Performance Model for Distributed Sonar Tracking

Transcription

1 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig Sefao Coraluppi Ai-Submarie Warfare eparme NATO Udersea Research Cere Viale S. Barolomeo a Spezia ITAY coraluppi@saclac.ao.i ABSTRACT The objecive of his paper is o moivae he wo-sage approach o arge racig for a field of sesors wih variable deecio performace, ad o provide a modelig ool o predic racig performace. I he wo-sage approach, each sesor performs is ow arge racig, followed b a auomaed rac fusio capabili. We moivae he wo-sage approach b developig a Marov chai model for he mulisage racig process, ad aalzig is performace. Earlier Marov chai modelig wor applied o arge racig ca be foud i [-]. I his aalsis, all sesors share a commo coverage area. Thus, his paper does o address he beefi of muliple sesors i erms of area coverage, or does i address improved localizaio accurac. INTROUCTION This paper moivaes he wo-sage approach o arge racig for a field of sesors wih variable deecio performace, ad provides a modelig ool o predic racig performace. I he wo-sage approach, each sesor performs is ow arge racig, followed b a auomaed rac fusio capabili. This paper is orgaized as follows. I Secio, we iroduce he arge ad sesor models used i his aalsis. I Secio 3, we iroduce he sigle-sage racer model. Secio 4 discusses he wo-sage racer, which icludes a cocaeaio of models for he firs sage (he sigle-sesor racer model), ad he secod sage (he rac fusio model). Secio 5 provides a simulaio-based performace evaluaio of he racig approaches. Coclusios ad recommedaios for furher research are i Secio 6. I paricular, i is of ieres o compare our model-based coclusios wih resuls based o acual siglesage ad wo-sage racig algorihms. TARGET AN SENSOR MOES We model arge moio wih a sadard earl cosa veloci model. We model arge deecios o be ois posiioal measuremes, wih a cosa measureme covariace mari R. We assume ha for each arge-sesor pair, deecio saisics are govered b a wo-sae Marov chai, wih high-deecio ad low-deecio saes. This is a simple model for variable deecio performace due o he sourcearge-receiver geomer ad eviromeal effecs. The model is illusraed i Figure. We deoe he deecio sae b X. The deecio probabili is a fucio of he deecio sae X. aper preseed a he RTO SET Smposium o Capabiliies of Acousics i Air-Groud ad Mariime Recoaissace, Targe Classificaio ad Ideificaio, held i erici, Ial, 6-8 April 4, ad published i RTO-M-SET-79. RTO-M-SET UNCASSIFIE/UNIMITE

2 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig λ X X p λ p Figure : Marov chai model for deecio sae. Noe ha he deecio model is defied i coiuous ime. The probabiliies of rasiio from oe sae o he oher from he h o he (+) s sca are give approimael b euaios (.-.) below, where is he ime bewee scas. + ( ) ( ) p ep λ, (.) ( ) ( ) p ep λ. (.) I addiio o possible arge deecio, i each daa sca we have uiforml disribued false alarms i measureme space, wih desi λ. 3 SINGE-STAGE TRACKER MOE I he sigle-sage racig approach, each sesor (i.e. each source-receiver pair) geeraes a ime-ordered seuece of scas, ad he seueces are merged io a sigle ime-ordered seuece of scas. The, he racig algorihm processes he scas seueiall. Sice for he same arge he deecio sae varies across sesors, i is simples o model he deecio probabili o be cosa ad eual o is weighed average based o he saioar probabili disribuio for he deecio sae X. This saioar disribuio is give b he followig: The, we have: π π ( ) λ r X, (3.) λ + λ ( ) λ r X. (3.) λ + λ π + π. (3.3) Our rac iiiaio is based o M-of-N cofirmaio logic. The followig fiie sae Marov chai describes he damics for he rac logical sae, for 3-of-4 cofirmaio logic RTO-M-SET-79 UNCASSIFIE/UNIMITE

3 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig Figure : Marov chai model for rac logical sae, wih 3-of-4 cofirmaio logic. Before a daa is processed, each (fuure) eaive rac is cocepuall i logical sae. Subseuel, deecio eves (idicaed wih a ) ad missed deecio eves (idicaed wih a ) lead o logical sae rasiios. If he M-of-N crierio is achieved, he rac is cofirmed. I he eample i Figure, his correspods o logical sae X 7. The daa associaio gaig (or validaio) parameer leads o a probabili feasibl associaed o a rac. Thus, deecio eves occur wih probabili ha a deecio ca be, ad o-deecio eves occur wih probabili G. Thus, for he arge prese case, he probabili rasiio mari A for he logical sae Marov chai model is give b he followig (i he case of 3-of-4 cofirmaio logic): G G G G G G A G G. (3.4) G G G G ( ) ( ) ( ) We defie o be he epeced probabili disribuio o logical rac saes π π... π afer he h sca; is he umber of rac logical saes. All racs sar i sae, so ha π ( ). The probabili disribuio evolves as follows: The average rac cofirmaio ime is give b: ( ) π ( ) A π +. (3.5) RTO-M-SET UNCASSIFIE/UNIMITE

4 ( ) ( ) ( ) ( ) K T K j j K π π π lim. (3.6) I addiio o he average rac cofirmaio ime i he case of arge presece, we are ieresed o deermie he average umber of ew false racs per ui ime (hours), which we deoe as. This ca be deermied approimael as he produc of N FT Λ, he average umber of false alarms per sca ( S λ Λ, where S is he size of he surveillace regio), he probabili ha a false alarm leads o a cofirmed false rac ( ), ad he sca rae r (he average umber of scas per hour). (Noe ha we eglec he fac ha, due o daa associaio, i ma be ha o ever false alarm is reaed as a poeial sar of a ew rac). Below, we discuss he deermiaio of. FT FT The size of he validaio regio will deped o he deecio seuece. I paricular, he rac sae covariace is defied recursivel as follows: ( ) ( ) R & & σ σ, (3.7) ( ) ( ) ( ) ( ) ( ) Q + Φ Φ +, (3.8) ( ) ( ) ( ) ( ) H I δ, (3.9) where ad reflec a priori ucerai o arge veloci, σ & & σ + δ deoes a deecio eve, + δ is a o-deecio eve, ad R σ σ, (3.) ( ) Φ, (3.) ( ) Q , (3.) 37-4 RTO-M-SET-79 erformace Model for isribued Soar Tracig UNCASSIFIE/UNIMITE UNCASSIFIE/UNIMITE

5 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig ( + ) H( + ) H R S +, (3.3) ( + ) ( + ) H S ( + ), (3.4) H. (3.5) Noe ha, for simplici, we use a plaform-idepede posiioal measureme model. The cosas ad are process oise parameers. Usig he geeral epressio i [, p. 96], he validaio regio a sca + is give b: ( ) S( +) V δ γπ. (3.6) The choice γ. 9 leads o G.99. I (3.6), we mae eplici he depedece of he validaio regio o he deecio seuece δ ( δ,..., δ ). The umber of false alarms i he validaio regio a sca + is oisso disribued wih parameer validaio regio is give b: FA () λ V ( δ ). Thus, he probabili of a false alarm i he ( δ ) ep( λv ( δ ). (3.7) For each false alarm, we have π, i.e. we cosider he false alarm o have occurred i he firs sca. The, for he 3-of-4 rac cofirmaio logic illusraed i Figure, we have he followig probabili rasiio mari for he rac logical sae: A FA FA FA FA FA () FA () FA (, ) FA (, ) (, ) FA (,) (,, ) FA (,, ) (,, ) (,, ) FA. (3.8) The, we have: FT π ( N ), (3.9) from which we obai: N λs ( N )r. (3.) FT π Noe ha our racer model does o use a slidig-widow M-of-N, i ha he firs deecio defies he ime eed of he widow ad if rac cofirmaio is o achieved, he eire rac segme is discarded. A more effecive approach (bu more difficul o model) is used i our MHT racer [3]. RTO-M-SET UNCASSIFIE/UNIMITE

6 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig 4 MUTI-STAGE TRACKER MOE 4. Sigle-Sesor Tracer Model I he simplified sigle-sage racer model described above, we use a average ha is based o he saioar probabili disribuio give b euaios (.3-.4). For a sigle-sesor racer, we esimae he deecio probabili as a fucio of he deecio seuece. We use sadard recursive calculaios for fiie sae Marov chais [4]: ( δ ) [ π π ] π, (4.) π + + ( δ ) π c ( ) ( ) ( ) ( ) + ( δ )( ) + + ( )( ), δ + (4.) where c is a ormalizig cosa. The, he esimaed deecio probabili is give b: + + ( ) π ( δ ) δ +. (4.3) We use euaio (4.3) i euaio (3.4), raher ha a cosa deecio probabili. This resuls i he followig probabili rasiio mari: A () G () G (, ) G (, ) (, ) G (, ) G (,, ) G (,, ) (,, ) (,, ) G G. (4.4) G G Noe ha rasiios from sae ( δ ) are based o he saioar deecio probabili, sice. Euaios ( ) are uchaged. Also, for sigle-sesor racig, here is o chage o N FT he calculaios for he umber of false racs rae give i Secio 3. (Noe ha geerall he sca rae r will be slower i sigle-sesor racig, relaive o he sigle-sage muli-sesor approach). 4. Muli-Sage Tracig As a simple model for he muli-sage racer, we assume ha all racs geeraed b a umber of siglesesor racers are combied as follows: rue racs are fused, ad false racs are o. Thus, if a ssem has cofirmed arge deecio, he overall ssem has cofirmed deecio as well. The probabili of ssem cofirmaio a sca ad he average rac cofirmaio ime are give below. S is he umber of sigle-sesor racers. We have added a superscrip o ideif he i h racer s probabili disribuio i i, ( i) over logical saes, ad he probabili associaed wih he cofirmaio sae is deoed π. π 37-6 RTO-M-SET-79 UNCASSIFIE/UNIMITE

7 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig Ideed, hese uaiies ma var across racers as idividual racer parameers ad ipu daa characerisics var. Π S ( () ( ) i, i ( ) π ), (4.5) i T K j ( Π ( ) Π ( )) j lim. (4.6) K Π ( K ) The false rac rae is give b: where i N FT i N FT N FT, (4.7) is he false rac rae for he i h sigle-sesor racer. i 5 SIMUATION-BASE ERFORMANCE EVAUATION Our simulaio is based o he simplifig assumpios ha all sigle-sesor daa raes are he same, ad ha he overall daa flow has uiforml spaced ier-arrival imes. (For eample, if here are four sesors each wih a mi pig arrival ierval, he sigle-sage racer receives oe daa file each 5sec). We se arge, sesor, ad racer parameers as defied i Table. Our choice of Marov chai rasiio raes leads o a saioar probabili disribuio wih π.9,. 9. π Table : Simulaio parameers. rior veloci parameers σ &, σ & [m/s], Moio ucerai parameers, [m s -3 ] -5, -5 Marov chai rasiio raes λ, λ [s - ] -, -3 Surveillace regio S [m ] False alarm desi λ [m - ] Number of sesors S 4 Measureme covariace parameers σ, σ [m ], ig repeiio ime [sec] 6 Trac iiiaio parameers M, N 5, 6 Associaio gae probabili.99 Marov chai seps K,, 4 (for covergece) RTO-M-SET UNCASSIFIE/UNIMITE

8 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig We will cosider hree selecios for he high-deecio ad low-deecio probabiliies. Firs, we se hese o be he same:.5 (case ). Secod, we have. 3 ad.9 (case ). Third, we cosider a lower low-deecio probabili: ad..9 (case 3). For each of hese cases, we are ieresed i he average rac cofirmaio ime (i miues) ad he false rac rae (per hour), for hree racig cofiguraios of ieres: sigle-sesor (S), sigle-sage muli-sesor (SS), ad muli-sage muli-sesor (MS). Resuls are give i Table, ad Figures -3 show he probabili of rac cofirmaio as a fucio of ime for all racig cofiguraios. Table : Simulaio resuls. eecio probabiliies Tracer cofiguraio Average rac cofirmaio ime [miues] False rac rae [per hour].5 Sigle-sesor Sigle-sage Muli-sage ,. 9 Sigle-sesor ,. 9 Sigle-sage ,. 9 Muli-sage ,. 9 Sigle-sesor ,. 9 Sigle-sage ,. 9 Muli-sage Noe ha, for each of he hree ses of model deecio seigs, resuls are based o varig Marov chai ime horizos; his is doe o esure covergece i he resuls. Also, oe ha he sigle-sesor performace provides isigh io iermediae resuls, achieved afer each sesor performs is ow racig ad before he rac-fusio sage. I he firs case, where he deecio probabiliies are he same, oe ha he sigle-sage rac cofirmaio ime is slighl beer ha he muli-sage cofirmaio ime. This is o surprisig, as here is a beefi o he high daa rae see b he sigle-sage racer. The real beefi of muli-sage racig emerges whe we cosider wo widel varig deecio saes, as i cases ad 3. Sigle-sage racig performace is based o a deecio probabili ha is he weighed average of he wo deecio probabiliies, ad his is a raher low value. This reflecs he fac ha, o average, we have difficul i deecig arge presece. O he oher had, each sigle-sesor racer eplois reds i he daa, sice a firs deecio of he arge idicaes a high probabili of beig i he high-deecio sae, which i ur leads o a high probabili of a secod deecio, ad so forh. The rac fusio sage furher improves upo he sigle-sesor rac cofirmaio ime, b eploiig iformaio from muliple sesors. Noe ha, if he low-deecio probabili drops low eough (as i case 3), sigle-sesor performace acuall ou-performs he sigle-sage muli-sesor racer. This red is 37-8 RTO-M-SET-79 UNCASSIFIE/UNIMITE

9 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig eve more dramaic as we lower he low-deecio probabili o. or less, which is a case of operaioal ieres. Of course, i addiio o average rac cofirmaio ime, i is impora o address he false rac rae as well. Noe ha he resuls obaied do o deped o he arge deecio probabiliies, ad so are he same for all hree ses of model deecio seigs. Also, oe ha while he sigle-sage false rac rae is greaer ha he sigle-sesor rae, i is i fac smaller ha wha migh be epeced due o he icreased daa rae: i our case, he icrease is b less ha a facor of wo, raher ha a facor of four. This resul is due o he fac ha a high daa rae leads o small daa associaio gaes i he racig filer. As a resul, we see ha he muli-sage racer, which combies all false racs from he sigle-sesor racers, has a larger false rac rae ha he sigle-sage racer. Give he muli-sage racer s sigifical beer average rac cofirmaio ime, i is eas o ue he rac iiiaio parameers M ad N so ha, for he same false rac rae, we sill achieve a sigifical smaller average rac cofirmaio ime. Thus, we coclude ha, faced wih arges ha fade i ad ou of favorable deecio saes due o source-receiver-arge geomer ad eviromeal effecs, here is beefi o be gaied b pursuig a wo-sage racig process. I remais o address he addiioal beefi ha his archiecure ma achieve i erms of icreased robusess o daa regisraio errors. Figure. robabili of rac cofirmaio over ime (.5 ). RTO-M-SET UNCASSIFIE/UNIMITE

10 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig.3. Figure. robabili of rac cofirmaio over ime (, 9 )... Figure 3. robabili of rac cofirmaio over ime (, 9 ) RTO-M-SET-79 UNCASSIFIE/UNIMITE

11 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig 6 CONCUSIONS AN RECOMMENATIONS eceralized racig is a effecive archiecure for muli-sesor udersea surveillace ssems wih limied power ad badwidh. I addiio, he muli-sage approach o sesor fusio ad arge racig has he poeial for improved robusess i he face of daa regisraio errors ad variable deecio performace, whereb arges fade i ad ou of view depedig o eviromeal codiios, sourcearge-receiver bisaic rage, arge aspec agle, ec. This paper iroduces simple models for sigle-sage ad muli-sage racers. These models are useful o predic racer performace as a fucio of parameers ha relae o arge damics, deecio performace, ad racer seigs. Furhermore, hese models sugges ha variable deecio performace is bes addressed hrough a muli-sage racig approach whereb sigle-sesor racs are combied i a subseue auomaed rac fusio sage. These coclusios are based o he model simulaios described i he paper. I fuure wor, we pla o cofirm hese fidigs wih acual racer performace based o simulaed as well as sea rial daa. Our racer models, as well as our muli-hpohesis racer described i [3], do o as e address sesor regisraio errors. We pla o develop effecive real-ime approaches o reduce he derimeal impac of bias errors, as hese ma sigifical reduce he effeciveess of muli-sesor fusio. 7 REFERENCES [] Y. Bar-Shalom ad X. i, Muliarge-Mulisesor Tracig, YBS ublishig, 995. [] J. ezer ad T. Kirubaraja, Trac formaio i cluer usig a bi-bad imagig sesor, i roceedigs of he 3 rd Ieraioal Coferece o Iformaio Fusio, Jul, aris, Frace. [3] S. Coraluppi ad C. Carhel, rogress i Mulisaic Soar ocalizaio ad Tracig, SACANTCEN Repor SR-384, ecember 3. [4]. Kumar ad. Varaia, Sochasic Ssems, reice-hall, 986. RTO-M-SET UNCASSIFIE/UNIMITE

12 UNCASSIFIE/UNIMITE erformace Model for isribued Soar Tracig 37 - RTO-M-SET-79 UNCASSIFIE/UNIMITE