Ship-Track Models Based on Poisson-Distributed Port-Departure Times

Transcription

1 Naval Research Labratry Washingtn, DC NRL/FR/ ,122 Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times RICHARD HEITMEYER Acustic Signal Prcessing Branch Acustic Divisin January 20, 2006 Apprved fr public release; distributin is unlimited.

2 Frm Apprved REPORT DOCUMENTATION PAGE OMB N Public reprting burden fr this cllectin f infrmatin is estimated t average 1 hur per respnse, including the time fr reviewing instructins, searching existing data surces, gathering and maintaining the data needed, and cmpleting and reviewing this cllectin f infrmatin. Send cmments regarding this burden estimate r any ther aspect f this cllectin f infrmatin, including suggestins fr reducing this burden t Department f Defense, Washingtn Headquarters Services, Directrate fr Infrmatin Operatins and Reprts ( ), 1215 Jeffersn Davis Highway, Suite 1204, Arlingtn, VA Respndents shuld be aware that ntwithstanding any ther prvisin f law, n persn shall be subject t any penalty fr failing t cmply with a cllectin f infrmatin if it des nt display a currently valid OMB cntrl number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (Frm - T) Frmal Reprt 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times 6. AUTHOR(S) Richard Heitmeyer 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Research Labratry 4555 Overlk Avenue, SW Washingtn, DC b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 62747N 5d. PROJECT NUMBER UW e. TASK NUMBER 5f. WORK UNIT NUMBER PERFORMING ORGANIZATION REPORT NUMBER NRL/FR/ , SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) Office f Naval Research One Liberty Center 875 Nrth Randlph Street Arlingtn, VA SPONSOR / MONITOR S ACRONYM(S) ONR 11. SPONSOR / MONITOR S REPORT NUMBER(S) 12. DISTRIBUTION / AVAILABILITY STATEMENT Apprved fr public release; distributin is unlimited. 13. SUPPLEMENTARY NOTES 14. ABSTRACT This reprt presents tw mdels that describe the tracks f ships during an arbitrary time interval. Bth mdels cnsist f a track functin that describes the tracks f the individual ships and a prbability law n the ttal number f ships en rute during the interval, the initial psitins f thse ships, and their nminal speeds. The prbability law assumes that the ship departure times are Pissn-distributed with a time-varying departure rate and that the ship speeds and the ship rutes are statistically independent. Fr bth mdels, it is shwn that the ship-track parameters are distributed as a multidimensinal, nnhmgeneus, Pissn prcess with a time-dependent rate functin. In the first mdel, the tracks are deterministic functins with the cnstraint that a track can neither duble back n itself nr intersect any ther track. In this mdel, the track functin and the track-parameter rate functin are determined frm prbability densities describing the cncentratin f ship psitins alng the rutes. In the secnd mdel, the ship tracks are btained as realizatins f a Markv prcess withut any cnstraints n the track crssings. 15. SUBJECT TERMS Shipping mdel Shipping distributins Ship tracks Pissn distributed ship psitins Prt-t-prt traffic 16. SECURITY CLASSIFICATION OF: a. REPORT b. ABSTRACT c. THIS PAGE Unclassified Unclassified Unclassified 17. LIMITATION OF ABSTRACT UL 35 i 18. NUMBER OF PAGES 19a. NAME OF RESPONSIBLE PERSON Richard Heitmeyer 19b. TELEPHONE NUMBER (include area cde) (202) Standard Frm 298 (Rev. 8-98) Prescribed by ANSI Std. Z39.18

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4 CONTENTS 1. INTRODUCTION KINEMATICS The Rute Functin The Ship-Mtin Functin The Ship-Track Functin THE PORT-DEPARTURE AND TRACK-PARAMETER PROBABILITY LAWS A DETERMINISTIC ROUTE FUNCTION MODEL The Rute Crdinate Prbability Density The Rute Functin The Track-Parameter Prcess A STOCHASTIC ROUTE FUNCTION MODEL The Rute Set Prbability Law The Rute Set Functin The Track Parameters SUMMARY...18 ACKNOWLEDGMENTS...19 REFERENCES...19 APPENDIX A Transfrmatins f Pissn Prcesses...21 APPENDIX B Derivatin f the Track-Parameter Prbability Laws...27 APPENDIX C A Simplified Deterministic Mdel...31 iii

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6 SHIP-TRACK MODELS BASED ON POISSON DISTRIBUTED PORT-DEPARTURE TIMES 1. INTRODUCTION A number f applicatins require a statistical descriptin f the lcatins f ships in a regin, e.g., ambient nise mdels and prbability-based ship tracking algrithms. Fr many f these applicatins, it suffices t specify the mean number f ships n a grid f elemental lngitude-latitude areas. Such descriptins, ften referred t as shipping distributins, have been btained frm bth direct measurements f individual ship psitins and frm shipping mdels based n prt-departure times and ruting data. A well-knwn example f the latter is the Histrical Tempral Shipping (HITS) mdel [1,2]. This mdel specifies shipping distributins fr different classes f cmmercial ships and different time perids ver a large prtin f the wrld's ceans. The reslutin f these shipping distributins, i.e., the dimensins f the lngitude-latitude grid, is 1º 1º. Many applicatins, hwever, require a descriptin f the tracks f the individual ships ver sme time interval ( t, t rather than simply the mean number f ships in elemental areas. Fr sme f these applicatins, the duratin f the required bservatin perid T may exceed the time interval ver which many f the ships are within the regin f interest. Fr these applicatins, the ship-track mdel must be capable f intrducing new ships int the regin in a manner cnsistent with the underlying shipping distributin. The requirements f the ship-track mdel als depend n the types f ships that are relevant t the applicatin. Fr mst cmmercial shipping, the ships simply transit frm ne prt t anther alng rutes that cnstitute a mre r less well-defined shipping lane. Fr this type f shipping, it may suffice t prvide simple smth apprximatins t the actual rutes traveled frm the departure t the destinatin prt. On the ther hand, shipping traffic such as fishing, recreatinal and military vessels can travel alng cmplicated rutes as dictated by the specific missin f the vessel. Fr this type f shipping, mre cmplicated descriptins f the rutes may be required. In either case, the accuracy requirements f the ship tracks will depend n the specific applicatin f the mdel. This reprt presents tw mdels, a deterministic and a stchastic mdel, each f which describe the tracks f ships en rute in a regin during an arbitrary time interval ( t, t. Bth mdels cnsist f a track functin that describes the tracks f the individual ships and a prbability law n the ttal number f ships en rute during ( t, t, the psitins f thse ships at the initial time t, and their nminal speeds. The prbability law is btained under the assumptin that the times at which the ships depart each prt are Pissn distributed with a time-varying departure rate and that the ship speeds and the rutes that the ships travel are statistically independent. Under this assumptin, it is shwn that the ship-track parameters are distributed as a Pissn prcess with a time-dependent rate functin. The rate functin that specifies this prcess differs fr the tw ship-track mdels. The tw ship-track mdels prvide alternate descriptins f the ship s tracks. In the deterministic mdel, the ship tracks are deterministic functins derived frm a prbability density n ship psitins in Manuscript apprved September 8,

7 2 Richard Heitmeyer manner analgus t that f the HITS mdel (see Ref. 2). As such, the individual ship tracks can neither duble back n themselves nr intersect any ther ship tracks. This may be an acceptable cnstraint fr ship traffic that simply transits frm ne prt t anther, prvided that detailed representatins f the individual tracks are nt required. In the stchastic mdel, the ship tracks are btained as realizatins f a Markv prcess withut any cnstraints n the rute crssings. Accrdingly, this mdel is suitable fr shipping where the tracks are determined by peratins mre cmplicated than prt-t-prt transiting. Furthermre, the stchastic nature f the tracks may be mre realistic than the smth track apprximatins f the deterministic mdel. This reprt is rganized as fllws: Sectins 2 and 3 prvide the backgrund fr the mdel definitins. In Sectin 2, we present the definitins and assumptins that are used t describe the kinematics f the shipping fr each rdered pair f prts that supprt shipping in the regin. Specifically, fr each prt pair, we define the ship-track functin in terms f a rute functin and a mtin functin. In general, these functins depend n a track parameter that specifies the distance that the ship has traveled at the initial time t, the specific rute that the ship travels between the departure and the destinatin prt and the ships nminal speed alng that rute. T accunt fr thse ships that are nt en rute at time t, but depart the prt during the interval ( t, t, we allw these parameters t take n virtual values. In Sectin 3, we state the assumptins n the prt-departure prbability law and describe the prbability law n the track parameter that results frm these assumptins. It is seen that the track-parameter prbability law can be viewed as the cmpsitin f tw Pissn prcess, ne representing the ships that are present at time t and ne representing the ships that enter the regin in the interval ( t, t. The ship-track mdels themselves are described in Sectins 4 and 5. Fr each f these mdels, we first specify the rute set prbability law that frms the basis f the mdel. We then specify the track functin and the track-parameter prbability law that results frm the rute set prbability law. Fr bth mdels, the track parameter can be expressed in terms f the ship psitin at the initial time t and the nminal speed. This leads t the ntin f the shipping density frm which the shipping distributin can be derived by integrating ver elemental areas. The derivatins f these results are presented in Appendixes A and B. Fr reference purpses we have als prvided a definitin f a multi-dimensinal Pissn prcess in Appendix A. In Appendix C, we present a simplified versin f the deterministic mdel. Finally, the results f the reprt are summarized in Sectin 6. An example f an ambient nise mdel that draws n a deterministic ship-track mdel is described in Refs. 3 and 4. Reference 5 presents an ambient nise applicatin t the regin arund San Dieg, Califrnia. 2. KINEMATICS A realizatin f ship tracks must represent nt nly thse ships that are present in the regin during the entire time interval f interest, but als thse ships that either enter the regin r exit the regin during that interval. The latter d s by either departing r arriving at a prt r by crssing the bundary f the regin. Figure 1 illustrates this prcess. The regin f interest is bunded by the castline segments indicated by the heavy black lines and by the dtted line segments acrss the access areas t the regin. The prts, labeled P 1 thrugh P 9, are classified as either real prts r pseud prts. The real prts are thse that are physically lcated within the regin. The pseud-prts, lcated alng the access bundaries f the regin, represent the shipping frm the aggregate f the prts lcated utside the regin that prvide traffic t the regin. The prts P 6 and P 9 are pseud prts; the remaining prts are real prts. The ship tracks are indicated by the line segments. The dts n the segments shw the ship psitins at time t ; the arrwheads shw the psitins at time t + T. The ship tracks fall int tw grups, ships that are en rute at time t and ships that are in prt at time t but depart the prt during the interval ( t, t. Ship tracks in the first grup have a dt at the beginning f the track; ship tracks in the secnd grup d nt have a dt at the beginning f the track. Fr tracks in the first grup, there are a number f pssibilities: (a) the ship remains en rute fr the entire time interval (indicated by a track with

8 Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times 3 bth a dt and an arrwhead); (b) the ship arrives at a prt during the time interval and remains there fr the rest f the time interval (indicated by a track with a dt but n arrwhead); (c) the ship arrives at a prt and then departs frm that prt during the time interval (indicated by the dashed track at P 2 ); and (d) the ship arrives at and then departs frm mre than ne prt during the interval. The same pssibilities exist fr the ships that are at a prt at the beginning f the time interval nce thse ships have departed the prt. P 2 P 1 P 3 P 4 P 9 P 8 P 5 P 7 P 6 Fig. 1 A pssible set f ship tracks fr a regin The ship-track mdel presented here des nt describe the tracks f ships that pass thrugh a prt during the time interval f interest. Instead, it represents such tracks as tw tracks ne assciated with a ship that arrives at the prt and remains there fr the rest f the time interval and ne assciated with a ship that is at the prt at the beginning f the interval and then departs at sme time during the interval. Thus, the track f the ship passing thrugh P 2 in Fig. 1 is represented as the track f ne ship that enters P 2 plus the track f anther ship that departs frm P 2. This apprximatin simplifies the mdel cnsiderably since it bviates the need t relate a ship s departure time t its arrival time. Furthermre, it allws the shipping between any pair f prts t be cnsidered separately frm the shipping between any ther pair f prts. This apprximatin is reasnable if the duratin f the time interval is shrt cmpared t the mean time that ships spend in a prt since, in this case, the number f tracks that pass thrugh the prt in the time interval will be negligible. It is als be reasnable fr lnger time intervals if it is nt necessary t preserve the identity f the individual ships that pass thrugh the prts. As a result f the independent prt-pair apprximatin, the reginal shipping mdel can be btained as the cmpsitin f ship-track mdels fr each rdered pair f prts ( Pn, P m) that supprt shipping within the regin. The first prt in the prt-pair is the departure prt; the last prt is the destinatin prt. We als include the prt-pair ( Pn, P n) since fr certain kinds f shipping (e.g., fishing and recreatinal) the destinatin prt and the departure prt can be the same. Furthermre, we include bth the prt-pairs ( Pn, P m) and ( Pm, P n), since the traffic frm P n t P m is nt necessarily the same as the traffic frm P m

9 4 Richard Heitmeyer 2 N p prt- t P n. Thus, if there are pair mdels fr the reginal mdel. N p prts fr the regin, including pseud prts, there are as many as T describe the kinematics fr an individual prt pair, it is first necessary t specify the rutes that the ships travel and the mtin f thse ships alng thse rutes. These quantities determine a ship-track functin that specifies the tracks f the individual ships during the perid f interest. 2.1 The Rute Functin The rutes that the ships travel are specified by a rute functin that describes the set f all pssible rutes frm P n t P m by assciating each pssible rute with a rute parameter ω Ω and expressing that rute as a functin f the distance x measured alng that rute frm the departure prt. The rute functin has the frm ( ) ( ) ( ) ( ) Rλφ, x; ω = Rλ x; ω, Rφ x; ω, (1) where Rλ ( x; ω ) and Rφ ( x; ω ) are the crdinate functins fr the lngitude crdinate λ and the latitude crdinate φ, respectively. Fr each value f the rute parameter ω, we dente the length f the rute by L( ω ) and assume that as x increases frm 0 t L( ω ), the rute functin traces ut the rute ω frm P n t P m as a cntinuus path in the set λ ( π, π ] and φ ( π/ 2, π / 2). This set represents all psitins n the Earth, except fr the nrth and the suth ples, with the cnventin that negative values f λ crrespnd t lngitudes east f the Greenwich meridian and negative values f φ crrespnd t latitudes suth f the equatr. T accunt fr ships that depart frm P n after time t, we allw the distance traveled x t be negative and define the rute functin by Rλ ( x; ω ) =λ π (2) Rφ ( x; ω ) =φ=ω With this definitin, rute segments determined fr negative values f x d nt describe actual rutes since the crdinates ( λφ, ) take values in the set R v = (, π ) Ω. We refer t this set as the virtual ship crdinates and use it t implicitly determine the departure times f the ships that leave P n during the time interval ( t, t. We refer t the set R r = ( π, π ] ( π/2, π /2) as the real ship crdinates and t the unin f bth sets R e =Rv R r as the extended ship crdinates. In the sequel, it is cnvenient t define the rute functin and the track functin in an auxiliary crdinate system and then map the results t the latitude-lngitude system. The definitin f the auxiliary crdinates, which are dented here by ( θ, γ ), is illustrated in Fig. 2. In this definitin, it is assumed that the rute set is such that there is a nminal rute ω n with the prperty that, fr each pint ( λ, φ ) n ω n, there is a great circle arc thrugh that pint that is rthgnal t the nminal rute and that intersects every ther rute in the rute set. The auxiliary crdinate θ is defined as the distance alng the nminal rute frm the departure prt t the pint ( λ, φ ). This distance increases frm zer at the departure prt t L n at the destinatin prt, where L n is the length f the nminal rute expressed in radians. The crss-sectinal crdinate γ is defined as the signed distance alng the great circle arc frm the pint ( λ, φ ) n the nminal rute ω n t the pint ( λ, φ ). This distance is taken t be psitive in the directin f the upper rute envelpe, dented by eu ( θ ) in Fig. 2, and negative in the directin f the lwer rute envelpe el ( θ ). Fr rute segments that lie in the virtual crdinate set, ( λφ R, ) v = ( π Ω,, ) we take θ=λ+π and γ equal t a functin f φ that is specified belw.

10 Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times 5 upper rute envelpe rute ω P n x (λ'',ϕ'') γ Pm θ (λ',ϕ') VIRTUAL ROUTE SEGMENT lwer rute envelpe crss-sectinal cut at θ' nminal rute Fig. 2 The auxiliary crdinate system With the auxiliary crdinate system s defined, the lngitude-latitude crdinates are related t the auxiliary crdinates by a ne-t-ne transfrmatin. Cnsequently, the rute functin Rλφ, ( x; ω ) can be btained by first determining this functin in terms f the auxiliary crdinates (i.e., as the functin Rθγ, ( x; ω )) and then using the crdinate transfrmatin t express the results in the lngitude-latitude crdinate system. Reference 5 prvides an example f a ne-t-ne transfrmatin between the auxiliary crdinates and the lngitude-latitude crdinates fr a particular chice f the nminal rute. In the deterministic mdel presented in Sectin 4, all f the rutes in the rute set prgress frm the departure prt t the destinatin prt withut dubling back n themselves. Fr this case, there can be nly ne value f the crss-sectinal crdinate γ fr each value f the nminal rute crdinate θ. Cnsequently, fr each rute ω, the crss-sectinal variable can be expressed as a functin γ= a( θω, ), s that the rute ω is traced ut as the pints ( θ, a( θω, )) as θ increases frm zer t L n. Furthermre, the distance traveled alng the rute is given by X 1/2 2 a( θ, ω) θω ; = 1 + dθ, (3) θ 0 ( ) which is an increasing functin f the nminal rute crdinate θ. It fllws that θ can be expressed as a functin f the distance traveled by the inverse functin ( x; ) X 1 ( x; ) Θ ω = ω. (4) With these definitins, the rute functin fr prgressive rute sets is given by 1 ( ; ω ) =Θ( ; ω ) = ( ; ω) ( ; ω ) = ( Θ( ; ω), ω) Rθ x x X x Rγ x a x. (5)

11 6 Richard Heitmeyer 2.2 The Ship-Mtin Functin The ship-mtin functin describes the distance traveled by a ship alng a rute as a functin f the elapsed time after its departure frm the prt. Since this mtin depends n the nminal speed that the ship travels and pssibly n the specific rute that the ship fllws, we include these quantities as parameters f the ship-mtin functin. Furthermre, we als allw the ship-mtin functin t depend n abslute time t accunt fr envirnmental effects (e.g., strms). With these cnventins, the distance traveled frm the departure prt at time t by a ship that departs at time τ and then travels the rute ω with nminal speed v is represented by x = Mt ˆ ( τ; t, ω, v). As defined, the ship-mtin functin takes n distance values in the interval [0, L( ω )] as t varies frm the departure time τ t the arrival time at the destinatin prt η. As with the rute functin, it is cnvenient t allw the distance traveled t take n virtual values by extending the definitin f the ship-mtin functin t the whle time axis. T this end, we assume cnstant speed mtin fr t [ τη, ] and define an extended ship-mtin functin by ( ) fr t< ˆ ( ) [ ] ( ) L( ) fr t> vt τ τ ˆ Mx ( t τ; t, ω, v) = M t τ; t, ω, v fr t τη, vt τ + ω τ. (6) With this definitin, the extended mtin functin M ˆ x ( t τ; t, ω, v) is a mntnically increasing functin f elapsed time ς= t τ. It fllws that given any initial time t, there is a unique initial distance x btained by setting ς=ς = t τ in Eq. (5). Furthermre, given an initial distance x, there is a unique value f the initial elapsed time ˆ ς x, ω, vt ; = t τ= M 1 x ; t, ω, v, (7) ( ) ( ) x crrespnding t x. Cnsequently, it is always pssible t express the distance traveled as a functin f the initial time t and the initial distance x. T this end, we redefine the ship-mtin functin by M ( t t ;,, ) ˆ x ω v = Mx( t t +ς( x, ω, v; t) ; t, ω, v). (8) Figure 3 shws an example f these definitins. In this figure, we have illustrated the ship-mtin functin fr tw ships, each f which travels the same rute ω with the same nminal velcity v. The first ship departs at time τ 1 and arrives at time η 1; the secnd departs at time τ 2 and arrives at time η 2. During the perid when the secnd ship is in transit, a lcal strm causes it t reduce speed with the result that its ttal transit time η2 τ 2 is lnger than the transit time fr the first ship. Bth ships are present during the bservatin interval [ t, t. The first ship is en rute at the initial time t since t τ ( 1, η 1). This ship has an elapsed time ς( x, t; ω, v) 0 and a real initial distance x [0, L( ω )]. The secnd ship in nt en rute at the initial time since t ( τ2, η 2). Hwever, it is en rute during the bservatin interval since t + T ( τ2, η 2). This ship has an elapsed time ς( x, t; ω, v) < 0 and a negative initial distance x [ vt,0]. The first ship arrives at the destinatin prt in the bservatin interval; the secnd ship des nt.

12 Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times 7 Virtual x L ω η 1 η 2 M( t - τ 1, t ; ω,v) Real x ο, 1 M( t - τ 2, t ; ω,v) τ 1 ζ 1 ζ 2 τ 2 t ο t ο +T t x ο, 2 Virtual Fig. 3 The ship-mtin functin Nte that in this example, the nminal speed parameter specifies a characteristic speed f the ship rather than the actual speed made gd. Other attributes f the ship relevant t a particular applicatin f the mdel (e.g., type f ship) can be included in the mdel by interpreting v as a vectr parameter in the subsequent develpment. Als nte that the definitin f the mtin functin des nt preclude the secnd ship frm vertaking and passing the first ship. Strictly speaking, with the rute functin defined by Eq. (1), this wuld amunt t the secnd ship passing thrugh the first ship if bth ships were traveling n the same rute. Hwever, with the specific rute functins cnsidered in this reprt, this is an event f zer prbability and is nt cnsidered further. 2.3 The Ship-Track Functin The ship-track functin G λφ, is btained frm the rute functin R λ, φ by using the ship-mtin functin M t express the distance traveled by a ship as a functin f the elapsed time and the ship parameter. It fllws frm Eqs. (1) and (7) that the lngitude crdinate functin G λ and the latitude crdinate functin G φ f the ship-track functin have the frm Gλ( t t; x, ω, v) = Rλ( M( t t; x, ω, v) ; ω) Gφ( t t; x, ω, v) = Rφ( M( t t; x, ω, v) ; ω ). (9) Nte that nly the rute parameter cmpnent f the track parameter ( x v), ω, is needed t identify the particular rute, whereas the cmplete track parameter is needed t specify the mtin alng that rute. A ship-track realizatin is btained by specifying the number f ship tracks in the realizatin n and the track parameters fr thse ships {( x, ω, ); 1,..., } k k vk k = n and then cmputing the track f each ship using the track functin. An example f a ship track realizatin is shwn in Fig. 4. The rutes that the

13 8 Richard Heitmeyer ships travel are shwn by the thin curves. The tracks f the ships alng thse rutes fr the time interval [ t, t are indicated by the heavy lines n these curves. Fr the tracks alng the rutes ω 1 and ω 4, the ships are en rute during the whle time interval [ t, t. Fr the track alng the rute ω 3, the ship is en rute at time t but it arrives at the destinatin prt prir t t + T. Thus, fr these three tracks, the initial-distances are psitive and the initial psitins, indicated by large dts, lie in the set f real ship crdinates. Fr the track alng the rute ω 2, the ship departs frm P n after time t. Therefre, the initial distance fr this track x 2 is negative and the initial psitin lies in the set f virtual ship crdinates. At time t, the ship starts at the initial distance x 2 and mves alng the rute segment in the virtual ship crdinates with cnstant speed v 2 up t the departure time τ 2 = t x / v 2 2. At this time, it actually departs the prt and travels alng the rute segment shwn with its mtin determined by M ( t t; t, x, ω 2 2, v2). ω 1 P n ω 2 ω 3 P m VIRTUAL ROUTE SEGMENT ω 4 Fig. 4 Example f a ship track realizatin fr a single prt-pair T determine individual track realizatins, it is necessary t have a mechanism fr determining the track parameters fr each realizatin. This is dne in terms f the prt parameter prbability law defined in the fllwing sectin. 3. THE PORT-DEPARTURE AND TRACK-PARAMETER PROBABILITY LAWS T an bserver lcated at the departure prt, the shipping is described by the sequence f parameters {( τk, ω k, vk)}, where τ k is the departure time f the k th ship t leave the prt, τ k is its rute parameter, and v k is its nminal speed. In this reprt, we assume that: (1) the ship departure times are described by a Pissn prcess with a time-dependent rate functin μ τ() t that represents the mean number f ship departures per unit time; (2) the rute that each ship travels and its speed are independent f its departure time and independent f ne anther; (3) different ships select rutes independently f ne anther and each ship selects its rute frm the same prbability density p ω ( ω ); and (4) the nminal speeds f all ships are independent f ne anther and are described by the same prbability density pv ( v ). The rate functin μ τ( t) can be estimated frm ship departure data; the ship speed density pv ( v ) can be determined frm the ships register data given the identities f the departing ships. Hwever, in general, the rute parameter prbability density p ω ( ω ) cannt be determined independently f the prbability law describing the rute set. Accrding t the Pissn departure time assumptin, the number f ships leaving the prt during the time interval ( t, t is a Pissn randm variable N τ with a prbability mass functin given by

14 Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times 9 where n Mτ ( t, T) Pr[ Nτ = n] = exp { Mτ( t, T) }, (10) n! t + T M τ( t, T) = μ () t dt t τ (11) is the mean number f ships t leave the prt in the interval. Furthermre, given that there are n ship departures during the interval, the ship departure times { τ k ; k = 1,..., n} are independent and identically distributed with cmmn prbability density. ( ) ( ) (, 1 μ ;, τ τk Mτ t T) fr τk [ t, t + T τ ] pτ k t T = 0 therwise. (12) Nte that since the departure rate μ τ() t depends n time, the mean number f ships Mτ ( t, T), the departure-time prbability density pτ( τ k; t, T), and the prbability mass functin depend n the chice f the time interval [ t, t. We allw fr this time dependence t incrprate variatins in the number f ship departures with time f day, day f the week, etc. Fr hmgeneus Pissn prcesses, the departure rate is independent f time ( μ τ( t) =μ τ ); hence, the mean number f ship departures is simply 1 Tμ τ and the departure times are unifrmly distributed n the interval [ t, t (i.e., pτ( τ k ) = T ). It fllws frm the prt-departure assumptins that the prt-departure parameter ( τω,, v) is described by a multidimensinal Pissn prcess that is specified by a rate functin μτωv,, (, t ω, v ). Mrever, frm the definitins f Sectin 2, there is a ne-t-ne transfrmatin between the prt-departure parameter (, τω,) v and the ship-track parameter ( x, ω, v). It fllws that the ship-track parameter is als distributed as a multi-dimensinal Pissn prcess that is specified by a rate functin μx,, (,, ;, ) ω v x ω vt T. The specificatin and derivatin f these rate functins is presented in Appendix B (see Eqs. (B1), (B4), and (B5)). The definitin f a multidimensinal Pissn prcess is presented in Appendix A. The Pissn prcess n the ship-track parameters describes the number and the distributin f the parameters fr all ships present in the regin during the interval [ t, t. As seen in Sectin 2, the ships present in the regin during this interval are thse that are en rute at time t and thse that depart during the half pen interval ( t, t. The frmer have psitive initial distances; the latter have negative initial distances. Cnsequently, by restricting the track-parameter rate functin μx, ω, v( x, ω, vt ;, T) t psitive values f x, we btain a Pissn prcess that describes the ships that are en rute at time t. Similarly, by restricting μx,, (,, ;, ) ω v x ω vt T t negative values f x, we btain a prcess describing the ships that depart during ( t, t. The ttal number f ships en rute during [ t, t is the Pissn prcess btained as the cmpsitin f the tw cmpnent prcesses. Fr each f these prcesses, the number f ships is a Pissn randm variable with a prbability mass functin given by Eq. (10) and the track parameter fr thse ship are independent and identically distributed with a prbability density given by an equatin analgus t Eq. (12). The mean number f ships is btained as an integral f μx ( ), ω, v x, ω, vt ;, T ver the apprpriate vlume. The definitin f these prcesses and explicit frmulas fr the relevant quantities are presented in the fllwing sectins.

15 10 Richard Heitmeyer The track-parameter rate functin μx,, (,, ;, ) ω v x ω vt T cmpletes the mdel if the rute set prbability density p ω ( ω ) and the rute functin f Eq. (1) are knwn. In a simple apprximate mdel, where the traffic is described in terms f, at mst, a finite number f rutes, the rute functin might be described analytically in terms f great circle rute segments and the weightings p ω ( ω ) might als be available. Hwever, in the mre realistic case, where there is a cntinuum f rutes, the rute functin and the rute parameter density must be determined frm data n the rutes that the ships fllw. In principle, these data can be used t cnstruct a prbability law n the ensemble f rutes frm P n and P m. In general, such a prbability law cnsists f the jint prbability density n the crdinate psitins {( λ( x1), φ( x1)),...,( λ( xn), φ ( xn))} fr all pssible samples f the distances traveled { x1 <... < xn; n> 0}. T determine such a prbability law requires mre data than are usually available. Cnsequently, it is necessary t intrduce apprximatins and assumptins that allw the rute functins t be defined in terms f simplified prbabilistic descriptins. In the fllwing sectin, we describe a deterministic mdel where the rute functins are determined frm a prbability density that can be estimated frm any sample f ship crdinates, regardless f whether thse crdinates are rganized int specific rutes. In Sectin 5 we describe a stchastic mdel which assumes that the ship rutes are described by a Markv prcess. 4. A DETERMINISTIC ROUTE FUNCTION MODEL In the ship-track mdel presented here, bth the rute functin and the rute parameter density are determined frm a prbability density n the crss-sectinal ship psitins γ cnditined n the nminal rute crdinate psitins θ. This is dne in such a way that each rute is uniquely determined by any pint n that rute. As a cnsequence, the Pissn prcess describing the distributin f ship-track parameters can be expressed in terms f the initial crdinates f the ship at time t, rather than the initial distance and the rute parameter. This simplicity is btained by limiting the rute structure t prgressive rute sets with the additinal cnstraint that the individual rutes can nt intersect ne anther. T describe the mdel, we first define the underlying prbability densities. We then specify the rute and the track functins that are determined by these densities. Finally, we present the prbability law n the track parameter which results frm these definitins and the assumptins f Sectin The Rute Crdinate Prbability Density The rute crdinate prbability density describes the cncentratin f the γ values f the rutes as they intersect the cut in the rute set at θ. We dente this prbability density by p γθ ( γθ ; ), where θ specifies the lcatin f the cut, and refer t the crrespnding cumulative prbability distributin ( ; ) ( ; ) Pγθ γθ = pγθ γ θdγ (13) ε ( θ) l γ as the rute crdinate distributin functin. Figure 5 shws an example f a rute crdinate density and distributin functin fr a specific cut in the rute set. In this example, the rutes fr the cut θ =θ, are cncentrated near the lwer envelpe el( θ ). The rute crdinate density at ther cuts in the rute set need nt be the same as the ne shwn. Fr example, the distributin f rutes shwn in the figure suggests that the rute crdinate density fr the cut at θ =θ 1 wuld reflect a mre disburse cncentratin f the rutes clser t the center f the rute envelpes. A methd fr estimating these densities frm ship rute data can be fund in Ref. 2 alng with a number f examples.

16 Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times 11 upper rute envelpe, e u (θ) rute, ω nminal rute (θ 1, γ 1) (θ ο, γ ο ) P n θ γ Pm lwer rute envelpe, e l (θ) crss-sectinal cuts at θ 1 and θ ο p γ θ ( γ ; θ ο ) rute set density and distributin functins at θ ο P γ θ ( γ ; θ ο ) e l (θ ο ) e u (θ ο ) γ Fig. 5 Example f a rute crdinate prbability density It is imprtant t emphasize that the rute crdinate densities, in themselves, d nt specify the rutes that the ships travel since many different pssible rute functins can satisfy a given rute crdinate density. 4.2 The Rute Functin Fr prgressive rutes, the rute sets are specified by the crss-sectinal functin a( θω, ) and the rute parameter prbability density p ω ( ω ). Fr the deterministic mdel, we define these quantities in such a way that the resulting set f rutes is cnsistent with the rute crdinate distributin functin P γθ (; γθ ). T this end, we restrict ω t the interval [ 0,1 ] and define the crss-sectinal functin a( θω, ) as the inverse f the rute crdinate distributin. Specifically, fr each ω [0,1], a( θω, ) is the functin satisfying ( a( ) ) Pγθ θ, ω ; θ =ω. (14) Clearly, a( θω, ) is well defined since fr each value f θ, P γθ (; γθ ) is an increasing functin f γ 1 taking values in the interval [ 0,1 ]; hence, fr fixed θ, there is an inverse functin Pγ θ that maps ω [ 0,1] t the set f all pssible crss-sectinal crdinate values γ. Fr a fixed ω, the γ value crrespnding t the θ value n the rute ω is the value given by this inverse functin. Fr ntatinal cnvenience, we write 1 a( θω, ) = Pγθ ( ωθ ; ) fr θ> 0. (15) Figure 6 illustrates the definitin f the crss-sectinal functin. As seen in the figure, the tw rute envelpes are als rutes in the rute set. The lwer rute envelpe el ( θ ) crrespnds t the rute parameter ω= 0 ; the upper rute envelpe eu ( θ ) crrespnds t ω = 1. Fr fixed θ, the crss-sectinal

17 12 Richard Heitmeyer functin a( θω, ) yields γ values that increase ver the interval [ el( θ), eu( θ )] as ω increases ver the interval[ 0,1 ]. Fr a fixed value f ω, a( θ, ω ) traces ut the γ values alng the rute ω as θ increase frm 0 t L n. Nte that the rutes are defined s that n tw can intersect ne anther. (If they culd, Eq. (13) wuld nt be well-defined.) Als nte that since the lwer rute envelpe and the upper rute envelpe are assumed t be distinct, the interval [ el( θ), eu( θ )] cannt degenerate t a single pint even at the departure prt θ= 0 r at the destinatin prt θ = L. upper rute envelpe, e u ( θ) ω ο rute ( θ, a ( θ ; ω ο ) ) γ θ (θ ο,γ ο ) P m rute distributin functin, P γ θ (γ ; θ ο ) P n lwer rute envelpe, e l (θ) nminal rute crss-sectinal cut at θ ο Fig. 6 Definitin f the crss-sectinal rute functin and the rute parameter Equatin (15) describes the segments f the rutes that lie between the departure prt and the destinatin prt. These rute segments lie in the set f real ship crdinates R r = [0, L] [ el( θ), eu( θ )]. Fr the rute segments that describe the fictitius mtin f the ships befre they leave the prt, we take a( θω, ) t be the inverse rute distributin functin evaluated at the departure prt (i.e., 1 a( θω, ) = Pγθ ( ωθ= ; 0) ) fr θ< 0. These rute segments lie in the set f virtual ship crdinates R = (,0) [ e (0), e (0)] and have γ crdinates that are independent f θ. v l u The definitin f the crss-sectinal functin suffices t specify the rute functin Rθγ, ( x; ω ). The θ 1 crdinate f Rθγ, ( x; ω ) is Θ( x; ω ) = X ( x; ω ), where the distance traveled functin X ( θ; ω ) is determined frm a( θω, ) by Eq. (3). The γ crdinate f Rθγ, ( x; ω ) is a( Θ( x; ω), ω ). The track functin Gθγ, ( t t; x, ω, v) is btained frm the rute functin using the mtin functin M ( t t; x, ω, v) t express the distance traveled as a functin f time. An imprtant cnsequence f the definitin f the crss-sectinal functin is that there is a ne-t-ne transfrmatin between the track parameter cmpnents ( x, ω ) and the initial crdinates ( θ, γ ). In

18 Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times 13 particular, the initial crdinates are determined by θ =Θ( x; ω ) and γ = a( Θ( x; ω); ω ). The initial distance and the rute parameter are determined by and ( ; ) ( ; ) ω = P γ θ = P γθ γ θ (16) θ 2 1 Pγθ ( ω; θ ) x = X( θ, γ ) = 1 + dθ θ. (17) 0 1/2 Equatin (16) fllws immediately frm Eq. (14) by replacing a( θ, ω ) by γ. Equatin (17) fllws frm Eq. (3) by setting θ=θ and by substituting fr ω frm Eq. (15). Nte that by virtue f Eqs. (16) and (17), the rute functin can be expressed in terms f the initial psitin and the mtin functin and the track functin can be expressed in terms f the initial psitin and the characteristic speed. Cnsequently, the ship track realizatins can be btained frm the track functin by specifying the track parameter sets {( θ, γ, ); 1,..., } k v k k k = n. The interpretatin f a ship track realizatin is the same as that f Fig. 3 with x and ω replaced by θ and γ (i.e., the initial psitins (the dts) are the pints ( θ, γ ) and the rute parameter in the ship-mtin functin is determined frm these pints by Eq. (16)). The prbability law n the track parameters is described in the fllwing subsectin. T cnclude these definitins, we shw that the crss-sectinal functin is cnsistent with the rute crdinate distributin functin if the rute parameter is unifrmly distributed. T see this, we nte that Pγθ ( γθ ; ) = Prb{ γ [ el( θ), γ ]} = Prb{ γ [ el( θ), a( θω ; )]}. But γ [ el ( θ), a( θω ; )] is equivalent t ω [0, P γθ ( γθ ; )], s that Prb{ γ [ el ( θ), a( θω ; )]} = Prb{ ω [0, P γθ ( γθ ; )]}, which is the same as the cumulative distributin f the rute parameter evaluated at ω = P γθ (; γ θ ). Thus, Pγθ (; γθ ) = Pω( Pγθ (; γθ )), where P ω ( ω ) is the rute parameter cumulative distributin functin determined frm the rute parameter prbability density p ω ( ω ). This equality hlds if and nly if the rute parameter is unifrmly distributed n [ 0,1 ], in which case P ω ( ω ) =ω. 4.3 The Track-Parameter Prcess The ship-track prbability law describes the number and the distributin f the parameters f all the ships present in the regin during the interval [ t, t. Fr the deterministic mdel, this prbability law is btained frm the prbability law f Sectin 3 thrugh an invertible transfrmatin that maps the parameter ( x, ω, v) t the parameter ( θ, γ, v ). It is shwn in Appendix B that this results in a Pissn prcess with a rate functin μθγ,,( v θ, γ, vt ;, T) defined n the extended rute set R e =Rv R r. As nted in Sectin 3, this prcess can be represented as the cmpsitin f tw Pissn prcesses, ne that describes the ships that are en rute at time t and ne that describes the ships that depart during ( t, t. We refer t the frmer as the en rute prcess and t the later as the entry prcess. T define these prcesses, it suffices t specify their rate functins. The number f ships and the prbability density n the parameters fr thse ships are then given by Eqs. (10) and (12), where the mean number f ships is btained as an integral ver the rate functin. T cmpute this integral, it is cnvenient

19 14 Richard Heitmeyer t define a shipping density μθγ, ( θ, γ ; t ) as the integral f the rate functin with respect t ship speed. The mean number f ships is then btained as the integral f the shipping density with respect t the ship crdinates. The shipping distributin is the integral f the shipping density ver element areas. The specific equatins are as fllws. The derivatins f these equatins are presented in Appendix B. Fr the en rute prcess, the nly ships that are en rute at time t are thse that have real ship crdinates. Cnsequently, the rate functin is btained by restricting μθγ,,( v θ, γ, vt ;, T) t the real ship crdinates R = [0, L] [ e ( θ), e ( θ )]. The rate functin is given by r l u (,, vt ; ) μθγ,, v θ γ = ς( x, ω, v; t) X( θ, γ) μτ( t ς( x, ω, v; t) ) pγθ ( γ; θ) pv v x x= X( θ, γ) θ ω= P( γ, θ) ( ). (18) As nted abve, the shipping density fr the ships that are en rute at time t, μθγ, ( θ, γ ; t ) is the integral f the rate functin with respect t ship speed; μ ( θ, γ ; t ) = μ ( θ, γ, v; t ) dv; ( θ, γ ) R. (19) θγ, θγ,, v r 0 The mean number f ships with initial crdinates ( λ, φ ) in any set Cr R r is the integral f the shipping density with respect t the ship crdinates ver the set C r ; ( ; ) (, ; ) M t C = μθγ θ γ t dθ dγ. (20) r r C, r It fllws frm the definitin f a Pissn prcess, that the number f ships en rute at time t with crdinates ( θ, γ) Cr R r, Nr( t; C r), is a Pissn randm variable with a prbability mass functin given by Eq. (10) with M τ replaced by M r( t; C r). Furthermre, the track parameters fr thse ships are independent and identically distributed with the cmmn prbability density functin _1 r(,, ; ; r) θγ,, v(,, ; ) r( ; r) (, ) r p θ γ v t C =μ θ γ v t M t C θ γ R. (21) The ttal number f ships en rute at time t is determined frm these equatins by taking C r =R r. An interpretatin f these equatins is as fllws. The first factr in Eq. (18) describes the mean number f ships per-unit distance alng the rute determined by ( θ, γ ). This factr incrprates the tempral variatins in the rate at which ships depart the prt thrugh the time dependence n the ship departure rate μ τ( t) and the mtin f the ships alng the rute thrugh its dependence n the departure time functin ς ( x, ω, vt ; ), and its first derivative. Fr the special case f cnstant speed mtin, ς( x, ω, vt ; ) = xv, this factr simplifies t μτ( t X( θ, γ ) v ) v. Thus, fr cnstant speed mtin, a change in the rate at which ships depart the prt simply prpagates alng the rute with the nminal ship speed. The secnd factr in Eq. (18) describes hw the ships are distributed acrss the rutes. This factr is equal t the rute crdinate density weighted by the magnitude f a derivative that represents the rate f

20 Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times 15 change in the distance traveled with respect t the nminal rute distance. Fr mst rute sets, the rutes d nt depart significantly frm the nminal rute s that this derivative is apprximately unity and hence, the secnd factr is apprximately equal t the rute crdinate density. Fr these narrw rute sets and fr cnstant speed mtin, the track-parameter rate functin simplifies t ( ) ( ) 1 1 ( ) ( ) ( ) ( ) μθγ,, v θ, γ, vt ; =μτ t X θ, γ v pγθ γ; θ v pv v ; θ, γ R r, (22) and the shipping density becmes 1 1 μθγ, ( θ, γ ; t) = pγθ ( γ; θ) μτ( t X( θ, γ) v ) v pv( v) dv; ( θ, γ) Rr. (23) 0 Nte that if in additin the departure rate is independent f time, μτ() τ =μ τ, then the rate 1 functin is μθγ,,( v θ, γ, vt ;, T) =μτpγθ( γ; θ ) v pv( v) and the shipping density is μθγ, ( θ, γ ; t ) = 1 μβ τ pγθ( γ; θ ), where β is the mean value f the reciprcal speed, β= E[ v ]. Fr the entry prcess, the nly ships that depart in the interval ( t, t are thse that have virtual ship crdinates. Cnsequently, the rate functin fr the entry prcess is btained by restricting μθγ,,( v θ, γ, vt ;, T) t the virtual ship crdinates R v( T) = [ vt,0] [ el( θ= 0), eu( θ= 0)]. The rate functin is given by ( ) ( 1,, vt ;, T t v ) p ( ; 0 ) v 1 p ( v) ; (, ) ( T) μ θ γ =μ θ γ θ = θ γ R, (24) θγ,, v τ γθ v v and the shipping density is ( ) ( ) ( 1, ; t, T p ; 0 t v ) v 1 p ( v) dv; (, ) ( T) μ θ γ = γ θ = μ θ θ γ R. (25) θγ, γθ τ v v θ / T Nte that Eqs. (24) and (25) are the same as Eqs. (22) and (23), except that p γθ ( γ; θ ) is replaced by p γθ ( γ; θ = 0) and the lwer limit f the integral in Eq. (24) depends n bth ( λ, φ ) and the time interval duratin T. This is nt surprising since, by the definitin f the rute functin n the virtual crdinates, the ships mve at cnstant speed and the derivative f X( θ; γ ) is unity. Als nte that fr the entry prcess, the rate functin and the shipping density depend n the duratin T f the interval [ t, t, as well as the initial time t ; whereas, fr the en rute prcess, these parameters depend nly n t. Equatins (24) and (25) determine the entry prcess. The mean number f the ships that depart in the interval ( t, t with crdinates ( θ, γ) Cv R v( T), M τ ( t, T; Cv), is btained by integrating the shipping density ver the set Cv R v( T). The prbability mass functin n the number f these ships Ne( t, T; C v) is given by Eq. (9) with M τ replaced by Me( t, T; C v). The prbability density n the parameters f these ships pe( θ, γ, vt ;, T) is given by Eq. (11) with μτ( τ k ) replaced by μθγ,,( v θ, γ, vt ;, T) and Mτ ( t, T) replaced by Me( t, T : C v). Finally, the Pissn prcess describing the ttal number f ships in the interval NT( t, T ) is the cmpsitin f the en rute and the entry prcesses. Thus, fr the clsed interval[ t, t, the ttal number f ships is NT( t, T) = Nr( t) + Ne( t, T), the mean number f ships is MT( t, T) = Mr( t) + Me( t, T), and the track-parameter prbability density is pt( θ, γ, vt ;, T) =μθγ,, v( θ, γ, vt ;, T)/ MT( t, T).

21 16 Richard Heitmeyer The rute set assumptins fr the deterministic mdel impse ptentially imprtant restrictins n the rutes in the rute set. In particular, the rute set apprximatin can hld nly if the rutes have n cmmn pints and nly if individual rutes d nt have multiple pints (i.e., the rutes cannt intersect ne anther and cannt crss ver themselves). Fr thse ships that simply transit frm ne prt t anther, the apprximatin that results frm these restrictins may be adequate fr thse applicatins that are limited t prt-t-prt traffic and d nt require precise descriptins f the tracks. There is, hwever, shipping traffic which des mre than simply transit frm ne prt t anther (e.g., fishing vessels, recreatinal, and military). Fr applicatins where this traffic is imprtant, the rute set restrictins f the deterministic mdel may nt be acceptable. 5. A STOCHASTIC ROUTE FUNCTION MODEL In general, a stchastic descriptin f the ensemble f rutes requires the jint prbability density n the crdinate psitins {( θ x1, γ1), K, θ( x n, γ n )} fr all pssible samples f the distances traveled { x1 < K < xn; n> 0}. In this sectin, we present a stchastic mdel based n the assumptin that the rute set prbability law is Markv. We first specify the rute set prbability law, describe hw the prbability law is used t generate the ship tracks in terms f the track parameter ( θ, γ, v ), and cnclude by presenting the rate functin that specifies the Pissn prbability law n the ship-track parameters. 5.1 The Rute Set Prbability Law Fr a Markv prcess, the jint prbability density is cmpletely determined by the first-rder prbability density and the prbability density transitin functin. The first-rder prbability density pθ( x1 ), γ( x) ( θγ, ; x ) describes the crdinates f all rutes in the rute set that have a fixed value f the distance traveled x. In general, this density depends n x (e.g., fr small x the distributin f the ship crdinates is cncentrated near the departure prt); as x increases, the distributin f the crdinates migrates twards the destinatin prt and becme mre diffuse. Nte that at the departure prt ( x = 0 ) the nminal rute crdinate θ is zer, s that the first-rder density has the frm pθ(0), γ(0) ( θ, γ ; x= 0) = pγ ( γ ) δ ( θ ). (26) d d In the fllwing, we refer t γ d and p γ ( γ ) d d as the departure crss-sectinal crdinate and prbability density, respectively. The transitin prbability density, dented by pθ( x), γ( x) θ( x ), γ ( x ) ( θ, γθ ;, γ, xx, ), is the prbability density n the crdinates at the distance x > x, ( θ( x), γ ( x)), given the values f the crdinates at the distance x, ( θ ( x ) =θ, γ ( x ) =γ ). The first-rder density fr the crdinates at x is determined frm the first-rder density f the crdinates at x by p θ γ θγ x = p θγθ γ x x p θ γ θ γ x d θ d γ. (27) θ γ ( x), ( x)(, ; ) θ( x), γ( x) θ( x ), γ( x ) (, ;,,, ) ( x ), ( x )(, ; ) Fr the special case f x = 0, the transitin density can be written as ( ) ( ) ( ) ( ) ( ), 0, 0, ;, d,, x x 0 θ( x), γ( x) γ (, ;, d, ) ( ) p θ γ θ γ θγθ γ =γ x x = = p θγθ γ x δ θ, (28) d

22 Ship-Track Mdels Based n Pissn-Distributed Prt-Departure Times 17 where pθ( x), γ( x) γ ( θγθ, ;, γ, ) d d x is referred t here as the departure transitin prbability density. Substituting frm the last equatin and Eq. (26) int Eq. (27) results in 5.2 The Rute Set Functin p θ γ θγ, ; x = p θ γ γ θγγ, ; ;. d d x p γ d d d γd (29) γ ( x), ( x)( ) ( x), ( x) ( ) γ ( ) As with the deterministic mdel, the rute and the track functins are specified in terms f the track parameter ( θ, γ, v ). The set f real ship crdinates R r and virtual ship crdinates R v have the same frm as fr the deterministic mdel, where the rute envelpes are determined by the supprt f the firstrder prbability densities. The definitin f the rute functin is as fllws. Fr the real ship crdinates, the rute functin is determined recursively in terms f the transitin prbability density. The prcess is as fllws. At time t, the initial crdinates are specified by the track parameter ( θ, γ, v ). After a time δ t, the ship has traveled a distance δ x as determined by the shipmtin functin. The ship crdinates crrespnding t this increment in the distance traveled are distributed accrding t pθ ( x ), ( ) ( ), ( ) (, ;,,, ) +δx γ x+δx θ x γ x θ γθ γ x +δ x x. An applicatin f a randm number generatr fr this prbability density yields specific values f the ship crdinates ( θ ( x +δx), γ ( x +δ x)). The prcess is repeated t yield the sequence f ship crdinates crrespnding t the sequence f distances traveled determined frm the sequence f time increments. Fr the virtual crdinates, we take γ=γ d and θ = x in analgy with the deterministic mdel. At the departure prt ( x =θ= 0 ), the ship crdinates are (0, γ d ), s that the recursive prcess described abve is initiated with the transitin density pθ( x), γ( x) γ ( θ, γγ ; ; ) d d x. As such the crss-sectinal crdinate γ d and the density p γγ ( γ ) crrespnd t the rute parameter ω and its density p ω ( ω ). 5.3 The Track Parameters d d The track parameter ( θ, γ, v ) is related t the track parameter ( x, γ d, v) = ( x, ω, v) by the firstrder prbability density f Eq. (29). It fllws frm the prbability law f Sectin 3 and the stchastic transfrmatin prperty f Appendix A, that the track parameter is described by a Pissn prcess (see Appendix B). Fr the real ship crdinates, the rate functin is given by eu ( 0) L( γd) μθ,, (,, ; ) ˆ,, (,, ; ) ( ), ( ) (, ;, ) γ θ γ vt = μ γ x γ vt p d θ x γ x γ θ γ x γ dxdγ d e ( 0) 0, (30) v x v d d d l where ( ) ( x v t ) t ( x v t ) μˆ x,,,, ;, ; ; γd v γ d =μτ ς γd ( x, γ ; vt ; ) ς d x. (31) Fr the virtual ship crdinates, 1 1 ( vt T) p ( ) p ( v) v ( t v ) ( ) ( T) μθ, γ, θ, γ, ;, = γ γ μτ θ, θ, γ R. (32) v d v v