Effects of Antenna Radiation on Coherent Back-Scattering from the Ocean Surface

Transcription

1 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT Effc of Annna adaon on Cohn Back-Scang fom h Ocan Sufac Jmmy O Alah ( Wayl Waylkwkyj ( ( Th US Naval ach Laboaoy Wahngon DC 375 (Emal: jmmyalah@nlnavyml ( Th Gog Wahngon Unvy Wahngon DC 5 (Emal: waylkw@gwudu Abac Th analycal modl pvouly dvlopd by h auho fo backcang fom a om adlcc nfac ud o nvga h ffc of cohnc n h backcad fld on h monoac ada co-con (CS of h a Th a-ufac pnd by val mpcal wav-numb pca fo a fully dvlopd a namly h Pon-Numann Spcum Pon-Mokowz Spcum h Elfouhaly Unfd Spcum A Gnalzd Nomalzd ada Co-Scon (GNCS noducd a a mc of h combnd ffc of cohnc annna adaon chaacc By mployng a mall-lop appoxmaon (SSA of h a ufac h lav pow of h gnal un a compud compad o h gnal-pow un bad on daa-dvd nomalzd CS X-b (NCS ( σ valu fo h copondng mulad a a Th objcv of h analy o a h ffc of paal cohnc on backcang ung a GNCS hough valdaon wh σ oband fom maumn of a backca Indx Tm Gnalzd Nomalzd ada Co Scon (GNCS Annna cpocy Ocan Backcang Annna adaon Pan I INTODUCTION Nomally h amn of ada-clu un fom a dbud ag (ha l a c bad on modl ha dcz h cang conbuon fom whn h annna foopn [] Th ufac whn h man bam foopn ad a a om dbuon of pon ca wh no muual nacon Th ndvdual conbuon fom h pon ca a akn a popoonal o h pcv ncdn gnal pow ummd To manan conncy h vcal dplacmn of h ufac modld a a om poc wh no condaon fo h paal colaon whn h bam foopn Bad on h aumpon h lav pow of h ada clu mad by valuang h ufac ngal of h gan-quad dvdd by h fouh pow of h ang o h local gon ncompad by h annna foopn mulplyng h ul by h nomalzd ada co-con (NCS [] o σ a commonly known Thu ad xplcly P P λ G ( 4π 3 4 ( θφ ds Th NCS h mo mpoan mc n h chaaczaon of ada clu pop a aumd o b ndpndn of h annna adad-fld fau whn h llumnad gon of h ufac Fo h mo pa σ condd o b paabl fom h vaabl nng no h ufac ngal dmnd by d nvon Obanngσ wh h chnqu qu ha h llumnaon whn h aa conbung o backcang b conan Howv whn h clu ogna fom a nonunfomly llumnad pach of h a ufac (ay h n annna foopn h valdy of h nvon chnqu quonabl nc nly gno any paal colaon among h clu un fom dffn gon of h annna foopn In addon h aumpon aocad wh h nvon chnqu fal o ak no accoun of h acal phycal pop of h ufac a wll a of h annna adaon pan Th pnd n a numb of xampl n h lau Fo nanc n [] [3] [4] [5] h gan polazaon w aumd o b unfom houghou h annna foopn H h gan polazaon conbuon fom h cn of h man lob of h annna pan a akn o b h only valu of n In [6] [7] a dcud n [8] h uncolad cad fld aumpon nfocd bad on h dcolaon m of h ada un I ha bn dmonad ha h maud ada cocon (CS p un aa of a ough ufac dpnd on boh h knmac ochac pop of h ough *Th u of h wok cd olly fo acadmc pupo Th auho of h wok own h copygh no poducon n any fom pmd whou wn pmon by h auho*

2 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT ufac a wll a on h pop of h vng (mng annna adaon polazaon pan [9] In addon fomulaon of σ whch ncopoa h ffc of paal cohnc hav bn nvgad n [] [] A mo mpoan fau lluad n [] wh h auho dnguh h dffnc bwn h NCS gnad by a plan wav an annna llumnang a ough ufac Hn h auho ablh a con fo h NCS bng qual fo boh yp of llumnaon; povdd ha h ad of cuvau of ufac a mall compad o h annna foopn Howv h amp aum an ncohn cang modl /o only cond h ada un fom h bam cn of h annna pan An appoach o modlng backcang fom omly pubd ufac pnd n [8] capabl of ncopoang h ffc Tha modl wa ubqunly appld n [] n a udy of paal-cohnc ffc of h lcomagnc fld backcad by an gula dlcc-a nfac In h pap an xnon of h fomulaon of [] appld o h analy of ada ocan clu ung a modfd von of h mall-lop appoxmaon (SSA Th wav-numb pca of h ocan ufac a pnd by h Pon-Mokowz h Pon-Numann h Elfouhaly unfd wav-numb pca fo a fully dvlopd a A a mau of back- cad pow σ placd by h Gnalzd NCS (GNCS p un aa whch ncopoa h ffc of colaon of h backcad fld a wll a dpndnc on h lvan annna paam a obvd whn h foopn Sad xplcly mbddd n h fomulaon of h GNCS a: h acal phycal pop of h ufac; h knmac dcpon of h ufac; h gan polazaon gnad by h annna Numcal ul a pnd n m of h avag lav vd pow a a funcon of wnd pd wnd don annna adaon pan polazaon compad wh h copondng lav vd pow bad on maud σ a X-B Laly w ablh h poduc of h annna gan GNCS a a mc fo h combnd ffc of paal cohnc of h ufac annna adaon chaacc whn h foopn Th objcv o a h ffc of paal cohnc on backcang ung h GNCS hough valdaon wh σ oband fom maumn of a backca II CS BASED ON AN INCOHEENT SCATTEING MODEL An annna a man hgh H abov h ocan ufac hown n Fg Th Caan coodna ym vng a a fnc fo h phcal coodna θ φ ndcad by h d ax Th annna can b bam d n lvaon o θ θ n azmuhφ φ Th annna gan wll hav maxmum aθ θ φ φ copondng o h poon vco Th poon vco dd fom h pha cn of h annna o a gvn pon on h ough ufac wh magnud x y H ς ( xy Fo ( xy ς x y H ( Fg Scang gomy annna foopn Th old angl Ω Annna dmn h aa of h bam foopn A on h ufac A bam-unolvd ca dfnd a a ca ha ubnd a old angl Ω ca much mall han Ω Annna A a ul h fld ncdn on h ca can b appoxmad by a plan wav h v-o-m f funcon can b xpd by [8] : b ( π j G θφ jk a k 4π 4π π π ( wh a h mulu fom h m b h pon du o h mpngng plan wav h cang co con of a ca σ appoxmad by h mall pach of h llumnad ufac G ( θφ h avalabl annna gan Th p p a h (complx polazaon vco pcvly of h md cad fld all valuad a ( θφ Whn Ω Annna Ω ca w pak of a bam-olvd ca fo whch h calculaon of h v pon qu an ngaon of ( ov Ω Annna ada clu ma a gnally bad on an ncohn ngaon (ummaon Th amoun o ummng h quad magnud of ( wghd wh h dffnal co d p p ds wh ds h dffnal con ( aa A a ul on oban b a λ G ( 4π 3 4 ( θφ ds (

3 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT whch ju h avag of h ao of vd-om avalabl pow Th uual appoach n fndng σ fom ( o aum ha whn h man bam (fo all paccal pupo a h bam pak σ conan mply nv ( On poblm wh h pocdu ha fal o ak accoun of conbuon ognang oud h man bam howv dfnd Whn h man bam no accoun akn of ncohn addon a a ponal ouc of o I appa ha h jufcaon fo ncohn pocng manly on lav mplcy Anoh common dalzaon o nly gno h nnc dpndnc of σ on h annna adaon fld Th dpndnc ha bn dnfd n [8] [] [] ncopoad no h GNCS dcud n h qul III COHEENT SCATTEING AND THE GNCS A gvn n [8] laboad on n [] h vm f funcon lnkng h m oupu a wh h pon o h backcad fld b can b pu no h fom: Bad on (3 h avag of h ao of vd-om avalabl pow λ b a G ( θφ ( θφ j k d d A 4π π S π To a h conqunc of appoxmang cohn cang by ncohn cang w fom (5 no a fom mblng ( To h nd w f pn h dffnal d n m of pojcon on h xy plan ( θφ ( θφ + ς( d dxdy ds x y ds dfn a nw cang vaabl ( θφ ( θφ ( θφ Q p S p j k( θφ d Facong (5 avagng only h poduc of h cang vaabl yld (5 b π ja G ( θφ jk π Sd ( θφ π d A 4π a ka (3 G( θφ G( θ φ b λ Q( θφ Q( θ φ ds ds a 4π 4π A A Af aangng h m n h f ng w g wh S ( θφ a d cang max dmnd by h back-cang chaacc of a ough a/dlcc nfac Th cang paam (n h nwok n a b a nomalzd uch ha b a pn pcvly h vd pow h m avalabl pow Th m polazaon vco p dfnd by ( θφ p F F ad ad ( θφ ( θφ F ad pn adad pow p adan Accodngly h avalabl annna gan G ( θφ ( θφ Fad 4 π a No ha h xpon n (3 h annna pocy laonhp fo a gvn annna adad by an abay lcomagnc fld Equaon (3 wa dvd fom h now clacal amn whch ulz nwok paam o gv a full dcpon of h lcomagnc phnomna of any annna Th nclud paam lk gan polazaon apc ao adaon ffcncy cang c Fuh dal gadng h dvaon of (3 a addd n [3] [4] [5] (4 3 b a λ ( θφ ( 4π λ ( θφ ( θφ ( θ φ G 4π G ds Q( θφ Q( θ φ ds G 3 4 A A ( 4π G 3 4 A S ds Th la quaon ha h fom of ( wh ( θ φ σ placd by 4π G σ( S Q( θφ Q( θ φ ds G ( θφ A whch w dfn a h GNCS Th GNCS no only a funcon of h adaon fld (ncludng polazaon of h annna bu alo of h knmac ac of h ufac A can b n fom (6 n gon wh G ( θφ conan can b movd fom h ng canclld wh h gan funcon n h dnomnao In ha ca h GNCS bcom ndpndn of h annna gan Howv h cancllaon appl only o h gan no o h polazaon Thu n gnal h GNCS polazaon dpndn In vw of (5 h dpndnc n a an (6

4 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT nacon bwn h annna polazaon h cang chaacc of h ufac σ θφ whn No ha (6 duc o ˆ θf θ f γ ( θf δ( ρ ρ Q Q f whch copond o ncohn cang Tha f all of h ufac pop a known o b acally uncolad hn ( S fˆ σ σ θf σ θf 4 πγ θf (7 Only h SSA o ah a modfd von of ud n h pap follow cloly h xpoon n [] Th convnonal fomulaon ung h SSA fo cang by an gula bounday pnd n [6] Unlk n h pn pap h mpha on cang p wh h annna playng only an nd ol Und h modfd SSA h cang max ad dly fom [] Eq 4 θφ S d ( θ k coθ Γ n kς co θ π Γ ( θ (9 Th paam γ om mau of h oo-man quad (MS hgh lop /o hozonal colaon danc of h ufac compad o k On phycal gound fˆ ( θf mu ncompa h polazaon of h quvaln fld conuv pop of h ufac n A Fom an mpcal pon h σ oband fom h ocan (o fom any ough ufac ju h avag o man valu of σ S gvn by σ S ds ; A A A σ A ds (8 In xamnng (6 o (7 n ha pop of h ufac a gnally no paabl fo d nvon fo compung σ I h opnon of h auho ha h b way o chaacz ocan clu hough h vd pow a h po bcau ha wha acually maud no σ IV SSA OF THE SCATTEING MATIX AND THE GNCS Th pdng ah gnal appoach o ufac cang follow cloly ha pnd n [8] ( Appndx A Th ul a puly fomal fo hy om h cux of h poblm whch dmnng h cang max ( Appndx B Agan a oulnd n [8] n full gnaly w a dalng wh a complx bounday-valu poblm o ha analycal ul nvaably mploy om fom of appoxmaon Th pap mploy h mplfcaon noducd n [] ung an appoxmaon cloly lnkd o wha ha bn fd o a h aylgh hypoh [6] In ou npaon amoun o appoxmang h fld cad by an gula bounday by h Fou fom of h fld cad by an quvaln plana ufac Th appoxmaon dffcul o jufy hocally o ha valdaon mu com ndly fom xpmnal daa Ung h fomulaon on can fnd h cang max ung pubaon h fo mall amplud (ha k ς o fo mall lop ( xy ς ( xy ς ς ( xy ; ς ( xy x y 4 wh Γ ( θ ( θ Γ a h Fnl flcon coffcn fo paalll ppndcula polazaon pcvly a h man lvl of h ufac H n Fg No ha unl k ς ( xy h cang max a nonlna funcon of kς ( ρ Th nonlna dpndnc copond o lag vcal ufac dplacmn (wav hgh compad o h ada wavlngh I woh nong ha ς ( ρ a lav dplacmn wh pc o h man hgh H no affcd by a dplacmn of h gula ufac n h vcal don n h amn Small lop foc θφ + ς xy n (6 o uny nc h annna uffcnly fa away fom h ufac ς ( ρ coθ (6 aum h fom ( S G ( θ φ jk 4k πg ( θφ jk jkς( coθ ( π Sd ( θφ π jkς( coθ ( π ( θ φ S π A d ds ( Th m n h xpcaon opao n ( a d complx-conjuga pa n whch h poduc can b mplfd o p Sd ( θf p n k ς( co θ f ( θf ( wh f ( θf h polazaon faco und h modfd SSA whch qual o f ( θf co θ pθ ( θf Γ ( θ pf ( θf Γ ( θ ( jkς ρ coθ ρ coθ n h nmbl xpcaon opao n ( alo pnd by Th poduc of n k ς

5 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT a of complx conjuga pa Th poduc xpd a k ( ρ jkς( ρ coθ j3k ς( ρ coθ jk ς( ρ coθ Ψ θφ n ς coθ j (3 4 n+ jk jk α ς ρ n β ς ρ n Ψ( θφ Ψ ( θ φ ( 4 { kα k } { } α k α k 3 α 3k 4 co θ 3kco θ k co θ k co θ ; { kβ kβ kβ kβ } { k θ k θ k θ k θ } 3 4 n 3 co co co 3 co (6 A mplfcaon achvd whn h naow-bam appoxmaon appld o ( wh h ul n + ρ θ (4 Th un vco ρ dfnd a ρ x coφ + y nφ wh φ h annna azmuh-bam ng angl If h appopa ubuon a mad n ( ung ( (3 (4 hn h GNCS und h SSA xpd a 4 k f ( θf ( S πg ( θf A jkn G f θ ρ ρ θ f θ f Ψ θ f Ψ θ f ds (5 Equaon (5 la h maal pop of h ufac h lcomagnc pop of h backcad fld whn A o h paal cohnc of h ufac a chaaczd by h quan Ψ( θφ Ψ ( θ φ jknθ ( ρ ρ Th m ( Ψ θφ Ψ ( θ φ h xpcd valu of h Hman poduc of h pha-faco Ψ θφ whch lad o h mpdanc n a h annna po du o h vcal dplacmn ς ( ρ Th nam "pha-faco" wa lcd bcau ς ( ρ n no h agumn of h complx xponnal n (3 An xpon fo Ψ( θφ Ψ ( θ φ can n gnal no b gvn xplcly unl h jon pobably dny funcon (PDF of ς ( ρ pcfd Fo h ocan ufac h PDF can b conucd mpcally hough dou xpmnaon Th ul of h mhod hav no y poducd labl PDF a no mployd n h nvgaon Ψ Ψ xpd a h um of h chaacc funcon fo h dffn paal pcal wgh k α k n β whch a dly n dvd fom h nn ou algbac poduc m of Ψ θφ Ψ θ φ o Connung onwad ( θφ ( θ φ Fo h pupo of h nvgaon h PDF of ς ( ρ chon o b a Gauan om poc wh auocolaon funcon ( ψ κ wav-numb pcum ρ ρ Equaon (6 hn aum h fom 4 Ψ Ψ 4 ( θφ ( θ φ ( n n+ χn wh T k α k n ςς ρ ρ α n cn kβ ( k n ρ ρ ςς βn (7 ( ρρ ( n co n co n nco co ςς k α θ+ β θ αβ θ θ ςς (8 wh { α α α3 α 4} { 33} { β β β3 β 4} { 3 3} σ ςς h qua of h MS vcal dplacmn ς ( ρ Subung (7 (8 no (5 yld ( θf ( θf 4 k f jknθ ( ( S G( θ f f ( θ f pg A co co n ς α n θ βn θ + + m ( xp ( ds 4 αβ n ncoθcoθ ςς (9 wh ς m kςς Equaon (9 pn h GNCS of a Gauan om ufac llumnad by a naow-bam annna No ha h ng nclud h faco jknθ ( ρρ whch a hown n [] n h lm of conan gan ( polazaon ov h annna bam foopn lad o Bagg cang call ha Eq (9 vald only fo ufac ha oby h conan ς ( xy a vy pon on h ufac Thu (9 do no apply o paally uncolad ufac nc dvg whn ( ρ ρ placd by δ ( ρ ρ To fuh llua h known ha fo uncolad ufac h man-qua lop ( ρ canno b mall ρ 5

6 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT Hnc σ ( S canno b bad on an ncohn cang modl On h oh h whn n addon o mall lop on alo aum mall amplud ha wh k ς ( ρ h xponnal n (9 can b placd by a lna funcon of ( ρ ρ o ha (9 bcom compabl wh an ncohn cang modl A hown n quaon (36 of [] h compably a gnal fau of h mall amplud appoxmaon (SAA ndpndn of h conan on h ufac lop Th fau do no mply ha h SAA qu ncohnc Th choc of h colaon funcon abay can b chon o modl ufac cang wh cohnc nal a fo xampl n Bagg cang [] V SIMULATION OF THE GNCS AND COMPAISONS WITH PUBLISHED DATA Plo of b / a a a funcon of h (lvaon bam-ng angl compud ung (5 (6 wh σ ( S pnd by (9 a hown n Fg 3 Th plo a fo h h afomnond wav-numb pca a dnfd n h lgnd wh an "S" fo mulad Alo hown fo compaon a plo of b / a (o ( dvd fom publhd daa of σ fo 4 m/ wnd pd a cownd Th plo fo h hozonal polazaon a hown n Fg ho fo h vcal polazaon a hown n Fg 3 Smla ul fo wnd pd of 5 m/ a hown n Fg 4 5 Th mulaon w conducd fo h gvn a-ufac condon a X-B ( GHz dlcc conan ε 5585 j377 Th annna a ccula dh unfomly llumnad on mon wh a 6-wavlngh dam (48 m placd wavlngh (3 m abov h a ufac (Fnl numb 64 Th avag lav vd pow a h annna po wa compud fo a gvn bam-ng angl angng fom o 75 n ncmn n lvaon wh h azmuh ng angl fxd a 9 Th mulad cuv σ S n ach fgu a plod n old ln wh fo magna fo h ocan ufac modld by ψ ( κ fo h Elfouhaly Unfd pcum [7] lgh gn fo h Pon-Numann pcum [8] blu fo h Pon Mokowz pcum [9] Dal of h copondng wav-numb pca can b found n [8] [9] [7] Th daa pnng h mdan CS p un aa of h a ufac σ ud fo h compaon (fd o a baln cuv w oband fom vaou publhd accoun [] [] [] Th mdan valu of h NCS w maud a wnd pd of 4 o 6 m/ fo a gvn lvaon annna bam-ng angl fo cownd wnd don a dnfd n h lgnd wh an "E" fo xpmnal n ach fgu blow Th daa ud o dv h b / a ung Long [] a dpcd n Fg 3 wh a old-black ln wh qua Th cuv gnad fom daa publhd by Daly al [] a hown wh a dahd black ln wh ccl h cuv gnad fom Plan al [] a hown wh a dahd dak-gn ln wh nvd angl A mla foma mployd n Fg 4 5 wh h addon of a dffn daa a povdd by Long dpcd wh a dahd black ln wh qua h daa gnad fom Plan fo 6 m/ u gn ln wh angl Th daa fom Plan w ud n Fg 4 5 o confm h nd of h mulad b / a cuv Th cod ud o compu b / a fo ach wavnumb pcum wa nally dvlopd n MATLAB wh a mhod of ngaon dvd fom h D ccula convoluon wa ud o dmn h GNCS und h SSA Th auo-colaon funcon of h ufac ( ρ ρ fo Fg hough 5 wa compud fom a D fa-fou fom (FFT of h ocan wav-numb pcum ψ ( κ fo a gd z of To accla h calculaon h cod wa pod no FOTAN 9 h ngaon poon wa codd ung h Mag Pang Infac (MPI o ak advanag of h hghly paalllzabl nau of h algohm Th calculaon w pfomd ung Cay XE6' mad avalabl hough h DoD Supcompung ach Cn a h US A Foc ach Laboaoy a Wgh-Paon AFB h US Amy Engnng ach Dvlopmn Cn a Vckbug Mpp Th mulaon cuv n Fg povd o b pomng fo hozonal polazaon howng a ll a a 4-dB dvaon n h avag lav vd pow a compad wh h aocad baln valu fo mall lvaon angl ( o 4 Fo h lag lvaon angl (5 o 75 h dvaon fom h baln lav pow cuv gnad fom Long [] fo h Pon-Numann Pon- Mokowz modl w a lag a db Th Elfouhaly modl appa o povd h b ovall pfomanc wh a maxmum dvaon of 5 db ablhd bwn baln cuv fo Long [] Daly [ ] I hould b nod ha h baln cuv fo Plan [] dva fom h manng cuv n Fg 5 In h ca Plan' baln would pn a low bound on h σ o b / a valu wha h baln valu aocad wh h daa fom Long would pn an upp bound 6

7 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT Fg lav cvd Pow v Elvaon Bam Sng Angl of a hozonally polazd ccula dh abov a fully dvlopd a ufac wh wnd pd of 4 m/ dvaon w a mall a 4 db fom h baln cuv gnad fom Long fo h Pon-Numann Pon- Mokowz modl Th Elfouhaly Spcum agan povd h b ovall pfomanc wh a mnmum dvaon of db fom h baln cuv fo Long Daly No h baln cuv a 4 m/ fo Plan down gnfcanly fom h manng cuv n h 5 o 75 ang by a much a db du o h paal llumnaon ffc of h 5-n pul Howv Plan' cuv a 6 m/ compa vy wll wh h b / a cuv fo h Pon-Numann Pon-Mokowz modl wh a maxmum dvaon of 3 db h b / a cuv fo h Elfouhaly Spcum ha an ovall dvaon of 3 db Th ul fo vcal polazaon hown n Fg 5 a alo mla o ho hown n Fg 4 n ha h mulad cuv xhb good agmn wh h cuv fom Long whn h ang of lvaon of o 4 Fg 3 lav cvd Pow v Elvaon Bam Sng Angl of a vcally polazd ccula dh abov a fully dvlopd a ufac wh a wnd pd of 4 m/ Th ul fo vcal polazaon hown n Fg 3 a mla o ho of Fg wh h mulad b / a cuv alo xhb good agmn wh h baln cuv fom Long whn h ang of lvaon angl of o 4 In -xamnng Fg 3 n ha h a lag dvaon fom h baln cuv of Long a wll a much a db fo all of h mulaon cuv Agan h b / a gnad fom h Elfouhaly pcum appa o b h ovall b ma fo h hgh lvaon angl wh a mall -db dvaon compad o h low bound an 8-dB dvaon compad o h upp bound Th mulaon cuv n Fg 4 povd o b accpabl fo hozonal polazaon howng a much a a 7-dB dvaon n h avag lav vd-pow pon whn compad wh h aocad baln valu fo lvaon angl of o 4 Fo lvaon angl of 5 o 75 h Fg 4 lav cvd Pow v Elvaon Bam Sng Angl of a hozonally polazd ccula dh abov a fully dvlopd a ufac wh a wnd pd of 5 m/ In xamnng Fg 5 n ha h lag dvaon fom Long' cuv a much a 7 db fo all of h mulaon cuv whch du o h pon whn h nhancd poon Agan h b / a cuv gnad fom h Elfouhaly pcum appa o b h ovall b ma fo h hgh lvaon angl wh a mall -db dvaon compad o h low bound a 5-dB dvaon compad o h upp bound 7

8 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT Fg 5 lav cvd Pow v Elvaon Bam Sng Angl of a vcally polazd ccula dh abov a fully dvlopd a ufac wh a wnd pd of 5 m/ Fg 6 ang Exn v Bam Sng Elvaon Angl cuv fo h annna foopn dmnon (co ang down ang pul lngh (5 n 5 n A pcal no abou h mulad daa ha h ngaon o ook pdnc whnv h gd pacng wa no mall nough (a majo cau fo h appan ocllaon n h cuv Anoh no ha fo mall angl of ncdnc a fw of h daa pon a mng whch pobably du o h quung of h paalll poco VI THE GNCS AS A METIC FO SPATIAL COHEENCE In Fg hough 5 h lav pow vd fom h a cho a pdcd by h GNCS wa n ubanal agmn wh xpmnal baln valu Th cuv dcbd h laonhp bwn h ufac lcomagnc pop whn h foopn Howv gvn h z of h foopn h hozonal colaon lngh of a ufac h cuv w xpcd o convg nc n mo ca h foopn wa lmd by h annna whl h ang xn aocad wh pulwdh wa lmd by h bam' xn Hnc h foopn cond of uffcn numb of colaon lngh uch ha h GNCS wa nally dncal wh σ Cond Fg 6 wh h co-ang down-ang xn h ang xn aocad wh wo pulwdh a hown (n m of lcomagnc wavlngh a a funcon of lvaon I can b n wh h foopn pnd by a bam-lmd o pullmd llumnaon pofl Th pulwdh ang xn (down-ang xn cuv a hown fo 5 n 5 n Th 5-n cuv how ha fo naly all lvaon angl h foopn bam lmd pnng a fully llumnad ufac fo ach pul Th would man ha fo h 5 n pul h NCS ( σ a mau of h cohnc ffc of h n cang ufac dfnd by A Howv h 5-n cuv dmona ha h foopn pul lmd Thu h NCS valu gvn by Plan would gnfy ha wa only paally llumnad hnc h NCS could only b dfnd ad hoc by h aocad ang xn fo h appopa angula covag Ovall h cuv hlp o how ha f h numb of colaon lngh uffcn o cov h annna-bam foopn hn σ all on would nd o dcb h cang nau of h ufac Howv f h dmnon of A w mall nough o lm h numb of colaon danc whn would σ ll man a good cang mc? To add h quon cond h poduc of h gan h back cang co con G ( θφσ ( S wh h σ ( S gvn by (9 If ( S σ placd by σ hn h poduc would b nally lnaly popoonal o h gan Thfo h xn o whch h plo of h wo poduc vu gan dff can b akn a a mc of h ffc of paal cohnc on h backcang co con In oh wod f h wo poduc a dncal hn σ ( S σ Th poduc 8

9 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT ( ( S G θφσ compud n Fg 7 8 a a fxd bam- cn poon fo a ccula dh whn h nomalzd adu of h dh ( ka vad fom 5 o fo a gvn polazaon plod Th G( θφσ ( S cuv a hown ung σ ( S oband fom h Elfouhaly Spcum Fo ach polazaon h G ( θφσ cuv of Daly Long follow a lna nd Fo hozonal polazaon n Fg 7 h wo baln cuv a paad by 5 db wh a pad of o 7 db on h low nd 3 o 7 db on h upp nd Fo gan valu blow 39 db h G( θφσ ( S cuv dva ubanally fom h lna nd bu naly lna wll abov h baln cuv fo gan valu abov 39 db Fg 8 Th poduc of Gan wh ( S polazaon (wh a wnd pd of 5 m/ σ ( σ v Gan fo vcal σ ( σ v Gan fo hozonal polazaon (wh a wnd pd of 5 m/ Fg 7 Th poduc of Gan wh ( S Fo vcal polazaon n Fg 8 h h baln cuv a paad by a much a 4 db wh a pad of 3 o 7 db on h low nd 3 o 8 db on h upp nd Th G( θφσ ( S cuv h follow a mla nd a fo h hozonal polazaon Boh of h fgu how ha a h gan nca o h annna-foopn dmnon dca G ( θφσ nca lnaly Howv h G( θφσ ( S cuv ha a non-lna nd ha wll abov h uppbound baln cuv whch mpl ha h cang cong con gan dpndn Hnc h numb of hozonal paal colaon lngh ablh h lav ngh of h backcad fld Th dmona ha h GNCS h b mc fo chaaczng ada clu VII CONCLUSIONS Th cohnc ffc on h backcad fld du o a om a ufac a ablhd by h pop of h adang annna w nvgad ung h annnapocy laonhp o pdc h mono-ac gnal un fom bam-olvd dbud ag fo a-clu analy Th a-ufac backca wa xamnd fo a fully dvlopd wnd-dvn a modld by val mpcal wav-numb pca n h pnc of a unfomly llumnad ccula dh a X-b Th lav pow un w dmnd fo vaou annna paam (hgh abov ufac polazaon bam foopn fo a gvn a a Th lav-pow cuv gnad by h GNCS compad vy wll wh pon dvd fom maud daa Th dvaon oband fom h daa-dvd baln nd w a mall a 3 db Oddly nough h mulad pon pfomd unxpcdly wll a h hgh-ncdnc angl a majo ndcaon ha ough-ufac cang analy hould nclud h full nau of h ncdn fld a wll h ufac knmac Ovall h ul how ha paal-cohnc ffc can gnfcanly nflunc ada-clu ma A pcal no abou h mulad daa whch wa pvouly mnond ha h ngaon o ook pdnc whnv h gd pacng wa no mall nough whch a majo cau fo h appan ocllaon n h cuv ACKNOWLEDGEMENTS Th auho would lk o hank h vw fo h nghful commn Alo h auho would o lk hank h US Naval ach Laboaoy (pcally h ada dvon fo allowng u o u h HPMCP compu o un h xnv mulaon fo h pap Fnally h auho would lk o xnd a majo db of gaud o D C Kung fo ang u n lang h MATLAB cod no FOTAN 9 fo ablhng u on h HPMCP nvonmn 9

10 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT EFEENCES [] MI Skolnk ada Hbook: McGaw-Hll 8 [] G Bgnc "Small-lop appoxmaon mhod: A fuh udy of vco wav cang fom wodmnonal ufac compaon wh xpmnal daa" Pog n Elcomagnc ach vol 37 pp 5-87 [3] OM Phllp "ada un fom h a ufac - Bagg cang bakng wav" Jounal of Flud Mchanc vol 56 pp [4] WJ Plan "A ochac mulcal modl of mcowav backca fom h ocan" Jounal of Gophycal ach C: Ocan vol 7 pp 3- [5] WJ Plan PH Dahl WC Kll "Mcowav acouc cang fom paac capllay wav" Jounal of Gophycal ach C: Ocan vol 4 pp [6] D Hollday LL Daad GJ S-Cy "Fowadbackwad: a nw mhod fo compung low-gazng angl cang" Annna Popagaon IEEE Tanacon on vol 44 pp [7] D Hollday LL Daad J GJ S-Cy "Sapk backca fom a pnng wav" Annna Popagaon IEEE Tanacon on vol 46 pp 8-3 Januay 998 [8] W Waylkwkyj "A nw fomulaon fo cohn backcang fom dbud ag" IEEE Annna Popagaon Socy Innaonal Sympoum Honolulu HI; pp Jun 7 [9] JW Wgh WC Kll "Doppl pca n mcowav cang fom wnd wav" Phyc of Flud vol 4 pp {466-\&} 97 [] DM Lvn "Compaon of σ Oband fom h Convnonal Dfnon wh σ Appang n h ada Equaon fo omly ough Sufac" IEEE Tanacon on Gocnc mo Snng vol GE- pp [] D Hollday G S-Cy NE Wood "A ada ocan magng modl fo mall o moda ncdnc angl" Innaonal Jounal of mo Snng vol 7 pp [3] W Waylkwkyj "Gan Effcncy cvng Co Scon of Annna n a Dpav Mdum" ado Scnc vol 5 pp [4] W Waylkwkyj WK Kahn "Thoy of muual couplng among mnmum-cang annna" IEEE Tanacon on Annna Popagaon vol 8 pp [5] AC Galy J DJ Sock B Cho "A nwok dcpon fo annna poblm" Pocdng of h IEEE vol 56 pp [6] AG Voonvch Wav cang fom ough ufac d: Spng 999 [7] T Elfouhaly B Chapon K Kaao D Vmak "A unfd donal pcum fo long ho wnd-dvn wav" Jounal of Gophycal ach vol pp [8] B Knman Wnd Wav Th Gnaon Popagaon on h Ocan Sufac: Pnc-Hall 965 [9] WJ Pon L Mokowz "Popod Spcal Fom fo Fully Dvlopd Wnd Sa Bad on Smlay Thoy of S a Kagoodk" Jounal of Gophycal ach vol 69 pp 58-& 964 [] MW Long ada flcvy of L Sa: Ach Hou [] JC Daly JT anon JA Buk "ada Sa un - JOSS I" NL UP DEPT OF COMMECE SPINGFIELD VA 6 Fomal po AP 97 [] WJ Plan WC Kll K Hay "Smulanou maumn of ocan wnd wav wh an abon cohn al apu ada" Jounal of Amophc Ocanc Tchnology vol pp [3] W Waylkwkyj "pon of an annna o abay ncdn fld" IEEE Annna Popagaon Socy Innaonal Sympoum Wahngon DC; pp 39-4vol 3B July 5 [5] AG Voonovch "A wo-cal modl fom h pon of vw of h mall-lop appoxmaon" Wav n om Mda vol 6 pp [6] G Valnzula "Tho fo h nacon of lcomagnc ocanc wav - A vw" Bounday-Lay Moology vol 3 pp [] W Waylkwkyj J Alah "Cohn cang fom dbud ag" Mcowav ada mo Snng Sympoum Kv Ukan; pp 36-4 Spmb 8

11 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT APPENDIX A DEIVATION OF THE ANTENNA ESPONSE FOM A FIELD SCATTEED BY AN IEGULA DIELECTIC /AI INTEFACE USING A MODIFIED VESION OF THE SSA plan; k z k z a z-componn of k pcvly n a h dlcc mdum; k h f-pac wav numb; ubcp f o quan aocad wh ncdn fld ubcp o v componn; k k k κ z κ a un vco along h x y ax ρ h poon vco n h xy plan Z µ ε Fg A Dcpon of h cang mda Th boundng ufac bwn h dlcc mdum f pac n Fg A dfnd wh pc o h fnc plan H by H ς ( xy Th f-pac dlcc z H mn ς xy gon copond pcvly o j jkz z z H max ς ( xy A plan wav k ρ E ncdn fom f pac ogh wh h ougong wav pnd by jk jkz z jk + jkz ( zh fl jkz H jk + jkz( zh d k ( k k E E + E + (A In accodanc wh h aylgh hypoh h la m pnd by h Fou ngal fncd o h plan z H an appoxmaon o h connuou pcum dfnd a h cad fld xcludng h pcula j jkz ( z H m fl k E + Th lcc fld md no h dlcc gon ( jkz H jk jk z ( zh E ( Ef jkz H ( jk jk z( zh + d k ( k k (A H mlaly fo h copondng magnc fld ( ( H ( ( k k ( k k a lcc- fld cang amplud pcvly n a h dlcc mdum Th followng an ouln of h pocdu ud o fnd an appoxma oluon o (A fo ( k k (nally a Taylo xpanon o O ς la o h gnalzd NCS whch on of h objcv of h pap und vw Th ymbol ndcad h n h ubqun quaon a dfnd a follow: k h v wav numb ( h pojcon of h wav numb k on h xy Machng h x angnal componn of h oal lcc magnc fld on h mda boundng ufac ndpndnly h angnal componn of h plan wav on h hypohcal plana bounday H yld h followng of ngal quaon fo h cang amplud ( k k ( ( ( k k K k k K k k d k k k K k k K k k k k V ( k k + V ( k k (A3 wh h max lmn mn ( K k k a 3x3 dyadc ha dpnd on wav numb h ufac lop plan-wav Fnl coffcn Upon applyng h Numann aon pocdu o (A3 on fnd ha o pn h oluon o O ς h f m uffc Th quvaln o whn appoxmang ( k k by V k k n (A3 whch ( mpoan o no h appoxmaon do no conan h z of kς wh ( ( k k TE VE k k + TH VH k k ; k ε k k ε T k z kzε + kz kz + kz kzε + kz ( z z E ( κ κ + κ κ κ ( kk kk TH ( k + z k kz kz z z k kz kz ς ( z z z κ κ κ κ ε + k + k ( ε + κ j( k k (A4 (A5 ; ( ς( x y jkz x y fl E ( k k d jk x y (A6 E z + E z zς ( π ( Ef z (A7

12 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT ς j( k k ( ς( kz z H + z H d zς ( π ( z Hf (A8 x y jk x y z fl o o H ( k k jk x y cang h n phcal coodna w dno h angl of ncdnc by θ (lvaon φ (azmuh h cang angl by α β h lcc-fld cang amplud olvd along h phcal un vco α β aum h fom wh j( k d k ρ Ψ ( k k ρ ; ( k k (A9 ( k k { o Tαθ E } θ + T E αϕ ϕ + o Tβθ E θ + Tβϕ E ϕ Ψ ρ α β (A T αθ ( k k co ( β φ ( π ( n α + coα co n ρ jk zς ρ ( jk zς coθ n α ρ ρ ( θ ( θ α + + Γ T αφ ( k k n ( β φ ( π ( n α + coα n co ρ jk zς ρ ( jk zς n α coθ ρ ρ ( θ ( α θ + + Γ T βθ ( k k n ( β φ ( π ( n α + coα ( ρ jk zς( ρ ( jk zς α θ ρ ρ ( θ ( n α coθ + + n co Γ T βφ ( k k co ( β φ ( π ( n α + coα n co ρ jk zς ρ ( jk zς n α coθ ρ ρ ( θ ( α θ + + Γ (A (A (A3 (A4 Nx w cond h pon of h cad fld by an annna Fom h gnalzd pocy laonhp [3] w hav j ab Fad k Ek ( d k (A5 π k Z Wh h nomalzaon h lcc fa fld on mon Z F b h vd pow ( k ad jk wh h copondng adad pow 4π (A6 d k Fad ( k a (A7 k coθ Fo compably wh h dfnon ( Ek pn h Fou Tanfom of h cad fld a z jk Ek E ρ d ρ To u fomula (A5 w mu xp ( (A8 Ek n m of h cang amplud ( k k n (A Fo a ngl ncdn pcal componn ( a plan wav z z jk jk H jk z E k H ρ z d k k k Fo a gnal llumnaon w nd E ρ E ( k ρ d k ( π (A9 (A whch cd o h plan z h locaon of h annna Subung (A n (A8 w g j( kz + kz H j( k + k Ek k k d ρd kd k π z z ( ( z + z ( k k d j k + k H k k k k d kd k + j k k H d k (A Subung h la xpon n (A9 w g h v oupu n m of h paal pcum: j( kz + kz H j ad ( d ( F k d ab k k k k π ko Zo (A

13 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT A fom mo uabl fo aonay pha valuaon k k n (A wh (A9 oband by placng ( a b jd F k Ψ k k dk dk jk jk ad π k Z o o (A3 Fo k ((A good nough w can u jk aonay pha on copondng o h oh o j k wh aonay pon a α θ β φ θ θ φ φ Th la no k n k ρ k θ n (A9 α θ β φ A ul w g a b jk jd co θ F ad ( k Ψ ( kn θ kn θ π k Z (A4 Th max lmn n (A (A (A3 (A4 valuad a h aonay pon a: j Tαθ ( k n co ; k ρ Γ θ k ς θ ( π ρ j Tβφ ( k n co ; k ρ Γ θ k ς θ ( π ρ Tβθ ( k ( k ρ Tαφ k k ρ (A5 o ha Ψ ( ρkn θ ρkn θ ρ j n k ς co θ{ Γ ( θ E θ + Γ ( θ E φ} ( π ρ θ φ (A6 H a n (A9 E θ Fad φ h componn of ad ( E φ a popoonal o Fad θ F k n (A6 W can fnd h popoonaly conan by ognzng ha (A6 h aympoc fom of h lcc fld Ead ( a k ha can b compud fom jk Ead E ( k d k (A7 π A aonay pha valuaon gv k π j jk Ead co θe ( θj ( π k Fad ( θj jk Z (A8 Wng (A8 n componn fom k π j θfad + φfad co ( E E θ φ θ θ + φ Z ( π k θ φ (A9 w oban h dd popoonaly conan: Z π F Z π F adθ adφ E θ E φ jk coθ jk coθ Ung h n (A6 (A3 Z n kς coθ ( kn θ kn θ Ψ πk coθ { θγ ( θ Fadθ + φ Γ( θ Fadφ} (A3 Th fnal fom of h v oupu oband by h ubuon of (A3 no (A4 π jk n kς coθ θ F ( θφ ( θ θ π θ φ ( θ jk π k coθ ρ n ς co θ{ ( θ adθ ( θ adφ} π ρ θ φ a b j d co Γ F + Γ F ad ad adφ o j d k Γ F Γ F k ( θ (A3 T jk π k coθ Fadθ Γ Fadθ ab j d n k co k ς θ π F adφ F Γ θ adφ jk π j d ad ( θφ d ( θφ ad ( θφ k F S F APPENDIX B BIEF DEIVATION OF THE OUGH- SUFACE FIELD INTEGAL EQUATION FO THE MODIFIED SAA (A33 A ynop of h dvaon of om of h ky fomula ud o oban h ngal quaon gvn n Appndx A h cang max gvn n (9 of h pap a xpland hn Th fomalm gvn n Appndx A wll b adopd n h documn W wh o mphaz ha h fomulaon n ou pap an alna von o h fomulaon gvn n [6] alhough ou nd ubanally dffn xpon wh unqu funconal pop Th dffnc n h xpon a pmaly du o h applcaon of h ufac conan fo mall lop ς ( xy mall amplud k ς ( xy h pha fnc aocad wh h man ufac dplacmn (whch wll b xpland n h qul Th mhodology pnd n [6] [5] ummazd n [6] ulz h ufac conan mulanouly fo h SSA Howv h fomulaon gvn n Scon VIII hn mploy h conan paaly Hn h appoxma fom of h ough-ufac cad fld oband by a conucon ha aum h valdy of h aylgh hypoh (S Fg B Bgnnng 3

14 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT wh h dcpon of h afomnond conucon of h lcomagnc fld n boh mda w hav h followng pnd n h pcula m of (B (B3 W hav alo aumd ha h ough ufac cad fld a xpd ung h Fou xpanon hom a vald whn h gon ( h aylgh Hypoh Hmn ς xy z H max ς xy Th bounday condon fo h pculaly flcd angnal lcc-fld componn a H ( ς ( xy gvn a ( E + Efl Ef z (B5 Fg B Dcpon of ough-ufac cang phnomna z H ς xy (Mdum lluad n Fg B can b wn a jk jkz z jk + jkz ( zh E E + Efl jk + jkz( zh d k ( k k Th lcc fld n h gon dfnd by jkz H + (B j jkz z Th ncdn flcd plan wav k ρ E j jkz ( z H fl k E + pcvly a chaaczd a h hypohcal plan z H ( jkz H j jkz( z H ca d k + E k k k k (B h ough-ufac cad fld whch alo valuad a h plan z H Smlaly h fld quan n Mdum a ( jkz H + jkz H jk jk z ( zh E Ef ( jk jk z( zh d k ( k k (B3 jkz H j jk z ( z H H f k E a facd plan wav avlng n Mdum h fld cad no Mdum wn a jkz H j jk z( z H ca d k E k k k k (B4 jkz H No ha h pha m wa ncludd n h xpon fo h pcula fld a z H o ablh h pha fnc a ha locaon Th pm h flcd facd plan wav componn (a wll a ( k k ( ( k k o b popoona o h ncdn fld va h Fnl flcon Coffcn Th pupo of jkz H can b xpland aly xpland n h conx of mon-ln hoy wh h annna (npu po h fnc mnal plan h load plan locad a z H H h flcon (mon coffcn a load plan fncd o h npu flcon coffcn a 4 Th copondng magnc-fld xpon fo boh mda hav h gvn fom of jkz H + fo Mdum jk jkz z jk + jkz ( zh + fl jk + jkz( zh d k ( k k H H H jkz H + jkz H jk jk z ( zh f ( jk jk z( zh k ( k k H H d (B6 (B7 fo Mdum In h abov quaon h plan-wav pcal pnaon fo h cad magnc fld a wn a ( jkz H j jkz( z H ca d k + H k k k k (B8 jkz H ( j jk z( z H ca d ( k H k k k k (B9 Smlaly h bounday condon fo h angnal componn fo h magnc plan wav fld a h man lvl z H alo gvn a H + H H z (B ( fl f In xamnng h pha m n (B (B4 (B8 (B9 jkz H ad a an abay conan ubmd n h xpon o gv o a can ymmy I can alo b ad ha ach mmb of h pcal nmbl fo h cad fld pha fncd o h plan z H Agan h p yld a ymmc of oluon fo h ( ( cang amplud ( k k ( k k whch vanh whn ς ( xy No ha h cal of ς ( xy ha y o b pcfd; only h aylgh Hypoh wa nfocd Th cang amplud hav a of mpdanc laonhp whch w dvd fom Maxwll' Equaon a gvn a ( k k z ( k k ωµ ( k k z o (B

15 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT k z k k k k + kz ωµ o (B (B7 a addd ubacd fom boh d of (B3 pcvly Equaon (B3 hn -xpd a Havng dfnd h ncay paam fld quan n ach mdum h oluon o h cang amplud fo ( k k hnc ( k k can b dmnd by ung (B (B3 (B6 (B7 a z H ς ( xy wh h bounday condon fo h angnal fld componn whch a gvn by (B5 (B (B (B Th p a hown a follow Th angnal lcc-fld componn mu b connuou a z H ς ( xy Thfo h bounday condon mpod on h ufac pnd by h followng xpon ( E ν ( ς( + E ν x y jkz x y jk fl jk jkzς ( x y + d k ν j + jkz x y jk + jkzς( x y d ν ( k k k ς ( ( E f ν k ( k k + (B3 fo h lcc fld ( ν H d ( ς( + ν H x y jkz x y jk fl jk jkzς ( x y + kν j + jk x y jk jk ς( x y ( k k ν k ς ( ( ν Hf k ( k k z + d z (B4 fo h magnc fld Th ouwad nomal dfnd a z + ς ν (B5 + ς Nx h cancllaon of + ς ς on boh d of (B3 (B4 pfomd h nomal ν now o ν z + ς Fo convnnc h ngaon vaabl wa changd fom k o k A fuh modfcaon qud o xplo a pacula fau of ς ( xy n h xpon abov n ha k d jk ( k k ν jk ρ ς ( ( jk k k k kz k k d + d (B6 jk ( k k ν ( jk ρ ς ( ( ( d k d k k k + d k z k k jk 5 ( E ν ( ς( + E ν + x y jkz x y jk fl jk jkzς ( x y ( k k ν jk d k ς ( k k jk d kz ( k k jk + jk zς ( x y ( Ef ν ( jk jk zς ( x y ( k k ν ( jk d k ς ( k k ( jk d kz ( k k d k + d k ( (B8 Copondngly h magnc fld ha a mla fom o h abov ( ν H ( ς( x y jkz x y jk + fl ν H jk jkzς ( x y + d ( kz k ν k k k jk ( d k ς kz k k k ( jk d ( kz kz k k k jk ρ+ jk zς ( x y ( ν Hf jk jk ς ( x y + d kz ( jk ( z kν k k k d k ς kz k k k d kz kz k k k jk (B9 wh k h f pac wav numb Z Th vco wav numb k k a dfnd a k k kzz; k k k z z µ ε (B Th mpdanc laonhp gvn n (B (B w ud o oban (B9 No ha h m conanng h ± jkzς ( x y jk zς ( x y faco can b hown o b O ς xy W now ( nglgbly mall compad o mulply boh d of (B8 wh π jk nga ov h n xy plan Snc j( k k ρ d ρ π d k k (B (

16 Foum fo Elcomagnc ach Mhod Applcaon Tchnolog (FEMAT w oban d ( E k k ν j j( k k d k d ( ς( x y jkz x y fl jk zς ( x y ( E f ν jkzς ( x y ( j k k ν k k + d k d + ς ( k k ( π + E ν z ( k k ν ( + ς ( k k jk ς ( x y ( { z ( k k z ( k k } fo (B8 A mla p pfomd on (B9 yld ( ν H k k j( k k d k d (B ( ς( x y jkz x y j fl o jk x y kz d zς ( ν Hf jkzς ( x y ( j ν k k k k k + d k d + ς ( k k k + ν H ( jk zς ( x y ν k ( k k ( + ς ( k k k ( ( π { z k k k z k ( k k } (B3 Wh h gvn xpon h fnal oluon fo h vco cang amplud can b oband by ung a of dou algbac p (whch wll no b hown h Th fnal ul a oluon o h Fdholm Ingal quaon of h cond knd ( k k ( ( ( k k K k k K k k d k k k K k k K k k k k V ( k k + V ( k k (B4 wh h f-od oluon fo h cang amplud a ( ( k k ( ( ( ( E E ( + H H ( ( + ( V k k V k k k k T V k k T V k k TE VE k k TH VH k k dvng funcon V k k a dfnd a E H (B5 ς ( k k ( x y ( x y ( E ν + E j fl ν k k zς ( Ef ν d ( π E jk x y ς ( ν H j( k k ( ς( x y jk x y z fl H ( k k jk x y (B6 kz + ν H d zς ( π ( ν Hf (B7 call ha und h mall-lop conan ς ( xy << ν ; a a ul (B6 (B7 duc o (A7 (A8 (n Appndx A pcvly In xamnng (B (B3 on hould b mndd ha h no conan on h cal of k ς ( xy hu h connuy of h angnal lcomagnc fld componn hold a z H ς ( xy a long a h aylgh Hypoh vald Anoh condaon ha ς ( xy complly ndpndn of H hu h cad fld lack any pha connuy lad o h dffnc bwn ς ( xy H Thfo any lvl hf pubaon of ς ( xy (ay by om conan ± h would b complly abobd no H h fomulaon gvn n Appndx A hn would ll apply Jmmy O Alah wa bon n Wahngon DC n 975 H vd h BS MS PhD dg n Elccal Engnng n 998 pcvly fom Th Gog Wahngon Unvy n Wahngon DC H pmay n a n h aa of ough ufac cang mcowav ccu H ha bn mployd a a ach ngn by h ada Dvon a Th Naval ach Laboaoy n Wahngon DC Snc H cun ffo nclud ach on h dvlopmn of ada v m modul advancd gnal pocng chnqu Wayl Waylkwkyj (F'97 vd h BEE dg fom h Cy Unvy of Nw Yok n 957 h MS PhD dg n lccal ngnng fom h Polychnc Unvy n pcvly In 984 af a 5 ya ca n pva nduy wh h la poon wa vc pdn gnal manag of Phycal Dynamc Inc n La Jolla Calfona h jond h Gog Wahngon Unvy Wahngon DC a Pofo of Engnng Appld Scnc H ach xpnc n h acadmc nvonmn a wll a a conulan o nduy cov a boad pcum of appld lcomagnc Th nclud mcowav componn chnqu phad-aay Annna EM popagaon cang ada co-con modlng a wll a modlng gophycal ocanogaphc lcomagnc phnomna In 997 h wa lcd Fllow of IEEE fo ognal conbuon o lcomagnc popagaon cang o fundamnal undng of phad-aay Annna H lavly n ach wa n don fndng algohm wh pacula mpha on h ffc of muual couplng on algohm pfomanc Th focu of h cun ach n n annna fo gophycal applcaon cang fom om ufac 6