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
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