.13 76.05

N2 0.0789992 0.0117 0.002 248.00 7.90

NU2 0.0792016 0.0014 0.001 234.88 61.64

GAM2 0.0803090 0.0007 0.001 349.66 97.50

H1 0.0803973 0.0051 0.001 325.63 18.43

M2 0.0805114 0.0715 0.001 241.97 1.17

H2 0.0806255 0.0018 0.001 152.75 44.94

MKS2 0.0807396 0.0024 0.001 348.49 26.64

LDA2 0.0818212 0.0008 0.001 207.42 105.83

L2 0.0820236 0.0049 0.002 247.02 30.76

T2 0.0832193 0.0042 0.001 195.95 18.95

S2 0.0833333 0.0429 0.001 257.45 1.86

R2 0.0834474 0.0002 0.001 244.70 179.65

K2 0.0835615 0.0118 0.001 261.65 5.04

MSN2 0.0848455 0.0006 0.001 164.00 150.46

ETA2 0.0850736 0.0004 0.001 247.41 124.01

MO3 0.1192421 0.0008 0.000 352.58 12.05

M3 0.1207671 0.0016 0.000 12.58 7.55

SO3 0.1220640 0.0009 0.000 351.98 11.09

MK3 0.1222921 0.0008 0.000 26.58 11.48

SK3 0.1251141 0.0007 0.000 342.09 16.24

MN4 0.1595106 0.0008 0.000 307.26 16.28

M4 0.1610228 0.0016 0.000 327.87 8.66

SN4 0.1623326 0.0004 0.000 339.40 35.45

MS4 0.1638447 0.0017 0.000 347.50 07.10

MK4 0.1640729 0.0008 0.000 290.77 14.39

S4 0.1666667 0.0005 0.000 13.24 23.77

SK4 0.1668948 0.0004 0.000 311.84 25.78

2MK5 0.2028035 0.0003 0.000 260.11 21.15

2SK5 0.2084474 0.0000 0.000 321.66 215.51

2MN6 0.2400221 0.0002 0.000 209.00 27.55

M6 0.2415342 0.0002 0.000 202.19 30.95

2MS6 0.2443561 0.0003 0.000 194.65 19.81

2MK6 0.2445843 0.0002 0.000 222.71 23.27

2SM6 0.2471781 0.0003 0.000 216.94 22.16

MSK6 0.2474062 0.0002 0.000 222.00 29.10

3MK7 0.2833149 0.0001 0.000 41.37 75.52

M8 0.3220456 0.0001 0.000 353.48 33.22

53

Table 9: Harmonic analysis tidal constituents of Amasra

Tidal harmonic

constituents

Frequency Amplitude Amplitude

error

Phase Phase

error SA 0.0001141 0.0333 0.008 100.02 15.75

SSA 0.0002282 0.0089 0.009 143.37 60.56

MSM 0.0013098 0.0024 0.006 191.79 159.40

MM 0.0015122 0.0039 0.006 292.93 128.92

MSF 0.0028219 0.0027 0.006 274.30 166.15

MF 0.0030501 0.0017 0.006 349.42 194.75

ALP1 0.0343966 0.0001 0.000 229.17 228.49

2Q1 0.0357064 0.0001 0.000 103.38 191.89

SIG1 0.0359087 0.0003 0.000 90.74 113.58

Q1 0.0372185 0.0006 0.001 16.52 57.26

RHO1 0.0374209 0.0001 0.000 97.56 193.27

O1 0.0387307 0.0012 0.001 29.79 24.33

TAU1 0.0389588 0.0002 0.000 142.86 156.44

BET1 0.0400404 0.0021 0.000 145.08 159.88

CHI1 0.0404710 0.0002 0.000 183.95 126.63

PI1 0.0414385 0.0005 0.001 17.75 59.14

P1 0.0415526 0.0022 0.001 175.44 16.14

S1 0.0416667 0.0037 0.001 1.64 9.27

K1 0.0417807 0.0008 0.001 94.23 37.26

PSI1 0.0418948 0.0009 0.001 356.16 35.37

PHI1 0.0420089 0.0006 0.001 31.30 50.59

THE1 0.0430905 0.0001 0.000 331.39 198.39

J1 0.0432929 0.0001 0.000 109.84 204.01

SO1 0.0446027 0.0001 0.000 238.23 228.59

OO1 0.0448308 0.0002 0.000 45.18 130.96

UPS1 0.0463430 0.0001 0.000 289.58 190.41

OQ2 0.0759749 0.0001 0.000 151.60 126.85

EPS2 0.0761773 0.0002 0.000 60.48 131.11

2N2 0.0774871 0.0002 0.000 324.12 153.57

MU2 0.0776895 0.0001 0.000 84.06 203.76

N2 0.0789992 0.0008 0.000 291.48 31.03

NU2 0.0792016 0.0000 0.000 56.77 214.64

GAM2 0.0803090 0.0002 0.000 251.65 115.68

H1 0.0803973 0.0014 0.000 128.52 17.52

M2 0.0805114 0.0035 0.000 295.60 7.64

H2 0.0806255 0.0007 0.000 234.31 35.05

MKS2 0.0807396 0.0006 0.000 312.70 37.78

LDA2 0.0818212 0.0002 0.000 20.69 100.72

L2 0.0820236 0.0002 0.000 255.09 102.43

T2 0.0832193 0.0008 0.000 345.67 32.95

S2 0.0833333 0.0014 0.000 258.27 14.62

R2 0.0834474 0.0008 0.000 215.26 29.43

K2 0.0835615 0.0006 0.000 286.64 42.33

MSN2 0.0848455 0.0000 0.000 200.49 241.96

ETA2 0.0850736 0.0002 0.000 238.41 149.69

MO3 0.1192421 0.0001 0.000 157.38 150.61

M3 0.1207671 0.0002 0.000 183.09 74.92

SO3 0.1220640 0.0001 0.000 83.35 121.28

MK3 0.1222921 0.0001 0.000 109.25 99.69

SK3 0.1251141 0.0000 0.000 24.53 178.37

MN4 0.1595106 0.0001 0.000 39.88 95.22

M4 0.1610228 0.0001 0.000 24.88 89.95

SN4 0.1623326 0.0001 0.000 91.81 146.56

MS4 0.1638447 0.0000 0.000 348.36 156.17

MK4 0.1640729 0.0000 0.000 299.46 191.40

S4 0.1666667 0.0002 0.000 4.34 38.57

SK4 0.1668948 0.0001 0.000 295.62 66.69

2MK5 0.2028035 0.0001 0.000 82.02 99.26

2SK5 0.2084474 0.0001 0.000 50.40 111.43

2MN6 0.2400221 0.0001 0.000 69.67 114.24

M6 0.2415342 0.0001 0.000 136.84 63.78

2MS6 0.2443561 0.0000 0.000 254.01 206.60

2MK6 0.2445843 0.0000 0.000 87.12 241.51

2SM6 0.2471781 0.0001 0.000 214.11 148.45

MSK6 0.2474062 0.0001 0.000 262.59 71.60

3MK7 0.2833149 0.0001 0.000 141.69 149.00

M8 0.3220456 0.0001 0.000 126.73 101.73

54

Table 10: Harmonic analysis tidal constituents of Bozyazi

Tidal harmonic

constituents

Frequency Amplitude Amplitude

error

Phase Phase

error SA 0.0001141 0.0704 0.023 259.49 17.79

SSA 0.0002282 0.0472 0.021 241.35 26.10

MSM 0.0013098 0.0058 0.016 245.61 166.02

MM 0.0015122 0.0029 0.015 210.73 235.56

MSF 0.0028219 0.0028 0.014 265.58 221.97

MF 0.0030501 0.0141 0.020 302.83 92.30

ALP1 0.0343966 0.0000 0.000 126.97 170.34

SIG1 0.0359087 0.0000 0.000 217.47 173.28

RHO1 0.0374209 0.0000 0.000 355.94 228.58

TAU1 0.0389588 0.0000 0.000 42.66 69.96

BET1 0.0400404 0.0006 0.002 310.83 190.23

CHI1 0.0404710 0.0013 0.002 315.95 141.18

PI1 0.0414385 0.0027 0.003 290.54 69.56

P1 0.0415526 0.0000 0.000 338.83 67.27

S1 0.0416667 0.0263 0.005 283.11 47.48

PSI1 0.0418948 0.0058 0.003 120.21 31.31

PHI1 0.0420089 0.0033 0.003 263.54 61.12

THE1 0.0430905 0.0000 0.000 59.00 218.77

OQ2 0.0759749 0.0010 0.006 252.98 227.42

EPS2 0.0761773 0.0008 0.005 255.36 230.81

2N2 0.0774871 0.0018 0.005 239.50 182.28

MU2 0.0776895 0.0004 0.005 265.71 273.18

N2 0.0789992 0.0100 0.007 195.05 41.98

NU2 0.0792016 0.0030 0.005 177.08 122.79

GAM2 0.0803090 0.0127 0.007 115.07 29.51

H1 0.0803973 0.0193 0.007 181.68 20.96

M2 0.0805114 0.0544 0.007 166.40 7.80

H2 0.0806255 0.0323 0.007 252.99 13.49

MKS2 0.0807396 0.0072 0.007 138.36 56.08

LDA2 0.0818212 0.0017 0.005 198.67 188.13

L2 0.0820236 0.0015 0.004 201.75 173.28

T2 0.0832193 0.0138 0.006 7.93 29.55

S2 0.0833333 0.0297 0.007 180.92 13.95

R2 0.0834474 0.0146 0.005 66.87 23.64

K2 0.0835615 0.0118 0.006 170.00 28.88

MSN2 0.0848455 0.0022 0.006 17.47 150.64

ETA2 0.0850736 0.0009 0.004 164.00 249.10

M3 0.1207671 0.0007 0.000 0.36 40.55

MN4 0.1595106 0.0000 0.000 221.05 167.81

M4 0.1610228 0.0018 0.000 162.67 95.31

SN4 0.1623326 0.0000 0.000 224.07 239.08

MS4 0.1638447 0.0000 0.000 248.83 183.06

MK4 0.1640729 0.0001 0.000 211.70 152.55

S4 0.1666667 0.0017 0.000 56.28 115.60

SK4 0.1668948 0.0000 0.000 275.27 196.33

2MN6 0.2400221 0.0001 0.000 69.08 129.10

M6 0.2415342 0.0028 0.000 62.97 66.65

2MS6 0.2443561 0.0001 0.000 29.10 141.79

2MK6 0.2445843 0.0000 0.000 315.80 187.91

2SM6 0.2471781 0.0000 0.000 234.10 215.63

MSK6 0.2474062 0.0000 0.000 292.82 197.96

M8 0.3220456 0.0000 0.000 257.92 155.49

55

Table 11: Harmonic analysis tidal constituents of Trabzon

Tidal harmonic

constituents

Frequency Amplitude Amplitude

error

Phase Phase

error SA 0.0001141 0.0663 0.010 144.51 09.04

SSA 0.0002282 0.0240 0.009 219.81 22.03

MSM 0.0013098 0.0020 0.007 74.63 179.72

MM 0.0015122 0.0033 0.007 52.75 130.61

MSF 0.0028219 0.0006 0.006 288.82 248.13

MF 0.0030501 0.0021 0.006 267.03 211.38

ALP1 0.0343966 0.0002 0.000 251.55 113.44

2Q1 0.0357064 0.0003 0.000 356.33 60.85

SIG1 0.0359087 0.0001 0.000 16.57 151.19

Q1 0.0372185 0.0012 0.000 222.92 16.51

RHO1 0.0374209 0.0005 0.000 246.82 39.03

O1 0.0387307 0.0067 0.000 213.22 2.70 4.

TAU1 0.0389588 0.0003 0.000 327.21 93.19

BET1 0.0400404 0.0001 0.000 207.70 139.06

NO1 0.0402686 0.0005 0.000 219.24 31.35

CHI1 0.0404710 0.0003 0.000 216.36 58.44

PI1 0.0414385 0.0043 0.000 154.98 62.13

P1 0.0415526 0.0350 0.000 215.32 5.94

S1 0.0416667 0.0244 0.001 199.41 7.21

K1 0.0417807 0.0109 0.000 215.17 2.00 8.

PSI1 0.0418948 0.0002 0.000 114.90 130.Tem

PHI1 0.0420089 0.0004 0.000 246.59 52.81

THE1 0.0430905 0.0001 0.000 208.72 189.04

J1 0.0432929 0.0005 0.000 185.43 44.73

SO1 0.0446027 0.0003 0.000 186.64 57.04

OO1 0.0448308 0.0002 0.000 251.44 51.12

UPS1 0.0463430 0.0001 0.000 291.17 142.62

OQ2 0.0759749 0.0001 0.000 182.86 196.35

EPS2 0.0761773 0.0001 0.000 176.37 165.52

2N2 0.0774871 0.0006 0.001 172.81 49.72

MU2 0.0776895 0.0008 0.001 171.44 33.91

N2 0.0789992 0.0040 0.001 178.28 8.32

NU2 0.0792016 0.0009 0.001 182.69 36.00

GAM2 0.0803090 0.0008 0.000 230.24 38.93

H1 0.0803973 0.0014 0.001 278.16 24.38

M2 0.0805114 0.0223 0.001 181.29 1.51 1.

H2 0.0806255 0.0014 0.001 67.17 22.56

MKS2 0.0807396 0.0004 0.000 283.36 61.04

LDA2 0.0818212 0.0005 0.000 118.53 69.52

L2 0.0820236 0.0015 0.001 173.87 36.47

T2 0.0832193 0.0011 0.001 138.95 28.16

S2 0.0833333 0.0105 0.001 188.62 3.00 3.

R2 0.0834474 0.0003 0.000 225.15 104.24

K2 0.0835615 0.0025 0.000 188.28 8.95

MSN2 0.0848455 0.0001 0.000 289.42 229.96

ETA2 0.0850736 0.0000 0.000 121.94 241.76

MO3 0.1192421 0.0001 0.000 190.74 58.87

M3 0.1207671 0.0003 0.000 143.03 25.75

SO3 0.1220640 0.0001 0.000 125.98 138.43

MK3 0.1222921 0.0000 0.000 231.54 196.85

SK3 0.1251141 0.0003 0.000 7.96 32.04

MN4 0.1595106 0.0011 0.000 135.84 86.57

M4 0.1610228 0.0017 0.000 31.15 1 24.85

SN4 0.1623326 0.0001 0.000 251.88 63.71

MS4 0.1638447 0.0001 0.000 133.95 63.14

MK4 0.1640729 0.0001 0.000 322.04 53.74

S4 0.1666667 0.0015 0.000 85.36 11.32

SK4 0.1668948 0.0001 0.000 305.47 44.22

2MK5 0.2028035 0.0001 0.000 290.87 96.64

2SK5 0.2084474 0.0001 0.000 248.16 50.49

2MN6 0.2400221 0.0001 0.000 22.04 62.97

M6 0.2415342 0.0010 0.000 194.14 173.95

2MS6 0.2443561 0.0001 0.000 212.44 90.65

2MK6 0.2445843 0.0000 0.000 319.48 160.75

2SM6 0.2471781 0.0001 0.000 256.81 79.35

MSK6 0.2474062 0.0000 0.000 24.85 119.28

3MK7 0.2833149 0.0000 0.000 328.69 82.60

M8 0.3220456 0.0001 0.000 42.86 46.96

56

The comparison of the maximum and minimum amplitudes calculated by both

methods for all frequency domains in terms of annual, semi-annual, Lunar Monthly,

diurnal, semi diurnal and shallow water constituents is shown in Table 12 and Table

13 (See also Figure 26 to Figure41). Diurnal tidal constituents of the tidal data have a

usual frequency domain between 0.0372 and 0.446 Hz. The maximum amplitudes of

the diurnal tidal constituents which have been calculated by HA for Antalya, Trabzon,

Amasra and Bozyazi tide gauge stations are 0.0213m, 0.0244m, 0.0037m and

0.0263m. Also the maximum amplitudes of the same station which have been

calculated by HHT are 0.1241m, 0.1143m, 0.046m and 0.1417m respectively. Result

shows that the magnitudes of maximum amplitudes obtained by HA are lower than the

magnitudes of maximum amplitudes calculated by HHT.

Table 12 shows the comparisons of minimum and maximum amplitude of diurnal,

semi diurnal and shallow water constituents which have been calculated by HA and

HHT. Also Table 13 shows the minimum and maximum amplitude comparisons

annual, semi-annual, Lunar Monthly of 4 tide gauge stations which have been obtained

by HA and HHT.

From all comparisons can be concluded that the both minimum and maximum

amplitude from HA analysis have lower magnitudes than the HHT. The reason for this

fact may be referred to the difference of the parametric and nonparametric estimation

methods used. Parametric method is represented by HA based on the least square

fitting procedures while the HHT is the kind of nonparametric adaptive method.

57

Figure 26: Corresponding frequency and amplitude of the Diurnal tides of the

Trabzon.

Figure 27: Corresponding frequency and amplitude of the semi-diurnal tides of

the Trabzon.

Time (year)

Time (year)

Time (year)

Time (year)

Sem

i-d

iurn

al t

ide

amp

litu

des

(m

)

Sem

i-d

iurn

al t

ide

freq

uen

cies

(H

z)

Diu

rnal

tid

e am

pli

tud

es (

m)

Diu

rnal

tid

e fr

equ

enci

es (

Hz)

58

Figure 28: Corresponding frequency and amplitude of the shallow water of the

Trabzon.

Figure 29: Corresponding frequency and amplitude of the sub-tidal of the Trabzon.

Time (year)

Time (year)

Time (year)

Time (year)

Su

b-t

idal

am

pli

tud

es (

m)

Su

b-t

idal

fre

qu

enci

es (

Hz)

Sh

allo

w w

ater

am

pli

tud

es (

m)

Sh

allo

w w

ater

fre

qu

enci

es (

Hz)

59

Figure 30: Corresponding frequency and amplitude of the diurnal tides of the

Bozyazi.

Figure 31: Corresponding frequency and amplitude of the semi-diurnal tides of

the Bozyazi.

Time (year)

Time (year)

Time (year)

Time (year)

Sem

i-d

iurn

al t

ide

amp

litu

des

(m

)

Sem

i-d

iurn

al t

ide

freq

uen

cies

(H

z)

Diu

rnal

tid

e am

pli

tud

es (

m)

Diu

rnal

tid

e fr

equ

enci

es (

Hz)

60

Figure 32: Corresponding frequency and amplitude of the shallow water of the

Bozyazi.

Figure 33: Corresponding frequency and amplitude of the sub-tidal of the

Bozyazi.

Time (year)

Time (year)

Time (year)

Time (year)

Su

b-t

idal

am

pli

tud

es (

m)

Su

b-t

idal

fre

qu

enci

es (

Hz)

Sh

allo

w w

ater

am

pli

tud

es (

m)

Sh

allo

w w

ater

fre

qu

enci

es (

Hz)

61

Figure 34: Corresponding frequency and amplitude of the diurnal tides of the

Antalya.

Figure 35: Corresponding frequency and amplitude of the semi-diurnal tides of

the Antalya.

Time (year)

Time (year)

Time (year)

Time (year)

Sem

i-d

iurn

al t

ide

amp

litu

des

(m

)

Sem

i-d

iurn

al t

ide

freq

uen

cies

(H

z)

Diu

rnal

tid

e am

pli

tud

es (

m)

Diu

rnal

tid

e fr

equ

enci

es (

Hz)

62

Figure 36: Corresponding frequency and amplitude of the shallow water tides of

the Antalya.

Figure 37: Corresponding frequency and amplitude of the sub-tidal of the

Antalya.

Time (year)

Time (year)

Time (year)

Time (year)

Su

b-t

idal

am

pli

tud

es (

m)

Su

b-t

idal

fre

qu

enci

es (

Hz)

(()

(Hz)

Sh

allo

w w

ater

am

pli

tud

es (

m)

Sh

allo

w w

ater

fre

qu

enci

es (

Hz)

63

Figure 38: Corresponding frequency and amplitude of the diurnal tides of the

Amasra.

Figure 39: Corresponding frequency and amplitude of the semi-diurnal tides of

the Amasra.

Time (year)

Time (year)

Time (year)

Time (year)

Diu

rnal

tid

e fr

equ

enci

es (

Hz)

Sem

i-d

iurn

al t

ide

freq

uen

cies

(H

z)

Sem

i-d

iurn

al t

ide

amp

litu

des

(m

)

Diu

rnal

tid

e am

pli

tud

es (

m)

64

Figure 40: Corresponding frequency and amplitude of the shallow water of the

Amasra.

Figure 41: Corresponding frequency and amplitude of the sub-tidal of the

Amasra

Time (year)

Su

b-t

idal

am

pli

tud

es (

m)

Su

b-t

idal

fre

qu

enci

es (

Hz)

Time (year)

Sh

allo

w w

ater

fre

qu

enci

es (

Hz)

Sh

allo

w w

ater

am

pli

tud

es (

m)

Time (year)

Time (year)

65

Table 12: Minimum and maximum amplitude of the diurnal, semidiurnal and

shallow water of the tides in each tide gauges stations.

AMASRA 2005-2010

Sub-tidal and Tidal

Constituents Frequency Domain IMF

EMD

Amplitude (m)

HA

Amplitude(m)

Min-Max Min-Max

Sub-tidal 0.333< frequency =<1.000 IMF1 0.0011-0.0932

Shallow-Water 0.125<= frequency <=0.333 IMF2 0.0002-0.0536 0.0001-0.0002

Semi-diurnal 0.0775<= frequency <=0.0835 IMF3 0.0005-0.0745 0.0001-0.0035

Diurnal 0.0372<= frequency <=0.0446 IMF4 0.0072-0.0456 0.0001-0.0037

ANTALYA 2003-2010

Sub-tidal and Tidal

Constituents Frequency Domain IMF

EMD

Amplitude (m)

HA

Amplitude(m)

Min -Max Min -Max

Sub-tidal 0.333< frequency =<1.000 IMF1 0.0057-0.1883

Shallow-Water 0.125<= frequency <=0.333 IMF2 0.0045-0.0018 0.0001-0.0017

Semi-diurnal 0.0775<= frequency <=0.0835 IMF3 0.0014-0.0900 0.0002-0.0715

Diurnal 0.0372<= frequency <=0.0446 IMF4 0.0019-0.1241 0.0003-0.0213

BOZYAZI 2008-2010

Sub-tidal and Tidal

Constituents Frequency Domain IMF

EMD

Amplitude (m)

HA

Amplitude(m)

Min -Max Min -Max

Sub-tidal 0.333<frequency=<1.000 IMF1 0.0081-0.2160

Shallow-Water 0.125<= frequency <=0.333 IMF2 0.0011-0.1053 0.0001-0.028

Semi-diurnal 0.0775<= frequency <=0.0835 IMF3 0.0016-0.0770 0.0004-0.0544

Diurnal 0.0372<= frequency <=0.0446 IMF4 0.0011-0.1439 0.0006-0.0263

TRABZON 2003-2010

Sub-tidal and Tidal

Constituents Frequency Domain IMF

EMD

Amplitude (m)

HA

Amplitude(m)

Min -Max Min -Max

Sub-tidal 0.333< frequency =<1.000 IMF1 0.0035-0.0767

Shallow-Water 0.125<= frequency <=0.333 IMF2 0.0059-0.0924 0.0001-0.0015

Semi-diurnal 0.0775<= frequency <=0.0835 IMF3 0.0012-0.0828 0.0005-0.0223

Diurnal 0.0372<= frequency <=0.0446 IMF4 0.0011-0.0914 0.0001-0.0244

66

Table 13: Amplitudes of the Lunar Monthly (Mm), Solar semiannual (Ssa) and

Solar annual (Sa) of the tides in each tide gauges station

AMASRA 2005-2010 Sub-tidal andTidal

Constituents Frequency IMF

EMD

Amplitude (m)

HA

Amplitude(m)

Lunar Monthly(Mm) 0.0015122 IMF9 0.0139 0.0039

Solar semiannual (Ssa) 0.0002282 IMF11 0.0238 0.0089

Solar annual(Sa) 0.0001141 IMF12 0.0460 0.0331

ANTALYA 2003-2010 Sub-tidal andTidal

Constituents Frequency IMF

EMD

Amplitude (m)

HA

Amplitude(m)

Lunar Monthly(Mm) 0.0015122 IMF9 0.0449 0.0054

Solar semiannual (Ssa) 0.0002282 IMF11 0.0717 0.0340

Solar annual(Sa) 0.0001141 IMF12 0.0567 0.0849

BOZYAZI 2008-2010 Sub-tidal andTidal

Constituents Frequency IMF

EMD

Amplitude (m)

HA

Amplitude(m)

Lunar Monthly(Mm) 0.0015122 IMF8 0.0523 0.0029

Solar semiannual (Ssa) 0.0002282 IMF9 0.0645 0.0472

Solar annual(Sa) 0.0001141 IMF10 0.0162 0.0704

TRABZON 2003-2010 Sub-tidal andTidal

Constituents Frequency IMF

EMD

Amplitude (m)

HA

Amplitude(m)

Lunar Monthly(Mm) 0.0015122 IMF9 0.0227 0.0033

Solar semiannual (Ssa) 0.0002282 IMF11 0.0497 0.024

Solar annual(Sa) 0.0001141 IMF12 0.0829 0.0663

67

7.2.4 Analysis of sub-tidal phenomena using HHEMD

As explained before HHT is also an appropriate tools for detecting sub-tidal

fluctuations.

Sub-tidal phenomena are predominantly caused by atmospheric pressure and wind

stress. Roden (1960), Denbo and Allen (1987), Spillance et al. (1987) and Strub et al.

(1987b). (Percival and Mofjeld 1997) used multi resolution analysis with maximal

overlap discrete wavelet transform to analyze non-stationary sub-tidal sea level

fluctuations occurring over a very broad range of a longer time scale. Also Persival

mentioned that the tide gauges which measure water level suppress the higher

frequency fluctuation resulting from the observation period. By contrast, in this study

after analyzing of tide gauge data and extracting the instantaneous frequencies of each

IMF, high frequency fluctuations which are higher than shallow water were associated

with IMF1 for each 4 tide gauge stations. As can be seen in table 12 the maximum and

minimum amplitudes and velocities of IMF1 are higher than the tidal constituents in

terms of IMF2, IMF3, IMF4, IMF9, IMF11, and IMF12. In view of the fact that the

amplitudes and velocities of tides are low at Turkish coasts, these high frequencies and

amplitudes may probably be caused by wind or other meteorological factors.

Sub-tidal fluctuations triggered by atmospheric effects can also be available in the low

frequency part of the tidal data. Figure 42 shows HHT analysis of Antalya 2003-2010

tide gauge records consisting of a smooth wave (bottom of the figure), and three detail

series. The smooth wave (Figure 42d) corresponds to variations on a scale of 64 day to

182 day while the three detail series are associated with changes on scale of 6 hours

(shallow water, Figure 42a), 12 hours (semi diurnal, Figure 42b), 24 hours (diurnal,

Figure 42c), respectively. The smooth wave (Figure 42d) contains the annual

variations in sub-tidal fluctuations (also seasonal) at Antalya. As can be seen in figure

42, there are considerable variabilities from year to year. The lowest portions of the

annual variations usually occur in late winter and spring that may be attributed to high

atmospheric pressure. High portions occurring in winter may be attributed to low

atmospheric pressure and wind stress. Also low summer fluctuations may be

connected with the high atmospheric pressure. However for a precise verification it is

necessary to compare all fluctuations with coastal pressure and wind data which are

not available yet.

68

Figure 42: Vertical axis is the sea level fluctuation (m) and horizontal axis is time

(year). a) Is the Sea level fluctuation of Antalya tide gauge station with change on

a scale of 6 hours (shallow water). b) Is the Sea level fluctuation of Antalya tide

gauge station with change on a scale of 12 hours (semi diurnal). c) Is the Sea level

fluctuation of Antalya tide gauge station with change on a scale of 24 hours

(diurnal). d) Is the annul Sea level fluctuation in sub-tidal fluctuation at Antalya.

Tid

al a

nd S

ub

-tid

al F

luct

uat

ion

(m

)

Time (year)

a

d

b

c

69

By comparing the sub-tidal phenomena between Antalya and Bozyazi tide gauge

stations have the same behavior for the both stations can be observed as expected (Sea

Figure 43).

Figure 43: Sub-tidal difference between Antalya (blue color line) and Bozyazi

(green color line).

At the Black Sea coastline also Trabzon and Amasra stations have the same sub-tidal

fluctuations. Seasonal sub-tidal fluctuations of Trabzon and Amasra are given in

Figure 44. As can also be seen there the lowest portion of the annual variations

measured at both stations happen between December and February which can be

attributed to high atmospheric pressures. The highest portions of the annual variations

occur in summer and maybe associated with seasonal warming and low atmospheric

pressure. But at any rate more additional meteorological information is needed in

order to give a definite statement about the annual variations.

Figure 44: Sub-tidal difference between Amasra (blue color line) and Trabzon

(green color line).

Su

b-t

idal

Flu

ctu

atio

n (

m)

Time (year)

Su

b-t

idal

Flu

ctu

atio

n (

m)

Time (year)

70

In the final analysis of sub-tidal phenomena the sub-tidal fluctuations of Amasra,

Trabzon, Antalya and Bozyazi tide gauge stations are shown in Figure 45. According

to these results all 4 tide gauge stations indicate the same behavior between the years

2008 and 2009 and inverse behavior from 2009 to 2011, while The amplitudes

difference between both seas are considerable large.

Figure 45: Sub-tidal difference between 4tide gauge stations

Sea level measurements have been carried out for more than a century at many tide

gauge station. The results of previous studies of sea level measurements at the Turkish

coasts are given in Şeşeoğulları et al.(2007), Sezen and Baybura (2010), Demir and

Gürdal (2000). In these literatures linear secular trends in annual mean sea level data

are calculated using the least squares linear regression. In literature, the trend for

Antalya (1986-2004) has been calculated and the results were given as 3.78mm/year

in Şeşeoğulları et al. (2007), 4.4mm/year in Sezen and Baybura (2010) and

7.3mm/year in Demir and Gürdal (2000). In this study trend has been calculated for

each station using the both methods namely HHT and HA and the result is shown in

table 14. As can be seen in Table 14 the yearly trend of Antalya (2003-2010) has been

calculated as 4.41 ±0.3 by EMD and 4.73 ±0.4 by HA. However for a relible trend

estimation the length of the time series is recommended to be 60 years or longer.

Table 14: Sea surface difference height for each tide gauge station during

measurement time.

Station HA trend (mm/year) EMD Trend (mm/year)

Amasra 0.67 ±0.02 0.76 ±0.02

Antalya 4.73 ±0.4 4.41 ±0.3

Trabzon 5.57 ±0.3 4.88 ±0.4

Bozyazi 13.26 ±0.2 18.46 ±0.4

Su

b-t

idal

Flu

ctu

atio

n (

m)

Time (year)

71

7.2.5 Tidal velocity

Table 15 shows the maximum and minimum velocity of the Diurnal, Semi-diurnal and

shallow water tidal constituents of 4 tide gauge stations. As can be seen in table 15,

the maximum velocity of the tidal constituents achieves roughly 50 cm/hour. On the

other hand it is known from the literature that the suitable velocity for tidal turbine

generator has to be at least 1m/s (Haas 2011). This result leads to the conclusion that

Turkish coasts do not have appropriate tidal amplitudes and velocities to produce

electricity by using tidal turbines.

Table 15: Minimum and maximum velocity of the diurnal, semidiurnal and

shallow water of the tides in each tide gauges stations.

AMASRA 2005-2010 Sub-tidal and

Tidal

Constituents

Frequency Domain IMF Velocity(m/h)

Min -Max

Sub-tidal 0.333< frequency =<1.000 IMF1 0.0027-0.2124

Shallow-Water 0.125<= frequency <=0.333 IMF2 0.0005-0.0506

Semi-diurnal 0.0775<= frequency <=0.0835 IMF3 0.0002-0.0365

Diurnal 0.0372<= frequency <=0.0446 IMF4 0.0002-0.0118

ANTALYA 2003-2010

Sub-tidal and Tidal

Constituents Frequency Domain IMF

Velocity(m/h)

Min -Max

Sub-tidal 0.333< frequency =<1.000 IMF1 0.0170-0.4078

Shallow-Water 0.125<= frequency <=0.333 IMF2 0.0004-0.0173

Semi-diurnal 0.0775<= frequency <=0.0835 IMF3 0.00074-0.0446

Diurnal 0.0372<= frequency <=0.0446 IMF4 0.0005-0.0321

BOZYAZI 2008-2010

Sub-tidal and Tidal

Constituents Frequency Domain IMF

Velocity(m/h)

Min -Max

Sub-tidal 0.333<frequency=<1.000 IMF1 0.0199-0.4615

Shallow-Water 0.125<= frequency <=0.333 IMF2 0.0016-0.1277

Semi-diurnal 0.0775<= frequency <=0.0835 IMF3 0.0008-0.0392

Diurnal 0.0372<= frequency <=0.0446 IMF4 0.0003-0.0377

TRABZON 2003-2010

Sub-tidal and Tidal

Constituents Frequency Domain IMF

Velocity(m/h)

Min -Max

Sub-tidal 0.333< frequency =<1.000 IMF1 0.0074-0.1701

Shallow-Water 0.125<= frequency <=0.333 IMF2 0.0012-0.1039

Semi-diurnal 0.0775<= frequency <=0.0835 IMF3 0.0006-0.0427

Diurnal 0.0372<= frequency <=0.0446 IMF4 0.0003-0.0221

72

73

CHAPTER 8

CONCLUSION AND FUTURE WORK

8.1 Conclusion

HHEMDD has successfully been applied to the tide gauge records collected from the

Turkish coasts. From the results can be concluded that HHEMD is an effective and

suitable non-parametric adaptive method to analyze tidal and sub-tidal phenomena

directly. The analysis given in this study illustrated how the Hilbert Huang transform

(HHT) provides an appropriate way to detect the non-stationary character of the tide

gauge data on the basis of a MATLAB based toolbox which has been developed and

implemented in this study. Similar results could be taken by applying a set of short

time Fourier transforms to the time series in terms of tide gauge records if these

transforms are utilized in a nonstandard way (Percival and Mofjeld 1997). However,

HHEMD method provided a more refined analysis of tide gauge data than the result

which was delivered by harmonic analysis. This may be referred to the adaptivity of

the EMD and the capability of HHT to find instantaneous frequency.

The completeness test of the whole signal used for the analysis in this study proved

that the algorithm has been implemented correctly and is consistent with the theory of

HHT.

The application of EMD to the data records from 4 tide gauge stations provided 13

IMFs for Amasra, Antalya, Trabzon stations and 12 IMFs for Bozyazı station. IMF1,

IMF2, IMF3 and IMF 4 indicated high frequencies, shallow water, semidiurnal and

diurnal tidal constituents of the 4 tide gauge stations respectively. Also the results of

the IMF9, IMF11and IMF12 showed the lunar monthly, solar semiannual and solar

annual for Antalya, Amasra and Trabzon. But these constituent occurred for IMF8,

IMF9 and IMF10 due to the short time interval of the Bozyazi. The comparison of the

maximum and minimum amplitudes of the tidal constituents calculated by HA and

HHT lead to the conclusion that the first one showed lower magnitudes than the HHT.

The reason for this event may be caused by the difference of the non-parametric

(HHT) and parametric estimation method (HA) which is a least squares fitting

estimation procedure to a trigonometric polynomial.

The HHEMD analysis of sub-tidal sea level fluctuations at Turkish coasts showed that

these fluctuations are stronger during the summer regime (May- August) for data from

Antalya and Bozyazi tide gauge stations. These stations have the lowest portion of the

sub-tidal fluctuations in winter (December-February). Also higher sub-tidal

74

fluctuations usually occur between March and August for Amasra and Trabzon. The

low portion of the annual variations of the both Turkish seas may be attributed to the

some meteorological factors such as high atmospheric pressure and temperature. The

high portion of the sub-tidal fluctuations may be connected with the low atmospheric

pressure and wind stress.

Finally, the maximum and minimum velocities of the tidal constituents have been

calculated in one direction because the tidal currents have been assumed as a plane

wave. By inspecting the maximum and minimum velocities calculated from tide gauge

data one can conclude that the magnitudes of the velocities are not sufficient to

generate tidal electricity power at the Turkish coasts.

8.2 Future work

The following items included future works on EMD and HHT method.

To use Navier-Stokes equations for the calculation of tidal velocities after

HHT

To refine HHT mathematically.

To convert MATLAB codes to the Python so that the tool box can be

integrated to a GIS software package.

75

REFERENCES

Baker, C. (1991). Tidal power. Energy Policy, 19(8), 792-797.

Brockwell, P. J. & Davis, R. A. 1991 Time series: theory and methods. New York:

Springer.

Defne, Z., Haas, K. A., & Fritz, H. M. (2011). GIS based multi-criteria assessment of

tidal stream power potential: A case study for Georgia, USA.Renewable and

Sustainable Energy Reviews, 15(5), 2310-2321.

Demir, Ç., and Gürdal, M. A., 2000: Sea level monitoring activities in Turkey. IOC

Workshop Report No. 176, Annex IVa.

Denbo, D. W., and Allen, J. S. (1987), “Atmospheric Forcing of Shelf Currents and

Coastal Sea Level offthe West Coast of North America: May–July, 1981 and 1982,”

Journal of Geophysical Research, 92, 1,757–1,782.

Desplanque, C., & Mossman, D. J. (2001). Bay of Fundy tides. Geoscience

Canada, 28(1).

Doodson, A. T. (1921). The harmonic development of the tide-generating

potential. Proceedings of the Royal Society of London. Series A, 100(704), 305-329.

Ezer, T., & Corlett, W. B. (2012, October). Analysis of relative sea level variations

and trends in the Chesapeake Bay: Is there evidence for acceleration in sea level rise?.

In Oceans, 2012 (pp. 1-5). IEEE.

Forrester, W. D. (1983). Canadian tidal manual. Department of Fisheries and Oceans.

Gorlov, A. M. (2001). Tidal energy. Boston.

Haas, K. A. (2011). Assessment of Energy Production Potential from Tidal Streams in

the United States (No. DOE/GO/1817-8). Georgia Tech Research Corporation.

Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., ... & Liu, H.

H. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear

and non-stationary time series analysis. Proceedings of the Royal Society of London.

Series A: Mathematical, Physical and Engineering Sciences, 454(1971), 903-995.

Jay, D. A., & Flinchem, E. P. (1999). A comparison of methods for analysis of tidal

records containing multi-scale non-tidal background energy. Continental Shelf

Research, 19(13), 1695-1732.

76

Magrin-Chagnolleau, I., & Baraniuk, R. G. (1999, October). Empirical mode

decomposition based time-frequency attributes. In 1999 SEG Annual Meeting.

Oppenheim, A. V. & Schafer, R. W. 1989 Digital signal processing. Englewood Cli_s,

NJ: Prentice-Hall.

Pattullo, J., Munk, W., Revelle, R., and Strong, E. (1955), “The Seasonal Oscillation

in Sea Level,” Journal of Marine Research, 14, 88–155.

Pawlowicz, R., Beardsley, B., & Lentz, S. (2002). Classical tidal harmonic analysis

including error estimates in MATLAB using T_TIDE. Computers &

Geosciences, 28(8), 929-937

Percival, D. B., & Mofjeld, H. O. (1997). Analysis of subtidal coastal sea level

fluctuations using wavelets. Journal of the American Statistical Association,92(439),

868-880.

Pidwirny, M. (2006). Earth-Sun Geometry. Fundamentals of Physical Geography, 2nd

Edition. Formatado: Fonte:(Padrão).

Pugh, D. T. (1996). Tides, surges and mean sea-level (reprinted with corrections).

John Wiley & Sons Ltd

Ro, Y. J. (2007). Tidal and sub-tidal current characteristics in the Kangjin Bay, South

Sea, Korea. Ocean Science Journal, 42(1), 19-30.

Roden, G. I. (1960), “On the Nonseasonal Variations in Sea Level along the West

Coast of North America,”Journal of Geophysical Research, 65, 2,800–2,826.

Ross, D.A. 1995. Introduction to Oceanography. New York, NY: HarperCollins. pp.

236- 242.

Rudi, J., Pabel, R., Jager, G., Koch, R., Kunoth, A., & Bogena, H. (2010). Multiscale

analysis of hydrologic time series data using the Hilbert–Huang

transform. UNKNOWN, 9(4), 925-942.

Sezen, E., & Baybura, T. (2010). The investigation of long-term sea level variations at

Antalya-I (1935-1977) and Antalya-II (1985-2005) Tide Gauge Stations

(Turkey). Scientific Research and Essays, 5(10), 1100-1110.

Shiono, M., Suzuki, K., & Kiho, S. (2002, May). Output characteristics of Darrieus

water turbine with helical blades for tidal current generations. InProceedings of the

Twelfth International Offshore and Polar Engineering Conference, Kitakyushu,

Japan (pp. 859- 864).

77

Simon, B., Lemaitre, A., & Souchay, J. (2013). Oceanic Tides. In Tides in Astronomy

and Astrophysics (pp. 83-114). Springer Berlin Heidelberg.

Simpson, J. J. 1991 Oceanographic and atmospheric applications of spatial statistics

and digital image analysis. In Spatial statistics and digital image analysis. Washington,

DC: National Academy.

Spillane, M. C., Enfield, D. B., and Allen, J. S. (1987), “Intraseasonal Oscillations in

Sea Level along the West Coast of the Americas,” Journal of Physical Oceanography,

17, 313–325.

Strub, P. T., and James, C. (1988), “Atmospheric Conditions during the Spring and

Fall Transitions in the Coastal Ocean off the Western United States,” Journal of

Geophysical Research, 93, 15,561–15,584.

Strub, P. T., Allen, J. S., Huyer, A., Smith, R .L., and Beardsley, R. C. (1987b),

“Seasonal Cycles of Currents, Temperatures, Winds, and Sea Level Over the

Northeast Pacific Continental Shelf: 35 N to 48 N,” Journal of Geophysical Research,

92, 1,507–1,526.

Sumich, J.L. 1996. An Introduction to the Biology of Marine Life, sixth edition.

Dubuque, IA: Wm. C. Brown. pp. 30-35.

Temam, R. (1995). Navier-Stokes equations and nonlinear functional analysis (Vol.

66). Siam

Thurman, H.V. 1994. Introductory Oceanography, seventh edition. New York, NY:

Macmillan. pp. 252-276

Titchmarsh, E. C. 1948 Introduction to the theory of Fourier integrals. Oxford

University Press.

Torge, W. (2001). Geodesy. Walter de Gruyter.

Şeşeoğulları, B., Eriş, E., & Kahya, E. (2007). Trend analysis of sea levels along

Turkish coasts. Hydrology Days, 19-21.

Zeiler, A., Faltermeier, R., Keck, I. R., Tomé, A. M., Puntonet, C. G., & Lang, E. W.

(2010, July). Empirical mode decomposition-an introduction. In Neural Networks

(IJCNN), The 2010 International Joint Conference on (pp. 1-8). IEEE.