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1 On sonobuoy placemen for submarne rackng Mchael A. Kourzn a,davd J. Ballanyne a, Hyukjoon Km a,yaozhong Hu b a MITACS-PINTS, Deparmen of Mahemacal and Sascal Scences a he Unversy of Albera, Edmonon, Canada b Deparmen of Mahemacs a he Unversy of Kansas, Lawrence, Kansas ABSTRACT Ths paper addresses he problem of deecng and rackng an unknown number of submarnes n a body of waer usng a known number of movng sonobuoys. Indeed, we suppose here are N submarnes collecvely maneuverng as a weakly neracng sochasc dynamcal sysem, where N s a random number, and we need o deec and rack hese submarnes usng M movng sonobuoys. These sonobuoys can only deec he superposon of all submarnes hrough corruped and delayed sonobuoy samples of he nose emed from he collecon of submarnes. The sgnals from he sonobuoys are ransmed o a cenral base o analyze, where s requred o esmaed how many submarnes here are as well as her locaons, headngs, and veloces. The delays nduced by he propagaon of he submarne nose hrough he waer mean ha novel hsorcal flerng mehods need o be developed. We summarze hese developmens whn and gve nal resuls on a smplfed example. Keywords arge rackng, sonobuoy, submarne, nonlnear flerng. INTRODUCTION Alhough passve maneuverable sonobuoys are a very effecve counermeasure agans oday's sealhy submarnes, hey have crcal drawbacks n use. Measuremens from he sonobuoys' hydrophone arrays are very sporadc due o sonobuoy deploymen paerns and lmed baery lfe. Moreover, he daa sampled by he sonobuoys are dsored by he propagaon loss and corruped by amben nose facors lke emperaure, curren and pressure varaons. Fnally, boh localzng relave drecon from he sonobuoys and processng of he hree dmensonal measuremens are nherenly complcaed. Our soluon o resolvng hese problems s o employ mahemacal models and mehods. In parcular, he poson-dependen propagaonal delay of he sonobuoy sound pressure forces us o develop a novel hsorcal flerng approach... Waer body We model he ocean as he negave halfspace R fο (x; y; z) R z<0g; () so pons n he ocean are consraned o have a negave vercal componen. Moreover, for exhbon purposes, we assume ha he ocean s compleely homogeneous and ha sound pressure hng he surface of he ocean wll reflec back n a lossless manner. Furher auhor nformaon (Send correspondence o D.J.B.) M.A.K. E-mal mkourz@mah.ualbera.ca D.J.B. E-mal dballan@mah.ualbera.ca H.K. E-mal hkm@mah.ualbera.ca Y.H. E-mal hu@mah.ku.edu PINTS web page hp// Sgnal Processng, Sensor Fuson, and Targe Recognon XIV, eded by Ivan Kadar, Proc. of SPIE Vol (SPIE, Bellngham, WA, 005) X/05/$5 do 0.7/.6009 Downloaded From hp//proceedngs.spedgallbrary.org/ on //0 Terms of Use hp//spedl.org/erms

2 .. Submarne sgnal We consder an neracng mulple arge rackng problem of a random number N of submarnes, where each submarne s consraned o reman under he ocean surface. For smplcy, we ake each submarne o be a sphere wh radus ">0. Ths means ha he cener of mass of he submarne s consraned o be n R ", where fο (x; y; z) R z<±"g () R ±" We also force he submarnes o say above a deepes value z mn < " o reflec he realy ha submarnes can no go arbrarly deep due o pressure consrans. The hree dmensonal poson, orenaon, and forward speed of he h submarne s modeled by a dffuson process desgned o keep he poson of all submarnes n R " wh her deph above z mn,herveloces v whn physcal consrans v mn» v» v max for some 0» v mn» v max, her angles of aack whn physcal consrans ff mn» ff» ff max for some ß collsons. In parcular, suppose X x y z» ff mn < 0 <ff max» ß ; and her orenaons adjused o avod 5 ;v,, and ff [ff mn ;ff max ] represen he hree dmensonal poson, forward speed, horzonal bearng, and angle of aack of he h submarne. Then, we defne he neracon erms for he h submarne wh respec o he oher submarnes o be NX» (y y n )cos( ) (x x n ) +(y y n ) +(z z n ) (x x n )sn( ) (x x n ) +(y y n ) +(z z n () ) and n;n6 NX nn6 " p # (z z n )cos(ff ) (x x n ) +(y y n ) +(z z n ) (x x n ) +(y y n ) sn(ff ) (x x n ) +(y y n ) +(z z n () ) In parcular, and are used o conrol he bearng and angle of aack, respecvely, of he submarnes n such away asoryo avod collsons wh oher submarnes. We also use conrol of he angle of aack o ensure ha he submarne does no leave hewaer or go oo deep. The oal conrol on he angle of aack s hen gven by fl 8 >< > fl ffmaxff ff maxff mn fl 0 fl ff ff mn ff maxff mn fl < 0,whfl and he dynamcs of he h submarne are gven by» dx dv 6 c z z z mn + dx v cos( )cos(ff )d + ff x dw x; dy v sn( )cos(ff )d + ff y dw y; dz v sn(ff )d + ff z p (" z )(z z mn )dw z; d c d + ff dw ; dff fl d + ff ffp (ffmax ff )(ff ff mn )dw ff; dv c v (v AV G v )d + ff v p (vmax v )(v v mn )dw v; c z z + " + c ff (5) 7 5 ; (6) Φ where W x; ;W y; ;W z; ;W ; ;W ff; ;W v;ψ are all ndependen Brownan moons, he ff's and c's are all posve consans, and v AV G vmn+vmax. We are really neresed n he counng measure ha keeps rack of he sae of all submarnes bu does no dfferenae whch one s whch, so we defne E as he collecon of fne counng measures over R 6 and S E by S NX j ff X ;V ; (7) Proc. of SPIE Vol Downloaded From hp//proceedngs.spedgallbrary.org/ on //0 Terms of Use hp//spedl.org/erms

3 where ff x pus a pon mass a he locaon x R 6. Now, snce he sound pressure n he sysem depends upon he curren and pas saes of he sgnal S,we defne he pahspace verson of hs process S [0;] C E [0; ) by ρ S [0;] (f) Sf f < f S (8).. Nose emsson Each submarne s assumed o f n a hree dmensonal "-ball B(X ;") cenered a he locaon X of s cenre of mass. I hen ems nose pressure dsrbued over B(X ;") accordng o some nfnely connuously dfferenable R-valued funcon of R,.e. C (R ). We assume ha s even wh respec o z,.e. (x; y; z) (x; y; z), and sasfes 0, B(0;") C 0, and where B(0;") C s he complemen ofb(0;")andξ (Ξ) ( x y z B(0;") c expn " jξj o jξj <" 0 oherwse,wherec (Ξ)dΞ, (9) 5. For example, could be RB(0;") exp n " jξj o dξ s chosen so ha R R (Ξ)dΞ R B(0;") (Ξ)dΞ. Then, he nose emed by heh submarne a me s assumed o be (Ξ X ). To approxmae he nose generaed by all N submarnes, we le T > 0 be chosen so ha all nose generaed pror me T has lef he area of neres by me 0 and defne he nose generaon funcon NX f(; Ξ) (Ξ X) (Ξ ß X (x))ds (x) n(t;) R R ; () 6» where ß X denoes he dmensonal projecon ono he posonal componen,.e. ß X X ( ) X. Ths V funcon f(; Ξ) represens he superposon of nose emed from all submarnes a me and a he locaon Ξ... Nose propagaon By our assumpon ha he ocean s homogeneous and wh perfecly reflecng surface boundary, he sound pressure u(; x) from he N submarnes follows he wave equaon d u(; x; y; z) d c u(; x; y; z)+f(; x; y; z) n(t;) R () u(t ; x; y; z) 0; d d u(t ; x; y; z)0nr d dz u(; x; y; 0) 0n(T;) R ; where c s a known speed consan. The ocean s consdered o be nfnely deep o avod furher reflecons, whch s a usable approxmaon n he mddle of he oceans. We le he unque soluon o hs equaon be denoed u(; Ξ; S [0;] ) o hghlgh he dependence of he nose on he hsorcal saes of submarnes hrough he forcng erm f(; x; y; z). (0) 6 Proc. of SPIE Vol Downloaded From hp//proceedngs.spedgallbrary.org/ on //0 Terms of Use hp//spedl.org/erms

4 .5. Sonobuoy locaons Φ The sonobuoy posons Ψ M f ff f are maneuverable whn he consrans ha d d f» K v and f d ff d» Ka. For smplcy n hs exhbon, we ake he sonobuoy pahs o be known,.e. deermnsc. Fuure work wll consder he case where he sonobuoys can be conrolled based upon he observaons ha arrve from prevous sonobuoy measuremens..6. Sonobuoy observaons A each measuremen nsan 0 < < <, here are M measuremens, one from each of M sonobuoys. The measuremen Yk ( k ) receved n he command cener from he h sonobuoy a he poson k and a he me k s a funcon of he sound pressure dsrbued over he sonobuoy and hen corruped by nose Y k 6 Y Y k ( ) k ( ). Y M k ( ) 7 5, Y k ( ) ff (Ξ k )u( k ; Ξ; S [0;k ])dξ+vk ; 0»» M; () R where fv k g s a R M -valued ndependen ±N (0;I) sequence. Here, he fff ( )g M are some gven generalzed funcons ha represen he densy of he hydrophone array as well as s sensvy o sound pressure, and f k g M ρ R are he sonobuoy locaons a me k. For smplcy, heff can be aken o be for some small f>0, n whch case Y k ( ) LX l0 ff (Ξ) LX l0 ff (0;0;fl) () u( k ; k (0; 0;fl) T ; S [0;k ])+Vk ; 0»» M (5) Ths would represen he suaon where we are sensng he sound pressure a L equally spaced pnpon locaons drecly below each sonobuoy..7. Objecves The objecve s o calculae he bes mean-square esmae of he submarne pahs gven he sonobuoy observaons f E S [0;k f ] fffy ;;Y k g Λ ; (6) effcenly and accuraely on a compuer... Noaon. METHOD For any opologcal space X,weleB(X ), C (X ), C (X ) denoe respecvely he bounded, measurable; connuous; and bounded, connuous funcons f X! R. Nex, recallng ha E s our space of counng measures, we le C E [0; ) denoe he E -valued connuous funcons of (me) [0; ). Moreover, we le ß be he projecon funcon from C E [0; ) oe a me, meanng ha ß (S) S for S C E [0; ). Fnally, we le Cc k (R n )be he k mes connuously dfferenable R-valued funcons wh compac suppor on R n. To keep rack ofall of he nformaon generaed by he submarne sgnal and sonobuoy observaons, we defne F SY fffs s ;Y k s» and k» g and Fk SY F SY k. Moreover, o keep rack of he nformaon gven by he observaons alone, we defnefk Y fffy ;;Y k g for k ; ; f f f f Proc. of SPIE Vol Downloaded From hp//proceedngs.spedgallbrary.org/ on //0 Terms of Use hp//spedl.org/erms

5 .. Mahemacal equaon for he hsorcal sgnal To handle he reflecons a he surface, we lex ; V R " be he mrror mage of X ;V over he plane z 0, se S b P P N j ff X ;V + N j ff X ;V, and ake (D(L); L) ρ C (E) C (E) o be he weak generaor for he Markov process fs ; 0g, so ha Ψ M S (') '(S ) '(S 0 ) L'(S s )ds; (7) 0 Φ s a connuous F S -marngale for each ' D(L). Now, snce we mus work wh he pahspace verson 0 of hs process, s convenen o defne a pahspace varan of our operaor ha wll be he weak generaor for! S [0;] C E [0; ). Wh hs n mnd, we lei m ff m g m be such haim ρ I m+, I [ m Im s dense n [0; ), and 0 f m 0» f m»»f m m» fm m. Then, we defne D(L [0;s] ) o be he lnear span sp Φ ' (ß f m ) ' m (ß f m m )' D(L); m; ; Ψ (8) for s [0; ) and ake L [0;s] o be he operaors on B(C E [0; )) defned on D(L [0;s] )by L [0;s] ' (ß f m ) ' m (ß f m m )' (ß f m ) ' j (ß f m j ) L(' j;m)(ß s ) (9) for f m j» s<fj m, where ' j;m(x) ' j (x) ' m (x). Ths operaor s necessarly me-nhomogeneous even when he orgnal operaor s homogeneous. Snce hese domans are only measurable funcons, we can no mmedaely use he fac ha hese funcons separae pons on D E [0; ) o conclude ha hey separae probably measures. However, we can show ha f wo probably measures μ ;μ agree on D(L [0;s] ) hey agree on he cylnder ses and hence on B(D E [0; )). Wh hs pahspace operaor, we can defne he hsorcal marngale problem aconnuous Φ F S Ψ M S (Φ) Φ(S ) Φ(S 0 ) 0 L [0;s] Φ(S [0;s] )ds; (0) 0 -marngale for each ΦD(L [0;s]). In parcular, one can readly verfy ha M S (' (ß f m ) ' m (ß f m m )) mx j f m j ^ f m j^ ' 0 (S f0 ) ' j (S fj)dm S s (' j;m ) () Equaon (0) unquely characerzes he dsrbuon of he submarne pahs and wll be useful n characerzng he opmal esmae of he submarne locaons gven he observaons... Explc soluon for he wave equaon To process he sonobuoy observaons, s convenen o derve an explc soluon for he sound pressure u n erms of he hsorcal submarne pahs. The soluon o he wave equaon () a me and pon (x; y; z) s where u(; x; y; z) ß g( + T j p c(x; y; z) (ο; ; )j; ο; ; ) j p dοd d ; () c(x; y; z) (ο; ; )j ρ f(; x; y; z) z g(; x; y; z)» 0 f(; x; y; z) z>0 () B( p c(x;y;z);+t ) In erms of he submarnes, he soluon s u(; x; y; z) B( p c(x;y;z);+t ) B( p c(x;y;z);+t ) u(; x; y; z; S [0;] ) NP NP h ((ο; ; ) Xj p c(x;y;z)(ο; ; )j ) + ((ο; ; ) X j p c(x;y;z)(ο; ; )j) dοd d ßj p c(x; y; z) (ο; ; )j h (Ξ Xj p c(x;y;z)ξj )+ (Ξ X j p c(x;y;z)ξj) ßj p dξ () c(x; y; z) Ξj 8 Proc. of SPIE Vol Downloaded From hp//proceedngs.spedgallbrary.org/ on //0 Terms of Use hp//spedl.org/erms

6 Ths explc represenaon wll ease he compuer evaluaon of he condonal dsrbuon for he hsorcal sgnal gven he observaons... Compuaon va Bayes' rule In order o calculae he dsrbuon of he sgnal gven he observaons s easer o frs calculae hs dsrbuon wh an arfcal (reference) probably measure and hen o conver back usng Bayes rule. Wh hs n mnd, we defne and A k ky j h k 6 h k h k. h M k 7 5,whereh k R ff k(ξ k )u( k ; Ξ; S [0;k ])dξ; (5)» j, where j exp h T j ± V j ht j ± h j exp»h Tj ± Y j + htj ± h j (6) In he case ha ff (Ξ) P L l0 ff (0;0;fl), wewould have ha h k LX l0 In eher he general or specfc case, we se u( k ; k (0; 0;fl) T ; S [0;k ]), for ; ;;M. (7) k A k ky j j (8) Now, fa k g Φ Ψ s a Fk SY -marngale and we can use Grsanov's heorem (and exenson) o defne a new k probably measure on F SY P () P (A k ) 8 Fk SY (9) for all > 0. Therefore, leng E;E denoe expecaon wh respec o P; P respecvely, usng he fac ha S [0;] s F SY -measurable and recallng Bayes' rule, we have ha E['(S [0;k ])jf Y k ] μ E['(S [0;k ]) k jf Y k ] μe[ k jf Y k ] (0) for ' B(D E [0; )). Snce he denomnaor and numeraor of (0) are boh calculaed from μ E[g(S[0;k ]) k jf Y k ] wh g andg ' respecvely, we only need a mehod of compuaon for μ [0;] ' μ E['(S[0;] ) k jf Y k ] for [ k ; k+ ) () over a rch enough class of funcons ' C E [0; )! R, such asd(l [0;s] )..5. Weghed parcle mehod By he mehod of he prevous subsecon, we only need o fnd an approxmaon for μ [0;] ' μ E['(S[0;] ) k jf Y k ] for [ k ; k+ ) () However, S [0;] s ndependen of Fk Y! S m wh he same law [0;] as! S [0;] and defne he wegh ofhem h parcle by m k wh respec o P. Therefore, we can consruc ndependen parcles ky j exp»h Tj;m± Y j htj;m ± h j;m ; () Proc. of SPIE Vol Downloaded From hp//proceedngs.spedgallbrary.org/ on //0 Terms of Use hp//spedl.org/erms

7 where n he general case or h k;m h k;m ff k(ξ k )u( k ; Ξ; S[0; m k R ])dξ, for ; ;;M, m ;;n () LX l0 u( k ; k (0; 0;fl) T ; S[0; m k ]), for ; ;;M, m ;;n (5) n he case ha ff (Ξ) P L l0 ff (0;0;fl). Then, usng he law of large numbers, one has ha and n nx m '(S m [0; k ] )m k.6. Resampled parcle mehod n nx m n!! μ E['(S[0;k ]) k jf Y k ]for' D(L [0;s] ) (6) ff S m [0;k ] ( )m k n!! μ [0;k ]( ) (7) The mehod n of he o prevous subsecon works n heory bu suffers n pracce due o he fac ha he parcle n locaons S[0; m k ] are no affeced by he observaons, bu raher only by he parcle weghs k m. Asa m resul, he parcles do no represen he submarne sgnal well and he approxmaon ends o break down n pracce. To counerac hs problem, our group has prevously developed he Selecvely Resamplng Parcle fler. Ths mehod does parwse resamplng of he parcles, sarng wh he wo wh he hghes and lowes weghs. I replaces hese wo parcles wh wo parcles boh havng he pah of he hghes weghed parcle wh hgh probably or he pah of he lowes weghed parcle wh small probably, and boh wh weghs equal o he average of he wo prevous weghs. Ths s done n a manner so as no o nroduce bas no he sysem. The process s repeaed unl he weghs of all parcles are whn a gven bound.. RESULTS Sofware mplemenng he above soluon has been consruced and shows promse n nal rals wh a sngle submarne. However, s no ye compleed o work wh mulple submarnes... Example problem The problem has been mplemened wh concree values for he many parameers nvolved. The radus of he submarnes " s aken o be 00, and he area of neres s he ocean n [0; ] [0; ] [z mn ; "]. Submarnes ha leave hs area of neres are consdered, as a smplfcaon, o have absoluely no neracon wh he remanng smulaon. The submarne consrans v mn, v max, ff mn, and ff max ake hevalues 000, 0005, ß,and ß. Consans of he submarne dynamcs c z and c v ake values and 00, whle c and c ff are unnecessary wh only one submarne. The consans for he nose erms n he submarne dynamcs are ff x ff y ff z 000, ff 008, ff ff 005, and ff v 0. For calculaon of he generaed nose, a value of " 00 mples ha he value for c mus equal approxmaely The speed consan c of ransmsson hrough he waer s aken o be, and hs means ha a value of T s enough o ensure ha boh he drec and refleced sounds made n he area of neres a me T wll have all lef by me0. As a smplfcaon, he sonobuoys are assumed o move along a known pah, aken o be a crcle of radus 0 cenered whn he area of neres. Four sonobuoys nally ake posons a he eas, norh, wes, and souh pons of hs crcle and each move counerclockwse abou he crcle wh a speed of one half degree of roaon per un me. No sonobuoys are added or removed n hs sage of he research. Each sonobuoy has L 5 hydrophones whch each descend a dsance f 00 below he prevous, wh he frs a he surface. In he 50 Proc. of SPIE Vol Downloaded From hp//proceedngs.spedgallbrary.org/ on //0 Terms of Use hp//spedl.org/erms

8 nose funcon V k for he sonobuoy observaons, he nose for each of he sonobuoys s aken o be ndependen and dencally dsrbued, so ha he marx ± k s a dagonal marx wh enres ff 05. Whle flerng, samples of he sgnal need only conan a shor hsory of sample submarne saes snce any nose generaed from earler submarne posons wll have lef he area of neres. ASelecvely Resamplng Parcle fler, prevously called a hybrd weghed neracng parcle fler, wh n 000 parcles s suffcen o provde excellen rackng resuls n an effcen compuaon. An example smulaon s shown n fgure and fgure. In hese fgures, he ocean ruh s a he lef, he observaons a he sonobuoys are dsplayed n he mddle, and he sae of he fler s dsplayed a he rgh. The submarne locaon s shown as a crcle ralng he pas pah of he submarne. Each sonobuoy sshown as a square. Noe ha he sonobuoys have moved counerclockwse from me 5 o me 00. The four lnes n he observaon dsplay ndcae he sound pressure Yk ; ; ; ; experenced a each ofhe sonobuoys. Posons of parcles n he (x; y) plane are shown as whe dos n he fler dsplay, and he rue locaon of he submarne s shown as a red crcle n order o faclae vsual nspecon of fler success. I can be seen ha by me 00, he fler has subsanally localzed he (x; y) poson of he submarne. Fgure. Example smulaon a me 5... Performance Fgure. Example smulaon a me 00. Whle he fler does no always hold a close localzaon of he arge submarne, does rack near o he arge afer an nal deecon phase. Also, he arge s almos always deeced successfully, meanng ha evenual localzaon does occur. Holdng a close localzaon a all mes s no o be expeced, snce he observaons are composed of nose pressures n he waer ha are deeced from pas posons of he submarne. If he submarne akes an unlkely maneuver, he fler wll provde a greaer lkelhood o saes o whch he submarne would have been more lkely o move unl such me as he observaons provde correcng nformaon. Tme consrans dd no allow he producon of graphcal analyses of he fler performance before publcaon.. CONCLUSIONS A promsng mehod has been developed o rack submarnes usng he daa from maneuverng sonobuoys. Inal ess wh hs fler usng he parameers gven above demonsrae a conssen ably o localze and Proc. of SPIE Vol Downloaded From hp//proceedngs.spedgallbrary.org/ on //0 Terms of Use hp//spedl.org/erms

9 rack one submarne wh random nal sae. Furher work s ongong o measure he resuls, o provde rackng of mulple submarnes, and o deermne opmal conrol of he sonobuoys. ACKNOWLEDGMENTS The auhors graefully acknowledge he suppor and sponsorshp of Lockheed Marn MS Taccal Sysems, he Pacfc Insue for he Mahemacal Scences, he Naural Scence and Engneerng Research Councl (NSERC) hrough he Predcon n Ineracng Sysems (PINTS) cenre of he Mahemacs of Informaon Technology and Complex Sysems (MITACS) nework of cenres of excellence, and he Unversy of Albera. Travel fundng was provded n par by he Mary Louse Imre Graduae Suden A ward. REFERENCES. D. J. Ballanyne, S. Km, and M. A. Kourzn, A weghed neracng parcle-based nonlnear fler," Proc. SPIE, Sgnal Processng, Sensor Fuson, and Targe Recognon XI 79, pp. 67, 00.. M. J. Berlner and J. F. Lndberg, Acousc parcle velocy sensors desgn, performance, and applcaons Mysc, CT, Sepember 995, AIP Press, Woodbury, N.Y., M. Fujsak, G. Kallanpur, and H. Kuna, Sochasc dfferenal equaons for he non lnear flerng problem," Osaka J. Mah. 9, pp. 90, 97.. M. Ikawa, Hyperbolc paral dfferenal equaons and wave phenomena," Translaons of Mahemacal Monographs 89, J. Schlz, Flrage de dffusons fablemen bruées dans le cas corrélé," C. R. Acad. Sc. Pars Sér. I Mah. 5(), pp. 996, Proc. of SPIE Vol Downloaded From hp//proceedngs.spedgallbrary.org/ on //0 Terms of Use hp//spedl.org/erms