th er

On The G r ound W ave P r opa ga tion Over A Cliff

With Finit e Ground P arameters

by

®Ha iyill Wang, M.Eng.

Athesissubmit te d to the Schoolof Grad uat e Studiesinpartialfulfillment ofthe

requirements fo rthe degre eof Master of Engineeri ng

Facultyof Eng inee ri ngand Applie dScience Memori alUniversit yof Newfound lan d

Decem b er199 2

St.Joh n's Newfound land Ca nada

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Abstract

nil',I,;orad,·rislj,·sf,f!lwJ.!;rolllld-W;l\"f·pr0l'ilgilliQIIove-rH.rli fTarcinl"('stigilu'<l lI_siJl~1,,,lhtill'1""sj,IIl";,udIlli 'compensationmet hods. TIll'1\\'0methods are re-

\·j,·w!·,1;111<1n'IIlI"lrl·,I.TIll'tIlI'oril"alill1<1lysisfortill' grOlllld1I'f1\"('pro pllgilt illg from Sr'ilI"";lf ll!ispn's" lIl"dill-h-tail.Theresiduemethodr-mployr-sFuu ril'Ttransform

I,'d llli'lll",TIlt';I1I<1 1.ysis oftll('two methods11.1'<'reilltroducedill,1meredarifil·J wily.

TIll'ditr"J'l'Il" I 'SIW(\\"'('1I1111' assumptionsofl'-'two methodsnreiden tifiednndcom- pari'lllllsof tIll'Tl'lill,j \" ,resultsere Illustrnted.

TIlt'll;l[·cJiIrapproxiumtion is uSI'1Itoobtain theterr ainroofflclcutso thatthe pfolol"lIl f'llllIll' solvedwhenthe antennas011 theclifftop arc dose to tileclilfedge.

'I'll<'finalr-xpn-s siousof te-rrain coefficientand fielddistributionOVNrhe cliffaswell

;IStill'seann-pn'S/·II11·(1.

..\11 approximutcanal ysisexpression, calledthetwoterms approximation.ispre-

S(' lI(" ,[illthiswork to f'llculat,.,terrai ncoefficient.Thetwoterm sapproximation

islo"s('ilon1111'lIat.-elif[approximation of (lrcentheoremmcthc-l.[tsimplifiesthe 1I11111"ri",,1calculationoftheproblem.

(;r.lp hsarl'showninthisworktodcscribthefield s cha nging over the oceanand

OH'rtill'dilf fo rthetran smitters011thecliffandon theoce anseparately. The

nuuu-rh-alresults areillagoodagreement withavailibleliterat uredata.

TIll'('Xl'tt'SSiUllSand n'Sultsgiveninthiswo r kcanbe usedto determinethe optimumlocationand11l'iglJtoranantenna overaclifT.

Prt.'s\'ntly.aprototype bislatifradarsystemwhichusesgroundwavepropa ga- non forO\'(' rthr-horizon target dete ctionisbeingbuiltonacliffatCapeRace,

:'\t·,,,rOUI\,Jliln,1.ThoTitJ;Uhas .,:u~I't'ri", li(oIrr,l yauu-uua;,~lilt"Iran~l1Ii tl"ran.l itlill"'.uarrayas a rco-ivr-r.TIl\.'results.,Ial.land1110'program.I:i\"o'liillthi-'h.r~

IITl'\-"'ryhelpfulill<I('jOi~nilll:11,..11 gttll1n d\\'ill"t·radarilUIt'lIl1ilStillIllO'dilrsin'"lllO' programcnu1,1'IIH,dtod<'l,'rlllillt' Ih,' opriruumlocariou IIlldIIO 'i.c,1I1"fIII, forboth thesourcealldtill'rt'n,jwT.

iii

Acknowledgement

First of all. I wouldlike to than k my supervisor Dr.Joh n Walsh for suggesti ngand support ing the research presentedinthisthesis.And I would liketo thankmyco- adviserDr. S. Saoudy{orbeing an inexhaustib le sourceofideasandencouragement throughoutthe researchdura tion.Itwasmypleasure towork under their guidance.

Iamgrate fultoDr.Sinha,for hiscourse,Antenn as and Propagation,and for encouragement tomeduringmystud y atMUN.

Itakegreat pleasure in acknowledging the instructionsI receivedfrom Dr.

venkateeen,and the assistance I gotfrom Dr.Son Le-Ngoc.

IamgratefultoProfessorKeqianZhang in Tsingbua University,China,Professor ZhenjunVan at Universit y ofScien ceand TechnologyofChina, andSeniorEngineer YunshangYe atChinese Acade my ofSpace Technologyfor theirencouragem ent duringmy graduate studyinCanad a.. Their timelyrecommendationsmade my studyat MUN possible.For these, I am deeplyindebted tothem.

I amgratefultothe Faculty ofEngineeri ngand Applied Sciencefor providing me withfinancialsupport in a formofGradua teAssistantship.

Than ksare alsodue tothe staffofCente r for ComputerAid Engineer ingfor their kind helpwhichmadethe researchpossible.

Lastbut not lust,I wouldlike to thank mymotherand myhusbandJianming for the encouragement and lovethatIreceived from them throughoutmy graduate st udies.Especially, IAmindebted to my daughtor Zhe for her understandinga long time separ at ing from herperen ta and myabsenceof a mom's duty.

Contents

Abstrad Aeknowled ge ment Cont ents ListofFigures Nome nclature 1 Int ro duct ion

lv

vii

1.1 ResearchRationale ....• •..•. .•... . . ... I 1.2 PreviousWork

1.3 Scopeof Tbelli,••... . ...•. . ..•....••.. ••... . . 2 Gene ralFor mu lae

or

Prop agationOver theIn homogeneousEarth 9

2.1 Geometry

or

TheCliffBound&/}' ..•.. ••.. .

2.2 Field ExpreseioneInDifferent Coordio" teSystems. ... 12 2.2.1 Earth CurvatureModifiedCutesia.DCoordinateSystem 12 2.2.2 SphericalCoordioilte!ll ..•• ... •• ..• •.• . ••..•. 14

2.3 Analysis ofAVerticallyPolarizedInfinitesima lDipoleoverGround

[HomogeneousCase) 15

2.4 TheInhomogeneous Ground Problem... Ii

3 ResidueMethod Approach 20

3.1 Terrain CoefficientforanInhomogeneous Problem. 20 3.1.1 WolveFunctionfor aHomogeneousEarth.

3.1.2 WaveFund ionfortheInhomogeneousEarth . 3.1.3 TerrainCoefficient

3.2 ExpressionforOne ShortSection

21 28 33 38

4 Com p ensa t ionMethodApproach 39

4.1 Analysis of TwoSectionProblem .•. •.... .... ... ..39

4.2 Analysis oftheInhomogeneousProblem 43

49 . . . .. . .... ... .. 49 5 FinalExpressionsCorClift'Case

5.1 Terrain ofLong Distance Sections

5.2 FlatCliffApproximat ion , , .•,

5.2.1 PropagationfromLand to Sea ,, ,, . , ..•. 50 5.2.2 Propagation fromSea toLand, , , .. ,. . •...•..•.• 52 5.3 TwoTerm sApproximati on.. ,.,,. ., . .. ..•. •,. .,." 52

6 NumericalRes ult s AndDisc ussion s ~4 6. 1 NumericalResultsforRadar.. • . .... •,.•• , , ,•••. 54 6.2 Two TermApproximationand FiveParamet ers..,.,••• ,,•. 56

T Conclusion

t.t TheDifferenceoftheTwo~t~thoc:ls. 7.2 TheSimilarityoftheTwo~It'thods.. 7.3 Two TermsApproximat ion.. Appendices

ACoordinat eExchangein VectorTh~ory B Derlveuo nortl.eFunct iong(t.r...l CDerivation for Computer Progr a m

C.l RootsI",.

C.2 ErrorFunction ... •.... . ..•. . .. . . • . • . • . . . 72 72 j:J

7,

7'

77 79 79 81 C.3 AiryFunction• .... .. .• •.. ... ...•. ...• . •• 82 References

vii

83

List of Figures

2.1 Form ofthree section terrain.. 10

2.2 Form of two sectionterrain. 16

2.3 Integrationdomain ofeq.(2.31).. 19

3.1 Integrationpath of.\jfor eq. (3.43)..., 30

3.2 Ceometryfor eq.(3.57) . , . . 34

4.1 Homogeneous sectionsfor compensationmethod . 40

4.2 R the distance between source and receiver... 4.1

4.3 Formofinhomogeneous sectionsforcompensat ion method 45

6.1 comparingEo in the two motheds . ..., . ... . . . .•. 58 6.2 variation of terrain coefficientwith the height ofreceiver onsea ., . 59 6.3 variation of terraincoefficient witb the height ofreceiveroncliff... 60 6.4 variationofterrain coefficientwithd.oncliff. •.. .. .... . .. . 61 6.5 variation ofterraincoefficientoverseawith d.. ..• . .. ... ... 62 6.6 variation of terraincoefficient over sea. withdp ••••••• 63 6.7 variation of terrain coefficientoverseAwit h clift' height .. 64 6.8 variation ofterraincoefficient overse&wit hh,... .... 65 6.9 variation ofterrain coefficient oversea.wit hhp•••••••••••• 66

viii

6.10 va riat ion of terraincoefficieDl0'"« cliffwithd... 6i 6.11var ialion ofterraincod ficientevercliff wit hd~. . .• . . 68 6.12 va.riationofterrainc~fficientover cliffwithc1iffhei ghl. 69 6.13 variationofterrainc~fficientoVl!r cliff withh... . ... iO 6.14variationofterraincoefficient over cliffwithh.. .• . .. .. ... iI

Nomenclature

A h h.

h.

'.

d, o E H Ht (: ) f A o Ai(: l,B i (z)

terrain coefficient cliffheig hl

theheightof thesourceantennaover earth surface theheight of thereceiverantenn a overeart hsurface the avera geradiu s of theearth (a =6 400km) the wave numb er ofthe free space thewavenumber ofsect ionj theradius of theearthinsectioni,inmet ers ... the dist anc e(Ifsectio ni,inkm

thenumericallengthof thesectioni(e;==(~)(¥)lJ3) theelec t ric field , volts/ m

themag netic field,empe re/ m

jt bkindHankelfunctionoftheorderiand argume ntz frequency,cycle/seorHz

the wavele ngthinmete rs relativ epermi tt ivit yof the ground conduct ivityof the ground,

vIm

Airyfunction of argume nt z error function diffgain

numericalheight(y,==ko(r, _aH*, )1/3) .. . z-ccrnpo ne nt ofthe magnetic vecte r potenti al

heightgain funct ion BUrfaceimpedance,ohm mutual impedanc e, ohm elect ric curre nt

scatt eri ng coefficie nt ofsectioniand sectio nj

Chapter 1 Introduction

1.1 Research Ra ti on al e

The problemof elecrremeg net icground-wa veproi'~gAlionOWTallilihollll)I;<:!l(,('Il~

eart hhasattractedmuch inte restforAlongtime,foritisrc!c li\llltorommunic«.

lions,navigatic u andappliedgeophysics.

Wit htheadvancement of cceenexploration.rndur isill'Tf',,,illl;l,'111M"!tfl,I.··

teett" tgel lSAt sea.and collect environme nta lly relat ed JOllaIrcmtheU(:" ,111surfac-.

Often,thelandbasedradu islocatedon acliff(Itbe achand~orksil thi~11Ire- quellC)'employinggroundwavepropagation mode.Thus. thestll.ly

o r

Itmllll'l waw- propaguion ever acliffbecomesofgreatimportance.

Presently..11.prototypebislll.l i<:raJAfsystem whirhll!>l'logrullil'1WAV"!'rlIplIAilt j' llI modeforever-the-horizon tu getdetectionis beingopera tedunitcHITatCapeIlil(f~, Xewlcundlsnd.Theradarhasa lugperiodicarrayantennaMllwIrlUiSmiUI:r,!t11<l,l lineararrayantennai\l!i\receiver.Boththe transmitterandthe receiver arelut::ttf;(J onthe cliff.Inthisthesis,ourattention will focusontheground-wavepropagation betweensea 4I1dlandinbothdirections across thecliff.

TIl"~r{}U11!1WIlI"-I'rr'f' Ilj.\illioll OI'f'r,1cliffis aproblemof ware propagation over illlillholll('J.\'!!W'IIIJ;,'arlltsurra",>whichconsis tsof two~ect i ol\swithdifferent heights

illld,'I,-,:Ir it'al l'rorw tt i,-s,TIll!WiI\"( :pathindlld('~la nd andSC.1,The landishigher

1.111I11 "r"'l ilal10~N·l,-\"(:I TIl!'landnndseahan' different electricpermittivitics itll<l '·oll,lllclivit i" s.

1.2 Previous Wor k

'l'llt! Ilenzporcut.ialdill'to a verticalelectric infinitesimaldipoleatthesurfaceofa splu'rical,l.omogc ucoa, carthwas deri vedbyvander Pol and Bremmerin19.1S[II.

\Vait [2J(k l'eloperitileseries expressionsof the elect romagne t ic field component s of lh,~aboveCilSC illterms or magnet icrector potentialin1962.

ManypAp':!"Shave been publishedon the problemor groundwave propagation over nsmoot h«ar t h(nover-ticaldiscontinuit ies) consistingofseveralsections with c1ill"l'I"('ulelccuical properties.Among thevarioustheoreticalapproaches (0the prohh-m waslinanalyticalapproachinitiated byGriinber.e;[:1)and developedby 1·" inlwrll:[-I.

.'jJ.

III Uriinbcrg'swork,theadoptionof approximate boundaryconditions anda st nudar d applicoticn orGreen's them-emledto an integral equationfor the normal component or E at the surface ofthe' earth.He solved the problem for thecase or two ci\rth media,whereoneofthemhas infiniteconductivity.The two media were Sl']lilf,,(cdby a~lri\ightboundary, The incident field was taken to be a plane wave, As an approximatetreatme nt,Grunbergconsidered that thedirectio nofthe wave propagation ata greatdistance beyondthebound a ryis the sameas thatof the in cidentwav e.

Grunberg'$workW;lS generalizedhy Feinber g

1·\.iiJ.

InI,,'i tl lwr!,'~11""1"1.,II\<'r"

was a differencein thatatrausminor was IO""tl'datnlinil",lisl;11lfl'from 1\l<' boundary. The expressionsthatwere derived too kiutoaccount1h,'variolls l'0sili" II' oftransmitt e rand receiverwithrespect10th ....bounda ry.

Feinberg[6) emphasizedthat ,forasmoothearth ,thel"ITdillillh"I Il"!<'-lU'ili,'s,I"

nothavethe sameeffectforallpartsofthewave pathbutan.'.I" pO'III],'111"11tln-ir locations,\Vhenthe inhomogeneityexte ndsonlyfor a short.r1i~'alln-"tJl.hal Llu- flal-earthapp roximat ion can be usedfor thatpart,itseffectis mosts,-dollswh" ll the inhomogeneityoccursinaterminalareaofalong wavopat h.whileits',-lr"cli:>l negligible whentheinhomogeneityoccurs inthemiddl ePII:t.('tJlI\'f-l'.s('I ~',110ru.uu-r where an inhomogeneous areaoccurs inthewnvcp ath,itseffecti~M("((tlllilal",1when it extends overa long distancealongthewavcp atu.

Clemmo w[71showed that thecaseof asinglelalld/S(~abO( l llrla r~..rIJril11;11"M!h was adequately treatedbyanintegralequationmethod appli"d toi~dilrr;u:ti"n theory.The land/s ea problemwas atwo-dime nsiona l problem wheT!'U",Sf',\lVa~

replacedbyaperfect ly conduct ing, semi-infinit e planesheet:Ilnollit,!tr,lllsllIiUf'l was,\vertically pola rize d line-source. He took it asagCll(~rali l.al iollofL1ll'(Llssj..

Som merfeldhalf-planediffracti onproblem.

Ina late r paper [8J,ClemmowgaveII.theoretica linVf~st igat ir'lIofwrtic" lIy 1'"

larizedrad io wavespropa gatingacrossa.boundarybetwee ntwo ('Ml h scct iuus with different com plexpermittivitlcs .The problemwastreatedMatWI).limcnsicn alo nc.

Theearthsurfa cewas assumed to beflat.Hesolvedthe bound ary value problem fora planewave incidentatan arbit raryangle;thescatt ered fielddue tosllrfa<:e currentswasexpressedas anangular spectru mofplane waves, and this formulatio n

led todual integralequations solvedby themethodsof contourintegration. After consideringthe case of both thetransmitt er and receiver at same ground level, he gave an approximat ionfor an elevatedtransmitte rand receiver.Theattenuation and phasecurves ina numericalexample weregiven.

Bremmer[91gave the extensionof Sommerfeld'sformulafor the propagationof radiowavesoveraRat eart hwithinhomogeneoussoilconductivities.The integral equationused by Bremmerwas based on Green'stheorem.The solutions for two adjacentregions of homogeneouselectricconstantsweretrea tednumericallywith theaid oftwo differentexpressionsfor the fieldnear theseparatingboundaryandfor the field farbeyondthisboundary.Therigoroussolut ionof theintegral equationwas provedtobe identicalwith thecorrespondingexpression derived ina verydifferent way by Clemmow[8J.

Therearetwo othermethodsto solve the problem ofpropagationover aninho- mogeneousearthtaking earthcurvatureintoaccount,which are the compensation method/101andthe residuemethod[11-15J.

Using aneigenfunctionexpansiontechniqueand surfaceimpedance approxima·

tio nbased on thecompensationmethod, Wait (10]obtained final expressionsfor propagatio nacross two or threesection earthwith thesamehe ights.He alsosolved radio wavepropagati on overaperfectlyconducting curvedgro und[16].In another paper (17],hepresented theresultsforpropagation overtwosectionearth withdif·

ferentheights.He alsoextendedthe method tothree dimentionalproblem.However, his methodand result s are onlyapplicable to long dist ancesections[101.

Theresid uemet hodWMdeveloped by Furuteu111-13].It givestheexpressions of the attenuationcoefficient andfield distribution for a.ninhomogeneousearthwith

different heights anddifferentelectricalproperties.

Since1955,Furutsudeveloped the theoryof propagation of electromagnetic wavesoverspherical inhomogeneous earth,wherethe waves aretransmittedacross one ortwoboundaries of discontinuityseparating differentearthmedia.Byemploy- ing Fourier transform,Furutsuobta.inedtheformulaecorrespondingto theordina.ry Watson formulafIS] fora.homogeneouseart h.He compared theresults with those he gotin the caseof a 8atea rth.Then he derivedthe generalexpressionfor the wave propagationacross any number of discontinuousboundarieswithGreen'stheorem.

Also,theformulaeoffieldstrengthin the case ofaOat earth, fordiffractio nor ground waves by a ridgehaving finitebreadthand finiteelectricalcond uctivitywere derived by Furutsu(121.

Furutsu',formulaeforaninhomogeneouseart h gavea multiplese riesto calcu- late theterrain(attenuation ) coefficient.When the propagationdistancesoverall sectionsofthe terrain werelarge enough, theserieswas so wellconvergentthat one coulduse the first term alone to calculatethe cliffgain. Whenthe propagation distance overany sectionwas so smallthat it couldbe regardedas a flat plane, serlea convergencebecame veryslow and theformulaWMnot available for practical use. Whenthe ter rain was reducedto ahomogeneousearthby letting all the heights and electricalprope rtiesof the section beequal, theseriesuniformlyreduced tothe well.knownBremmerseriesCorahomogeneousspher ical earth .Whp.n the lengt hsof theinternalsections weresufficientlylarge, the effectsofthe terminal Bec:tionsover which the transmitterand receiver werelocated became isolated,SOthe terminal effectsorterminal gaincouldbeevaluated indetail.The terminaJgain couldbe used in thesameway as the ordinaryantennaheight~ainfunction had beenused

f'.r tIl': gro llllflWit\·,)prop.'gari{mover11homogeneo us spheri cillearth.

TI,e·•.I,dlU·I,·g/l.inswerenum..riclllly iIlustrah....11I5Iunerie nsoftile<Jislillllcebe- rweeu Ill<:'j],slllc!c III1lI the path tcrrnin..1forthe typica lexam plesofa ridge.a.bluff nu allolnogl:'lI<"OllseArthond acliff atAecastlinr-,The ob..l/lclr>gaindefiner! inthe (MI.N WII.sthegain"'lus.:.1hy till'earthinhcmcgeneitylindthe gaincaused L,\'the antl'nlt:!.Iwight,The latterWiI~calledthe terminal gaintoo,fortheinhomogeneity inlilt' vir-init)'oftlu- JMtl1terminalwaspart icular lyemphasizedby theeffectof an- 1.omnaheig ht. These obs taclegains tended to becomeconst ants when thedistance incre ased .On the ether han d,atshortdistances, thegains weredirectlyaffected byi\,Iilfradiollloss,interfe rence between the direct and reflected waves,andorher r:lfedswhich werenor oplical incharacter.Aconvenient expression ofthe obstacle g;linWI\..,introdu cedforthepoor convergenceofthe relevantresid ue series,Itcould lit."Il~in thesame \\'ayas theordinary antennaheight gain over11.homogeneous r-atth.St>\'r ralfigllr t'S of rid gegain,bluffgain andcliffgain wereshown illFurutsc's work/14.15].

Based on Furutsu'sanalysis,Pielou,MilsonandHerringpresentedextens ivenu- Illl'rical r('Sult.!!incase ofa cliff(19].In thiswork.graphsof site gain contourswere given for\'arious cornbinerionsofcliffheigh t,path length andFrequency,The veri- OIti01l ofsite gainversus both cliffheight anddistan ce inla ndwere alsoshown,It ap p""red thatcliffheightandoverland losses cont rib utedin dependentlyto the site gai n,If thelosses fromcliffheightand from distan ceinlandwere givenseparate ly, summing thesetwo lossesprovidedan estimateof theexpectedoverallsite gain, These aut horspointe d out that therewere anumberofrestriction sinherentinFu- rut su'stheo ry:(i) a path cons isti ngoftwo ormore short sectio ns couldnotbemod-

Ilea:(ii} section-must haw'vruicnlbcundaru's:(iii)onlyIW,) ,li llw llt i,'I1ill t ' H ;lin heightvar-ietious arcallowedso thaI(o,\slalrefraction dr"l't"ml\l,ln"(II('\11, ,,111",1;

sea 10earthwhich \\,(>r(>computed ern[,!oyillgun[llll, lislw,l"'''l ,r!',.,i''l l ~

1.3 Scop e of T hesis

The purposeof thiswork is to de te rminetilefiel.ldisftibtuion fer ground11';1\'0' propagationeveracliff.whichisaspecialCII~"ofgroundW;t\'!'I'I""p' \J;at iollmw aninhom ogeneous earth. TheIWOdirectionsof prollagat iollan ussIhedill"111"1' considered. Anell'approximationisintrodureel to cakU!.11.1'til<"ll" rmnli/.•·dli"ld distribution.Basedontheresiduemethod, the expressions ofterrain cn..rlit-i"III, ,.Jin gainandfieldstrengthI, clifTC,W)arc give n.AuS"1"- fril'ndly(OllllH1t,'rIII'Ul!r alliis develop edto obta inthe nu me rica lresults for differe ntr:lirrIHlranwl"(S,

The general form ula for propaga ti onOV('T11 homogeneous earthis~iv" 11illr'lp,p, ter 2, Theprincipleto obtainthesolutionforallinhomogeneous l'arl.l,ll~in~h"lliU gcuecusearthsolut ions is described.

Theresi d ue method is reexaminedillChapter :\for groun d11';(\""1Il",ll', In Chapter-l,thecomp ensatio nmetho disre viewedandt'oll1 pi1.n~dwithtilt' resid uemethod.

Basedontheresidu e methodandthe compensationmethod , new upproxitua tcd expressionsforcliffcase are giveninChapter.5.Theseinclude fielddjstriblJlioll .~,difr gains andterrain (attenuati o n)coeffic ients . Thetwo directionsof wave propaga tion, se a-land andland-se a, areco nsidere d. Thenew approximat io n, two termsapprox-

illl;,liolli',,],ta!lwdII)simplifytlwll\lll1e ricalcalculationandredurethecompute r tiurr-.

\\',. ,h",','Ihill, 1\'11"11 Ihesour ceantennaislocat edontheclifftop.thegroundwave

!J10JIil ll;;ll " ~frem llalldtosua.TIl<'signalreturnedfromthl' targetgoesovertbecliff

<111.1readll'~1,111'p,,·/'i.-('l'antennaontheclifftop.Thewavereflectsanddiffractsat tln-dirf,~dg"1,,:\\\,('('1\Ia ndaud~C1\.Therehcct ionand the diffractionchangethe fi"lddborilmtionuverthecliffan dthesea. Ontheotherhand,thesurfaces ofland

<111,1,,';'withfill itpelectr-icalpa ra metersabsorbtheelectr omagn e t icenergyofthe wal',·,.\ lJ"fthe~('factscauseattenuation of the groun dwave.Sotheattenu atio n oftJIl~ground-waveeveracliffdepends onthe ele ctr ical paramete r softhesections.

llll'nntcnuaheight-thecliffheig htand th edistances betwee nthe antenna sandthe clifredA"·

In Chapter fo,numeric alresultsarepresentedanddiscussed.Graphsshowthe

infllll'l l'"I'"fdlitltgill gantennaheights,cliffheightsanddistances between ant enna s

;111,[c1ilfl'd~('Oilthe ter raincoe fficient. Approximateanddetailedexpressionsaloe

used toobtain Iill'propo r tiesofdifferentcliffpara mete rs.

L'haptt-rigivestheconclusio n.

:\computer programis developed to realize thenum er ical calculationforthe

~rl)lllillwan:propagatio novera dilLThe result s have good agreem ent wit h these published.

Chapter 2

General Formulae of Propagation Over the Inhomogeneous Earth

In this ch apter, thegeneralformulas fer radio wave propagationoveran inhomoge- neouseartharederivedfrom firstprinciples.Atfirst,we define the terraincoeffi- cient.Secondly, we presentthe express ionsof electric and magnetic fields

E

and

Ii.

respectively,in terms ofthe magnetic vector potentialAin two &eta of cocrdinatea;

spherical coordinatesand the eocrdinates which are based onCartesia ncoord inates tba t take into accounttbe curvatureof the earth.Thisisfollowed by using bound- ary conditions andapplyingsource conditionsforhomogeneous earth problemin thetwo coordinate systems. Atlast , we presentexpressionsfor the fieldsover an inh omogeneous earth geometry.

2.1 Geometry Of The Cliff Boundary

The present terrain structureconsistsof an aireectioubelowwhich twosections of both different earth radiirianddifferent propagation consta.ntsk;(i=2,3)exi8t as shown in Fig.2.1.Throughout the derivation, thesource S andreceiverRu e

a,fromlandtosea b.fromseatoland Figu re 2.1:Formof three se ctionte rrain

alw ;l.l' ~uvcr differentsectionsand theirheightsfromthecen t re ofth eearth arers andru,respectively.Theirdistancesfromthecliffedge ared2andd3measured alongilmennearth surfaceofradius a whic histake n at sea level.In fig.2. 1 (a), wave-propagatesfro mlandto sea,whileinFig. 2.1(b),wave direction is fromsea toInnd.

When radiofrequencywave propagates along the surfaceoftheeart h,itscharec-

teri sticsarcinfluenc ed mainlybythe profile and elec trical propertiesoftheearth's

sllrf'l'.:e.Theau onuatlcncoefficienthasbeen introduced{14]which is dep e ndent

Oilthe earth 's curvature,fin itegrou n d conducti vity, thesour ceand re ceiverheights abovethe ground,as wellasdistancestothecliffedge.

lf E isthe electric field strengthinthecaseofcliff,theat tenuation coefficient A hOI::;beendefinedsu chthatl1 0]

E=AEo (2.1)

theair._.isthe frequency ofthesource.0="1,1.1I'1i.~n'ilisth,'";HIllr.nliu-, Intilt,residuemethod.thedefinition oftheatte-nuationr",'lIi"i,'!!1.\is[I-I]

F:;::;2..\/.:.·~ (~.:l)

Inthis work, when:'1I"e usc twomethods:thert'si,luclIwlhodp I-J"-,JilllO l ll ll' ,'<l1lI pensat ionmethocl[IO).E~oftheresid ue method isdefinerli\~1J·II

1"2.ji

wherekoisthe propagat ion rousreutillfa ..: spar!!, 2r.1,\ u;11,,1,\!)i~til"\\';I\""I"II~111 infreespace .E&isconsideredtoh('theelectricalfieldstr<~I1~t hillfn ..'SliM " .11i~

half ofthe fieldstrengthabovep-rfectcond ucti ngground.

Althoughsomeauthors [10][1·1]call A all 1I,1l1:llllalioll'·o,· l!ir;j"lll..III";'1It.h"r disagrees withthisname.\\'hen :\incrcnses,th,: liddst.rl'llJ;lhirw rt' HM:S.So:\rI" ,·, notstand forthe anenuaticnofthefield.Tireauthor1,,·lir'Vf'stlllltitsl1l11l1.J1,,·

calle d terra incoefficientwhosevalueislessthan unity.Furpcrfcctclyr:IJtl.llIn ill~

homogeneous ground, the terraincoefficient Arises up to unity. lnthis work,Awill be denotedterraincoefficient.

For laterconvenience, the numerical distanceC,is employed, whichisdefinedby

12

j:?,,')l

wll"l"~":1:5Tjin Fig. 2.I(a )andr3:::Tlin Fig. :!.l(hj.

2.2 Field Expressions In Different Coordinat e Sys- t em s

2.2.1 Earth Curv a t ureModi fiedCartesia nCoor d inate Sys-

tem

Fo r.1vcrticallypolarizedcurrent earring conductorof surface currentdemit},J (11/1l'lm,tIll' magnetic vect or poten t ialIihas only one compo nentin thezdirection i.c.::l:~.Fromthe;\laxwell equations,weobtainthe equation[].)):

[IV'

+

FIA,I")

=

-J,(')

W!WL"l'

r

isthopositionvect or aJ th eobserv ed point.

furall infillitesito1il.1current dem ent loca ted atr.we set

(2.7)

(2.9) He re,~(jlis Diracdelta-fu ncnon inthreedimensiona l space .Also ,\lopera torin Carll~$iancoordinat esis'V=C/;,/;'

tl.

For any medium, thewave constantkis

13 definedas

wherefrandaarethe relativepermitti vity and the conductivityofthe rnedlum, respec t ively. tois the per miu ivit.y of thefree space and c is the lightspeed.

Accor ding to thevecto rtheor emin appendix A,andsett ingu.

=

^{r,111=II and}

UJ

=

z for coordinateelementsdx,dyand da, anycurved lineelem entdscanbe expreeeedee

where hi,h, and h3are defined as

hl%o

=~,

^h,yo

= ~,

^h3z0=

~

risthe position vector..:ro,Yoand %"0 are the coord inate unitvector s.

(2.11)

(2.12)

Ifthecurvatu reof the eart histaken into account,thevalu esofhi (i=I ,2,3) becom e

(2.13) andequation (11)gives

(2.14) V'ofequa tion( 2.9)can be obtainedin the earthcurvature modified Certeeie nC0- ordin atesystembysubsti t utinghllh ,andh3intoequat ion(A,6)in appendix A.

SinceEand

n

^&1'1.'relatedto magnetic vector potentialA

=

""z-;'bythefollowing relations

E

-;!-VxV xA JW'

ii

=

VXA ^(2.15)

14 U~i ngexpression! (A.S)in appendixA and (2.13),the components ofthe

E

andii are expressedinterms of1/Jforthe cur vedgroundas follows:

E, E.

H.

H. = 0 2.2.2 SphericalCoordinates

In spherical coordinates, the three parametershi,h2andh3become [23) (2.16)

(2.17) Sincethe currentdensityvectorj is in the0-;'direction.the magneticvector potentialAis

A,a ,.

Accordingly thehomogeneous Helmholz equat ion canbewritt en

(V¹

+ k2)~

⁼⁰

whereV'2canbefound using(2.17)and (A.6)ofappendixA.

LettingA..

=

^rU,theaboveequalioa becomes

Theco mponents ofEand

ii

canbeexp ressedinterms of U {2J

(2.18)

(2.19)

E. =~~

E. ^_-^{-.l- ~}j,.ocr ..d&rH

H. =0

H. =~'il'

H.

=

_-~'f ^(2.20)

2 .3 An alysis of a Vertically Polarized Infinitesi- mal Dipole over Ground (Homogeneous Case)

Thi. homogeneous problemi.atwoa«ti ODproble m.Section1i.airspace above section2;the groundwhichi.smoothearthwith radiusr... asshownin Fig.2.2.

The twosectioDs havedifferentwavenumbe rskoand1:.. ,wherem takesvalue20r3 toaccord forthe differentground par ameters inlaterchapters.Fot a verticaldipole abovethe curvedground, theequation. forthe magneticvectorpott'ntiaiA=A.i ineuth curvaturemodifiedCartesiancoord inatesbecome (21)

(2.21)

whe re :o=r.iJthezcoordinate of the bou nduybetweenthe earthand&it.

InthesphericalCQOtdinates,theequat ions for themagnetic vector pote ntial

A=

A,rbecomes12)

(V1 + ~)~

^=-6(r-;;)

r ~ r

^...

(V1 + k~)~

^{_0($r... (2.22)}

\L----~

'\

Figur e 2.2:Formof two sectionterrain rr

16

From the cont inuityconditionofthe tangenti al compone nt s ofthe fields atthe boundary,one can use the following boundar y condit ionsintheeart hcurvature modified CartesianCoordinate system [151,

(¢d.~I". ).

(k;ZtnWd.

=

(k;"tnt\n).

WhereiJ/Bn==8/8::.

and in sphericalcoordina tes[2]

lAd)••=(.1.. )••

(k; z8:;l)r...

= (k;'28~~"')r

^...

12.23)

(2.24) The source conditi on can he derived from equation (2-7) andequat ion (2·9)[21].

Now, letus consider an elect ricdipole orientedin z directio n.Because1/;1 is continuo

ousatt'\"er~'puiutof spaceandcontinuous at the pointofthe source, integratingtwo

17 sides of equation (2.21), we obtainthesource conditionsin Cartesian coordinates

(Ifoll.;,..,=(It'd.<.·

(~).>

^••-

(~I.<

^{••=-j(,') (2.25)}

The boundaryconditionsandthesource conditionsare usedtodecidetheun- known constantsin the solutionsof the equations.

2.4 The Inhomogeneous Ground Problem

Inthis case,there isone airsec tion overtwogroundsections.The two ground sections arcto havedifferen tlevel and differentparameters.HtP'(%) and"''' (x ) arc continuous functions,we have

From Green'stheorem,we havellS)

Having obtainedthe wave functionI/Imforthe homogeneous earthwith radius r...

and the propagationconstantkIlOas shown in Fig.2.3,where mtak~2 or 3, we define~2(rR, .s)as the wavefundionof the inhomogeneousproblem,while'h(.s,rs) is the wave functio nforthe homogeneous case at r

=

ra where the source exist.

Sett ing ""(,,)

=

tP32(rR,.s) and "'''(,,)

=

",~(",rs)inequat ion(2.26), andtaking the integrationvolumeas the domains or media.k.in the entire range:z>62inaddition tothek,domain.The leftside of equa.tioo(2.26)becomes

-TP32{rR,rs )+!h(TR,r S) -!lIo32(rR' ''3),tP2(''3.rS)]

18

(2.28)

where

" 3

^isthe surface enclosing the abovevolume.Itextendsfromz=a2,Z;?:0 toz:5Q2, Z=0as shown in Fig. 2.3.

We obtainthe wave propagating overacliffas

tP32(rR, TS)='herR,rs)

+

[TP32(rR. $3),TP2("3,rsl] Til,>T2

!,bJ2(rR,rs)=[!/J32(rR,J 3),!/I2(SJ,TS)J Til,<T2 (2.29)

where

" 2

^isthesurface including theregain (2),i.e.z:50,z:$;T2asshownin Fig.

2.3. Notethat ",,(rR.TS)

=

0ifT2<Til, since the second term of integration in (2.28) does not includethereceiverpointTR'

Now,we set!p'(s)

=

TP32(rR,s)and1/.1"($)

=

tP3(s,rs)inequation(2.26),i.e. the sourceis in regian(3).Accordingly,one may getthe following relation

Settingrs

=

$3in (2.30),we have the expressionof V32(TR,S3)' After we substi- tute(2.30) into(2.29) andomitthe higher orderterms,weget

!P32(rR.rs)

!P32(rR,rs)

¢2(rR.rs)

+

['h(rR,.s3),tP2(.s3.rS)! r/t>r2 [tP3(rR, .s3),'M.s3.rS)}TR<r2 (2.31)

Thisis a firstorderapproximati on. The higher orderterms are importantonly whenrs orrR exists inthe immediatevicinity orthe verticalboundarysurface. The amplit udes orthese termsdecrease rapidlywithdistancesfromtheboundary.

19

--·-···_-._-..- ~3. _._.

Figure2.3:Integra tion domainofeq.(2.31)

Now,ifwehaveobtained the wavefunctionstP2and Thforhomogeneousearth of radii ra and ra, respectively, wecan use equation(2.31)toderivethewavefunction

Vn(orthreesectio nproblem withradiiT1andr,1.

Chapter 3

R e sidue M ethod Approach

ThewnveInucuou for the inhomogeneousearth with two sectionswhich heebeen ,If·ri\·t·.llIsing Fouriertra nsformsAnd GCe(!n'stheor em 111-15)willberee xamin edfor I;rolln<l11'<\\'('propag ati onmode.Thete rrain (attenuat ion)coefficien tisobt ained for an Inhomogeneous ea r th of onelong distancesection(se a)while theother isa shlllt section{coast].Present workwillexaminethe derivationapplied 10 thegiven

~,ulIH'lrynnddetermine appropriate expressionstothegroundwave propagation 1I1m ll~lIfitCUiC>lalbasedgroundwaveradar.Presentanalysisisbase doncons ider ing theI.ooundaryof the'cliff is verrical tobot hthepath andthe earth surface,andit isinfinitely.\long y direction,sincethe methodisapplicabletoatwodimensional problem.

3 .1 Ter ra in Coefficient for an Inhomoge n eou s Prob- lem

Theterraincoefficientforthe inhomogeneousproblemisobtainedinthree steps.

first.the wavefunctionfor thehomogeneousearth is derivedbymeans of Fourier transform.Secondly,thewavefunctionfor an inhomogeneousearth is obtained

21 using the results ofahomogeneousearthby meansofGreen's theorem as illustrated in Chapter 2. Finally, the expression is arranged to get the terrai ncoefficient.

3.1. 1 Wave Function for a HomogeneousEarth Thewave functio n forahomogeneouseart his thesolutio nof thevectorpotential in Helmholtz's equations. FouriertransformiJ used to changethe three dimensional prob lem into aonedim ensionproblem andthesolutionintermsof Fourierintegral expression forHelmholt z'sequations isgiven.Then,the unknown const ants inthe solutio nare decidedby using boundary conditi ons.Thefinal solution isobta ined by performing inverseFou rier tran sform, i.e.by em playing complexcontourintegration ln the complex plane.

Taking thecurvat u reofthe earthiota account and using equations(2.9) (2.13) andequation(A.6)inappendi xA,wehave

Wherethe directionofz·axis is verticeltothe surfaceof the eart h whoseradius isa.

rand;' are positionvectors of the source aadtheobserver ,respectively.tP(r ,;')is the fieldradiateddue tothe verti cal infinitesimal linear currentsourceatthepoint

~.

Solutionof the aboveequationwillstart by int roducing thesetsofthefunction tPA(r)and¢.\(r )124)as possible solutions for the homogeneousequation

(3.2) Byconsidering three separate functions for thevariablex, y,z,thefunctionI/!can

22

bewritt en as

lbA(r )=e- iIA,"'+l'Ylb(z)

~A(r)::e;(A I"'H1~)h (z ) Where.V

=, q+,q.

AIand>.,arevariables infrequen cydomain.

Using eitherVJ.\or~).inEq.(3.3) and (3.4),weget

1 ~~(Z2i) ⁺

^{(k' -}

a:;'

^n/.\

=

0 The general solut ions forbare inthe form of a Hankel function ,l.e.

(kzrl/2niII121(b) (a'>.'

+

1/4)lI²

(3.3) (3.4)

(3.5)

(3.6)

Where

H 1

¹)(z)and

H 1

²)(r )arethefiutand second kindHankelfunctioosoforder n and argument x,respecti vely.

Defining

m»

as the wavepropagating along the positivea-directioninthe domainz>^a,andfi (:)ASthewave along the negative a-di rect ion in the domain z<a,equat ion(6)canbemodified togivethe following setoftwo equations.

(3.7) Wherekoandk,are wavenumbers in air(.II'>0)and ground (z<1:1)respectively.

Substit utingequation(3.8) into equation(3.3) and(3.4),¢f(r)and~(r)a.re given by

e- i("''''+.lou )t f (z )

ei(A,;r+Al~lrr(z) (3.8)

23

Substitu ti ng the aboveexpr~sion!into equation(2.27),and ulinstherela tion d~=(:/a)' dr dy,we get

(3.'1 and

where

6{.\1 -.\;}6(.\, -.\;)

(; j' I/,(,jW' f,Jt (,))-

Jt(')W'~/,(')))

^(3.11)

From the outwardpropagatingwavecondition,wCr,r')canbewritten as w(r, r')

L :

d),ljIt { r }a+(.\ ) z>z'

L:

d.\1,bi'(r)a-(.\) z<z' (3.12) Where z' isthe sourcebei«ht.

Usingequation (2.26),andequatiou(3.1) and (3.2),ODecan get

~.(r') f~.(r)6("-r'ldv

/k-'W.,(r )k'6(r-r'j

+

~(r, r').Old.

/k-'[-~,(r)(1l

+

k')~(r, r')

+

~(r,r')(1l

+

k')~,(r)1

1W.(,), ~("r')JI::;:;:: ^(3.13)

Substit.ut inzequa tion(3.12) into (3.13),onegels

In asimila r way,we canobta inthe expression for,¢;,(r')in termsofa+p.').So we deter minethea+(A)anda-tAlinequa tion (3.14)as

Substit ut ing abovea+(A)anda- p )intoequat ion(3. 12)leadsto:

-(2lr)-'1:

d.\.Tbt(r)c(A)~;(r')^z>z'

-v-rI:

d.ATbi'(r)c(A)~(r')^.I<.I' At theboundaryi.e.z= a, the constantc(.A)is obtai nedas

(3.15)

(3.16)

In theintegration domain ,A'_,CJ,andIk,-.Ala:>I,forvery large ordern, wecan write[;(.1)as

/i(z)

~ r/'(k,Z)-If2(k~Z2

^{-n')-1 /4}ei1v·P : ;L"e<»-'lft/k,.I-../4! (3.18) The aboveexpressionisstill valid underthe conditionh/kiz2-n'l ,. Iand arg(.jkiz'l-n1 )

1: -1f.

^{This leadsto}

';;/A(Z)=i~/A(Z)

^(3.19)

fi(a

+

Cu)1fi(a)~e;~A., l:!.z<0 (3.20) substituting(3.19)into (3.17)and realizing thatthewave numberabove.I=ais katwhile it isk,forz<a,c{A)becomes

f;:(')(kO'f,m,)-

ik:i'~ft(')J

-2if i (a)/t(a )k;-Ib,(.\r1 (3.21)

~:,

\\'Ilf're

h:(,\j ::!I'J/rl'~+.1:;11(1)1 11(.\)

j(~.q\{.:.tJ)j:;.

k;

ki (£·;_,\:,- In

Sllb~titutingthe above expressions(3.8).(:1.20)1111.1(:I.~:21illlOp.ltil.tb-~nlll

tion of theWAn 'functionin free spaceabovegrourul medium11111<'0'01111""

~'

^{...(r, r'J}

= -~ L : ^d.\ k:"b"'{'\)!1.\( :.r,~)

e-'[·\I(:-"'l+ \'(~-.'I+ Jij;::iit.-~ - "1l

Wherer", is the radiusa.tthesurface of ground mediumIll.

Tolindthe wave funct ion intll('region.:~.:'?r....1\'1'pr...1.."f"l l" willJ!.. Forthe case of .:>r",ande'>r""wedivide the-splice intothreerl'"i"l1~i.e.

(1)z2': :;',(2):;'2':z2:r",and (3) z:::;r",.Fortile! firstregiollof.::::.:'. propagates outward,so wavefunct ion of(3.16) r-anIll'r'-wrill "na"

",(r.r')=_(1 11" )-'L: JtCZ)A(Ak-'I.\'I:' -:h .\.h,-.JI,I.\ :;2::;';.,.... r1.21)

WhereA(A)is a pro pagationcoefficient.

Forthesecondregion ofz'2:^%?:^T....thereare direct waveIrorn thesource (inward) andreflectedwave from the groundsurface(outward)SQthattl\l~WllVI:

funct ion of(3.16) becomes

"'(r, r ' )='-{211"r'~^(D(A){;(z)+R{A)f:'(z))e-oI~'(:'-:I+~""-,lldA^z'2::^%2:r...

(3.25)

26

WhereDP)andno)aredenc ri veand reflect edwave coefficients.

For the third regionwhenz::;r",.wavefunctionof(3.16) becomes

Where

It{.:)=(~zr ll'lH~2)( koz) z>r", li(z )=(klzrt/2H~11(koz) z>r", f~- (z)

=

(k2Z rl/2H~ll(k2Z ) z<r", Applying theboundaryconditionsof(2.23) atz=r....wehave

(3.27)

I' (,. ID(' )

+

It(r. )R(')

=

1;-(r. )' (' )lt(,') (3.28)

kl'l';;f;(z)~

^..,...

D(~) + ki2#;ft(z)~

^...R()')=k;2c(),)!t (1:' )';;/i.-{z)z=,...

(3.29) and fromthe sourcecondition of (2.25) at z=z',wehave

Solving the above setofthree linear equationsfor three unknownsA(..\),D()') and R(..\ )finallygives,

t.=-m"I1/' ('.)k;'1;m , ).... -

m,. Ik;'1;m,),...

^(3.31

1

and

t.,= It(" )[lt(" )/;-(r.) !lt(r.)- 1;-(")+,(A)/t(,')r,- (r.) !!t(,. Ik;' (ft(r. l§;m ,),... -m,.) §;m, ),.,.)) (3.32)

2i Defining 9(1)(.:',r...)=:1;(z')/I;( r...), A('\Iisgiven by

A =

~

kJl~-(rml!9~ {z', rm) - 9\')(z', r...)J +(.\)r(~,)f.~- {rm ) Jt(r..,)f ;(r...)[f.9 \I I(z ,r...)~.r^.._f.9~(:,r^...)...

1

C A ·

n»...)

1"

; [f.~(.:"r )- 9\1)~,r

^...

)1 +c(A)9~(z'.r

...lf'i{ r...)(3.33)

f"(r"')[a.9~(z,r )...- ".9A(z, r...)••r..1

Subst itut ing e(A)=:tlft (r)f~-(r)J- l k; fq( A)into theabove equation,we have A

=

kJ{9~(.:'.rl-9t^l)(z' ,r)IJt(r...)- 1+!.k'~(A)g,,(z',r^{...) (3.34)}

i;9t^'J(z,r)... -t9"(.:, r...)... 2' ft (r",) The wavefunct ionis givenby

"'(r,r') -(2X'r'

L :

It(z)Ae-i[~I(;I'-..'l+"J(~-~·))d..\

.!'l j ""

^[g1I)(z',r...) -9"(': ',r",»)9,,(Z,r le-j{~'C..-r)+"Jlr-w'IJdA

411'" -""[i;9t^ll(Z,r...) -

:.9,,(Z,

r )I ...

-~ L :

^{k'b:z('\)g,\(z',rm})9"(z, r )e-,[ ),,C..- rl+"Jfv-^YlId..\ e>z'(jt atij (3.36)

Theabove expressioncan besimplified byusingthe following parameters bm(A) 2k:!lk,l

+

k~hm(A)J

h.(A)

ilf,g,(" ,.)1 •• ••

h~l(..\) = i[!;91

^{11(z, r ...})J••^r...

(3.37) Fineliy,expression(3.35)can be rewrittenas follows

29

the form

Ih,

i;A i : ^d.\k;gA(z, r2)1~(>')9A(z',r2)

^-

(2k~/k;)

(h, (>.)-h~I)p.))-1(9A(Z',r,)-gill(z'.r,))1

1/.>3 ~

l:

d,\k; b:J('\')9A(Z',r3)e-;[Al("'-r)+A'b'-~I+~('~-'1l Z>r3, z':5r3

(3.41)

(3.42)

SubstitutingThand1{J3into eq.(2.27),(2 .31),theinhomogeneouswave function(or a receiverinthe airsection above region3 (soil) due to sourcein air aboveregion 2 (sea),!P32is givenby

L ^JOO

¹⁸

1/13, e! _Olld;x_00dY!h (rR,r )!ij a;lh (r, rs)]

-I ij!;tP3(rR.

r)I¢1a(r,rS).ar3

( 8;;)2 f f:

d>'d),'k;k; b:J(>.')9,v (zR,r3) I(^Jkj_kj^-,V'

+ .!..2..)g.(,

_k~

az '"

',I)^'::·3

e;{.l.I:r$-.l.:'1l+1111",-.l.;~1l1 r~().)g.l.(zs.r2)

^-

¥!(h

^2P.)-

h~IJ(.\))-t(g.l.(zs.r2 )

^{-gi'l(zs,r2))]}

fO oo

e-i(.I.,-.l.:J'"dz

L :

^e-i(.l..- .l.H' dy (3.43) whereZ$andZRare the z components of the sourceand thereceiver positions.One shouldnotice thatinthe aboveintegratio nusingthesurface.'12.zis set tor3endx changesfrom-00to O.

28

(3.38)

Sincea.\>1And":::ka.expressionsfor g.l,(z,r...)Andg1¹'(z,r",)Ca.Dbesim- plified.u follows

y~(z.r...)

g1

¹)(z ,rm)

x(P.)/x(P. )

x"'(P. )/X(P. ) (3.39)

Thedet ailsofthismodification are includedin appendix B,where XUI) pI12Hm(~p3/2)

X(l)(P)

=

pl/2Hm(~p3/2)

P. (2/1".)"'(1",- ••) P. (2/1".)'''(1",.-

'.j

1'. (3.40)

H./₃(z)istheHukd(u[u::tiooofthe order 1/3and upment^It.Thesuperscript (I) and(2) represen tinwardand outwardprop~atiDSwave,respectively.

3.1.2 Wave Function for the Inhomogeneous Earth Forthe Inbcmcgenecce eaet b problem understudy wherethe terrainincludes three sections.Le.air,lOilandoceanshowninfig.2.1,thewavefunction canbeobt ai ned by substitutingthewavefunct ionsofhomogeneous earthmentionedintheprevious sectioninto expreaelen(2.27).Forthe case shownin Fig.2.l.bwherer,

:s

t"3and the sourceisabove orODtbetopofsurfaceriothewavefunct ionsthand:harein

30

Figure3.1:Integrationpathof'\~foreq.(3.43)

From thedefinationofthefunctionsb",(>'),g~and h",(..\)inequation(3.37),(3.39), wevbtain

and settingx'=-x,WI'have

t

e- ;P,-APardx=fOe- i(.\j-.\M dx'

= (,\~ = \ )-

1

^00e-i{),,-l ; llldy

=

I~coe-i{"i-A,lVdy

=

2"f(5P.~- '\'2) (3.45) where (At -'\I tdenotestheconditionIm(.\i-AI)<0,and thus the integr a tion patbof..\tisalong theinfinitesimal lower side ofAIpath (F ig.3.1). In Fig.3.1,a set ofpolesexistnearly alongthebroken line.

Substituting(3. 44) and (3.45) int o (3.43) and arranging , we get

1/132 .,.,

1 ;:3 1:L: d'\d,\lk;b31(,\)g'\(r3,r2) b:J(,\')g.\,(zR,r3)~1~~->.;~

3\

(3.46) Performing the integrations with respectto

>.;

and>.~ofEq.(3.46)using the residue methodleads to presentingtheter m~(..\)-1(A3,1-.\d-l, where ),3.1denotes theset ofpolesofthefunction~('\')in theA'.pl ane.Ontheother han d, we have polesofM>.)whenA\

=

^>'2.1in A."plane.SoW32is givenby

tP32

= 1.. L j ""

^dJ.1k; b:lp r1g,.,(r 3, T2)( '\'3,1-"\2,.) - 1

411'<:1,') _OXI

Re~AI..Au[~('\)g,,(zs, r2)I RelJ~;.AI.1[b:t(.\')g'1(ZR.Ta)1 (3.47) where

~(>,}~1 ~ [~ +k~h3P.;)1

~[l

^-

03g~(r3.

T,)/ 029., (r3. T,)]

>'3,1- ).2,1 (k~/k~)(koar2/3[ko( koa)-1/3(r3-r:d

+

T3-T21

g~

^{..(z,r...) o...}

kol(koa)1/3~gT

(z.r",) (3.48)

with0 ...=_;(.I::"/I:o)("oa)- 1/3,T_m

=

_2-¹/3p form=2,3andk~ =~'

Substi tut ing the terms givenbyeq.(3.47),1/;32reducesto W32(rR, rs) = _2ll'

~ E f. ~k>' 2 (211'kor3)-1/2A'(zRld31'l

'1,'1 -OQ r

~[oilg

...(r3. r2) -

oilg~(r3. r2)J

[ko(r2-r3)(koa )-1/3

+

T2- T3j- I(T2-1/2o~ )-1 ezp[- i kod2 (;'-1

+

(koar2/3T2) ]

g'r}(zs.r2)e-ⁱ[.l.JI'II- u )+.I:.lrll-r ' l+f l (3.49)

32

(3.50)

l/J3'lcanbe expressedinanother Corm{15J

(3.51) whereWo(rn, rs}is thewavefunctioninfree spaceandgiven by

(3.52) Whered=d,

+

d3•

AlsoA(zRtd3.d,l zs ) is calledterraincoefficient,this coefficientis not different than theattenua.tioncoefficient represantedin[10][11-15J.Itis givenby

A(.z,Rld3.d,lzs)

=

I:<dld3)1f2A'(zRld3)f') T.".~(r2 )g., (zs , r2 ) (3.53)

T:I,TI

where

TTl,,,

~loj'lg"ll(r3. r2)

^-

o2Ig~(r3,r2)1

[!:o(T, - r3)(koat 1/3

+

1"2

-"3r

^l

(1"2_1/2~) -tc: -d,.,j,I l'f -l+("'.J-1IS"'1 (3.54) Sim illU'ly, for the case ofr, ::::T3 ,T."r,isfoundin thefollowingform(11)

T..,,"l = i(O;lg~(r2,r3)-0i'lgT)(r2,r3)][ko(r,-T3)(koa)- 1/3

+

1"2-1"3]-1

(T2-J/2o~ )-l e-iko'M(.!tI-I+(l:o.1-~/J "J (3.55)

3J

3.1.3 TerrainCoefficient

To deriveoneexpressionof thewavefunct ion forboth"2:S"3(FiX.2.l.b) and '" 2:T)(Fi g. 2.1.a) ,we can add a section betweencliffand ocean,theheightof whichishigher than orequal tothe heightsof thecliffandthe ocean.andthenlet thewidthof thesect ion tendto zero. The newgeometryisshowninFig.3.2.In thisway,the terraincoefficientis expressedulll)

where

A(zRld),d2Izs }= ~(dl d3)1/2A'(%Rld3 )~ TI41(d2)f';l,"lg.,( zs ,Tt) zR~ r3. z.~"2

limit<4_~ ~T(d4).,.~.T{d2)~. ,'lI

i(oi'g~{T

••r3)9"J(r. ,T2}-

o;lg'1(r., r3)g~(T41 T tll

(1)-1/~tl(l:o(r2-"3)(l:or)-1/3+T) -T3)-1 (3.56)

(3.>7) where9'(z,a...) ""(x'P.)fx'(~...)l.l. 1_ and~'(,8)""~X(fJ).U"4

=

T3.thatmeaD.

T ,

^<"3,TJ~~becomestheesme uT'I'J•..,in equat ion(3.52). Thi.i,becaule 9'.,(r3'''4 )::; 1 and9.,("3, "">=1.Ifr..=

"2.

^{forthe aimilureason,expression}

T~4,J~reducestothAt ofequation(3.55 ).

Substi tuting1'......2- 1/3_t...and y",

=

ko(r..._0)((;)1/3iotoeq,(3.57),the generalized terrain coefficient isfinallywritt enas follows(14J

34

's

"0 "0

'.

^cliff

cliff r

\ "2 "3

I

"3 k2

a. fromla nd to sea b.from sea toland

Figure 3.2: Geometry for eq.(3.57)

!IJ(YR3)exp(-ic3(Y3

+

fJ)J [Q3f:.(YU)!I,(Y41)- q,fl, (YU)f:1 (Yn)]

(1/3-1/3

+

t3 -t,)-I(t,- q1)-1

f'I(YS2)ezp[-ic,(Y2

+

t,)] (J.58) where

q;

=

_i(koro/2)1/3(fr ;-1- .iO.018o';/f)1/2 (3.59)

tri -10.0180';/J

andris thefrequency,^f.ando are the relative permittivityand the conductivity of the ground,reapecrlvely,The values oftm(m=2,3) stand for the roots of the equation(141

W'(tm ) -

'm

^{W(tm )}

=

0 (3.60)

wet)

can beexpressedby Airy (unction

(3.61)

35

WhereAi and Hi are (IB]:

Ai(' )

= & v'iIL",(()-

I",((l!

Bill )~

,fi3IL" ,(()

+I", I() ]

(3.62)

(3.63) 1..ismodified BesselFunction of ordern,and argument{=JJ3/2.

The functionk.(y)withYij=Y;-Yjisthe ordinaryheight-gainfunction defined by

And

f:...

(y)isdefinedby

f,.(y )~W(lm -y)/ W( l. )

f:.(y )=W'('m-y)/W' (,.)

(3.64)

(3.65) Whenthe ground wave propagates(romeart b tosea,thegeneralized terra incoeffi·

dent may be writtenas [14]

where

F

'1 =

Lla[Q3f:,(Y43 I/'I(Y41)-Q2fl,(v4J)f :, (Y41)]

(113 -~2

+

h -t2)-I(h -qn-I/'J(Ys:!) ezp[- iC1(Y2 - 113+t2 - 'J ))

(3.66)

(3.67)

whenY1l3~1/43' One should notice that F

"

isa.functionof es,Y51,Yuand1/42,but independent of theother antenna height !/1tJandthe distanceC3of section 3.

36

3.2 Expression for One Short Section

Assuming<:2<:1istheparamet e r s settingforthisCMe.Itmeansthat the source isdosetotheedgeandd3islong andCJis large.When(2<I, theconvergenceor theresid ue seriesFj jbecomes veryslow. So,the flat-earthapproximation is used in theehcrtsection toobtain P1j,Defining thevariabletM[12]

(3.6S) andmodifyingthe functionh,"(~)(or theBa t-earthapproximationtobeintheform

giv~thefollowing relation sh ips:

i(;-ko)d.

d;

b...(A1 u.

.' +

iIl'.lo)-i1kod. 2(id...

r

^{1t dt}

2um/( l fu m ) (ko/k~).;;;;J:ii,-'·'·

( 2u:::r 'h/t2-

ikodN(a",- aN)a-^t

+ u:1

(ko/k~)~'-""

(A,- ~)- I =id...[{tf>:I-t~rl

(tr)~ ikodN(koa)-2/3[ko(r , -fN)(koar I/3

+

TIl (3.70) whereN=2,3,I==2,3and

9.\(: , r...) = e- ;2/ :' 1

/~

= ( , - ,. Ijk;fii:.,'·/·

^(3.7l)

37

The(unction,J?"j(u,f) and J",,(f)aredefinedas[12j

~

Ie

^{t[(tf )2- l' r l (t+}

^ur'e-

^{I '-i1/ldt}

limu _couJ..(u.1)

~

Ie

^{t((tr )1-} t' rl e- Il-i'/ ldt 00 the conditio niodN(r",-TN)/a<I, we have

(3.72)

~

r- '( d

N)1J'Y/I'g..,.,(I,rN)

=

e-;r(.".f.I-ll~,.(uf"9~(r••TL)-U1Y9...(r••TI») J..{UN,r ;

+

lit )

+ ~uf"'g~(r

.... rl){Jf'l(o.r;-1;'1-J...(O,/N

+

IN))

+lg,(r~,")(J"

⁽

m -/~)+ J,(r;; +/~»)

^{(3.n )}

Applyingthis to theexpressionorii,.well.nAilyhave[14) FCI= exp{th)[u2({~/q1 )fl,(Y43)-!.,(Y43»)J••(U2. J.1

+

152)

+~(t.Jlq1)U1J:I(Y43)(J,)(O, J41 -1S2)

^-11.(0./41

+

Isz))

+iJ~(Y43 )(J,,(f4:ll

^-In)

+

J.,(f.u

+

In))J ,n

~ ,n

F.. ezp(I~)(u2(("fq,lfc,(,'3)-!'.(Y43))JJ.,(U2,!tt +151)

+4("I.,)u,r.,(y~)(J,(O,fn

^-f,,)- J.,(O,/n

+

f,,1l +i"l,(Y43)(- JI.(fS1-/42)+JI,US'!

+

In»]

+ft,{yn) 'S2~Y.1 (3.74)

where

u, =

_~ f$'l

=

'iC;1/_'1yS1e.·/4

4ci'/'Y4,ei~/.

[iC2(13 + Y3- y, )]l/ 2

38

(3.75) and(, isthenumeri cal distance ofsection2 withdefinitiongivenin (2.5)and

J,I(Z,f)

=

_e/2[~(z

+

t3J

r

^l£(f - it32)

+~(l

- t32)- lf(f

+

^il32)

- :(: '-t~2rlf(j

+

iz)1 wit hf(Z)as theerror {unctiongivenby:

(3.76)

(3.77)

Usingthe residuemethod,the terrain coefficientforlong distance sect io nsor multiple sec t ions withone short section can be obtained.Becausetheground wave radar problemisaepecielcaseoftwo sectionswith one sh ortsect ion,itcanbe solved with theresidue met hod,

Whenwe usetheresiduemethod, sever al considerations should be taken into accoun t.Thefiratoneishowto differentiatebetweenthelong distance problemand theshortdistance one. Fromtheconditionc;<:I, we have(d;faHkaa/2)1/₃<:1.

It meansthat the dist anced;satisfiesd;<:>,:p(J2^f3/trwhen the problem is a short distance one,whereAois the wave length and a is the earth'.radius.Also, the method canbegeneralizedfor use inlong distanceproblems and multiplesection problemwithonlyODe.hort distancesection.

Chapter 4

Compensation Method Approach

Compensationmeth odis used toobtainthe terraincoefficient of ground wave prop- agationoverinhomogeneous earthwit h two long distan ce sections. Theboundary between thecliffand oceanis also vert icalto the wavepropagation path and the reflected modeisignored.

In thismethod,the definitionof multipleimpedances is used to derivethe ex- pressionfor the terrain coefficien t.Theresultca.n be extendedto three dimensional problemwithit.finitelong boundarybetweentheclift'and ocean.

The methodproceed sas follows;first,the wave function forground wave propa- gation over homogeneou searthis obtained .Then,theterrain coefficie nt for ground wave propaga.lionovertheinhomogeneous earthisderivedusingthe definitionof multi pleimpedance and theresultsor homogeneousproblem.

4.1 Analysis of Two Section Problem

In thiscase,epberieelcoordina.t es(r,9,,,)ereused andtbe sourceofthefield is an infinitesimally sbort electricdipolelocated in Ireespace atr :::band(J :::0, and orientedin ther&.dial direction asebown in Fig.4.1.

39

40 b

Figure4.1:Homogeneoussectionsfor compensationmethod Thevecto r masnetiepotentialsatisllelltheequ6lionl2J

(V2+ k2)~_ O Subs titutingA.

=

rU intoaboveeque rice,wet;et ('Q" +kl)U =0

(4.1)

(4.2)

Usin!the metbodollltparationof vari&bles , thesolutionoftbeaboveequat tce infreespaceis,independant andcanbeexpressedbysphericalfUDct iolllJinthe (orm[1l1

v = u.+ v.

(4.3)

where

V.("I)

= i~~,

^f:

, ..

⁽²ⁱ

+

l)hj"(kor )hl"lkobJP,(co,S ),

,~ b

^(4.4)

V.("I)

= _i~~,

f:(2i

+

l)hj"(kor)hj"(kob)P, (co,S) , ':!>b (4.5)

,"

._- -- --.

^-_...

- _._-- .---._-_._ --- - - - - -- -- - - - -

R

o

Figu re4.'2:R thedistance betweensourceand receiver

'I

whereCo=-Jd3/b iwand l}'I(korl&ndh}Z)(kor)arethefin t and thelC'C:ond ki nd! ofspherie&!Hankel functionsof orderjandargumen t!:or, respectively.AI'l)

Uk,S)=

i~o

^f:(2j

+

l)B,hl"(",)P,I""S) ('-6) j=O

whereU.haatheproper singulari tyuR- 0,andU.remainsfinite.As.hownin Fig.·4.2,R=ylr+r¹ 2rbcoJ9.

Fromthe boundaryconditionE,;::-ZH,atr...G,weget

~irU(r,8)._ =

ZitwU(r.8)....

Su bstitu ti ngU intotheboundarycondition,8Jcanbe expressedu (4.1)

8-=_hl"lloa) It'og%hl^ll(.) - il>]h'''(106) 14.1)

J h}2J(koa)ttogZh}21(Z)_ia ¹ wherel::J.

=

fJJZI/CO;::Z/Qa ead%=koo.

42

Usingthewatson transformationandthe residuemethodto U,the polesof the integr ati on

U ==-i

l ~

^J(v)P..[CO.'l(1l'-8)]dv arc located at thepointsV""v.whichare the solutionsof

wit hv==n-~andthen,usingthe rela ti on (Ap pendixB)

(4,9)

(4.10)

wherev::::r

+

.:r¹/ 3T,wehave

where

and

u.

h,

x

X,

=

(koa)" '(2h,/ a )'"

-i(koa~~/3Z

=

_2-¹/3 q -l Id"e-^ilod

b-a

(4.14)

The{unction,,(hi)iscalled.4height-gain(unct ionenditbecomesunityash;

approach eszero.

·...

_-_ ^{. .} _ -- - -- - --_._ - - - -

'3 WeCaIIfindthatit isthe sameexp ression of theseriesforterrain coefficientas thatintheresi duemethod,hereUis th asame as

"'Jr

inthe residuemethod.

4.2 Analysis of t h e Inhomogeneous Problem

Basedonthe solution(orhomogeneous problemand proportional relation between elect ri c fieldand magnetic vect orpotent ialforfa r field, tbe expression forthevertical electric fieldE.ata great circle distanced:=:rDmeasured along thehomogeneous earth'ssurface is givenby 171

where

E. =AE

o Eo = -i~:;ll

ez p(- ikod)(si: e )t

(4.15)

(4.16 )

A

~

^{e-i" /4(n )1/2}

~ e1::'(=~~') W~(~:s) W~(~.:R)

^(4.17)

where

WUi;.~d⁼l,(hs).

c

=

(koa/2)1/3(J , YS=(2/ ka)I/3kohs, YR ""(2/ !coa )I/3khR q=-.(1,, /2)" ' 1>. I>=Z/ 120.. ko

= w i'

(4.18 ) where~isthe vertical electric fieldof thedipole atdistanced (rom a vertica.l electricdipoleo( momentIIat heighth s where both areassumedto bejust above thesu rf aceofaperfectlyconducting ground plane.W is thesamerunctionaaW or expres sion(3.61)inCh apter 3.

I(thepath includes tine section with the impedanceZ~.the mutualimpedance

44

Z",betweendipoles at A and B of length sI..endl~ i~written as

where

A(d,Z2)

=

e-;"/4('II"Z) I/~

L:

exp(-iZ 2t2) l~ t2-q2 HereAistheterraincoefficient for aspecialcase,hs

=

hR

=

O.

(4,20)

Ifthe pathincludes twosect ions with the impedanceZ2 andZ3. the mutual im pedancez~between the dipolesA and Bover theboundary isexpressed

where A'isthe tena.in (att enuation )coefficient dependent ond,Z2andZ3.We have

in the aboveequation,Iiis a unit vector in the radial direction,

.e:.

andH~dueto currentfA are the field over the surface S with surfaceimpedanceZ3, E6 andH~

duetocurrentI~are thefield overthe surface S with surfaceimpedance Z2.

Usingtheapproximate boundaryconditionsfer the tangentiaJ.vectors (4.23) wechan ge eq.(4-19) as

Becausethe tangentialmagneticfieldshavethe form