IN GROUNO'WAVE RADAR

1",

-:.:..:.

" I

', '

THE

CONTRIB~TION

OF MUL)'ipATHING

EFF~TS

IN GROUNO'WAVE RADAR

_"

~RN

_,

^rROlI

_I- ^"

1'!lE

'SEA, SURFACE· .J by

_ ^.

"

.

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L·auto~1aat.ion,Ii Itte accord,'e .l la Blblioth~qu.e nati onale :d u Canada de ·. l cr o fil ll·e r

·c e t t.e thhe'etde prater ou

._

~~l:~:dre:" ~~II eu~p~~ir~lI

^.>dU

~

.,

. \>'

ISBN 0-31 5:-433 65-5

'". '

..

.

The"a uthor (copy'righ t owner) .L!au teur.'( ti tuldre.d u,droit·

h·as ee.eee.ve e othe r ·d~.Au t eu r ) ·... e' ,r ll a erYe,.I .a

J::~~~~~ -« . ^{r~:~:i;' ,~~~'} .:~~r~:'~~~·e denr~.~:ea:~~:~ .

.' :~e·:1v;riri::~~~~~ o:~::w-i~: , : ~~i::~tt.1I "atd r e. " ~1"~~1~~~ ' ~~" .

•teproduced without his/her.' ..autrellent:--reprodultaB.j,.1I0ri' '"

writtenperid. .lo~ ~· ' .. . autor l~~.~lO!,~.crlte.: "; " ",;

···:'.~m.i~·s10n

hubeen granted'"

"""'.::.'to the.Natioual Lib'rary.of

Canada. to".l c r o f ll lll th le',

:tl\es ia and to lend or sell cop~,e.•C?f~he'~ihl.

:.>

ABSTRAC~

" "

. .

^{' ", .}

," "

- -".'

The second-order

.qround.w~ve

spectral cross

seet ton of

.the ocean

~W:fac~

_.

^given

_{_..}

~y. exi~t!ng

_"

t~e'd~fes eerrespcnds

_{. :}J,"",

~~~

.\^t.

^he

,

^.caee

'

whe~e-

•.

both

^'.

,

theoretfcal scatterings occur withiQ the bb1Jnds or.a specljic.area

.cr

patch of

,.t he oC,ean

surface, Anothe~ ~ddit'i~nal'tem in

^the',

~econd·ordercross section

given by

W~lsh^.andSi:ivast~va"(1987),

• which

,represents.t h

e·, phenomenon ~here _a~ ^{Ieeat one}

"scatter,lng

~·CCU[s :

outalde

the

boundsof thepat ch'of the oeean

~I!rface,_

is examined\.

the properties and

'significanceof,this'off,..patc~.se'al ter (amul~l-

{

'pattil.ng

~ffe~~_l ~a.[e ^discussed.

^..'-

^.

^{: ..}

T~e Off~p~t~c~' .~p;ctr~i' c:ross'.s~c.ti9~ expre,~Hm f?~'

a,narrow

~~'a,ni

'r

7 ^Ge:!.vlng_

aritenil.a:-,

BY'us i~'g sult'b:le '"umerlc,'l

technt fiues,_·a".c,omput er,program is

tleV'ei~ped' ^for "''"'''''''9 ''''''

~ s'e~~~ron" :T~e' ~'r~gra~' ~is: ,a~Pli~atiie_ , ;~'r·',wige ti;a~' ^hansmii

.til1g'..,'.

.an tenna.

Theoret ical Doppler

-spect.re ~re' g'~nerated-_

^fO,i'different'.

·_rad~I.' fre~e~ies ~nd

^..sea·

^{st ates to stud;'lhe}

^..

tYPe· of-

s'catfe~~ ,

^.

~ c~mpar-is'on'

^ds

.c arrie~i' ~ut' ^be

^.tween

._ .'':'':---::...speet-r-a-Qf ; t- he above

,.~';"M~f-.o.f±p;rhrr-vrn-;.rn-:.,,,,,;,,iurri-;,;;,,--,-:-.,,----';

I., · ._ ';. " :,' •

"t o

infer the ef fect

-of"of f- pat ch

scatter in

ext rae~iofl'Qf

oce

an

surface

. .', ". ," . I

parane

ters.

Two

'P?ssi~le.ceses,

'.first

'when'ther'

ad.a t' Is

~6ca,ted;'

on

~

•

the

open

oc

ean andsecond'when-

it'

i~,)ocate~_on-the;beech,a,~_e;:

.' i t, .•' -,

conaidered

for

th,~' comP_~rison ^',··

^{' . .}

.

It'i~,d~'t~~'~ed

-'that

off':pat~h scat~e<._~~

not

S"ignifi~;mt in ~ Doppler regions near the first -order pea

ks;:shicb

are coemonly used.

to' ext~ract'

^the

^{.ocean surface par} srste ra... , Ho~~ver,' ^.at.... sO~ .Dopple-r

.frequ~·~cies it~ ~~ntributio'n ^{)nay be.} i.rnp?it~n~· ^{as contending} oc~an .

clutt er 'in ta~get''dete~tio;' probie~~ , .

-ACKNOIlLEllGEllElnS..

.

.: .

.. :~~

^{. " ,}

^{. -} - . . ~ - .:

'

..

\.

I wishtoexple~s_ .myls incereg[atitudetoDr,Jobn-Walsh,

'pr;oles;qr,: #

FaCUlty'

Of·.!£nglneelin~ a~d.

^APpliedSciihce,...fOr"his"

.

ex~ell~nt . quldance and ' ihsight_'dur~g- i:h~~P;ri~ of""thIs.

^{stUdY', The}

'assis ta,tice

a~~i1e~pful ~~9gestions_p~~vide.d by

D't.

S~ti~~" Sd~asta~a

r~.a~: alS0 -g~~a~1~ ,,~pp[e.~~at~:

^{;. "}

'~

^{..','.: .."..". . .}

. ._.Thi sW~I~w~s.suP~O[~edby,ast~ate9ip.r.:~e~[Ch'grant,from.~he.

.Natural,Scien~es.a~( Engin~er~_rig:,Res~afc~-c.o~ncil.:,{~SERC_Gl ~9~ 1_',to.

' Dr ; ~,Joh~

walsh::... ':.. )" ' . ' . :;'.'':'.'.

.' ·Sp.ecial t hanks are :dU~ '

^to·;

^Dean-F.

^..-ll;

^A l&iich,': ~choal

^of

^Graduate i .s:tudi~s: ~~an - d : :R: ·~ete~s.,. im~

^As.sociate

.~ari' ^{T .} R ~ Gh~I~.:~~~.~lty '

.,'of' En9·i~.e·~J~'~d' .\pph~ ·scie~~e/.. .: providlt9\bl~'

^{o'ppOitun'ity}

.:..

,an~: ~~~_c_~in~~/~~~~~a,nts~iPs.:

^{.\ ..:. .r.r-•. ' .}

',:~

^{."41 _, .}

Als,o,theauthor~ould·like

t o

^th.aniRandyHo~.llfor-his'

.1

"helP itS discussiC!~ ;.:Tha~s ar;'·_als,~.

^dlle,toMs..

Na~da

^{'Sut ler"}

f~~ ·' .

her-qu~~~'~nd

^{effic'lei\ttyping}

o f th'i~ ~nu'scriPt~

^,

--

-,

.

:

.

,.c '. ,"., -

,-;':~

, ;'. ,

-, -

v.~-'.' \>-

" ^{" , '}

. .

^';.~.

<,' ' ,i',' " .~I '

>,ci' > ,T<"''{ :;;' ,: ; '

,.~.:

. .

^.c

" '111

" ' , "'If'

Twopartsof-th e

secqQP-o[der ~cltscattered

⁵⁰

s~t.r~lcrosssection:'of~an'

oce an

surfaf epatch.

. 2~=;~~~ ~~ ~~~~h~l~das~ea5a~l~fior~~~a

.directionw~threS-pelt"t opatchdirect ion (open~eacondit i on) ~- .- ',' .'-.. .

i .

9 _

54

\ . . - • . J •

;~cf~~i~.c~as;h~e~~i~rid~~[~~ro~~~~S;~~~~~:d

^patch'

51.

, . , _~~~~~~~ ~~ '~~~~\I~aas,~JO ^{a: z} 05a~!~d"; ~

^f.

direct ionwit hrespectt~,patchdirec~ion (?pe~ s·ea-~?nd~tio.nl·:'.: .

r'o :

T~o

e rts

ofthe second-order backscettered

. ' " ~r~H~~a~~~S' ;~~~i~~~~f ar\ ~ce '

^patch

. --frequency,

30 ~not

w.in<!'·speed intl. "'

' . f ·

di rect i on,withreepect-to.pat 'o~n ^'J'"

,(open~,.sea<!'~ndit~onl,_ ',;'.:,'','

' , +. :.: '.

T~O' ^{pa 'rt, of}

^'the'

second4or.de~ '~cksc'attered'

^,"..' spectral cross sectionof'an'ocean surfacepatch

.'. _~~;;~~;~ jg . k~~i~wt~~~~~eaOa~Z9ij~d;fnd "

^•

1;~~~~~'~~~~.ia~gj~t , to-p~;ch'~irec~~~n

'~wo, parts' ^{Q f}

^the

S~Q~d.ord~r

backscat tered .

spectral cros$,. sect ionofan-cccan surracepatch .

~~rmal~~~~ ,~g T~~~h~t~aas~ea5a~l~o _;t~~r

"

'di~ion

_{_{open~ea..C?n}^with"_{di t}^{.respect t}_ionl ^o_"^{pitchdirecti on}_{• ~}^.

Two'part's-

of

the,s'econd-oiderbackscattered .55 spectral cross'sect i on,of an'ocean surfacepatch

, ~~~~~~~ ~g k~~~\t~a~i3~~a~~~l~;o[:1~a

^:

directlon withrespect:tc-patchddrect fcn.

(opense~_,.condi~~on)-,""". , '." . l .'Two part s.of.thesecond-Orderbackscat-te red' - '56 . spect ralerosesect .len of anocean surfacepat ch

normalized.to Jfcltcharea at 25.4'HHzoradar

_,~frequency,,_jO/knoLwind_speed_.and--:-90__alnd · di,",c\10."W

r' ,

^{11,r~,apect,to,patch},di recti on. ..

<-l open'sea'cndlt!on1 '.

~ . ' ~.

10

131:

'12

I

~,\"

..

',;,,,, .,,.,:'

~"vi H~~ (~,.~/'( I__~~.c·.· ..:!-I',; "" " .."<,y",,~~

, ' / , . , -J

" j ' "

Two parts of the seccnd-order/beckscattered 51 t

·

spectr~l

cross secttcn.

o~

an ocean surface patch

-. .,

, ~~~~~;~ ~g ~~~~~v~aas~eaOa~~Ooa~~~d'

^{i - . '}

~ , 1t~~at~~~ed~~~n~I~~~f to t~tc~ ^direction

· ;~cf;iiSc~~s;h;e~tIg~d~~r~~r~~:i~;~~i:~~dt1atch 58

~t~~~~~~~ ~g ~~~~hv!~jas~eaOa~Z4~6d:fnd

direction"dth·respect to patch" direction

(land based wnditlon)

I

Two

'part s'

~f' ^t'he ' sec~nd-order

:backscatte red - 59

'~ctra l

cross section of an ocean"Surface patch

t~~~~~~ i3 _~~~rh "tria;! _s~eaOa~Z 90 8d _:fnd

"di rect i on with,

(espect to patch direction

(land based,

condit.Ionl

. . . :. ^' -,/ .

i~c~~;I~C~~1~h;~~fIg~d;~[~~[~;~S;~~i:~~dpatch' .' 5 0

~~=~~~~'iO~~~~hI/t~a\~eaS~~d~O ;~~~~

directionl' i t h J;espec~ ^t

o",patch direction

fland baled ^conaitlOn)

^{, .: .;}

< .

Two parts

of the-second-order

beckscattered

61 SP~l

cross section of an ocean su'rfacepatch

·

f~eqtle~~~ Io k~~~~wl~das~:eaS a~l~50r~f~d

..

~f~at~~~edi~~ndI~~~f ^{to patch} dltecttcn

I ' " , ' , '

",

T'wo

parts

of theseco~d·order

backscattered

' '-.62 spect ral.

cross section' of .an ocean surface patch

~~~;~~~~ ig k~;~hw!~das~eaSa~d~5or:~~d

-'di r ect i on with respect to patch direction .

fl.andbased

.condit~onl ,_ . .:.-,

'i'wo parts of the

'second-order

backscattered

. 6~

spectral cross

'sect ion,

of an ocean

surface

patch

- _~~~~~~~, §g 'k~~fhw1~das~eaOa~zOoae~d

direction

wit~'r~s~ct

to.patch dire£t ion 11an~ based condltl'J

.' .:,,',

f 19

i

17

16. '

,IS'

, 18

'

. ... .

.7

22 Two~arts

of the,second-order.backscatt ered

.

64, spec ral cross sect i on of

~n

ocean surface

patch

\ ^{~~~_~~~~ '} ~~ ~;;~hwt~~as~eaOa~z4~8d:lnd

directiQn,wit

h respect to patch direc t ion

'-(l and basedcondit1on) .

. ...

23

'tw~-rarts

-'

of the

second-orderbackscattered

65 spec tal cross 'sectionof an ocean"surface patch

~~~~~~~~, ,~g k;~~~wi~aij.:~eaO·a~Z 9 ij8d :in d

direct1on'w ith respect to

patch'direc tion (landbased-conditlonl

-r '

^{."" . "." . \. ". .}

.

24 i~~cfarts .

^of

- the. se9ond-or~~ro~:~~s;~~}:~~ patch ⁶⁶

«

^lJomal

s~ea5a~d'O.;1~r- .

patch direction· . '

25

.(land~as.ed 9r~i~1~n)_;

, '" .

, ~

T

wo

erts

^{of ~e}second-aIde;backscat t ered 67

.spec ral er css

se ceren of

.an

ocean surface patch

wt~aas~ea5a~~50r:1~d -: .

eel

-tp patch direction \

on). '

. -

26

Twofart s

o f

tbe.second-cr derbackscat tered .

68

(itoec ral cross section ofan

ocean surface pat ch

^..

f~~~~~;~"§8 ,~~~~hwl~~as~e~5a~lIJOo~:t~d '

^t'

dir~tion

with

tes~ct to

patch.di rect Ion .' (lanabased

condit on)

,

"

, -

'

..

Ce,C_h Fp

Gt,Gr ^"

oj

K

k'0 p~o

~OPt 8("

Sa{x}

sqn(x)

* , ,\

,

~ 'A

&Ix'

" A

o.

r - v

Po

)<

: Second-order electromagnetlcand hydrodynamic -contributions ".""'., ^p ~ : One

way.~ound

wave attenuatioiTfunction

-between {he radar and the surfacepatch : Acc,eleratiOn)udo

.~ravity

^_

Freespace

qZ

^of-the transmittinganci

~~;~~~~n~a~gien.nasin,the.dtrectio~of Wave

n~r ve~~~r~~ ^Kx.bY

. -t.,

:

.ta~ni~.ucie

^{,Of K., .'} -Inctdent redlcwaven~er ,.,.

..

^,.(

R}CeiVe~power of the backs:attered.signal Average tranf>mitted.~ower.

Ocean dire\ctionaJ.

wav~height spe:~rUJll

Samplingfunction Sign

~unction'

^./

Wird speed . ' . "

One.jlalf the beamwidthof therece i ving

antenna \ .

:·\r-one-half';t he'patch width of.

t~~'ote6il

^surface Normalized surface impedance'...

D'irac deii:afunct i on

. ..'I' ,

' \,': Wavelength oftrensrdtted signal Doppler'frequenc,y,nortllali-zed to 'the Bragg.

.frequency'.' . Di!N;anceofth~.t a"rget

,I

aS1((Od )"O"S2 ((Od"

a

_S3 (llld)

"a

rod

"t

.~-// ~

.

Totalbackscattered Doppler frequencydependentI

~~~~gr~~r:~~~e~e~~i~~s0ir~~eocea\SUrface,.;,·.,"

First-orderbackacat.te red,peet ;.,

c "io,,':

sectionoftheoceansurfa ce patch normalized toit s area ' .

Three par t s of

second~orde~

b;cksca t t ered .

~~~~r~~~~~~e~e~~i~ts0ir~~e

^{"" ' surface} Braggfreqpency

Doppler shift

in

thereceived signal .Frequencx-of the transmitted signal

1.1 General

In recent years" ground wave Doppler.raaar s nave

,ga i~ed popul arltyi

n remote:

.se,nsi ng of ocean surface waves. T~e -theoret ic alfo~ula1:.j.on

requtred to',analytp_the radar sea'

ech~

is

-besed"onthe

interpret~tion ,~f

the eche power spectnin

in t:tms~ ^of

Barri~k 's

equation for the spectral

cr

oss' section. It is fromth

fs

·cross' sectio"rt

t

hat ocean surface parame'te-rs' (~ .g. , sig~ifica~t ^waye-:' height and'directim;al wav'eheight spectrum).may

be

extracted.

BarriCk;s ~.el f~r the.DoPPler·'~~?tr: of the r!'dar retu5n

fO,ra

small area

(pat c.hl of the,ocean.sur face ccnetst.s predominantly of-

fir st

-

and se~~nd-order scatt~r . : Ffist~oi:der'~c~tter ^is ,pro~uced

^by'.

th-e '~eflectioh ^o

^'fthe transmi'tted

~lgnai. ^from :o~ean ^waves '''hose~~~~e; ' ..

- ien~i:.~'

^is

o~~-half ^the r~da~'~s

^w

a; el engt tr and which, ar~

^'mo'ving'

either. tcsards

or,away

~~om th-e radar.

'

Fig~re l

^;

^{h'ows the first-}

order scatter fiomapatcti'of the.cceensurr ece. second-~'rder

·scat ter ton-petcbr is produced

when the tra;smLtted sig~ar iq,teract's

with.t wo oce

'an

wa

ves

that ~x{st ^{within the} bou~daries' of ^{the patch}

~nd'havediffer,~n;.:.ave.le~gth~i

n qeneral

e.

·,) In a differentepprcech;:

Sri-vastava

(1984) ha';

s

devel oped an .a~aly~ical

model

~or.therF'ba~~scattered ~o.ppler

spect

rumfor.

the

.

ccearrsurfece

.

!he first-?rdet cross sect Ion h

.the-,same

as derived.

':

by ~arric'k

^.but

^t

^he

.s~~on~-order cr~ss

^..sect.Ion

co~sists

^:of,

thr~e ~ ^•

.. .:

. . 'o" " . . . ,

'parts", The

firs~

^part

~s "~:J:i:~t

^t.othe second-orde.r

c~os~

^section':,

derivedby-Barrick.T~esecondpartcorrespond!_to thecasewherethe,

·first scatteroccurs atthesourcepointandthe secondscatter occurs_{' .}'on thepatch_.,This

additi~~al

^tena_\ "ilL be

pres~n;-~!llY _.

^when

, thecada"ri,ssurroundedbythe openocea\ (e,g., ashiporan,

,offs hore platfoIlllbased cad,ar ). and...t,hls fO(lll of scatte rdces-o ot

·et rectthe

reqi;;n~ '0 :t- Doppl~r

spectr umnear.

th~

^{first-orderpeaks}

wb.~ch'

are commonly :usedforestimatin g ocean

wav~

par ameters . The thirdpart,correspondstothe'case whereatieastone scatteroccurs -f ronthe 'Ocean waves

out~ide

the boundaries of thepatch.Thiskind

ot

^{sca.tteewhich may}b~.viewed.asa~ltipathingeffect'willbe i, referredto as the

Off .pat~h

..scat t er.The\hree pu t s oftlte....

·

~eCOnd-O[dec,scatte~' ~re 'Sh~wn'

^{in figu;e}

i. '. :.' .

" ~n 'o-rd~/tC?:'~tu~Y 'the c~nt~ibutiC?n

^,of

o~~pa'tcb ~~atter:

^a

S.Of.t".~.e

^'rodel:

~~'s'

^been..

d~V~~O~ ~~~ e~t'~ti~9';th~-:~pect.tai;

^cross'

section , the lOOdelincludes anUlllE!ricalevaluatioJl of anint egral.

':

~his

^s:lngle;

variable~ inteqrai.

is achieved

f~~

^a

f~urth

^order

inte~ral

^..

. . • . ... ,' . . ~ '. I ' •

bytaking advantageof,two Dira c deltafu.nctionsalldassumi.ng a-eide beamtrimsmit'ant enna

,.;;t

narrowbeam~e"ceive'antenna, This-eodel may be

~sed t~'

^{generat e'the}

s~tr~l'~~~~s se~tion ~f Off~pa~Ch

^scatterfor'

:'dif ferent'radarfrequencie sandseaconditio ns.

A . c'~mpariioi: is

^{'carri ed}

·out

pet~e~n, 'th~ otf-patch

and.on-pat chcross sect.i on.t o

in~er

^the

~rJIP?rtance

^"'of

·off~patc.h

^{scatter in}the extr action'of ocean surf ace'....

~ar~~t~rs:'and

^othe;

·reiev~n.t.·i~f~rmat~~n. 'Th~

results.lead t ; 'the:

ccricluaion~hatalt~ough'qff-pat'c,~ s~~tterisnot sig nific ant i~'

ext'r?ctingthe oceansurface'par amete rs" at'some DopplerIrequencles it naybevery importan tas-cont endl nq-oceenclut t er inta~get"detectio~

problem;.

1.2 Liter atu reReyiew;-

.The problem

of renecrtcn

of wavesCrDID arough

s~rface

has

, .

^{,' ' . "}

.. .

beencarriedout by many~nves tigatOql. Most interes tlies inthe

propaqatlcnJltradio

.wa~

^{over rough}

grou~d

^orover

th~ ~ ,

^{_ ___ ---c-}

Reyl ei gh Introduced a perturbationtechnique in'1896 tostudythe. refl ect i on of'acousticwaves from rough walls. Basedontheebcvev- .appr oach

Ri~e

⁽¹⁹⁵¹⁾

'~reate,d

^theprctnee- retetedtothe-

·~elIe·ction

'of

efectromaqneulc w~v~s from\Bli9~tlY ^rouqh SUr-f~ces; '

^He

~odelled,

^\_c;-

^_

^_

the surface as two

dimenBi~n~'

^and

pe'riodi~

and,'assumed,\he'incident

':~

^" ,.fi el d,t obe,a

:pi~ne w~ve , '

^He,

obt~ineithe

first-order'

a~d

^{the" ,}

;~-"~: ' ~

^" hi qber- order

expression";_: ~or,the scat,te~e~

^fie{d''and

dedve[l,,~h~

"'.

0<tht~

Bcattered."!ield'·componentsfor horlzont:al'an'Q've;t ical pOlarl:-

" " eat.Ions, ,'In1964Waitappliedtheabove technique.inthe electro- . magnetic problemfor ground'wave propagation'ove r,'flat earth. Wait (19711 andBarrick(19711.der dvedtbeexpressionfor the modified

, --

s.urfac,~impedance a,ndsca~tering,results ,,wai,t.assumedthe sur fac:

,to,beone.dimenS~Oi'lal,and periodiqwhile

Barr

lck concent ratedon .qroundwave propaqatlon ove r therough sea,'

Crombie

.11955)

'conducted,an'experiment'on

~adar,

^{returnfromthe}

oceansurf ace'at'13.56MHz and fiefoundtwo,domi nant peaks Inthe 'voppi erspectr urn','

He"investig~ted, th~se

^tso.peaka andceneto

t,he' conci~sio~ t~at

^they.were

,pr~duced.'

^bytwo

~i;a~av~(~::ac,~

".

i ·

. . ' ,- .

.having awavel ength equal to one-halfth~[ada-r'swavelength, one . mo'vlngtceardsand.the -flther'movi ng'awayfromthe'radar

~.

^Basedon .this

~b~ervation

^heesta biished'

th~t

^the.

~F

^radars

~y

^{beused in}

r~te se~sin9 ~f.

^ocean

~av~s~

^.Lat er'ear.rick.(1910)

Illiliz~-

^the' -.pert urbat.ion.t echnique-

~of~nd.~~e

^radarcross sect.Ionofthe.ocean

.s uira·ce. He(19721 derfvedan averagefi;sl,,:, and second-o rderbaci.-.

scatte~ed

^radarcross sect i onfOra'patchof theocean

~uriece

^{. ..The}

~'..,- ' --- , ' ---. -' - -' -." , " ~I :-

firstorder.resurt'offered atheoreti~al'expla nat~on f~rCrombieIS.

'experimental-ccnchialon.

.. "In

.'~. ~i'fferent appr~aCh." ^{W alstr} (1 980~ 'p~e~e.~te~:a ~\mUlat~~n . .

^.

I' for"rough surface pr9pagatio n.and-scatteringproblems.,'His . ,,:t.ecbolque

i~

^{'applicable.t o}

anY',.t~~iriVari~t

^rough

surfa~~';- ^{and,,1S·'· . ."}

.. bpen tc ariY,Hni"te

, ;~u.rc~ ins~'ead

^of'planevave Incfdence ..Based

anT ~ ... '

'.thi s',

t~~h~ique;.

^a

the~t~ti~al an:alysi~

^was.

mad'eby'-g~iv~st'~va :'it9:a4J .

.

' , .. ...

, . . '."

- '. .

. . .-,

for..Hr~s.cattedn9from theoceansur face" ,Srivast ava derived

i~;~ag7 'fi;st'~ ..a'~d "~e~~~~~der ~f baCks~a'~~;red' ~pecti~~, c·r~~s.

'sect fon..

:·for na£.~owlbe~~. reception

the'sca'U er iAq area'ofthe'

".

~ir~'t-o.rder ba~ks~~t~e'r~

^{fieldis shownto}

'hi!

apatchor'

~mall

^area-"

of.theocean's·urf~ce. ,B;utfo:rthe'second-ordercase,tbesiqnals .

a~~ recei~ed' flom

^thepat ch·! S.vel .l-asfrom.

~he s~r~ou;dill~

^-

l~i~ns

^..

.. " . . , /• . 1.. '" " " ,' :' : .

.

~ll. ~hes;: " s,ignal~ ~r:~.ve· ~t th~

..

sa~. ~imr"

^His,

se~on~~o~der'

^{,C.I:;OSS :.:.f"(}

sectioncontai nstwotermsin a4dit io n'tothatprovided

b y

eXist ing' .

. ~ ^. theorle~.: ^. .iiat~;,.· ~;jl~~~' ~t.' a:~

^..·.·

(.19'~, ,extend~~, th~:.abov~, ~~~~~ts

^.

.f6lelde~,amIe~e~:i~n'and_{~i n~~}theexpreaaionfQ.:'·~.ff.-~atc~· \ scatter,' The_'la~stwo~k..on_th~sprobletllMs,beenpresented~y,.wal.sh,

and

sri vasta va_{. - '. ',}'

i t ; ;1) .: T~e' first ~ddit~o~a'i

_{, " ..>}

'teim

_{'.' .}

has

_.d_.readl,:'been

"stUdi~d':

_{' ..}^.:.

::~

-'

- . .

.c.::

c. ,

.

~".

andnotfound signific"ntin proble mofe'xtraction of

oceiii'surf~ce '

^.

paramete~-

(Srivasta va11984),Nals h and

Sriva~tava

^{1198·7) 1. The}

s~cond

addit ional

t1111

whi ch

~o·r7esPODds. to

off-patch scat.terhasnot recei ved compl et estudy by oth er researchers, and, act ua:lly,hasbeen accnt roveralajissuetor several yearsIecjacdingitsimportance

t o.

theprobl em. ofextr act ingoce anwaveinformationfromradardata . In an effortto set tl e'this issue, its'properties and signif icance .forocel

' .

wave"neasureeents have

e .

~beencarried

o~t .

^.In

.

^{thi s}~.^-

the~iS :

".- .

1. 3.~co~^'0,[Thesi s (.... __ .

. . This thesisisprl~rilyconcernedwi~h'~he exam~na~ionofthe .,

·ef f~ct Of. seco~.d-order 'Of~-P~tc~ :ss:.t.te[ .

,"

~.he.

ener qerce.

tlr~hiS '

. .off-patch-termis.tot ally basedon

.tfie

analyticalmodel.developed by

sriv~st~v-;J~984) ,

and'Wal sh

':a~d. Sri,vasta va

^-

((9~7; (~i th~~-

^{HF" ."' ,}

,

_ba~kscattere(f

DOppler':'sp;c tt"Um-for'the"

~ean

surbee."The

~."

"

"dis~rep,~ney

^between'

ae~~a{

^radar

d~t~:"a~

the"theoreticai

~ie;

,spect rum

~rovided .bY

^eXi\t1ri9

'theori~~

^{ref sed .a}

SUSPicion·~t~·t~e

interferen ce"pf·,s ome.other ef f ecta.,

79~

abletos.eeifthe -'

~ff~~tCh

s"catler

i~_

^{thecause ofth"is}dfserepa"ncy,';jcompletestudy

_ l. ,": ". " ": , ' -.

Isdone to,u_nd~_rstandtnephysica lpropertiesand siqni fi cance of

thi~

^Icm

;of: s~atter

^.

^{, Fo r}

^this

pu~pose,

.e softwilre:

I ~l

^has.been,

~ ~femented

^..

~~' ge~;;,~,t~ _~~e ~heoretic~l Off~pa,tc,h

^ee

o~d ~.o.rder

^.cross,

seetio~fo.~ ~Hfere~~~ada r'.freq~~ncies.and:sea sta.S~' The" . .

resu~_ai:~ t~en' ~omp~,r~~

.wi th t he·en-patchcro.sss

'c~io~,

^{in?rder, .}

"',to yraw a,

~olH~~lusion.

^aboutits

impot;~ance;",

^{l,tis oundtha ttho4gh'}

O'f f-P~t.C~::~.ca~te« ; no~ s~_gni~ica.nt i~_

^theextra

t~on

^of

ocea~ "~'~~'~' "

. Pi

surface'param

et ers,

.Lt is

in some cases,

veryIepo r t ent in

target

detectio,

n.

pr?bl~ms.

cnapter, 2 coetadnstne radar

rangeequat ion and

the cross

secti on

'~~pres~~on.s ^{for fi}

^rst-;rder

andthe three ~arts pt

the second-order scatt ers.

The

definition ofall the variabies

and

para'~ter9 ^{. ;} give~/

^'./.

ChaPt,e: ...3'deals wit hth

e,

reduct Ion'of

the spectral cross

sect ion

~xpre~sion

^:of'the off -patc h

s~atter. ^to

^a

coT\futat ional f Ol;rn.

The reduction

is.achieved

byt

aking

.Intcacccunt

t

he

presenceof two

"

,nirac

,dei,ta""tunctio~s ' ^and' ap~roxi.ine.tiii~

^'an

inte~r.al u~in9' ^'the,

^,

re~fariguhr-.r~i~

^,for'

!1arro.~ be~lll

recepti on.

" Thi~ , ~ross ~ection

-e

xpre.ss·idn19

obtained.'from,the'sImilar'expressi on of Walsh

and'

Sriv~~,~'~v,a,"

^,(19871:

f 9r _{',~ "~id~} beam 'rei,ei~in~ '~n~,~~na .

~Ch~pter

⁴

·'copsl'sti . ~{ res~~t~

^{and discusaions.}

^A

^:

co~arison , is

.' , ', ' .', ' , ,

" , .

"

~ad~

j ir ^this

,ch~p~~rbe.~\lee.n.t he,af~:-p.atch'a~~ on~pa~~hs~<;tra,fo~

differe9t,~a~arhequenci~san~ ~east etee.•·

' "

pos,sibl e cases:

are

et

udledr. f~r-~~lY the case when, t~e ·r~dar. is' locate~ ·on t

he

open' ocea n

. (e.g ., a:

,shipor

an of fshore

p

latf orm based'

rad-ari and secondlyw

hen

'

the '~&,da~.>~ ^locate a 'on , th e

,'be ach.

" R~~Ults

^are

.d'iS~USS~

^"ror

^all

^-,

thesecaaea.. ' .' .

:

- ."' ,-

,,:

co~ciusip~s' ~nd

::r:ecormiendatia.n

s fo( f~tU(e, wOF~ ^are. ~reserited'

^hi.'

Chapte~

5.

ComputeJ."o'programs aie'given-~i,ri appen 'Six .

(2.1) CHAPTE~2

~ACKSCATT£REDRADARC:ROSSS~CTION 2.1 Radar'Range Equation

Thestatld~rdmono~ta,ti?rada r.r anqeequationforthe rece i ved' powerfromatarget maybewri tte n as (Ba:rick(1972)1

po PtGt

Gr ).~

^tFI400•

r (41U3P~

whe r e

P~,....recei vedpo~er,(wat t s )

Pt ,'"

t~ansmit~~d~wer .iwat\SI

Gt'".9a i~ o,~t~ansmittJ.~g,.antenna

Gr ,;,,'ga~n'^.o.f..re~eiving anten'n~ ^. .~,.).o•waveleng~h'oftr<in~irti.ttett's~9riall(met'eri ..F =oneway9r0tinijwaveett .enuetfo nfunct io nbet ween

..

~

^:.-'.

t~e

^{.sadaf aDdfhetarget. ". '.}

-V

^r

po.'"dist a.nee'o~the_t.arqet;.(meter)

~o=~~dar.cross sect ionof the~a..f.Qet'·(nieter2l·

. G_t and Gr a r~diineo $ionl e s spar ameters: .

We

·a,r~.con'S~.deIing

^{only a.smallpatch,.Of}

,.'t~e

^{oce,ansurf acefor .}

.

t~m~.te.

^{sens in g}and.,sinpe th.a,t cons ists,of

.~n~ , oce~n .wave~ ,m~~in?

~.,~i~~di.ff~:ent..v~locitie~, there,:,i~.l beaband.ofD~opp~rSh!fts~t(

.thereceived.'; i9na1: If 0\; is

-cb e

·tra~smi.tted^{.frequencyand.}

'rot

.:is.the

,recfdved".freqU~ncy;.

.t henth; r>ePP.let;'Shift {(bd )·.inthe,

receiv~dsi gnalmaybe.9~~inedas

(2.21

. . .

..~r<'

.~ Ifinsteadof thereceivedpower,'we use the receivedpower :spec.t rum then the

~ac~scatt·ered powe[ -spect[~

in terms of spect ral

cross section' norma+lzed to

t~e ~

patch area.maybe

gi~en

^as.

(2. 31

• where

12.51.'

. J. ..Pr(ffidl.is'th,ebackS,catteredpower

sp~tr~ ~nd

:G(IDdl isthe

spectrum cress section,nornialized to,patc~area Ap.

'+~_~ _c~_

^The

:"d ime nsio n s .·~f ^{'- th( patC " h} ~ePe(;d-on 1netfne ,delay,

betveen

.t he

trans~tt~: ^and 'receiv~d

sisnals , transmitted pulsewidthand

thebeamwidth of tberecetvtnqantenna•

.T.he

baCkssa~tered -spe~t.lai

^cross

~ectior/normali1:'ed

^tothe

•patc h,a:ea'asgi~en'inWa1s~ andSrjvastava'(19871'consists of

! four,

parts

and

may

bewri tten as,':

r.

( f

·.h~re m

^>-1

and"-1 a re- for summation,

.!

?h;!:.'\;!<f" 'j:," ~t"""l: ,.-"',"'c v'·~('o:P'l"'<';'",{"~\;?ii::!(""i"I~~'t'«'lj' 1 ",,~;:;;1;':1"::,,"l.;',~~...'t-'o-r.'1~8~:,& l}"i

e,r:r- ' ' ^~ -

^{f " :' ": , 9<}

^{"' , ,''',:~''~}

^{"". " ,}

^y::~~

."I. of these cress sectio ns'for.wi

di

beamtransmission and any, .

I

rec~ivim;f ^{;ntenna (e.g.', wi}

^de

be~m ^{or nar}

rowbeam) aregi ven in

'f , ' \ " ,

Wal shandSriva~tava(1987). F

a,

narr owbeamreceprtcn

their

expres s ions may esimpli fi edfuAh erand

are

^{presentedhere.}

'." . . .\ ~ I

2.2 First..:order Cross-ect .Ion'

..

\

^,

.

,

The first-order cross.

'sect i~:fO.L.na:towbeam.r,~cei~ing antennas maybe given ea ; )-^I• ~ . ^."'

8l1p,I rodI5 .'[od , ' ,\]2 " 2,[ ( wa ' I I

' 0tlro.) •

~

- - " Sa A,- - 2 k '

u "

'9 1 ', 9

''0,

P

g o , •

: S I~2q2sgnl~)iOI\' ^{. ' "} ' :i2..1).,

'.

wbe.~:. 'T]. '~ ·~dl<aa·~:.· ,~. ^{,. (2}

,9<111.2,

.I" the B:~;;

^{'f; ;>!uen,y,}

^A p .' Is

one-half,the patch wi dt h of the ocean surface: 9

Is

'theqrevl.ta- -/ '

:tio nal'

ecceler etfon.

'.

~~= k~~', 'is ~he .-incident

^radar

'wav~n~~r

v~~t~r:' ^S(..), r.ep~esents ^thi ocean'bir~ctional w~vehe·ight. 9pe~trum

as

deq~~ i; _~als;

and.Srivastava

A987) In ~ formaimi.Iar t~·

^{that ' \'}

given

by

Lipa

and Barrick(1982) . \ " -

-- 0 I · /'

A

ssuming a

l

arge

.~p

; . the

li~i~~fth; squared

sampling function [sa 2

_{(x))/ -In}Eq. '"(2.7)may

be' taken tio

,the iOira C.delta

funct·ion, In

thi~

process Eq. "(2,7)

~e,{!.uces

to

l (

i, ...

O

(lrod)

·"

16k~ m~l

51

rod

^t,

mOj,IS

I2mk

o i

.. ' I

I. ,

..19

I

. 2.3 second-OrderCros~section

'the secemr-ordercross seetfonconsists of three partsand may

be'

given as follows: ( , , " .

2.3.1 First Part

The first part.of the,second-ordercross section which corre- sponds.

to on-patch scatter-.,'ll'1Y

be.

given

as.

(2. 91'

~.wh~'r.e nr~aJd !lI1 ma(t:a.k.e

^the..

~a~~,s

^tl

~nd ' -1, defining four

..

.Po.s'sibl~.collibinations'ofd1.reciio'\'for'

the two scattering ocean

.

. ... " -+

-l . ' ,"

wa.venumber

vec~ora'._K1 and.'K2;'T~e,,spatial

wavenurrber

p-He s along

t

heradar,

beani"

with.'q,perpendicular, ()ther·variables:and

func;l'~ns : ¥e. . d~:fin~(i''a~

^{'fo11o;s: '}

.. ' ' ~ ^{... V " ,' " ..}

' < \Xi ".; (p , kolx + qy, .

~1=fKll

(2.10)

-+ "

A ,.: , -I

\ ,=

(-p

\kolX - qy" K2

'=;-I~.1

; Ik~ ^- 'pi , 2il . ^{i 2l}

C'= 1 4 ' -"iJ2 '

. .r

^,.(Kl" ,K2) . ,

" fT ·"

^{"' ,}·IKIK; . il1·

i ;

J

(~+~1

Ch ,~ ~1(I +Ki+'n~'.IKIK21 ~2(~ ^' ~~

.

(

Ce and Ch are the second-ordereIectrcmaqnetlca~~hydrodynamic

!

ccntdbuttonsrespectivel yas given

tJl.

^{Srivastava (1984).}

.•2:3.2 s~~~~~ _Part

f

These~pdpartof the second-o rdercross sectionmaybe.

writt enasf2:!-' , -, ,

1.- ~ ~ -I

K1

=

px

t- qy ..

K1

=

IK1¹

'.

f

where

, . I. FoIIb'!!Vi 1,) FoII"Vc r I, '+' 1 I'

2.3.3 ThirdPar t

The'thirdpartof the second-ordercross section derivedby WalshandSri va st ava(1981)

fo~

^a

g~al

^{receivingentenne has been}

s~,lified f~r

^'narrowbeam. recepti onand is presented'in.

HOw~n

^etel.

11?81). Thisthird part corresponds to off -p atchscatter andmay be give~as follows

/

ATes.trict ion ontheaboveequati onis: l~c l

s Ae. othe5

var iabl,e§and"tuncti ons'aredefi nedasfo llows:

L

\.

(2.201

12.211

,.r...

,.

. .

Kex .~

2Klx {K2x

t

1+coshPocOS{lol·t

1\2~sinPosinao '

(2.19) I

Key-2K1y

+

K

2 y

⁽¹

⁺

^coshPocosao)+K2xsinPosin~o

!

-i -I

-i-

s, -l A -I

Q

e

(K

l' K i) • t,

lK_l .K

el

[2k, IK_J

Kel + k,

IK2 •

K

e"

of . A : A .-1

~c·eKex

x +

KeyY, Kc ""IKel,

f ·

f/ .

'.r .

^det

.I s

defined

as

(

.

. .

($.01 •.~, 12Kl

.KC

+

K2vnhp,lna '

. •

" ~(~2 ' :'~Cl

⁺

(~2 ' ~cl,(1

⁺coshpcosalJ (2.2ll .

det shObld

~ 1!'Jalu~ted

at

P

=

Po

and

a · 11

0,

P o

isthe

sotutton,

of theequation, ,

13~

where

where

v

•suchthat

Po

is

real , non-terd

and sat isfies (he following-

eqton. " ~ .-

~ ~ 12K 1cosl~ ' + ~c~~~2JcOSI2COS~Po[1

^{• tanh2jo}

.r

t

an

2+

211n •K

2

:yn(pot,nI21Is1n2+2+co,hPo

! .

t (2.24i

{2.211

"

.;.

.

Once.

P o ,

i~obtain~~,,

a

o may be

derived

f~m"

tal\Clo • t' , nhP

ota

n'2

,

sqnlx

l

is

the

si~n

funct ion,def ined

~s

[

1 , x >

0

sgn(xl •. .

. -1I!x<0

.' -

Th

.

e Eqs.^-(2.71,^.

^{- ..}

^-.12.91;,(2.14)^,·and {2.1Si^{- --} tbuacepresent.^{.'. . -}the^{\ ,} Hrat-cr oerandthethree part of_thesecond-orde ruoppterfrequency dependent

-baekscat~er~d cr~s ' se~tion "

^ofIthe

o;~an sU~face: .Th~se

^.

expreesfonsareappHeabie or'widebeam tr.ansmis~ion-a~d na~row

'beam.reception. e.

.8~\tha~

O·S " o:$n. ASis equa.lE o one- half

ot.

the

bea.mwi~th of

the18cei vin g ant enna. Thefunction Folp, 9 ) isthe,s ame as-the

· SQrrvner fe l datte nuationfunctionexceptthat,the nume ric al distance contains

4

₀

IV.

Ineteed ofthenormalizedsurfaceImpedance !J.

·[~a rrick(1970)J. 'A~(S) is theaveragemodified~urfaceImpedancein the propagat i ondi rect i on 9 andittakesIntoaccountthe surface rough'nes s; The expressto n

fo~ '

^themodified sur'i.aceimpedance is given;

•in Walshand~civastava (1987';^{0 •}F

il'=..Fa(Po, Solistheone way"ground wa veatten uat i on functIcnbetweenthe'r ada r and the patch',

.r_b Is'the dis ta ncebetweenjhesource and fi rs t ecet.tertnq

poiilt. r c is thedistancebetwet. thefir,s~.and.s econdscattering p~ints . ra-represent sth~distan~ebetweenthe-second scattering poin tandthe

re;etvil~g

(or's;urce) point .".The

cor~esp"onding

.t hree di recti.ons.wit h respect to the',x.-axi sare,t~'b

+

4'c l . (Vc

+..

,~tn) 'and (' c.-+.;q

"~espe'~ti'v;iy .

^These

dist~nces

^"anddire,?tions

eeybe

olitaimidfrom

. , ' . " .2p

o '

ra;=~

. . ' . , ' I

Po(cosh~ot cosCto),_ _

- I[

sinh~osina~ ] . rb'", .11

+

cosh~ol,:':,'g~·."b⁼tan, COSh~oCO~llot 1 ,(~ . 291

_

Po(COSh~o'

^{-eosCtol _}

-I[ ' sinhp~sinao '

^{]. .}

~c- _ll.

+

coshpo' ' 'Ve-"tan "COShao?O~ao-1 .(2.30)

.

^"

3.1 Simplicatio~of Cross Sect i on Expres s io n .

The second-o rderoff - patchcross sectionexpression as givenin zq. (2.15)maybesimpl~fiedby using some of its mathematical properties. Wewi ll flrst,transfonn-thethree variabl es Ki t~1. an~K2 toneJvar~ables U,V and

t

_e'respectively such that,Eq.

(2.15)mayberevrit.ten as ,

13.41 13.31 13.21 •

, K

~

^;I-t'

c~shpo ~

2ko V'"~d+m(gK

1)1/2 tnil(gK2)1/2

'c='c (Kl' K2

" 1" iJ

·s3 1.,1

=~ ^f r ,' ^{J 'fe.}

• 7tAeltpl

m ,m ~~"; i"-1t R (U,VI 'c=-Ae

.

'Fo([b'~ +JcIFo~rc'~c ⁺

^!lie

⁺ 1tIFo(~:';c

^-t

Itl.l\ .

20t -I. '

• Qc·(K!"·}{2"c)Kt K2 ^-l .-t

•Ke,(1.' f

cosh~oj.l~et.l IJ.l S(mKl )~ lm' K21 S (U) O(V)

" diedOdV di2 .CHAPT~RJ

,

,

oSECOND:'O!IDEROFF-PATCHSCATTER

where

',J-

,

16

.3

^{.1.1 Reduction}

^{of Int egral s}

The cross section

expressi onas given-byEq. (3.1)

consis ts of

four

inte9r~ls . ^It

^{is noted,}

howe~er,

^{,t itatthi s}

^expressdon

^{MS two}

Dira

c:-delta

functions

wh~ch

may

beused to reduce

the number of Intecretions.

Since w

e

kno

w that.- -

t t !IX,YIOIXIOI~I ^{d y dx • ! IX'Yl lx.o}

a c

, .

y"'O

provided a'.

$

O

'

s: b

c

$0 S d

Using

zq,

13.6),we maywrite

sq.

13.5)as

Ifwe

repf~~

tbefunctionwith i n

J

}.'DYf('c"2)'we.J¥Y rewriteEq. (3.9)as

lU I

~'3IWdJ~· ~

'/\a IFpl m,m

^t

·

f

1 i2'"

r )

ie'./\a^{r {I'olrb''I\,}

⁺

^iel

,"olre'o/e

~

ⁱe

+ 'J '~lr.,ie ⁺

^',112

2-1'-i ' .

QeIKr'2,lel'lK:1

SI~lJ Slm:~2})

Ke ldet llJI

.,3IWd)

, ~ ,

^{,.E •}

r r flie, I1.)~~e di~

^13.10)

'/\al'pl m,m.' .fl'i2'" ie'-/\a " ', ' - ','

1

• : ' , ' 2

'f li e, I,1• I'ol rb''!\,

+

lel' olr""o/e

+ Ie ⁺

"' ol[.,le^t

'1.1

24 4 ' - \ ,

gc

^{Qe lKl' K}

^'~ldetllJ

^{2,l elK}

^I

^l'2,SlmklSI1

^{>r;.~2 ,~}

^{I ,' 13,111}

,':..rcr narrow be eption, theinFegra~att hrespectto 'c .

can be'approx~ted

by mUltiplying"thevalue of theintegrand at the' I

centr~

^{poi nt'}

^by th~.

^widthGf'the

li~t.s:

^.For.a

be~ter approxi~tion;

Uie-liJnitson

t

_{c c}an be di videdin:n pert sand sothegq.1(3.10)

may

again,

be

wdttenas

,t

-CH) <\! -In -61<\!

f -·--f C9c.92I d 9c

t

f ^{- n-fC 9c.92 Jd 9 c,-}

-In-2 )<\! '- In-4 1 <\!

- n - - n -

"

t....t

' r f(9c'92Jd9c] dj2 ' . (n-2!<\!

~

Usi ng.the·re,ctangulaY

r.Ul~·

^to.approximatethe.

~boV~ il)~egra-l,

^we

nay

write .f. .,

. 2<\! '

. 2ko f:-n l . . ' r' [ ^{{-cn -l l<\! . }.}

.\ . J

· sl.( '\jl · ,;'A_IF I" 'ut=±i_ , '

f.:-.'

- n . ,92

., r .: ."

.P •

9 2--'

t}

f ln; I , <\! ' 9 2}

t

ifn-n 51<\!

.92}

3.1.2 'JacobianofTransfoDllation

The 8acobian of the transformatlon.may be given as Il.lll

•

J=

aU au au

aKI 'dK2 'dft

a v a v _13.HI,

~.

'dK2

a~c

a~c a~c .

. _{~ dK2 dft}

13.. 161 ,

'(3.15"

\

... ....

De rivat i vesof V

,From Eq. (3 . 3) we~ypbtainthe following.part Ietderivati ves

of

vasav _ \ ; t dKi- p)

^I

~lm,:g:

01\2!..

)h 'av '

dft =

⁰

\ . ' .

,Usi.ng.Eqs•.(3.15,I.to(3.1-1),_~emay-:rewrite the Jacobian as'

Using Eqs.(2.1 6) , (2..17)and (3.22)we may write the~

fOll~wing,

.

.

^{- ~ .}

derivatives as

...

"I ' • ',p

'h.'K e

=

[C6S'c{2Sin'1 ⁺ K2-sin'2(sinh~Ocosa{) ~

" I , '

oK

1

, ,a ' " ,

^,~

-

COSh~~S!inao .~).

^-

K2cos~2.(COsh~osin~o .~

+ s-inh~ocasao ~J) - 'Sinl;\2eO~il~ ^K 2eosl 2

, , ' , " '~ '.,' ", ,a '

(s~n~~or;~,ao ~ .'~ cosh~osin(lo ~) ..+ K2sin'2['~OS~PoSi'naci ~

^t

SiIlh~o~~1!.ao

^.-.'

,', . , I" "

a .o l ) ]

¹

' .~, "c

'.= [2Si~lil ^{- l ei ; K}

²

^{;:' II 2 •} le)[SinhPoeo~a"~ ^.

a. , ,

- 'eosh~osinao di )- KlOOS 1 1 ₂ - lei

, I . '

', .i"

, [eOSh~oSi~ao~ ⁺ ':inhlloe~sa~J]i:: ^{• '} 13,231 ,~

'~I ^{.' ',} ·: . )(L,e. ' ^»;

S' dt" [eosle\Sinl2 + si"2eosh~oeosao ^{+ K2sinl2 \.}

'J

² , , . ,;

'..: -,-~

.".- '.

13.241

,I

, ' a~ ' " a.

"

(sinh~ocosao ~

^-

COSb~o~in~o ~]

, ' , ,~

- 'cos.2si~~osina.o

^-K1COS9i(COS

^{osin a}

o

~

C- ~- .

²

- . .-' + Si~h~o~~~ao ~)}. ~

^{.ti nQc{COs 2}

⁺ COS.2cosh'oeosa~

^.

, " . ^,~o " ^{" '·0 "}

+ YOS,!2(Si~~_oCO"'0 lKz - s~osln·o dilt ,

:

sin.2s!~posinao

_{,,' , "}

⁺

^{K.2S·int 2}_'

[;OS~oSi?Oo

_{,. ,}

~

_2'

~ ^sinhP

_,^{o ,.''"}

, i,'""}] Ii"

',cos·o dill ^X; - !

. [;;nl~ ^{'- I c} )(; '+ cOs~ocos.6! ^{- cOS112 :} 'cl' im:~o~in.o "

+ ~Si;I'2./lcl (~i~PoC~:.o ~- Cos;~osln.o ~J

, . "

' ,~

- IS co' i!2 -

_,

Icl(COSh~o'ln"

_{" ,}

di

₂

'·oJ '] 1

+Sinh~oCOS~o dKi ~ ,

Deriva~tivesof

K {

^{-l " '} f.The

ma~nitude

^of.

K~

^{a,sdifined in}

~q.

^(2.18)may'be

~ri~ten

as

r .

\

"',[{2K1X

+ ~2x(l ⁺ cosh~oco~ao)

^t,

K2y9inh~osinao}.~

~ ^{... . +}

^(2K1y

+ '~2y (1

^it

COShPo~OS~~)

^-

~2XSinh~oSinao}2p/2

^.

.

/ .

, .- -

_[ 2 · 2 2 2, 2 2 2,.2

- 4Klx

+

K2x^·tK2xcoshl"oCbS0.0

+

K,2

y

^{si nh"O~l~0.}0

f' + 2K~xCOSh~o~0~ao ,~, 2K2xK~ysinh~~sinao

.

~

.

+ 2K2xK2yCOSh~ocosaosinh~osinao .

.(

⁺

^{4Ki y}

⁺ K~y + ~~yCO~h2~~c~.~o ⁺ K~xsinh2~o9~~2ao ⁺ 4Kly~2y

.

.

.

. (

.+ 2K~yCOSh~ocosao

-

2K2xK2ysinh~O~inao-

.

'714Ki

~ 'K~

^t

KlCOSh2~oc.os2ao't K~slnh2~ osin\

/

:. ....

",

/ f

'''\, (3,30)

(3,311

13, 3')

13,33) (3:30)endsoare

, :.. ,

IsinlllKjsln~, ±

'1Kl1iK; cos'j , K,cosl, l ,

IKIC,J"'I 'K,co'hCOS',)} !']

,'

sinaoc~SI~, ^- ~I)

^t

'KI~2"jPl', ^- 'y [COSh~oSin~~

aa _{o J]}

, slnhP

ocosa

n d+;

13,'91

(

Derivat.ives of ~o " /' "

.

~o

is the solution of tn'e quadratic

equ_ati~n

^.•

^(2.2iand

^{w'il l}

have two values for eachval ueof _cosh~o and_~ybewritten. as ) .

Out

o f

^{fourroots of} ~Olwehav~totake.~nlY

one

rootfor which Cis,real' and- C,>1 andwhich satisfiesthe Eq. (2.26). Further

rroa sq,

(3.30)"wemay.writ.li,.- ../

a~o ,;-;

If , _ I

~'

., {IC"J,' • 1)1/2 - SIiiIifo;

' /' ' ,J

Toobtai~the

derivatives 'of

',~o' wemayuse Eq.

able tow

r i t e the:derivatives

as follows .where'

.. .. .

.13.351

where

~

is defined inEq.

·~(3.32) ."

^.

De ri vatives of

c±

Inthe

all~ve

equations we still have

t~

^findthe

deriva~ives

^of.

'ct

which maybewri tt e n fromEq.13.31)as

f,~.lll?WS

^.

,ct _ { '..

2 2, . ,

i ' 2

i

~i .'

~-. (4.Kl~OS 1.+.4.K1K2.COSlCOS 2 -:K2s^~ 2]

1±2Sgn{2Kl:os9l+

K2.~OS92H3~costl.t ~

K2eos, ;)

. I

. {'lco. 2i

1

⁺

';COS!.lCOSi2 }112 ;"KI·I2K'CO"1

+

K2cos'2ICOsOl{Klcos29~

^{+'Jt2COS.1Cost2r l/.fJ}

•

1"~ISi~t21

^±

2-1Ki1 2~lCOS'1

⁺

K~~~S~I

^.

/

\.

Simflarly.

a~

^t:

{[4.K~eos2.1

^-+4K1K2COS'lCQs'2-

K~sin2'21

'"2

· .:

{KICO~2~1

⁺

K2COS'lCOS~2 }1/2 ~ ~-JKi1 2~stl

^tK2

·•COS'2IcostlcOS*2{KlC092'1'tK1COS'lCOS'2}-1/2l

-

.tK~ lsint~1

^±

~~12KICOS' 1

^t

K2COS~2 1{Klcos2'1

t

K2CO~'lC~·S.2}1!~]

^"'.4Klcostlcos'2 -

'2K~si~2'2l }

·.

co~.'~S~~'l .t -K2sin'1i:os'2}{ Klcos2~

^,+

K

₂eos' l

~

. ..

13 .411 13 .401 13.391

'", j,

. 's;n~l cos~;I } Isin~21

14Kicos2~1 + 4K1J(2~OS~ lCOS~;

^-

J(~sin2t21 2

nerfv etieesof ao

UsingEq.(2.271the derivati vesof ao withrespectt~Kl'

K2. and '1 ~ybeobtainedasfollows 11

aa o t;n~2sech2~; a~~

~ ^"

^I

⁺ t.nh2~otan2~2 ~ aao

._

tan~2 ^.. eh2~o a~o : fK2 -

^I

+ tanh2~otan2., fK2

aao

' .

tan~2seeh2~o"

^,

a~~'

dIl"

^I

+ tan~~otan2~2 dIl .

Derivativesof"; .

Inorder tofindthe derivativespf ; asrequired-to calculate'det' inEq.(2.2~1, we mayuti lize

sq;

(2.23).w

hi ch"may,

alsobewritt en as

I,

\ \

\ I~,a) " ~

^{[2KI, os}

I~I

^-

~e) ^, K2sinh~ 'Intt _,sinl~2

^{- '}

^e)

I ! '

^K

211' eosb] eOS~~CPSI~2 -. ~cl l

^"

^13.421

To obtain aboveequati on,we haveusedthe,followingrelati onshi ps

4 •

KI

.

K e • K

1K

eeo ' l' l - ' el \

v •

4 •

K1 . K e • KI co, 1'1 - iel , -

,

Similar ly

1 3.4 3J

J , a

²

e '

I (K" h" .

.

'

-

. n •

\ 'att =

~

•

2⁵¹⁰II

smu

S^lnlY2-'tel-K2coshl-'

~ ;, '~ ^{(2leo' h p -} 2lKleo'l~1 ~ I e' ^{t 'lc;, hP -'21}

C p'

II

t co,h pl , •

l .

K

2

^{,II -}

co , alc08ll 2 ,- Ic,l - K 2'i nhp 'ina

, .

1 3.441

maybe writte nas Now fromEq. (3.421 derivative of

. e osu c08 ll2 " cll 13.451

' ~ . ^{II t :08hp , 2} IK2 I1t ~ •.,inl'2 - Icl" .K;:i:h P'

1 3.461

Swtmlarizing all the

requi red dertvettveeae

given in Eqs. {3.151 to i3.41jrnaybeused to calculate the ' Jaco~ianl,

.whil e Eqs.

(3.441 to

(3. 46) may be used to calculate

the

'

det'

.

',.

CHAPTER4

RESULTS ANDD~CUSSIONS .

t>•• • To qenerete the off pafch

.

^. spectralcrosssectionatanyDopple r:;

.

frequency, the'sum ofInteqrala asgiven in Eq:(3.13)has tobe

." ·eva~ted. The"funCtion ' f(~C"2)

^{isdefin edInsq. (3.11): The}

~

^beamwidthofthe. receiv ingantenna isassumeJ..tobeas6⁰inall !he conputationa.

So ~ "" 3 °' 9£ .ie "" m

^{radian . Als o for}

~onvenience

n'isassumed as

e.~ua~

^{to;.''therefor e the}

cro~s

^{sect ion}

e~pres~ion

^•

as givenbyEq.(3.13) :maybereducedto.

, .

a { mi . 4k o ' r. r [fl~,n

53

a

S; 21Fp14m,m'=±l;2""-'X " " 2

t f(':~ " 21

^+.fie"21

+f{~, l~i .

{4.1I

, Inthe aboveequation,

f

_c has assumedthevalues as

~ ',

~

_represent

. "

^0,_these

~

_{val ues,}^.and,'/ then it may

~.

^Ifthe,vari ablebe\oIIliUen'

-

as.'91 ^{is used( to}

(4.21

.

~her; ,

^W

fG ,a 'su~3 titution: r~w

Eqs:"(3.61,.l3.'(',and (4.21

J!!a~ be

'';":.. solved to obtaiq KI,K2:an~,^(...~L^{' __for',} .gi ven values9f.

'2 '

^{~, m',ling'}

,

^.

ok

But these~olutionswillberadar'frequency-dependent. 'Doavoid.. this.problem., all.the three equations arenormalized toradarfreq~ency andin thisway the obtained solutionsmay beused.ttlgeneratethe t.

erose sectionIcr any

'rad;r:fr~quency .

^'

Afternormalization,sqs. (3.£), (3.1}and(4.2)become

"

(4.3t

-Ji'

⁰

.

^{. '}

.

-. .

\ W=tan

_ir2K~sin~1+K~siri~2 11tCOSh~oCOSa~) -K~COS'2Sinh~o~,lna.o]

^.

.

l2K~COS~1+K~cos'2 (l+cosh~ocosaoJfK~Si~h~osinaosin$2 ,

where K~⁼Kl/2ko·and'-K~⁼K2/2ko arenormalizedsavencroers.

11. ;;:Qld/Olg 15the'n'o~lized.freque~cy .

Eqs. (4 .3) , (~. 4)

1ii

^{(4.5)ll}a¥now}besolved. Asitis ...

.evieenefrom'

E~.

^(4.4.

~t

^{_Ki and}

K~_

have a direct relationshi p, 'wemay substitutethe expressionof

K~

from"Eq.t(4.4)info"Eqs.

~

^.

(4.3"jand,(4.5)

~

So ultimately

ther~

^wil,I

~:

^only

t~o

^equat.Icneto

<be'solved. • :...

Toobt ainthe.

's~ lutions ~;

^Eqs.

^{i· 1.31}

^and

~ .s) 1

^a

tech~ique

^{is chosen.}

~he

^ranges

for K~,

^{'1 and}

For

~ach

^{valueof *2' the w}hoLe,range'of

'ti

issca.llOe'!y.lmlm.LY

for each/va lue of

'Ii

the cgmpleterangeof

' Kt

is >Ji;annect to find the acceptable solutions ..The

rang~

of

~2'

ischosenas _179°,

S ~;.

S ;179⁰ wH,hthe incrementof 2°', The range of

h

is taken as -1800to'l 80o'wf eh 1°int erval.

K1

is'allowedtovary between 0.01!nd'S.owith varyingincrement. The increment for Kt between0,01

~nd

^{0.1 is0.01 and,between0.1and 5.0is}

e.r.

Someimport antsymmetricpropertiesareinve stigated alllOngthe equationswhichmaybevery.usefu l.-insa vi ngcomputa tiont~m~whil e obtain~ngsolutions;.These tbreesynir.etricpropertiesmaybe'

s~rized.

as foll ows. 'iI

(1) Letus:c~ns~der.Eq.t4_.4). Wemay also.writeitas

.

_J(2

o

""~

·(-i- m .jK1 .1~

e' ~y.beeither'+~ or -1,'w~mayalsowrite.the above equation"as' " .

.K~^=''(1\'

+

m~Ky12

rnt jieabove'equation,Ifthe sign'of 11 and m are changed

at.

^the

..same time

~hen'

^{,liI'te.value'of}

':",K~,

^wilt:

r~~in

^{the.same.,This.mea ns}

)hat the set ofSOlut ~o~,s.r c c -11 wil lremain the~n\easfor +11 or viceversa exceptthat-

tJ:le:

~ign.of'-m.willchange,. . (2)

~t

us'e xamine Eq:'i·4.•:3 )'.,'.In,

t~H ,eq'u,a~'ion if

thesign of ,';, '1 and

Po

are changedat thesame tunethenthere

-n i

notbe

anY·.Cha~g~ 'in

^U,

Th~S indi~~tes t~,at

if

the

signol'

~2 i's .

.

'~ ChaOged'~he~ , ^t1 an~' ', ~o ~ili .alSO c~a~g~' th.~~~

^s'1go

wit~

^other

values,remainingthe saee. . .- . .

. • .

;'.

. '

~. ^,"

(3) Lastly if~inve s t i gat eEq.(4.51.it isevident thatif .2'

¢II' Po and.:91 changetheIr

sig.n~itmJltaneous1y ~hen w'

willalso ChangeInsdqn, In conjunct ion wit~t~e.above:s}'llllietty this will implythat"ifwehavesol utions f-or"¢Ie-8i'"0,thenV(!can generatethesol ut ion s for .c+81⁼0' orvice versa'by changing the s{gn of '2'.1 and"

Po'

Itis ~rrportantto remember

in

all theabOveS).mnelryanalyses that 110 stnalways.~'positive'as tiefi~edin "tq. (2.2fI)."

Therange of ~ ^ischosen"f rC(ll-3.08 to3.08wit hthei~ciement

Of ~O"04 . SOlut·ions· az:eg~ner.-ated ~fO~ · l\ va[yin~ from.~3 . 08 - to

^0.0

''r~d

^{ot he r}

h~if ' S~iutiOi1~ a~e

etu£ddY,obta i nedbyUS!nqjhe

·fJ.;~t

symmet rypr ope rty . Also the sol utions

n ave

^{to.begenerated}

for

^{five .}

,9i!fete~t ~ases '

corresponding

.to suinmat~on

^offivediff ere nt'

-f~nctio~~

·in thecr~ss.sect~onexpressit?n~_.Soluti onsaregen~uted'onlyfor

·three cases.and forthe other two cases theaclut icnsare obtained by useoft~ethir dsymnetry~rty. .

'.The strateqyinsolving:the:twaequatJ.onsis to take a value .

.

,.~

each for .,TJ and'

t

₂ within the designatedrange and then search for

pai~s

^{of +1 and}

~

whi -eh IIi11 satiSfY'

th~

restriction-on

~Q

as giveninEq•.1.2.?6). As~as.p~ev'iou!llybeendiscussed_~each

·value of

. 1

^weare scanning thecomplete rangeof ~l' s~atone

• • .,- • >0' .

particul ar

.1

^{theremaybemany K}l whi chw111satisf y the :r equi redrest ric tions.

F~rther ~~een t~ c~nsecutively cibt~lneci

·values

M Kl · ·~e

test th: cha ngeof

.~lgn in.

^{Uand W. -Thi swill} ensur~,thattll~re^{exists asolut io.nbetween}.th~t:-,ovaluesof K~.·

•~ncethe s econdi~iori~'aremet,anI~SL8~routineisu.se~tominimiz~.j