The Radar

Section for Ocean Swell: Derivation and Inversion

by

@Chengxi Shen , B. Sc.

A thesis submitted to the School of Graduate Studies in partial fu lfillment of t he requirements for the d egree of

St. John's

Master o f Engineering

Faculty o f Engineering and Applied Science

Memorial University of Newfoundland March, 20 13

Newfoundland

Over the last four decades, the application of hig h frequency (HF ) rad ars to t he monitoring of ocean surface h as emerged as a vibrant field of study in the remote sensing an d oceanographic communities. These HF radars , operating in the surface wave mode, can provide accurate and real-time information regarding surface currents and waves, which greatly aids in the planning and execution of oceanographi c projects, search a nd r escue events, a nd commercial fisheries. However, most present HF radar techniques are restricted to the measurement of sea state parameters associated with wind waves only, while the underlying swell component, which may severely distort the inversion results a nd pose certain hazards on offshore activities, is usually neglected.

In this thesis, the first- and second-order HF radar cross sections are derived for the random , time-varying, swell-contaminated seas. The analysis originates from t he electric field equations for the scattering of HF radiation from the ocean surface, with the source being a vertical d ipole with a pulsed sinusoidal excitation . T he various field components are then a utocorrelated and Fourier transformed to give t he power spec- tral density. Finally, the expressions of the cross sections can be obtained using th e radar ra nge equation. By introducing appropriate directional wave spectra to specify the ocean s urf ace as a mixture of wind waves a nd swell, t he derived cross section models are calculated and depicted. Essential characteristics and major differences from conventiona l cross sections for purely wind-driven seas are discussed.

11

con tinuous wave source (FMCW) , because such a waveform is often employed in practical HF rad ar systems. The ma thema tical expressions for t he FMCW cross sections of s well-conta minated seas are first presented, and their prop erties are then a ddressed . Only trivi al differences can b e observed when comparing the cross section model for the pulsed and FMCW wave forms, which indicates that an inversion routine

may be developed a nd applied simultaneously for both cases.

Finally, an inversion a lgorithm is proposed for the ex traction of swell parameters from HF radar Doppler spectra. These include the swell domina nt p eriod, propagating direction, frequen cy spreading, a nd significant wave height. The method involves t he identification of swell p eaks, the processing of swell p eak positions , the measurement of swell peak ha lf-power widths, and a maximum likelihood calculation. The procedure is then tested against simulated data, and promising inversion res ults are obtained . It is concluded that fine Doppler resolu tion is required to ensure the retrieval accuracy, and dual-radar systems are highly recommended to eliminate the directional ambiguity in swell direction.

Overall , the analysis presented h ere may provide a solid foundation for fut ure research on other types of ocean surfaces. Additionally, t he propert ies of the scattering as ma nif es ted in this thesis should be relevant to the u nderstanding of th e complicated hydrodyna mic interaction between swell and wind waves.

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I would like to thank my supervisors, Dr. Weimin Huang and Dr. Eric Gill , for their exceptiona l guida nce and enthusiastic support throughout t he research period , b oth academically a nd fina ncia lly. You gra nted me the ch an ce to pursue my years- long dream of studying abroad , you led m e into this fan tastic world of ocean remot e sensing , and you provided crit ical comments a nd constru ctive suggestions when I was lost . I a m indebted to you , always.

I would a lso like to express my sincere respect to Dr. J ohn Walsh , whom I do not have the fortune t o meet in p erson. Your brillia nce has always ama zed and inspired me whenever I read your work regarding t his topic. I hope that one day I can grow int o a great scholar just like you , independent a nd passiona te. Rest in peace.

I am also grateful to a ll my friends a nd colleagues in St.Joh n's . Forgive me that I cannot fit all your na mes into this short acknowledgme nt. St .John 's is n o pa ra dise, especially when t alking a bou t its ha rsh wea th er a nd lack of entertainmen t. It 's you t hat ma ke this place much wa rmer , much brighter, an d more like home. Be well.

Fina lly, this work could not h ave b een completed without t he patience a nd encour- agement from my wonderful family - my pa rents, Ha nde Shen and Xia nfa ng Zha ng, and my lovely wif e, Wenli Hu. In p articula r, dear Wenli has always b een tolera nt and supportive. Thanks for your unyielding d evotion and faith in me. T hank you .

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Abstract

Acknowle dgme nts

Ta ble of Conte nts

List of Tables

List of Figures

Table of Symbols

1 Introduction

1.1 Research Rationale 1.2 Literature Review .

1.2.1 Fundamental Concepts of HF Radars

1.2.2 R esearch on the Development of Cross Sections 1.2.3 Present Inver sion Algorithms

1.3 Scope of the Thesis . . . . .

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lX

xiv

XV

1 1 5 5 8 12 14

2 The HF Rada r Cross Sections of Swe ll-contaminated Seas for a

Pulsed Source 17

v

Ocean Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 C ross Sections with No Cou pling Effects I nvolved b etween Swell a nd

Wind Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 S pecification of the Ocean Surface and t he Corresponding Backscat-

tered E-field . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 The D oppler Power Spectr al Density of t he Received Electric

Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.2.1 The First-order Doppler P ower Spectra l Density . 29 2. 2.2 .2 The Second-order Doppler P ower Spectral Density 32 2.2.3 Deriva tion of the Cross Sections for the "No Coupling" Case 34 2.3 C ross Sections Involving Coupling Effects . . . . . . . 37 2.4 Calculat ion and In terpretation of the Cross Sections . 42 2.4.1 Choice of Spectra l Model for Swell Conta min ated Seas 42

2.4. 1.1 The Swell Comp onent . . 42

2.4.2 2.4.3

2.4.1. 2 T he W ind-sea Component T he First-order Cross Section An alysis The Second-order Cross Section Analysis

2.4.3 .1 The Eff ects of the R a da r Operating Frequency . 2.4.3.2 The Effects of the Swell Significant Wave Height 2.4.3.3 T he Effects of the Swell P ro pagating Direction 2.4.3.4

2.4.3.5 2.4.3.6

The Effects of the Swell Dominant P eriod T he Effects of the Local W ind Sp eed . . The Effects of the Local Wind Direction 2.5 Gener al Chapter Summary .

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44 44 48

50

52

53

54

55

56

58

FMCW Waveform

3 .1 The Derivation of t h e RCS for an FMCW Waveform 3.2 The Interpretation of the RCS for an FMCW waveform

3. 2.1 The First-order Cross Section Analysis 3.2.2 The Second-order Cross Section Analysis 3.3 General Chapter Summary . . . . . . . . . . . .

60

61 63 63 64 67

4 Extraction of Swell Parameters from Noisy HF Radar Signals 68

4.1 Simulation of the Doppler Spectrum in a Noisy Environment - Pulse R adar Op eration . . . . . . . . . . . . . . . . . . 69 4.2 Pre-processing of t he Received Doppler Spectrum 73 4.2. 1 R emoval of the frequency shift induced by surface current 74 4.2.2 Incoherent Averaging of Doppler Spectra . . 76 4.3 Extraction of Swell P eriod and Dominant Direction 78 4.3. 1 Defining Frequency Windows for Swell P eaks . 80 4.3.2 Swell Peak Ident ification .

4.3.3 Processing of Swell Peaks 4 .3.4 Applying Inversion Formulas . 4.4 Extraction of the Sha p e Factor . . . 4 .5 Extraction of the Swell Wave Height 4 .6 Test Results . . . . . . . . .

4. 7 General Chapter Summary .

5 Conclusions

5 .1 General Summary and Sig nificant Results 5 .2 Suggestions for Fu ture Work . . . . . . . .

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82 86

87 88 92

94

100

100

103

A Calculation of the Ensemble Averages 113

A.1 The Ensemble Average Involving Two Random Surface Va riables 113 A.2 The Ensemble Average Involving Four Ra ndom Surface Va riables 118 A.3 The Cross-correlations of the First- and Second-order F ield Components123

B Miscellaneous of the Swell Inversion Algorithm B

The Theoretical Pos itions of Swell Peaks . . . . .

126

126 B.2 The Derivation of t he Inversion Formula for Swell Period and Dominant

Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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4. 1 Frequency windows conta ining possible swell peaks 82 4.2 Inversion results for th e ref erence Doppler s pectrum 95 4.3 Inversion results for cha nging the swell significant height , s Hs 96 4.4 Inversion results for cha nging the swell p eriod , T

96 4.5 Inversion results for cha nging the swell direction , B s 96 4.6 Inversion results for cha nging the swell shaping factor, N 97 4.7 Inversion results for cha nging the local wind direction , Bw 97 4.8 Inversion results for cha nging the local wind speed , Uw 98

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1.1 A picture of swell dominated sea [ 1].

1.2 Sample backscatt er Doppler spectrum fr om the ocean surface showing prominent Bragg peaks due to waves advancing toward and recedi ng from the receiver. The t iny Doppler deviation from th e theoretical Bragg peaks, D..j , is due to ocean surface currents. T he second-order continuum is also identifia ble in this example. The d at a was collected

4

at Placentia Bay, NL, Canada , J a n . 2013. . . . . . . . . . . . . . . . 6 1.3 Bragg scatt ering of the incident EM wave (thin) by ocean waves (thick)

with wavelength

/2 (top), a nd the cance llation of EM energy for arbitra ry ocean wavelengths (bot tom). Figure taken from [ 2]. . . . . . 7 1.4 Electromagnetic double-scattering. The incident rad ar wave vector is

indicated by k

and the backscattered wave vector is denoted as - k

9 1.5 Hydrody na mic scatt ering. Two first order waves (black) interact t hrough

non-linear effects to produce a second order wave (red dashed lines).

T he new formed wave has a wave vector KB

K

+ K

and Ka ^must

satisfy the Bragg scatter co ndit ion. . . . . . . . . .

2. 1 The general geometry of the scattering ocean patch

X

9

35

s uremen t, showing both swell and locally wind-genera ted waves . (Fig- ure ta ken from [ 3]) . . . . . . . . . . . . . . . . . . . 45 2 .3 Simula ted wave spectrum combining a Wallop spectru m a nd a P M

spectrum. 45

2.4 The genera l geometry of pa t ch scatters on swell-contamina ted seas . 45 2.5 An example of the first-order rada r cross sections when the op erating

fr equency fo = 15 MHz. The ra dar look direction , e R, is 90 deg rees, the loca l wind direction is 180 d egrees, and the swell propagation direction is 60 d egrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.6 First-order cross sections with fo = 5 MHz. The other input pa ra meters

a re identica l with those in Fig. 2.5. . . . . . . . . . . . . . . . . 46 2 . 7 Firs t-order cross sections wit h differen t

The significant wave h eight

of the swell component is fixed to 1.5 metres. . . . . . . . . . . . . . . 4 7 2.8 Second-order cross sections with fo = 15 MHz. Four second-order swell

p eaks are clearly visible in the resul t under the coupling assumption . 49 2.9 Doppler sp ectrum measured along a south-westerly beam fr om So uth

Wales, t a ken from [ 4]. a±, approaching and receding firs t-order re- turns, indicating a n onshore wind. b, second-order continuum showing long waves predominat ely towards t he radar. c, very long wavelength swell peaks again propagating towa rds the ra dar. . . . . . . . . . . . 49 2.10 Second-order cross sections with fo = 15 MH z. The dashed curve is

obta ined by considering th e hydrodyn amic coupling effects only, while the solid curve accounts for the combined electromagnetic and hydro- dynamic effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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fo. The wind direction is 180°, the r ada r look direction is goo, the wind speed is 10 mjs, the swell wave height is 1. 5 m, the swell period is 14 seconds , and the swell direction is 60°. . . . . . . . . . . . . . . . . . 5 1 2.12 The effect on the cross sectio ns of cha nging the swell significant wave

height , s H

The remaining param eters are identical to those in the middle figure of Fig . 2.11. . . . . . . . . . . . . . . . . . . . . . . . . 53 2.13 The effect on the cross sections of cha nging the swell propagating di-

rection, B s· The intersection a ngle , ¢; in the figur e can be given by

¢; = IBR- B sl · The remaining parameters are identical to those in F ig.

2. 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2 .14 The effect on the cross sections of changing the swell dominant period ,

T

The remaining par a meters are identical to those in Fig. 2. 11 . . . . 55 2. 15 The effect on the cross sections of cha nging t he local wind s peed , U w.

The remaining parameters a re identical to those in Fig. 2.11. In the case where U w

15 m/ s, th e swell peaks are no longer visible.

2. 16 The effect on the cross sections of ch anging the local wind direction, B w·

The remaining pa rameters are iden tical to those in Fig. 2 .11. In the case where e w = goo (a ligns with the r ada r look direction) , the swell p eaks in the positive Doppler region totally vanish and may degrade the la ter inversion process. . . . . . . .

3. 1 A compa rison between the fi rst-order cross sections for pulsed a nd FMCW waveforms . The operating frequency is fo

15 MHz, the sweep ba ndwidth is B = 100 kHz, t he wind sp eed is 10 m/s, an d the wind direction is goo to the ra da r look direction. The patch width is

56

57

6p

= lOOOm , and the integr al limit 6 ,. =

64

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The remaining parameters are identical to t hose in Fig. 3.1. . . . . . 65 3.3 A full comparison between the cross sections of the FMGW a nd pulsed

waveforms . The operating frequency is fo

15 MHz. The remaining pa ra meters are identical to t hose in Fig. 3.1. . . . . .

4.1 An example of a t ypical 1 minute simulated time series received from swell-contaminated seas. Th e ra da r operating fr equen cy is 15 MHz,

66

and the SNR for this signal is 30 dB. . . . . . . . . . . . . . . . . . . 72 4.2 T he norma lized Dopple r spectrum of th e times series shown in Fig. 4. 1. 73 4.3 Evaluation of the Doppler shift b..f induced by surface current . . . . 75 4.4 An illustration of t he temporal averaging. Each fra me h as 512 points,

a nd the overla pping is 75 %. . . . . . . . . . . . . . . . . . . 76 4.5 An example of the spatia l averaging. The n umber of involved ra nge

cells are 1, 2, and 4, respectively.

4.6 A Doppler spectrum obtained from a two-minu te times series.

o av- erages are p erformed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.7 A Doppler sp ectrum averaged over ten minutes, wit h each frame being

two-minute long.

4.8 A Doppler spectrum both temporally and spatially averaged. A total

of four range cells a re involved .

4.9 Geometry of t he double scatter involving a swell vector Ks and a wind wave vector K w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 4.10 Receding Bragg p eak a t -0 .51 Hz, fla nked by two clean swell peaks

(asterisk). The dashed lines indicate the frequen cy window boundaries as presented in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . 83

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nulls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4. 12 The four points collected from Fig. 4.1 1. P eak A is clearly more t han

3 dB higher than the nearest null, Null B , and will be retained. On the contr ary, P eak C is within the 3 dB range of both Null B and Nu ll D a nd will be filtered out. . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.13 When a peak is 3 dB higher than either one of the nearest nulls, it is

a potential swell peak. In this case , both Peak A and Pea k

will be retained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4. 14 Meas uring distances of swell peaks away from th e respective Bragg

p eak. In this figure , swell peak B and C will be selected as a pair , while peak A will be elimina ted. . . . . . . . . . . . . . . . . . . . . . 85 4. 15 Application of the weighted mean algorithm t o swell p eaks when the

severe signa l degra dation occurs. . . . . . . . . . . . . 87 4.16 The inherent a mbiguity in swell direction calcu la tion 88 4.17 A typical swell spectrum with parameters notated in (4.16) 89 4 .18 Extraction

from the right-hand side peak of the receding Bragg

p eak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4. 19 Extraction of !Dhp when only one intersection p oint is found 91 4.20 The "reference" Doppler sp ectrum. Only t he frequency port ion t hat

contains Bragg peaks a nd swell p eaks is sh own her e . .

B.1 The contour of

against the wave height sp ectrum.

B.2 The contour of

against the wave height sp ectrum .

XIV

95

127

128

The page numbers h ere indicate the place of first significa nt reference. Although not all symbols a re explicit ly referen ced b elow, their d efini tions are obvious from th e con- text.

f s ,WB:

>-o:

g:

tJ.f :

fo :

c:

~ :

p:

ko : '170 :

Bragg frequency in Hz and radians (p.6).

Incident wavelength of the ra da r signals (p.6 ).

Gravitationa l acceleration (p.6) .

Doppler fr equency shift induced by surface curr ents (p. 7) . Ra dar frequency (p.7) .

Light sp eed (p. 7) . Bragg wave vector (p.8).

The scattered electric field norma l to t he surface and in the limit as the s urface is approached from a bove (p.l 9).

Elevation profile of the ocean s urf ace (p .l9 ).

Distance of a general point (x, y) on the surface (p .l9) . Wavenumber of the transmit t ed radar signal (p .l 9).

Intrinsic imped ance of free space (p.20).

Scattering patch widt h (p.20).

XV

s :

symmetric form (p.20).

Fourier coefficient of the ocean surface (p . 20) . Hydrodyna mic coupling coefficient (p.21).

Normalized surface impedance (p.25).

Autocorrelation of the electric fields (p .28).

Effective apert ure of the receiving a ntenna (p.28).

First-order directiona l ocean wave spectru m (p .30).

Doppl er fr equency in Hz and radians (p .31).

Doppl er power spectral den sity (p.31 ).

Combined coupling coefficient (p.32).

Radar cross sections (p.36).

Coupling coefficient between swell and wind waves (p.38).

Normalized directional distribution function (p.42) . Ra dar look direction (p.42) .

Directiona l s preading factor for a typical cardioid model (p .42).

U : Local wind speed (p.44).

s Hs : Sig nificant wave height of the swell component (p .52).

e s : Propagation direction of t he swell com ponen t (p .53).

cPi : Intersection a ngle between the ra dar look direction and the swell direction (p. 53).

T

Domina nt period of th e swell component (p.54).

B w : Local wind direction (p.56) .

XVI

a:

Sweep r at e of the FMCW signal (p.61 ).

B: Bandwidth of the FMCW signal (p.61).

T,. : Sweep interval of the FMCW signal (p.61) .

f s,

P eak frequency of the swell spectrum in Hz and R a dians (p. 68).

N : Sha pe factor of the swell spectrum (p.68 ).

Wsp :

Doppler frequen cy of the swell peaks (p.80).

K s : Dominant wavenumber of the swell component (p .80) .

whp:

Half-power width of t he swell peak (p .88) .

whp :

Ha lf-power width of the swell spectrum (p .88).

wDhp :

Ha lf-power width of the swell peak in Doppler spectra (p.88).

N(K,w): Random factor in surface Fourier coefficients (p.113).

£ (!{)) : Random phase variables uniformly dist ributed on [0, 21r] for each j( ( p.l1 5).

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Introduction

1.1 Research Rationale

The oceans have a lways been a key pa rt in ma ny cultures and a great influ ence on every individua l on planet Earth. Environmenta lly, oceans are t he sources for short-term weather activities and long-term climate change. Commercially, oceans teem with ri ch resources and provide indispensable routes for global shipment. In this sense, there has arisen a n unprecedented demand for effective ocean observation stra tegies, t echnologies and applications, especially in those coastal-based and marine de pendent countries such as the U.S ., Canada and China. Thanks to the increasing understa nding in hydrodynamics and ocean engineering enabled by modern technolo- gies, oceanographers a nd ocean engineers are able to estima te various parameters associated with directional wave h eight spectra and surface currents. Undoubtedly, such information is highly valuable for search and rescue operations , ocean-related nationa l defense, weather forecasts, the design of offshore structures , coast al fishery ma nagement a nd the control a nd tracking of surface-borne pollutants, to n ame just a few.

1

Conventionally, the measurement of the ocean surface has b een achieved by a num- ber of instruments including SLDMB (self-l ocating data marker buoys), Drifter Buoys, ADCP (acoustic Doppler C urrent Pro filers) , Waverider and other buoy technologies.

Although these technologies are basically very mature and are widely applied, the inherent dynamic nature of t he ocean and surroundi ng vessel traffic still render a cha lleng ing environment for these in situ sensors. For instance, moored and drifting buoys can be fr equently caught in fishing nets or damaged by shipping and, once broken , they are usually very expensive and difficult to r epair and mainta in. Besides, the d eployment, ret rieval and data transmission in rough seas, where the information may be most critical, can be both costly and impractical [5]. Moreover , most bu oys provide only limited spatial or temporal coverage that is far from satisfactory.

Against this backdrop , a variety of ocean remote sensing techniques have emerged m recent decades. Microwave radars , for example, are quite capable in providing ocean surface parameters with fin e resolutions. However, the line-of-sight operation of microwave radars severely reduces the observa tion range compared with in situ devices, a nd the measurement is likely to be a ffected by the weather condition (e.g., heavy ra in) so that t he accuracy cannot be guaranteed a t all times. More importantly, microwaves only interact with very small scale ocean waves (from several millimetres to metres in wavelength) , while most of th e wave energy is contained with in th e much longer gravity waves that have wavelengths of tens to hu ndreds of metres.

has proven to be a compli cated a nd indirect process to determine the complete wave energy spectrum based solely on the measurement of short waves , and this poses limita tions on the use of microwave ra dars in the a rea of ocean surface monitoring.

Another widely accepted remote sensing tool is the high frequency (HF) radar

opera ting between 3 and 30 MHz. Unlike microwave radars, radio signals in this

band, when guided by a good conducting medium like sea water, are able to travel

long distances along th e earth 's curvature. In other words, HF rada rs are no t restricted to line-of-sight operations and can , with moderate tran smitting power , actually see beyond the horizon. In addit ion, since the wavelengths associated with HF signals are o f the same order as those ocean waves that carry most of the spectral energy, the t ra nsmitted signals will interact very strongly with these waves. This reduces th e co mplexity in interpreting the backscattered signals for wave informa tion . F inally, HF ra dars are ra rely affected by weather, clouds or cha nging ocean condi tions , a nd are capable of providing n ear real-time m easurements with high temporal and reasona ble spatia l resolution.

As will be indi cated in the next section, several mature a lgorith ms already exist for the extraction of ocean wave spectra from HF return signa ls. However, most of t hese inversion rout ines a re for wind waves generat ed by the local wind field, whereas in fac t, the wave records often reveal a dditional spectral cont en t associated with waves of longer period . These waves are commonly called s well (see Fig. 1. 1). Basically, swell consists of waves tha t are generated by earlier storms at a distant location. After leaving the a rea of the active wind field , these waves propagate fr eely across the ocean s urface , dissipating energy dissipa tion and spreading laterally, and result in a decrease in the wave heights. T his effect is greater for the shorter period waves so that t he co mpon ents of swell a re generally long p eriod a nd n arrow-ba nd waves.

The coexisten ce of wind sea and swell , which may impose grave threats to sea-

keeping safety, offshore structure designs, a nd s urf forecasting [ 6] , has b een exten-

sively reported by various researchers globa lly. Titov [7] presented th e distribution

of the fr equency of wind waves and swell for the North Atla ntic d uring winter and

s ummer, showing that regions with heavy swell extend over open ocean as well as

coastal areas. T hompson [ 8] a nalysed wave records from nine locations a long United

States Atla ntic, P acific , Gulf a nd Grea t Lakes coasts a nd observed t hat multi-p eaked

Figure

A picture of swell dominated sea [1 ].

s pectra are common at a ll locations. Cummings et al. [9], using hindcast data from the North Atla ntic, determined tha t 25% of the sp ect ra were double-peaked , while Aranuvacha pun [10] reported 24% from the a na lysis of measured d ata in the same area. From these results, it is clear that combined wind wave an d swell systems can occur a t rela tively high fr equency both in the open ocean a nd in coastal s ites.

Predictably, in the context of HF rada r spectra, the und erlying swell comp onent will be reflected an d might , if not properly treated , resul t in inaccurate inversion for the local sea state. Meanwhile , since the co existen ce of swell and wind waves has already been proved to be t he " precursor" of some a bnorma l sea conditions [11] , knowl- edge of the incoming swell may help in providing early warnings so that operators of offshore units a re able to anticipate contingencies and int roduce accident prevention initiat ives. Thus , a new mo del accounting for t he swell effect must be established as a complement to the present HF rada r remote sensing techniqu es.

This thesis aims prima rily at developing a general HF cross section model for

swell-contaminated seas and an a utomated routine for swell parameter extraction .

Some practical eng ineering a pplications, including t he use of frequen cy mod ulated

(FM) waveforms , will be a lso considered here. Hopefully, t he model and algorithm formulated h ere will provide a foundation for future investigations that focus on other wave components, a nd ultimately benefit t he coastal-based industry at large .

1.2 Literature Review

1.2.1 Fundame ntal Conce pts of HF R a dars

An HF surface wave radar system (including antennas, transmitter, receiver, control and d ata processing units) is typically installed at a site very near the coas t because the signal can be severely attenuated by land. Moreover, for optimal surface wave propagation , the transmitted electromagne tic (EM ) signal is generally vertically po- larized and narrow band. As it propagates along the electrically conductive ocean water su rface , the EM wave is scattered off a ny object it encounters (inclu ding ocean waves) , and some of the reflected energy, which contains information regarding the target, is collected by t he receiving antenna. By analysing t his return signal through means of Doppler spectra, a variety of parameters can b e esti mated.

When HF ra da r technology initially appeared during t he second world war, it was

used to detect approaching enemy a ircrafts a nd vessels. Before long it was noticed

th at scattering from a n unknown source often masked t he targets being tracked. For

a relatively long t ime , there was no satisfactory explanation for this contamination

which was later referred to as "clutter" .

was not until 1955 that Crombie [12] first

suggested and confirmed the source of this clutter. Crombie correctly pointed out

th at t he two discrete spectral p eaks sitting at an equa l distance ab ove and b elow the

ra dar carrier frequency in the Dopp ler sp ectrum (see Fig. 1.2) are resulted from th e

Bragg scattering [13] and t hese peaks are thus named Bragg peaks. In the case of

monostatic radar configuration (co-located transmitting and receiving antennas) and

150

140

Ill

I

Bragg peak from ~f

receding waves \ ;

i I

; I 'I

::s.

I I I

Bragg peak from advancing waves

: -r

so~i~~~~~~~~~~a~~~~~~~~~~~~~~~~

-3 -2 -1 0 1 2 3

Normalized Doppler frequency

F igure 1.2: Sa mple backscatter Doppler sp ectrum fr om the ocean surface showing prominent Bragg peaks d ue to waves adva ncing toward and receding from the receiver.

The tiny Doppler deviation from the theoretical Bragg p eaks, !:::..f , is due to ocean surface currents. The second-order continuum is also id entifia ble in t his exa mple . T he data was collected a t P lacentia Bay, NL, Canada, J an . 2013.

grazing incidence, the Bragg waves refer t o those ocean waves having wavelengths one half to those of the incident EM waves, so that phase coherent reinforcement occurs (see Fig. 1.3). From Crombie [ 12], the Doppler shift caused by this first-order resona nt phenomenon is

f

= ~~ ⁼ * ^{= f!i =} · f£ ^{(1. 1)}

where f

is the theoretical Doppler shift (Hz), v is the ocean wave rad ial speed ( m/ s),

>. is t he ocean wave wavelength (m) , >.

is t he incident wavelength (m) and g is th e

gravity acceleration. Clearly, ( 1.1 ) directly expla ins the two dominant peaks in the Doppler sp ectrum.

Figure 1.3: Bragg scattering of the incident EM wave (thin) by ocean waves (thick) with wavelengt h >..

/ 2 (top), and the cancellation of EM energy for arbit rary ocean wavelengths (bottom) . Figure taken from [ 2].

Based on these findings, Crombie [12] realized that the small differences between the exp ected Bragg p eak positions and measured Doppler frequencies could be the co nsequen ce of surface current velocities. His conj ecture has b een extensively verified (see, for example, [14]). When t he frequency shift du e to surface currents is defined

as /l.f , the current velocity can be simply calcu lated from the relationship

!lfc

Vs = - - -

.f o (1.2)

where

is the radial mag nitude of the surface current component, c is t h e light speed ,

a nd .fo is the ra da r frequency. Since !lf is either positive or negative in a particular

Doppler spectrum , the inverted values for

can be positive or negative as well,

which represent ocean waves traveling towards or away from the radar look direction ,

respectively. This discovery a lone was a great asset t o t he field of oceanographic

remote sensing, and oceanographers and en gineers , encouraged by the pote ntial of

HF ra da rs, started to dedicate their time into this a rea .

At t his stage, however, th e an alysis is limited to short ocean waves b ecause th e Bragg waves a re mainly metres long in wavelengths, while an accurate descrip tion of the sea sta te relies on the mea surement of longer wave components. In 1972, Barrick [15] not iced th at th e continu um surr ound ing the first-order peak , which he referred to as the second-order r egion , is much higher tha n the rema ining noise fl oor. After close inspection , he concluded t h at the continuum is actually comprised of reflection s from ocean waves of all wavelengths a nd traveling directions. Therefore, by ex amining these second-order sea ech oes, the directional wave spect rum, wh ich fully describes the sea state, can be extracted.

According to [ 16 , 17], the second-order continuum is produced by two indep en- dent effects: a n electroma gnetic comp onen t arisin g from double-scatters from two distinct waves t ra ins, where t he geometry of th e waves causes coherent reflections (see Fig. 1.4), and a hydrodyn amic component corresp onding to a single scatter from second-order ocean waves resulting from th e n on-lin ear coupling effects between two first-order waves whose wa ve vectors must satisfy the relation R

+ R

R B, with

R B b eing the Bragg wave vect or (see Fig. 1. 5). Barrick [ 15] also d eveloped mat h- em atical mo dels to account for t hese second-order effects, a nd thereby establis hed a solid found a tion for future HF rad ar investigations.

1.2.2 R esearch on the Development of Cross Se ctions

While t he material present ed in t h e previous sectio n gives t he general ideas involved

in using HF rada rs to remotely prob e the ocean surface, further understanding of

the interaction between ra da r waves and ocean waves requires the development of a

Transmitting and receiving antennas

Figure

Electromagnetic double-scattering. The incident rada r wave vector is indicated by k

and the backsca tt ered wave vector is denoted as - k

Transmitting and receiving antennas

wave train

Figure 1.5: Hydrodynamic scattering. Two first order waves (black) interact th rough non-linear effects to produce a seco nd order wave (red dashed lines). The new formed wave h as a wa ve vector K

= i(

+ i(

and K

must satisfy th e Bragg scatter condition.

ra da r cross section model. The formal definition of the rad ar cross section is "the area which , when multiplied by the power flu x density of t he incident wave , would yield sufficient power that could produce by isotropic radia tion , the same radiation inten sity as tha t in a g iven direction from the scattering surface" [18].

Ba rrick was the first to derive a complete cross section model for the ocean surface

to second order. By employ ing the effective surface imped ance at grazing incidence

[19], he examined the exact propagation losses of EM waves due to surface rough-

ness in the

and

ba nds [20]. In his subsequent ana lysis , Ba rrick formul ated

a first-order HF scattering cross section by extending Rayleigh's p erturbation model [2 1] to include a slightly rough sea surface [22]. This model successfully verified Crom- bie's [12] experimental deductions. La ter in the same year , Ba rrick [ 15] modelled th e second-order backscatter from the ocean surface with a non-linear, two dimen sional Fredholm-type integral equation. As noted , the second-order backscatter actually contain one scatter from a second-order ocean wave and two successive scatters from two first-order ocean waves, which are indicated by the hydrodyn amic and electro- magnetic coupling coefficients, respectively. To illustrate the effects , two directional wave height spectra were used to originate the required wave components , and a Dirac delta fun ction was employed to constra in the manner through which the wa ve vectors are related, i.e. , K

+ K

Ks.

La rgely based on these initial efforts, Barrick published widely in refining and ex- tending the existing models. For example, in 1980 , Lipa and Barrick [23] noticed very na rrow sp ikes in the higher-order structure adjacent t o the first-order peaks.

These spikes are indica tive of ocean waves with very limited , high frequ ency sp ectral components, which match the features of long period swell. By assuming a cardioid distribution in direction and a Gaussian distribution in wave frequency for the swell spectrum, a general cross section model to account for such a mixed ocean s urface was proposed. Also , in 1986, Barrick and Lipa [24] produced a n ew hydrodynamic cou- pling coefficien t for shallow water a pplications and demonstrated its validity against measured data.

Although Barrick's models are t he most accepted and studied in t he a rea of HF

marine radar technology, it must be noted that a ll his techniques are based on th e

assumption of a plane wave as the incident wave field. Consequently, t he Bragg

scatter mech anism are accounted for by Dirac delta fun ctions in the cross section

equations , which suggests the Bragg peaks to be infinitesima l in width and infinite in

amplitude. However, this is not the case in real Doppler sp ectra . Such a discrepancy is conventionally att ributed to surface currents structures or external noise. While these factors do play a role in broa dening the Bragg peaks , they do not explain the underlying cause satisfactorily.

The first analysis which a ddresses the Bragg peak broa dening in a fun damental ma nner was conducted by Wa lsh and Donnelly [ 25]. They studied the problem of EM wave scatter at the boundary of two different media b ased on a generalized function approach . The solutions were shown to agree with classical methods , wit h the major difference being that the boundary conditions evolve directly from the formulation as a uxilia ry equat ions. Later, Walsh a nd his colleagues (Srivastava, Dawe, Howell, and Gill) further applied the generalized function a pproach to the problem of ocean surface scatter under the assumption of a pulsed rad ar waveform [ 26, 27] . As a result , t he Bragg scatter mechanism was shown to be better modeled as a finite squared si ne fun ction , rather t ha n a Dirac delta function. This new cross section model is more realistic than the traditional ones , as t he app earance of this squa red sine function accounts for a finit e width of the scattering patch on the ocean surface and a signa l of finite bandwidth. The corresponding cross section has a finite wid th for the Bragg peak just as in real practice.

By extending the techniques , Walsh and his colleagues successfully developed a

va riety of cross section models for different practical situations. During 2000-2001,

bistatic first- and second-order rad ar cross sections of the time-varying ocean surface

we re derived [ 27 , 28, 29] . These contain the earlier monosta tic result as a special

case with the bistatic angle set to zero . T he other significan t contribu tions include

t he HF radar cross section for a Frequency Modula ted Continuous Wave (FMCW)

waveform [ 30], the combined sea clutter and noise model [31], the study on Bragg

fluctuations due to t he ran domn ess of ocean surfaces [ 32] , the HF cross section model

incorporating antenn a ba rge motion [ 33] , and a new form of cross sections for swell- conta mina ted seas [34, 35, 36]. These works serve as a solid theoretical foundation for the primary content of this thesis.

1.2.3 Present Inversion Algorithms

W ith the first- and second-order backscatter cross section models est ablished [15], investigators started t o interpret the HF r ada r sea echoes u sing a va riety of meth- ods . Barrick [ 37, 38] first p resented general techniques to extract non-directional ocean wave da ta. An importa nt step in his inversion algorithm was to remove t he pa th gains a nd losses from the second-order region by normalizing to the firs t-order spectral power. As noted , th e t est results were proved to b e rela tively insensitive to direction . In 1978, Lipa [ 39, 40] showed t hat directiona l features of th e ocean spec- trum can be indeed d erived from th e second-orrler echo. She first red uced t he double int egral of the second-orde r mo del into a set of linear equations, and t hen ap plied the regula ri zation methods of Phillips [ 41] a nd Twomey [ 42] to solve the equations.

Ba rrick a nd Lipa [43] consolid ated this a lgori thm by applying a simila r procedure to a broa d beam system composed of a cross-loop a ntenna and a monopole, which they n amed the Coastal Ocean Dyna mics Application R ada r (CODAR). Reasonable results were obta ined from the exp eriments. Since t hen, t his linearisation scheme has been rigorously applied to CODAR systems.

An importa nt constraint of the techniques described a bove is that t he ana lys is of

th e Doppler sp ectrum is na rrowed down t o t h e frequency band s urrounding t he Bragg

peaks . This is due to the assumption of a one-on-one mapping relationship b etween a

certain ocean wave frequen cy and a radar Doppler frequency, which is on ly valid for

those Doppler fr equen cies close to the first-order peaks. Thus, t h e frequen cy range

for the extracted wave sp ectrum is severely limited , especially when low HF radar

fr equencies are used. In view of this, such inversion scheme was only applied t o high HF (i.e., 15-30 MHz) measurements. However, doing so could significant ly reduce th e range capabili ty of the instrument , or even smear the first- order Bragg returns and ma ke the calcula tion impossible.

Recognizing the need to extend the fr equency ra nge t hat can be used , Wyat t [44]

suggested an improved model-fi tting t echnique based on Lipa an d Ba rrick's work [ 45] . Basically, Wyatt created many simulated Dopple r s pectra for a variety of sea states , and real radar data are matched with the simulation using a least squa res manner.

This method gave a ccurate estimates as long as th e dominant wave direction i s not pe rpendicular to the incident wave direction. Also, tests using t his method revealed tha t a two radar system may provide higher accuracy than a single radar sta tion , a nd the use of two ra dars elimina tes the usua l left / right directional ambiguity associa ted wit h single rada r systems. However, a consequence is that a noticea ble over-prediction in amplitude of the wave heig ht is often observed .

The most r ecent inversion a lgorithms were prop osed by Gill a nd Walsh [ 46] a nd Howell and Wa lsh [ 47]. The basis of this technique, simila r to Ba rrick a nd Lipa's, is t o numerically approximat e the integra l equation representi ng t he second-order radar cross section as a equation in which the Fo urier coefficients of the ocean sp ectrum a re the unknowns. A novel singular value decomposition a pproach is t hen a pplied to invert th e kernel m atrix to y ield t he ocean Fourier coefficients. To valida te the a lgorithm , Howell implemented his technique on Ba rrick's classic model [1 5] and Walsh 's new one [26], a nd very good results were produced for both cases. P a rticularly, results based on Wa lsh's cross section model agreed even better with in situ da ta .

Up to this stage, it should b e noted that all t echniques intro duced sou ght to add ress

the extraction of wind wave para meters from Doppler sp ectra , while th e research on

long period swell was not so rigorously conducted . T he initial effort in t his area was

ma de by Lipa and Ba rrick [23] , who suggested that the positions a nd a mplitud es of th e higher-order swell peaks are closely related with the dominant swell p eriod , direction and wave height. Moreover , the swell inversion can b e viewed as independ en t fr om that for local wind wave pa ramet ers . Thus, by identifying the swell peaks in the Doppler s pectrum, various parameters regarding the incoming swell can b e extracted withou t applying the general inversion techniques to the whole Doppler spectrum.

Although Lipa and Barrick [48] conducted t hree separate narrow-bea m HF radar experiments on the Pacific Ocean in order to va lida te their swell inversion algorithms , it was later found by Bathga te [49] tha t th ese rout ines are not so rob ust since th e precise positions of swell peaks a re often difficult to resolve a nd are suscep tible to ex- t ernal noise. Bathgate then presen ted an a lternat ive approach based on the frequency modula tion effects imposed on Bragg waves by swell. He a lso cond ucted a case study a t Tweed Heads , Austra lia, and proved the method to be fast and effective. However, since the data set coll ected in [49] has little variation in both swell periods and wave height , the genera lity of this simplified method r emains untested.

1.3 Scop e of the The sis

In this thesis, a new monostatic HF rada r cross section model for swell-contaminated

seas is established based on the funda mental electric field equations appearing in [26] .

The non-linear hydrodynamic coupling effects between local wind waves and incom-

ing swell are particula rly investigated a nd are proven to be t he ma jor cause for th e

second-order swell peaks in the backscatter Doppler sp ectrum. A data interpretation

algorithm for the extraction of swell param eters is also develop ed and tested. As

noted , the primary content in this work is based on the theoretical ana lysis develop ed

by Walsh and his co lleagues over the past two decades.

Chapter 2 starts with electric field equations describing the scatter received from a general rough t ime-varying ocean surface. The current excitation is a pulsed sinusoid on a vertical dipole. Strictly, the first-ord er £-field contains two portions: 1) a scat ter fr om a single train of first-order ocean wave an d 2) a single scatter fr om a second-order wave train formed by two independent fir st-order waves. The latter portion is then considered in conjunction with the second-order £-field, which describes successive double scatters. To introduce th e concept of swell cont amina tion on local wind waves , a three-dimensional Fourier series, consisting of contributions from swell and wind waves, is used to represent the rough ocean surface. By assuming the surface to b e stationary and homogeneo us within each ra nge cell that is interrogated , one can easily obtain the auto- a nd cross-correlations of t he various £-field components, and the Doppler power sp ectra l density is then calculated through a Fourier transform.

The remainder of Chapter 2 is devoted to calculating th e cross sections for differen t sea states, a nd to examining the effects of changing the input parameters .

must be noted tha t for the purpose of demonstration , deep water is assumed for all cases, though the results can be easily extended to general depths.

In Chapter 3, the derived cross section model is extended to the FMCW waveform.

The results are then depicted a nd compared to those for the pulsed waveform. This is a crucia l step as t he field da ta to be examined in real world is collected by radars opera ting in FMCW mode.

In Cha pter 4 , the received radar time series with external white Gaussia n noise is

fir st simula ted based on t he ana lysis presented in Chapter 2, from wh ich the Doppler

spectrum is obtained as a p eriodogram. Next, following the techniques d escribed

by Bathgate [ 50] , all swell peaks in the sp ectrum are ident ified and processed with a

robust peak recognition routine, and the positions of those peaks are used to calculate

the period and d omina nt direction of the swell. The half-power width of each swell

peak is extracted in order to derive the frequ ency spreading of the unknown swell wave height sp ectrum. Finally, a maximum likelihood method is applied to the swell peak a mplitudes to determine the significant wave height and directional spreading of swell. The performance of the a lgorithm is tested on substantial simulated Doppler spectra.

Chapter 5 summa rizes the fundamental conclusions from the previous three cha p-

ters. Some constra ints related to this thesis , as well as a few obvious suggestions for

fu t ure research are a lso presented there .

Chapter 2

The HF Radar Cross Sections of Swell-contaminated Seas for a

Pulsed Source

The goal in this cha pter is to develop the monostatic HF ra da r cross sections of swell- contamina ted seas. The ini tia l s tep is to develop t he electric field equa tions received from a random time- varying o cean s urface, as has been done in [26]. Next, t he specific scena rio of swell conta mina tion will be characterized via a three-dimen sional Fourier series - two-dimensiona l in space and one in time. During this step , the hydrodynamic coupling effects b etween the incoming swell and loca l win d waves are h ighlighted via two different manners, which leads to two Fourier representations of the ocean s urface. Upon determinin g the mathema tical form of the ocean surf ace conta ining a mix ture of swell a nd wind waves , the associated equations of the scattered £ -field can be obtained , whose power sp ectral density (PSD) is then calculat ed through a n autocorrela tion a nd Fourier transform. The radar cross sections , which contain b oth first- and second-order portions , a re easily derived by examining the obtained P SDs

17

against th e monostatic rada r range equation .

Some of the key fea tures of the cross sections are discussed in Section 2.4. Effects of different local sea states a nd rad ar pa ra meters a re investigated . To do so, we simply change the input varia bles one a t a time (such as swell period , direction a nd significant wave height) a nd illustra te the responses in the Doppler spectrum. Overall, the simula tion results shown at the end of this cha pter clearly suggest the possibility of a fast and st able algorit hm for swell extraction , which will b e introduced in Chapter

2.1 The Electric Field Equations for Scatters From a Random Time-varying Ocean Surface

The funda mental ana lysis for the received electric fiel d components, scattered from th e

ocean surface due to a ra diation from a vertical dipole sour ce , can be f ou nd initially

in [ 26] a nd a lso a ppeared la ter in [27, 28, 29] . In these works, t he ocean surface is

defin ed to be random , rou gh , and time-varying, which means our study that involves

swell can be also included as a sp ecia l, less general case. Thus , it is p ossible for us

to estimate t he form of the E-field received from swell-contamina ted seas with ou t

having to re-d erive a completely new set of equations fr om first principles. Still ,

certa in assumptions regarding the mixed ocean surface should be invoked: 1) small

slopes, where the powers of the surface slope which are greater tha n unity (for a single

scatter) are neglected ; 2) small s urface heights, where t h e product of the ra da r wave

numb er and surface height is taken to be much less t ha n unity; 3) the ra ndom surface

can b e viewed as a zero-m ean Gaussia n process even after the swell comp onent is

incorporated. T hese assumptions , as well as t he specific simplifications they introduce

to th e mathematical a na lysis, a re discussed extensively in t he p reviously cited open

literature a nd a re not rep eated her e. Now , the scattered fi eld , En, normal to th e surface a nd in t he limit as t he surface is approach ed from above, m ay be cast as (e.g.

[27])

{

xy e-jkop} e-jkop

En-

\l( En) * F(p)--

CoF(p)--

2np 2n p (2. 1)

where C

= (I

6..lk

)j(jw

c

is a constant for a dipole of length D..l carrying a current

!

whose r adia n frequency is

and whose wavenumber is

in a space for which t h e permit ti vi ty is

denotes the distan ce of a general p oint (

y ) on t he s urface

m easured from the origin, a nd F (p) is the usual Sommerfeld attenuation function . The

operator

(2.1) indicates gr adients in t h e x-y pl ane, a nd x i indicates a two- dimensional spa tial convolu tion.

should be noted tha t En in (2. 1) actually contains electric field components of a ll orders, i .e.,

(2.2 )

where (En)o is the zero-order term representing the EM wave p ropagation over a smooth pla ne s urface, and t he r em a in ing t erms ar e scatt ers of higher orders indicated by the corresp onding subscripts. In t his t hesis, only t he first- an d second-order E- fields are ana lysed , while the higher orders ar e neglected due to t heir relatively sm all contribution to the received £ -field.

Following the same procedure as describ ed

earlier p apers (see [27, 28 , 29]) , it is straightforward to d educe the first- (i.e. for a single scatter ) and second-order (i.e.

for a double scatter ) backscatter electric field equa tions in th e time domain as

(En h(t) = -)7]o6..l6..plo k6 (~

;~} 12 e-j~ejko.0. p ~

~p ^{(J( - 2ko)]}

K,w

(2.3)

where

is th e int rinsic imp edance of fr ee sp ace, Sa(-) is t he usual sampling fu nction ,

is t he scattering "patch width" for pulsed signals , E r

is the electromagn etic coupling coefficient for pa tch scatters in a symmetric form [ 26], and K r ⁼ K

⁺ K 2

refers to a wave vector that lies along the radar look direction . T he key term in (2.3) a nd (2.4),

re presents the total Fourier coefficient of the general sea surface and will be sp ecified in t he next section . It should be p oint ed ou t that there is actua lly a second term in the a bove expressions conta ining factors in the form of eF f

but it is negligible because of the rapidly decaying sampling function [ 26].

2.2 Cross Sections with No Coupling Effects In- volved between Swell and Wind Waves

2.2.1 Specification of the Ocean Surface and the Correspond- ing Backscattered E-field

In order to model the backscat tered signal, the properties of the ran dom ocean surface must be first specified. As in [26], here the genera l surface (with or withou t th e background swell component) is represented by a three-dimensional Fourier series,

~(p, t)

L

(2. 5)

K, w

with K and

being the wave vector and angular frequency of a certain wave trai n . It must be understood that the Fourier coefficients,

fully describes the ocean surface,

t) , and can be inverted through

L L T

1 2 2 2

P j j jJ:(p-, t) e-j(pK+wt)dxdydt

T R,w =

£2T

(2.6)

where L is the funda menta l wavelength of the surface and T is the fundamenta l p eriod.

Meanwhile , the

defined in (2.5) is actually the sum of a ll orders of surface displacement so that we may expand it to second order as

~(p,

t ) =

t ) +

t ) ^{(2 .7)}

where

t) denotes the contrib ution from first-order linear gravity waves, and

t) accounts for second-order non-linear waves. Naturally, this expa nsion will be reflected in the Fourier series and, to second order , the Fourier coeffi cients will be of the form

rP1-< _,w = r1P1-< _,w

+

(2.8 )

In addition, by following the perturbationa l analysis proposed by Hasselmann [ 51], the second-order Fourier coefficients,

may b e conveniently written in terms of products of two first-order coefficients (nPR: , J and a hy drodyna mic coupling coeffi-

T2PJ<,w

= L

R!+f{Fj{

WJ+W2=W

(2.9)

This form emphasizes the relationship between the first- and second-order wave vee-

tors to indicate the fact that a second-order gravity wave actu ally arises from the sum

of two first-order components. The hydrodynami c co upling coefficient ,

r

accounts

for the manner through which the firs t-order waves interact with each other. The exact form of

r

was given explicit ly in [ 29] as

where K

and K

are magnitudes of K

and K

resp ectively.

is worthy of note that

is in a symmetric form.

Having specified the ocean surface in a general sense , we are at a position to examine t he potential und erlying swell component. Th e first thing to consid er, of co urse, is what might ha ppen when the long-period swell "merges" with the short- period wind waves. In fact , such non-linear interaction between swell a nd wind waves is quite controversia l due to its complexity. Even the most sophisticated wave models up to date cannot fully explain the mechanism behind t.he coupling process [52].

Therefore, some oceanographers suggested to ignore it when modeling t he sea surface [53]. To confirm if this interaction is truly negligible, we first assume th at t here is no coupling effects b etween swell and wind waves so that the total Fourier coefficient r Pr<,w can b e expanded linearly as

r PK -

= s P

+ wP

(2 .11 )

where s PR , w a nd w PR, w denote the cont ribution from swell and wind waves, respec- tively. As in (2.8) and (2.9), s PR ,w and w PR, w in (2.11) also represent the sum of all orders of surface displacement , which can be written to second-order as

(2.12 )