Radar Sensor Management for Detection and Tracking

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1 Raar Sensor Management for Detection an Tracking Krüger White, Jason Williams, Peter Hoffensetz Defence Science an Technology Organisation PO Box 00, Einburgh, SA Australia 0 Abstract Avancements in sensor technology provie new multi-sensor systems with increasing flexibility. The sensor management process aims to perform sensor actions that support the overall goal of the user of a multi-sensor system. Some sensors can support multiple functions. When the ifferent sensor functions utilise share resources then the sensor actions must be chosen as a compromise between competing actions. The problem of managing a multifunction Electronically Scanne Array (ESA) raar is consiere. The ESA raar is capable of performing surveillance (search) an tracking (revisit) functions. Two criteria are use to etermine sensor actions with revisit actions base on the expecte information gain an surveillance actions base on the probability of etection for previously unetecte targets. The characteristics of the resulting sensor management policy are examine in an illustrative scenario. Keywors: Sensor Management, Raar, Surveillance, Tracking. I. SENSOR MANAGEMENT Sensor management allows optimisation of the measurement process in sensor fusion systems. The manner in which the measurement process is optimise is often one of the relatively immature components in sensor fusion systems. The goal of a sensor fusion system is to etect objects in a region of interest an to form estimates of the state of such objects. Sensors can be regare as resources that can be taske to optimise overall system performance []. In many existing military an civilian sensor systems, the selection of sensor actions is performe by an operator who etermines how sensor resources are employe an even how the sensor ata are to be interprete. However, the consieration of all of the factors involve in using a sensor system effectively to support mission objectives is typically a complex task. A sensor management system that automatically tasks sensors aims to optimise the selection of sensor actions an to perform those actions in a timely manner. Management of sensors must be performe in a ynamic object environment with constraine sensor resources an account for the characteristics of the sensor system. When the number an state of unetecte objects is unknown, the sensor system may be taske to search for objects in a moe that is also known as the surveillance moe. Another common moe of flexible sensor systems is the tracking or revisit moe which permits the estimates of object states to be maintaine or refine by tasking the sensors using information about the objects that are alreay known. Effective employment of the search an revisit moes can enhance the etection an tracking of objects within a region of interest. Various techniques have been propose for sensor management []. Information-theoretic approaches base on the Shannon measure of entropy have been shown to perform effectively in a variety of sensor management an object tracking applications [], []. Such approaches strive to perform the sensor action that maximises the expecte gain in information. Other approaches have propose the use of permanently existing virtual targets to influence the search moe component of the sensor management functionality []. The sensor management of an active electronically scanne phase array raar is consiere where the aim is to etect an track targets within a surveillance region. The approach aopte is to partition the sensor management problem into two separate subproblems corresponing to the two raar moes. A single parameter is use to specify the ratio of sensor time resource that is allocate to the search an revisit moes which in turn controls the allocation of resources to the etection an tracking functions respectively. In the case of the search moe, the sensor action is taken to maximise the probability of etecting previously unetecte targets (base on an assume ranom istribution for the arrival of new targets). In the case of the revisit moe, the sensor action is taken to maximise the expecte information gain. The paper begins in Section II with a escription of the parameters of a phase array raar an outlines an approach for selecting between the types of raar moe an the raar beam to illuminate given the type of raar moe selecte. In Section III the sensor management approach is implemente for a simple simulate scenario consisting of a raar illuminating an receiving measurements from a set of targets that ranomly arrive an travel within the surveillance volume. II. PHASED ARRAY RADAR CONTROL An electronically scanne phase array raar can perform aaptive illumination of targets by steering a raar beam electronically. In this section an ESA raar system is consiere. The raar is locate at the origin of a surveillance area with raius R s. A set of iscrete beams m M = {,..., M} is use to illuminate the surveillance area. The beams are orere by

2 their azimuth centre positions such that η m+ > η m an each beam has an associate beam with η m an well time τ m. A raar search moe is use to scan for new targets in the surveillance region an to upate existing tracks with measurements. A raar revisit moe is use to illuminate selecte targets. It is assume that the two raar moes can be operate using the same beam set M an that any target that is illuminate by a beam may yiel raar measurements irrespective of the moe. In practice, operational raar systems may employ raar beams having ifferent performance characteristics for each moe [], []. In this paper the aim of the raar sensor management problem is to etermine a time sequence of raar beams m, m, m,..., m k in orer to satisfy requirements for target etection an track maintenance. Each beam may be use in the surveillance moe or the revisit moe so that there is a corresponing sequence of raar moes π, π, π,..., π k, where π k {S, R} enotes surveillance or revisit moe respectively. Sensor management schemes may be myopic (short term) so that the sensor actions are ecie for a single time step ahea. Non-myopic (long term) sensor management schemes plan multiple sensor actions in the future [], [], []. Typically, non-myopic schemes will yiel enhance performance with a greater computational eman. A myopic approach is aopte here where the sensor management actions are etermine one time step in avance. The surveillance beam inicator variable s m,k an the revisit beam inicator variable r m,k are efine for time step k so that { if mth beam use for surveillance s m,k = () 0 otherwise { if mth beam use for revisit r m,k = () 0 otherwise. At each time instant k, exactly one beam is use in either surveillance or revisit moes so that M m= {r m,k+s m,k } =. The ratio of well time spent on revisits to the overall well time up to an incluing time step k is γ k = T r k Tk r + T k s, () where the cumulative well times spent in the revisit moe an in the surveillance moe are given respectively by T r k = T s k = M τ m m= κ= M τ m m= κ= k r m,κ an () k s m,κ. () A esire revisit ratio Γ [0, ] is specifie by an operator, who can prioritise between revisit an surveillance tasks for the raar epening on mission imperatives for maintaining tracks on etecte targets an searching for previously unetecte targets. The revisit ratio γ k is compare with Γ to etermine which type of ESA raar moe to employ at the next time step k: { R γ k < Γ π k = () S otherwise. The effect of this moe selection scheme is to maintain the revisit ratio γ k close to Γ. Once a moe is selecte, the next step is to etermine which beam m k M to illuminate. A. Surveillance Moe Beam Selection The surveillance moe of the raar is use to search for previously unetecte targets. To o so, we maintain an estimate of the spatial ensity of previously unetecte targets. We assume that the number of unetecte targets in any region at any time follows a Poisson istribution, but the Poisson ensity may vary in time an space. We approximate the non-homogeneous Poisson ensity by iviing the surveillance volume into non-overlapping covering cells A,..., A A such that in any cell A i the Poisson arrival rate is constant (spatially an temporally), an the probability of etection is constant. We enote by λ Ai the Poisson arrival rate of new targets in cell A i, an by P Ai the probability of etection in cell A i. In each time interval, new targets may arrive in each cell, targets may move from one cell to another, an targets may be etecte (such that they are no longer unetecte, an hence are no longer counte in the ensity of unetecte targets). In what follows, we state upate rules from Λ(A i, t + k ), the expecte number of unetecte targets in cell A i at time t k after the observations performe at time (k ), to Λ(A i, t k ), the expecte number of unetecte targets at time k prior to the observations at time k, to Λ(A i, t + k ), the expecte number of targets which remain unetecte after the observations at time k. The first transition is mae by way of approximation, since realistic targets travel with a nominally constant velocity but our regions are only spatial in orer to keep imensionality manageable. We efine the transition kernel: Φ k (A i, A j ) = Pr{target is in cell A j at time t k target was in cell A i at time t k }. () This transitional kernel may be learne by simulating a large number of realistic target trajectories an observing what proportion of the targets that commence in cell A i en in cell A j (average over all time). Alternatively, a iscrete space Markov chain motion moel may be implemente irectly [, p.]. Our moel for target transitions is thus that the unetecte targets in a cell split inepenently among all cells with the probabilities given. Incorporating the number of newly arrive targets in the same time interval yiels Λ(A j, t k ) = λaj (t k t k ) + A Φ k (A i, A j )Λ(A i, t + k ), i= The superscript + enotes after observations at the specifie time interval, while the superscript enotes prior to the observations. ()

3 where λ A j is the average rate of arrival of targets in A j. The final transition captures the impact of observing a cell with the raar. If the cell is not observe, we simply have Λ(A i, t + k ) = Λ(A i, t k ). If the cell is observe, we have Λ(A i, t + k ) = ( P A i )Λ(A i, t k ), an the expecte number of newly etecte targets resulting from the observation on the cell is P Ai Λ(A i, t k ). The cells A i are assume to consist of slices of the surveillance region. The raar beam is forme from slices in azimuth with extent η i while the range extent of the slices is chosen to ensure that the constant P assumption is reasonable ρ i = ρ i (upper) ρ i (lower). We enote the set of cells within which the probability of etection of the m-th beam is nonzero as B m = i Im A i, an the probability of etection of the m-th beam within cell A i as P A i,m an so I m = {i : P A i,m > 0}. It is convenient to choose the cells A i such that the azimuth extent of each cell is equal to the azimuth beamwith η m for the corresponing beam m. The target arrival function λ A j in Eqn () can incorporate three ifferent types of arrivals: (i) arrivals from the bounary of the surveillance region, (ii) arrivals from known or likely entry points, eg. airports, an (iii) arrivals from anywhere within the surveillance region. Let λ B enote the average arrival rate for targets at the bounary of the surveillance region an it is assume that the arrival can occur with uniform probability aroun the outer range ring. The average arrival rate of targets within a region E l = i Il A i associate with the l-th entry point is enote λ l E, where I l is the set of corresponing cell inices that efine the region in which targets may enter. Finally, targets arrive on average at a rate of λ S uniformly within the surveillance region. Consiering each of the ifferent types of arrivals in turn an compensating for ifferent cell sizes yiels an expression of the form: λ S η i ρ i /(πr s) + λ B η i /(π), A i lies on the bounary in range λ λ Ai S η i ρ i = /(πr s)+ λ l E η i ρ i /( i I l η i ρ i ), A i E l, the l-th target entry region λ S η i ρ i /(πr s), otherwise () an ρ i in Eqn () is efine ρ i = [ρ i (upper)] [ρ i (lower)]. (0) In Ref [] the sensor management objective function use the posterior expecte number of targets (PENT), but here the posterior expecte number of newly arrive targets is use to etermine the expecte number of newly etecte targets. When a surveillance action is chosen at time step k, we select the beam which maximises the expecte number of newly etecte targets. This can be calculate as: m k = arg max m M P Ai,m Λ(A i, t k i I m ). () B. Revisit Moe Beam Selection The revisit moe of the raar is use to maintain track state estimates on etecte targets. Consier a target whose state at time k is represente by x k. The probability ensity function (pf) of the state given the full set of associate measurements Z k = {z,, z k } is assume to be available from a target tracking function an is expresse as p(x k Z k ). The entropy of a ranom variable x with a pf p(x) is efine by H(x) = p(x) log e p(x) x. () The information gain is efine by the change in entropy from the prior state to the posterior entropy I = H(x k Z k ) H(x k Z k, z k ), () where z k enotes the unknown measurement (incorporating a binary etection flag, an if true, a continuous value) at time step k. Conitioning upon z k implies an expectation over the values that the measurement may assume. For a measurement action at time step k, let ζ k {0, } be an inicator variable to enote the cases where the target is not etecte an etecte respectively. Expaning the expectation over ζ k allows the conitional entropy to be written H(x k Z k, z k ) = ˆP H(x k Z k, ζ k =, z k ) + ( ˆP )H(x k Z k, ζ k = 0), () where the expecte range of the target an the beam that illuminates the target are use to evaluate the probability of target etection ˆP = P (ˆρ, ˆm). If it is assume that a misse etection oes not provie any information about the target state (ie., that the etection probability oes not vary substantially within the region of the target) then H(x k Z k, ζ k = 0) = H(x k Z k ) () an so Eqn () can be expresse as I = ˆP ( H(xk Z k ) H(x k Z k, z k ) ), () where the inicator reference variable is implie an therefore omitte for the convenience of notation. If the target inex j J is introuce, an m k,j is the beam that woul illuminate the jth target at time step k, then the expecte information gain for each target I j can be compute to etermine the next revisit beam m k = m k,j with j = arg max I j. () j J III. EXAMPLE IMPLEMENTATION OF PHASED ARRAY RADAR CONTROL In this section a simulate scenario is consiere where the beam selection scheme escribe in the preceing section is implemente for a set of targets that are injecte an travel accoring to specifie ranom istributions.

4 A. Target an Raar Configurations It is assume that the raar position is fixe at the centre of the surveillance region which is epicte in Figure by the outer circle which has raius R s = 00km. A set of 0 targets exist at the beginning of the scenario, numbere to 0 in large text in Figure. During the scenario aitional targets are injecte ranomly into the surveillance region accoring to a Poisson istribution at a rate of λ S = 0.s until the en of the scenario which is of uration t = s. The target state is represente by a vector in Cartesian coorinates as x = [x, ẋ, x, ẋ ] T. Once a target is injecte into the surveillance area, its state evolves accoring to a iscrete time linear ynamical system: where the state transition matrix is F k = x k+ = F k x k + w k, () T T () an T = t k+ t k is the time interval between successive time steps. The ynamics noise w k is assume to be escribe by a continuous white noise acceleration moel with process noise covariance matrix [, p.] Q k = T / T / 0 0 T / T T / T / 0 0 T / T q, (0) where q is the intensity of the process noise [0, p.] Fig.. Distribution of targets positions overlai on the raar surveillance volume. Target velocity vectors emanate from the target positions. The set of raar beams is numbere in a clockwise irection m =,...,. The isplaye range of each beam sector correspons to the range ρ 0 (m) where the probability of etection is 0.. The set of M = iscrete raar beams is also shown in Figure an the range that is isplaye for each beam correspons to the nominal range ρ 0 (m) at which the probability of etection is P = 0.. A given beam has a value of ρ 0 = 0, 00 or km. The well time for each beam τ m is a function of the beam inex as shown in Figure. The cumulative beam well time M= m= τ m = s. Furthermore, the beamwith η(m) may take one of three possible values:., or. Dwell Time (s) Beam Inex m Fig.. Dwell times for each raar beam. The probability of etection as a function of range ρ an beam m is evaluate in the following manner: ) The require Signal to Noise Ratio (SNR) for raar etection with a probability P = 0. is etermine for a specifie probability of false alarm P fa [, Fig.]. For a P fa = 0, SNR(ρ 0 ).. ) The SNR is estimate for a given target range ρ SNR(ρ, m) = 0 log 0 ρ 0 (m) ρ Cumulative Dwell Time (s) + SNR(ρ 0 ) () ) Albersheim s etection equation allows the probability of etection to be compute [, p.]: P (ρ, m) = eb + e b, () where b = (0 SNR(ρ,m)/0 a)/(0.a+.) an where a = log e (0./P fa ). The range istribution for the probability of etection obtaine from Eqn () is shown in Figure. When a beam illuminates a target that lies within the geographical extent of the beam, a ranom sample is rawn accoring to the probability of etection to etermine if the target is etecte or not. If a target is etecte then a raar measurement z k is generate with range an azimuth components that are relate to the target state via the nonlinear equation [ ] ρk z k = = h(x η k ) + ν k, () k

5 Detection probability, P Fig.. of ρ range, ρ (km) Probability of etection as a function of range for the three values where h(x k ) is the nonlinear measurement function h(x k ) = [ x + x tan (x /x ) ]. () The measurement noise ν k follows a white process process, ν k N {ν k ; 0, R} an the measurement noise covariance matrix is given by [ ] σ R = ρ 0 0 ση. () an for the upate step: ˆx k k = ˆx k k + K k [z k h(ˆx k k )] () P k k = P k k K k Hk P k k () K k = P k k HT k [ H k P k k HT k + R k ], (0) where H k is the linearise measurement matrix efine by H k = h x. () x=ˆxk k Using the extene Kalman filter, the istribution of the target state at time k conitione on measurements up to time k is approximate as Gaussian with mean ˆx k k an covariance P k k, ie. p(x k Z k ) N {x k ; ˆx k k, P k k }. Likewise, the istribution of target state at time k is approximate as a Gaussian, p(x k Z k, z k ) = N {x k ; ˆx k k, P k k }. C. Sensor Management Function Selection of the raar beams follows the scheme escribe in Section II. Simplifying assumptions are use to explore a reuce feature set of the algorithm. A ecision for the type of beam to implement at time step k is mae by comparing the esire revisit ratio Γ with the cumulative well times spent in the surveillance an revisit moes up to an incluing the time step k. For the example consiere in this section the cells A i of the surveillance region are chosen with azimuth extents corresponing to those of the raar beams an with range extents of 000m. Figure illustrates how each beam region B m an target entry region E l is constructe from cells of the form A i. The istribution of the measurement error is assume to be inepenent of the beam use. Values for the stanar eviations of the range an azimuth components are chosen to be σ ρ = 00m an σ η = respectively. B. Tracking Function A tracking function is use to initiate an upate a track on each target base on measurements receive from the raar. A simple scheme is aopte whereby a track on each target is initiate base on the first measurement receive for that target. The implemente tracking function oes not account for ata association uncertainty when initiating or upating a track. An extene Kalman filter is implemente to upate track state estimates for each of the targets that are etecte. It is assume that the association problem is solve so that each of the raar measurements is available to upate the track state estimate for the corresponing target. The equations for the extene Kalman filter are as follows for the propagation step: ˆx k k = F k ˆx k k () P k k = F k P k k F T k + Q k () E l A i B m Fig.. Diagram illustrating how the surveillance region is partitione using the range an azimuth coorinates into cells A i, from which beam regions B m an target entry regions E l are constructe. The parameters for the surveillance beam selection are simplifie by assuming that the targets are stationary. This

6 assumption can be justifie if the targets are observe regularly enough an so the transition kernel in Eqn () is { i = j Φ k (A i, A j ) = () 0 otherwise. Targets that are assume stationary an are injecte in regions where the probability of etection is close to zero will have little influence on the selection of surveillance beams throughout the uration of the scenario. In practice, the movement of non-stationary targets from a region where the probability of etection is initially low may lea to a subsequent influence on the selection of surveillance beams. In this paper the assumption of stationary targets simply serves to illustrate the general concept of the sensor management scheme. If a surveillance beam is require, then the expecte number of unetecte targets Λ(A i, t k ) is compute from Eqn () an substitute into Eqn () to select the beam which yiels maximum expecte number of measurements for new target arrivals. Turning our attention to the revisit moe, we note that for an n-imensional Gaussian ranom variable, the entropy is given by the analytic expression []: H(x) = n log e(πe) + log e P, () where P is the eterminant of the covariance matrix P. Substituting the covariance matrix expressions from Eqns () an (0) into equations of the form of Eqn () allows the information gain in Eqn () to be expresse as I = ˆP log P k k e P k k = ˆP log e () H k P k k HT k + R k. () R k If a revisit beam is require, then the beam chosen is that which illuminates the track that yiels maximum expecte information gain accoring to Eqn (). D. Results The sensor management scheme outline in this paper has been implemente in orer to select the raar moe an beam to illuminate target configurations with specifie ranom istributions. Two ifferent values for the esire revisit ratio parameter are consiere. Firstly, the esire revisit ratio parameter was set to a value Γ = 0 so that the beams were selecte accoring to the surveillance beam selection process only. Figure shows the raar beam illuminations as a function of time for each raar beam m when Γ = 0 an when the target arrival function captures only the arrival of targets uniformly within the surveillance region, so that λ B = λ E = 0. The horizontal with of the bar correspons to the well time τ m for that beam. Each raar beam is initially illuminate in turn, but then the beams are illuminate at regular time intervals. Raar beams m = an m = are illuminate most often, while beams m = an m = are illuminate least often. Beam inex m Time (s) Fig.. Raar beam illuminations for each raar beam when Γ = 0. TABLE I SCAN TIMES FOR EACH BEAM WITH DESIRED REVISIT RATIO Γ = 0. Beam Nominal beam Beamwith Mean m range ρ 0 (m) (km) η m (egs) Scan Time (s) Table I lists scan times for the set of raar beams, with the mean scan time efine as the mean time interval between successive illuminations of a given raar beam. A comparison of beams that have the same beamwith but ifferent nominal etection range shows that the beams with the longest nominal etection range have the smallest time interval between illuminations. Similarly, for beams that have the same nominal etection range but ifferent beamwith, the beams with the largest beamwith have the smallest time interval between illuminations. This result arises because beams that have long nominal etection range an large beamwith typically yiel the greatest expecte number of new target measurements. Seconly, the esire revisit ratio parameter was set to a value Γ = 0.. A single target entry point with an arrival rate of λ E = 0.0s was efine with the entry region extening between the ranges an 0km an spanning the beams m = an m =. The entry region E l, where l =, is

7 epicte in Figure. For the first s of the scenario, each beam is illuminate in the surveillance moe only. After that time the type of raar moe is chosen to be either surveillance or revisit by comparing Γ with the ratio of the well time spent on revisits to the total cumulative well time (from t = s) accoring to Eqn (). If a surveillance action is chosen, the beam that yiels the maximum expecte number of measurements of new targets is selecte. If a revisit action is chosen, the beam that correspons to the track which yiels maximum expecte information gain is selecte. Figure shows the raar beam illuminations as a function of time for each raar beam m when Γ = 0.. As is the case for Figure, the surveillance beam illuminations are epicte in blue vertical bars, but in Figure revisit beam illuminations are interleave an these are epicte with re bars. Beam inex m Time (s) Fig.. Raar beam illuminations for each raar beam when Γ = 0.. The two raar moes are istinguishe by the blue (surveillance) an re (revisit) vertical bars. It can be seen in Figure that the illumination of beams is far more ynamic than is apparent in Figure. The coupling between the surveillance an revisit moes is evient because for some beams the revisit moe substitutes for the surveillance moe by illuminating the beam at regular time intervals. Several beams are each illuminate using the surveillance moe only (eg. m =,,, an ) an so any etecte an tracke targets that are illuminate by these beams eviently o not lea to sufficient expecte information gain compare to that of other tracke targets. The mean scan time for beams that are illuminate using the surveillance moe is typically greater for the case in which Γ = 0. compare to the case in which Γ = 0 because beam well time is allocate to revisits when Γ 0. For example the mean scan times for beam m = are 0.0s an.s for Γ = 0 an Γ = 0. respectively. In contrast, the mean scan time for beams m = an m = can be seen to ecrease with the presence of the entry region E which spans these beams. For example, the mean scan times for beam m = are.0s an.s for Γ = 0 (with no entry TABLE II TARGET LABEL FOR TRACK INDEX j. Track j... Target 0... Track j 0 Target 0 point) an Γ = 0. (with one entry point) respectively. Figure shows the beam illuminations for the set of tracks that are initiate using measurements receive from the raar. The target injection time is inicate by an asterisk for each target. An open circle inicates the track initiation time, which here correspons to the first measurement receive from the target. The track inex j is incremente with each new track initiate on a target. A list of the target labels for each of the initiate tracks is given in Table II. Track inex j Time (s) Fig.. Raar beam illuminations for each track on a target when Γ = 0.. The times at which the target is injecte an at which a track is initiate on the target are shown by an asterisk (*) an circle ( ) respectively. The black line in Figure correspons at each time step k to log e P k k, which is a measure of the entropy or uncertainty in the track state estimate as expresse in Eqn (). When a raar beam illumination yiels a measurement on a target then the entropy of the track estimate ecreases but an increase in the entropy is evient in the absence of any illuminations or when no measurement arises. Inspection of Figure, together with Figures an, an Table II, permits the following observations. Bursts of revisit beams typically follow soon after a track has been initiate on a target an when the entropy of the track has grown in the absence of subsequent measurements on the target. A comparison of the illumination sequence for beam m = in Figure an the illumination of tracks in Figure reveals that tracks j =,,,, an are each illuminate by beam m = only. Table II shows that track j = is forme on target an Figure inicates that the target moves from

8 being illuminate by beam m = to beam m = which is evient in the illumination sequence for track j = in Figure where the beam well times change at a time of about s. After this time no other targets are etecte in the coverage of beam m = an Figure shows that the illumination sequence for beam m = consists primarily of revisit beams until s when the illumination continues with surveillance beams only. IV. CONCLUSIONS In this paper a sensor management approach has been escribe for phase array raar applications where the aim is to etect targets an to maintain a track estimate on each target. The approach aopte treats the sensor management problem as a separate subproblem for each raar moe. In the case of the surveillance moe the objective function for selecting the next beam is the expecte number of measurements for previously unetecte targets. In the case of the revisit moe the objective function for selecting the next beam is the expecte information gain. Future work shoul consier extening the approach to look ahea multiple time steps instea of a single time step as presente in this paper. The selection from a fixe set of raar beams has been consiere but flexible sensors such as phase array raars typically have many parameters whose values may be optimise. A esire revisit ratio must be specifie in the sensor management approach consiere in this paper. Exploration of the impact of the revisit ratio on tracking accuracy an initiation time remains the topic of further stuy. While the esire revisit ratio affors some level of operator control in the istribution of resources between the ifferent raar moes, further investigations are require to etermine means of unifying the separate objective functions within a single framework. The objective functions presente in this paper o not account for the varying beam well time an so subsequent work may investigate the potential trae between the benefit from taking sensor actions an the time cost for those actions. The surveillance subproblem is treate by partitioning the surveillance region into a iscrete spatial gri of cells. A more complete representation of the target space woul exten the imensions of the iscrete gri to consier the velocity components. Dynamically choosing the size of the iscrete gri cells may lea to more efficient implementations []. In realistic environments targets such as low flying aircraft may be obscure from raar etection by terrain features such as mountains. Incorporating the state epenent aspects into the probability of etection permits the current sensor management framework to capture targets that isappear from the scene. Assessment of any sensor management scheme must aress the key performance aspects of importance to employment of the sensor system. Further work is require to assess the avantages an isavantages of the presente sensor management approach compare to alternative approaches. REFERENCES [] S. S. Blackman an R. Popoli, Design an Analysis of Moern Tracking Systems. Artech House, Boston,. [] A. O. Hero, D. Castañón, D. Cochran an K. Kastella (es.) Founations an Applications of Sensor Management. Springer series on Signals an Communication Technology, 00. [] J. L. Williams, Information Theoretic Sensor Management. PhD thesis, Massachusetts Institute of Technology, USA, 00. [] G. A. McIntyre an K. J. Hintz, An information theoretic approach to sensor scheuling, Proc. SPIE Conference on Signal Processing, Sensor Fusion an Target Recognition V, vol., pp. 0,. [] S. Suvorova, S. D. Howar an W. Moran, Beam an waveform scheuling approach to combine raar surveillance an tracking - the Paranoi Tracker, in Proc. IEEE International Waveform Diversity an Design Conference, Hawaii, USA, January 00. [] A. S. Chhetri, D. Morrell an A.Papanreou-Suppappola, Nonmyopic Sensor Scheuling an its Efficient Implementation for Target Tracking Applications, in EURASIP Journal on Applie Signal Processing, Volume 00, pp. -. [] J. Wintenby, Resource Allocation in Airborne Surveillance Raar. PhD thesis, Chalmers University of Technology, Göteborg, Sween, 00. [] B. F. La Scala, M. J. Rezaeian an W. Moran, Optimal Aaptive Waveform Selection for Target Tracking, th International Conference on Information Fusion, Philaelphia, PA, USA, July 00. [] L.D. Stone, C.A. Barlow an T.L. Corwin, Bayesian Multiple Target Tracking, Artech House,. [0] Y. Bar-Shalom, X. Rong Li an T. Kirubarajan, Estimation with Applications to Tracking an Navigation, John Wiley & Sons, Inc., 00. [] R. Mahler an T. Zajic, Probabilistic Objective Functions for Sensor Management, in Proc. SPIE Conference on Signal Processing, Sensor Fusion, an Target Recognition XIII, vol., pp.-, Bellingham, WA, 00. [] G. Morris an L. Harkness. Airborne Pulse Doppler Raar. Artech House, eition,. [] N. Xiong an P. Svensson, Multi-sensor management for information fusion: issues an approaches, Information Fusion,, pp. -, 00.