Radar Transmitter Geolocation via Novel Observation ... - DTIC

Radar Transmitter Geolocation via Novel Observation

Technique and Particle Swarm Optimization

John G. WarnerUS Naval Research Laboratory

Washington, DC 20375202-279-5060

Jay W. MiddourUS Naval Research Laboratory

Washington, DC 20375202-279-4338

Abstract— The ability to precisely determine the location ofradar transmitters can be crucially important in maintainingdomain awareness. This, however, may be problematic withtraditional methods when used with a distributed network ofdisparate sensors. A novel geolocation technique for circularlyscanning radar transmitters is introduced. This technique usesthe differenced central times of arrival (DCTOA) of the mainbeam as an observable. The solution for the transmitter’sposition and scan rate are given using a weighted least squaresapproach as well as a particle swarm optimizer. Experimentalresults show this technique is able to locate a radar transmitterwithin 11 meters, while maintaining minimal complexity. Thistechnique has the advantage of requiring orders of magnitudeless timing synchronization among receivers, an order of mag-nitude less data transfer, and it does not require simultaneousillumination of receivers.

TABLE OF CONTENTS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 THE SEARCHLIGHT METHOD . . . . . . . . . . . . . . . . . . 2

3 SOLVING THE LINEARIZED SYSTEM . . . . . . . . . . . . 3

4 SOLVING THE NONLINEAR SYSTEM . . . . . . . . . . . . 4

5 CONFIDENCE REGION CALCULATION . . . . . . . . . . 5

6 TESTING RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1. INTRODUCTION

The ability to accurately locate radar sources in a timelymanner is crucial in maintaining maritime and other domainawareness. Traditional methods of precision geolocation re-quire multiple receivers to be time synchronized on the orderof nanoseconds and often require simultaneous illumination.Further, receivers must exchange data that often occurs atthe pulse rate of the radar. This, however, may present achallenge when a distributed network of diverse sensors isused that have limited timing and data exchange abilities. Amethod of precision geolocation with less stringent timingand data exchange requirements would enable a disparatenetwork of sensors to accurately estimate a radar transmitter’slocation, as well as expand the state of the art of radargeolocation techniques.

There are a number of methods currently used for radargeolocation. Reference [1] provides a general survey of

U.S. Government work not protected by U.S. copyright.1 IEEEAC Paper #1582, Version 1.0, Updated 21/12/2011.

techniques, while Reference [2] discusses the theory of thesetechniques in detail. Many of these techniques require thesame radar signal to be observed by multiple receivers.A common example of this is Time Difference of Arrival(TDOA), where the difference in signal time of arrival ismeasured between two spatially distributed receivers. De-tails of this technique can be found in Reference [3]. Thistechnique may be problematic with arbitrarily distributedreceivers as it requires simultaneous illumination and precisetiming synchronization between receivers (as solution error isproportional to timing error times the speed of light). Othercommon techniques have similar requirements but observedifferent signal characteristics. For example, the differencein the phase of the signal between two receivers may be ob-served in interferometry or the Difference in the Frequency ofArrival (FDOA) may be used to estimate a radar transmitter’slocation. These techniques have similar timing constraintsand will further require data to be shared between receivers atthe pulse rate of the radar, which may be prohibitive.

Additionally, the doppler shift of the signal may be observed,if there is relative motion between the transmitter and thereceiver, and can be used to estimate a transmitter’s location.This technique may be problematic in practice as it gener-ally requires information about the emitted frequency, whichmay not be readily available and may vary over time in anunknown fashion. Reference [4] contains further discussionof this method.

The last common technique is based on the Angle of Arrival(AOA) of the signal. Reference [5] as well as Reference [2]give ample discussion of this method. This technique usesmultiple measurements of the AOA of a signal to triangulateits source. This method does not require precision timingsynchronization of receivers or high rates of data exchange.However, as determining the AOA of a signal to sufficientaccuracy may be challenging, precise geolocation results arenot often achievable.

A novel method has been developed that uses differencedtimes of arrival of maximum signal strength across multiplereceivers to locate a circularly scanning radar transmitter.The method allows for passive, precision geolocation withoutprecise timing requirements or prohibitive data exchangevolumes. The difference between signal times of arrivalis related to the angle the radar has swept out betweenobservation events, thus the radar transmitter location maybe calculated to be along a line of constant angle between thetwo receivers. Given sufficient data, the radar transmitter’slocation may be estimated. This method has the advantage ofbeing independent of platform and tolerant to timing error onthe order of hundreds of microseconds.

This technique has been developed and tested at the US NavalResearch Laboratory (NRL), and has been developed by other

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14. ABSTRACT The ability to precisely determine the location of radar transmitters can be crucially important inmaintaining domain awareness. This, however, may be problematic with traditional methods when usedwith a distributed network of disparate sensors. A novel geolocation technique for circularly scanningradar transmitters is introduced. This technique uses the differenced central times of arrival (DCTOA) ofthe main beam as an observable. The solution for the transmitter’s position and scan rate are given using aweighted least squares approach as well as a particle swarm optimizer. Experimental results show thistechnique is able to locate a radar transmitter within 11 meters, while maintaining minimal complexity.This technique has the advantage of requiring orders of magnitude less timing synchronization amongreceivers, an order of magnitude less data transfer, and it does not require simultaneous illumination of receivers.

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researchers in References [6], [7] and [8]. Major work at NRLinclude field tests in 2006 and 2009, the latter of which isdocumented in Reference [9]. NRL has been issued a UnitedStates patent for this algorithm, [10].

A complete description of the methods used to reliably imple-ment this technique, called SearchLight, are presented. Thedevelopment of the basic observable is presented along withthe necessary algorithm to allow the technique to be usedfor an arbitrary receiver configuration. Next, methods forestimating the radar transmitter’s position are presented forboth the linearized and the nonlinear system. The nonlinearoptimization process has the added benefit of being able toeasily resolve a location ambiguity that exists within thesystem. Last, experimental results are presented using datafrom the 2009 data collection. Ultimately, it is shown that theSearchLight algorithm enables precision radar geolocationusing distributed receivers without the need for precise timingsynchronization.

2. THE SEARCHLIGHT METHOD

The SearchLight concept makes use of the fact that a sta-tionary, circularly scanning radar transmitter will illuminatea receiver at a regular interval. If one envisions two receiversdistributed about a circularly scanning radar, the change intime of illumination between the receivers is equal to the an-gle the transmitter sweeps out between the receivers dividedby the scan rate of the transmitter. This can be expressed bythe following equation.

tj − ti =1

ωcos−1

(

(ri − r) · (rj − r)

|ri − r| |rj − r|

)

(1)

In practice, the position of the transmitter, given here byr, is the unknown quantity that must be derived along withthe scan rate, ω, given a series of differenced central timesof arrival (DCTOA), ti and positions at these times, ri.While the position vector may be expressed in either a localvertical local horizontal coordinate system or in an Earthfixed coordinate system, the implicit assumption is that theradar transmitters and receivers do not vary significantly fromthe local tangent plane. The central time of arrival for areceiver is defined as the time of arrival of maximum signalamplitude (corresponding to the arrival of the center of themain radar beam).

The method for determining the central time of arrival is de-pendent on radar signal characteristics as well as the numberof active radar in the collection area. When there are multipleactive radar being collected against, each signal must bedeinterleaved. There are several methods for accomplishingthis, all of which rely on the specific characteristics of theradar to properly sort the signals. Reference [2] contains ageneral overview of available techniques. Once each radarsignal has been isolated, the central time of arrival may becalculated. For pulsed radar, this time may be estimated byfitting a quadratic curve to the center of received radar pulsesand calculating the time corresponding peak of the quadraticfor each radar pass. A similar technique may be applied tocontinuous wave radar.

It should be noted that the angle calculated in Equation 1 is bydefinition of arc cosine between 0 and π; therefore, a check isnecessary not only to ensure the correct quadrant is resolved,but also that the calculated angle between receivers is indeedthe angle the radar has swept out between detection events.

θ

2π − θ

2

1

Figure 1. Diagram Showing Example of Relative Geometryof Two Fixed Receivers to One Fixed Radar Transmitter

Consider Figure 1. As the radar transmitter sweeps in acounter-clockwise direction, the angle it sweeps betweenconsecutive receiver illumination events will alternate be-tween θ and 2π − θ. That is, the angle that the radartransmitter sweeps from receiver one to receiver two is θ,while the angle the transmitter sweeps from receiver two toreceiver one is 2π−θ. However, the calculated angle betweenthe receivers will always remain θ. A set of geometricchecks must be implemented to differentiate these cases inorder to calculate the correct error residual and to extend thistechnique to arbitrary receiver geometry.

It can be shown that angle the radar sweeps between detectionevents falls into one of three cases. As shown above, thefirst case occurs when the radar has swept out exactly thecalculated angle θ between detection events. In the nextcase, the radar has swept out not θ but its explement (i.e.2π − θ). These cases are demonstrated in Figure 1. Thelast possible case occurs for one mobile receiver. Betweendetection events, the receiver may have displaced an angle θwith respect to itself. In this case, the radar sweeps out onecomplete revolution plus the additional angular displacement,which is 2π + θ. This occurs when the receiver moveswith the direction of radar sweep: when the receiver movesagainst the radar sweep the angle is 2π − θ. Assumingthat system uncertainty is small with respect to the timebetween detection events, the correct DCTOA error residualis implemented as the following.

θ = cos−1

(

(ri − r) · (ri+1 − r)

|ri − r| |ri+1 − r|

)

(2)

e = DCTOAobserved − DCTOAcomputed (3)

= min

[

DCTOAobserved − θ/ωDCTOAobserved − (2π − θ)/ωDCTOAobserved − (2π + θ)/ω

]

(4)

To reiterate, for each measurement a calculated differencecentral time of arrival will be computed based on the cur-rent estimate of the radar transmitter’s position and scanrate. This calculated DCTOA must be checked against theobserved DCTOA to ensure that the correct geometric caseis identified. From here, the error residual is computed asthe value that is the minimum of the three geometric casesdiscussed above and shown in Equation 4. This method isvalid as measurement error is at least an order of magnitudelower than time difference between the three geometrical

2

cases. It should be noted that without making this correction,the algorithm is constrained to geometric cases where eachreceiver is 180o or less apart from each other with respect tothe radar transmitter.

Note that this relationship holds true regardless of receiverconfiguration. The benefit of this method is that data fromany number of receivers in any distribution may be used toestimate the radar transmitter’s location without additionalimpact upon the SearchLight algorithm. Indeed, the onlyrequirement is that sufficient change in receiver bearing withrespect to the transmitter is observed; thus, a multitude ofreceiver platforms and configurations are supported.

Further, it can be seen that information regarding specifictransmitter characteristics is not necessary a priori. Thisgives SearchLight an advantage of competing geolocationtechniques such as geolocation via doppler shift observation,where the emitted signal frequency must be known. Also,only the arrival of the center of the main radar beam must beisolated when multiple transmitters are present, rather thanthe arrival of each pulse. This may decrease the complexityin the signal deinterleaving process and allow the SearchLightalgorithm to be used in applications where specific signalcharacteristics are not known a priori.

Next, SearchLight may be used to geolocate ContinuousWave (CW) radar transmitters. Many traditional techniques,such as TDOA, use the arrival of pulses to define a measure-ment event and cannot be easily implemented to geolocateCW radar transmitters.

Last, it can be seen from Equation 1 that this observationis much less sensitive to clock error than traditional timedifference of arrival (TDOA) methods. In TDOA methods,the resulting position error due to clock error is proportionalto the speed or light; however, for this DCTOA method,the position error is only proportional to the scan rate ofthe radar transmitter, which is several orders of magnitudeslower. Thus, much larger clock errors are tolerable. Thisfact enables this method to be used with distributed receiverswithout the need for precise timing calibration. Additionally,the requirements for data transfer with this algorithm ismuch lower than traditional methods. Traditional techniquesrequire data measurements to be taken at the pulse rate ofthe radar, while this technique only requires the sharing ofdata taken at the scan rate of the radar transmitter. Thistypically results in the need for one order of magnitude lessdata transfer between receivers to formulate a geolocationsolution.

3. SOLVING THE LINEARIZED SYSTEM

Given multiple DCTOA measurements, a stationary, circu-larly scanning radar transmitter’s position and scan rate maybe estimated. It is clear that the measurement equation (Equa-tion 1) is nonlinear, so a linearized least squares solutionmay be used. The least squares criteria was first posedby Gauss. For a more in depth treatment see References[11] or [12]. Here the state that is solved for is the radartransmitter’s position, r, and scan rate, ω. The measurementis the DCTOA, which is related to the state by Equation 1,which is repeated for completeness.

x =

[

rω

]

(5)

y =1

ωcos−1

(

(ri − r) · (ri+1 − r)

|ri − r| |ri+1 − r|

)

(6)

= h(x, ri, ri+1) (7)

To linearize the measurement equation, the first order vari-ation of its Taylor series expansion about a known constantstate x0can be considered. So then, the following equationshold.

y = h(x, ri, ri+1)− h(x0, ri, ri+1) (8)

y ≈ H [x− x0] (9)

H =∂h(x, ri, ri+1)

∂x(10)

In the weighted least squares (WLS) formulation the goal isto estimate the state, xo, that minimizes the following costfunctional, which is the sum of the squares of the error.

J(x) = 1/2 ǫTWǫ (11)

= 1/2 (y −Hx)TW (y −Hx) (12)

Here, W is a positive definite weighting matrix which is gen-erally set to the inverse of the measurement noise covariance.This has the well-known solution shown below.

xo =(

HTWH)

−1HTW y (13)

This quantity is often described in terms of the covariancematrix, P .

P =(

HTWH)

−1(14)

xo = PHTW y (15)

As this is a nonlinear WLS problem, an initial stateestimate,x0, must be generated and the solution must beiterated upon until convergence. An initial guess for the radartransmitter’s position can be generated by a rough grid search.To generate an initial guess on the radar’s scan rate, the timesbetween detection events for any individual receiver can beaveraged. This may then be averaged between all receivers.This time divided by 2π yields a fairly accurate guess of scanrate. This is because the time between detection events foreach individual receiver roughly corresponds to one completerevolution of the radar beam.

While an analytic form of the matrix H may be found(this is shown in the Appendix), analysis has shown thatcalculating numerical partial derivatives generally yield betterresults. Reference [8] contains an alternate formulation ofthis solution.

3

4. SOLVING THE NONLINEAR SYSTEM

As the system is nonlinear, it would be beneficial to estimatethe radar transmitter’s position and scan rate in a nonlinearfashion. Additionally, there is an issue that makes nonlinearanalysis preferable. It can be seen from Equation 1 thatany given DCTOA can be generated by estimating the radartransmitter at the true position and true scan direction or ata position mirrored about the line joining the measurementlocations and having the opposite scan direction. This falselocation will be referred to as a “ghost point.” This ghostpoint will move as the relative geometry of the receiverschange; however, the area in the vicinity of the ghost pointwill represent a local minimum of the sum of the squarederror. This may cause the WLS to erroneously converge.Thus, a method for locating a global minimum (despite theexistence of local minimums) is necessary. An example ofthis phenomenon can be seen in Figure 2 below. As can beseen, there is a distinct local minimum corresponding to theghost point mirrored about the mobile receiver position.

283.6 283.7 283.8 283.9 284 284.1Longitude

36.80

36.85

36.90

36.95

37.00

37.05

37.10

Latit

ude

True Location

Ghost PointFixed Receiver

Mobile Receiver

Figure 2. Plot Showing Error Contours of the Cost Functionwith Relative Receiver Geometery for F1 Dataset

In order to meet these goals, a particle swarm optimizer(PSO) was implemented. PSO was first theorized in Ref-erences [13] and [14]. For a more in depth discussion seeReference [15], while Reference [16] provides an excellentoverview of the process. The problem must be posed asfinding a minimal value of a cost function given a set of inputparameters. This applies exactly to the SearchLight problem:the sum of the squares of the residuals may be minimized byfinding the correct transmitter location and scan rate. Fromhere, the search space for the solution must be defined. Interms of this application, this means that appropriate boundsneed to be placed on the location as well as the scan rate forthe receiver. In this execution of the PSO, the initial boundson the radar transmitter’s position have been as large as twodegrees in latitude and longitude. The initial bounds on thescan rate can either be estimated by the end user or takenas within a reasonable percentage of the estimated scan rateusing the method described above. Next, the search spaceis randomly populated with a number of state guesses (inthis case receiver position and scan rate), known as particles.These particles are also given an arbitrary initial velocity.At each time step the cost function is evaluated for eachparticle, and the a number of best states are tracked. The firsttracked state is known as the individual best solution whichcorresponds to state with the lowest cost at any time step on astrictly per particle basis. So, for N particles, there will be Nindividual best states tracked. The next tracked state is knownas the global best solution and is the state with the lowest costfor any particle at any time step. There is typically only oneglobal best solution. From here, each particle is stochastically

accelerated along three directions: along its current direction,towards its best known individual solution and towards thebest known global solution. This process is iterated until astopping condition is met, which is typically a fixed numberof iterations.

The equations in this process are defined in the followingparagraphs. Here, the objective function is defined as thesum of the squares of the residual error: this is a functionof transmitter position and transmitter scan rate.

J(x) =N∑

i=2

(yi − hi(x, ri−1, ri))2 (16)

Here, yi and h(x, ri, ri+1)i are the ith observed DCTOA and

calculated (via Equation 1 and Equation 4) DCTOA, whichare summed over M observations.

Now, an initial population of particles can be created. Thisis done by uniformly, randomly populating the search spacewith N number of initial states. This is accomplished by

invoking the following equation for the jth particle.

Xj =Xmin + r(0, 1)d (17)

d =Xmax −Xmin (18)

Xj is the state of the jth particle, Xmax is the upper boundof the search space, Xmin is the lower bound for the searchspace and r(0, 1) signifies a random number with uniformdistribution between 0 and 1.

The following process is then iterated until a suitable stoppingcondition is reached. First, evaluate the objective function foreach particle.

Υj = J(Xj) (19)

Next, check to see if this current state represents either an

individual best solution for the jth particle, denoted by ψj ,

or a global best solution, represented by G. Here, only oneglobal best solution will be tracked. If these conditions aresatisfied then the following are implemented.

ψj =Xj if Υj < J(ψj) (20)

G =Xj if Υj < J(G) (21)

For M particles, the algorithm tracks M individual particlebest solutions and typically one global best solution. The in-dividual and global best solutions are not necessarily updatedat every iteration; rather, they are only updated when a betterindividual particle solution or global solution is found. This isknown as a global best topology. A formulation can be madewhere each particle is only aware of the best solution withinits group of neighbors as defined in state space, as opposedto each particle having knowledge of best solution globally.This formulation is generally regarded as allowing the PSOto be more resistant to premature convergence; however, itrequires more computation and was not deemed necessary forthis work. Further discussion is available in Reference [17].

Now that all particles have been evaluated and checked forthe individual or global best solutions, the particles mustbe moved to a new position for the next iteration. This is

4

accomplished in the following manner. First, the velocity ofeach particle is set in the following manner.

V kj = cI V

k−1

j

+ cC

(

ψk −Xk−1

j

)

+ cS

(

G−Xk−1

j

)

(22)

Here, j refers to the jth particle and k refers to the kth itera-tion. The initial particle velocity can be assigned arbitrarily.In this implementation, the velocity direction was randomlyassigned while the magnitude was set to be arbitrarily small.As can been seen, the particle’s velocity is impacted bythree main factors: the particle’s previous velocity, the vectordifference between its individual best state and its currentstate, and the vector difference between the globally bestsolution and its current state. These three quantities are scaledby inertial, cI , cognitive, cC and social, cS weights, which aredefined in the following manner.

cI =1 + r1(0, 1)

2cC = 1.49445 r2(0, 1)

cS = 1.49445 r3(0, 1) (23)

Notice each of these three constants involve an independentuniform random number, given by ri(0, 1). Each of thesestochastic scale factors were chosen from suggested valuesin the literature and may be adjusted in a number of ways.Both References [18] and [19] discuss the effects of alteringthese parameters. Once the particle velocity is calculated,the particle is propagated to its new location in the followingmanner.

Xij+1 =Xi

j + Vij (24)

This then concludes one iteration. This process is thenrepeated until an appropriate stopping condition is met.

5. CONFIDENCE REGION CALCULATION

In many applications, it is not sufficient to only estimate theposition of the radar transmitter: it is necessary to estimatethe size of the confidence region about the solution. Thereare two methods that were explored.

First, a confidence ellipse may be calculated from the lin-earized system. Reference [20] gives an excellent review ofcalculating a confidence ellipse given a covariance matrix.The covariance matrix, P , of the linearized system is relatedto the confidence ellipse through its eigenvalues, λi, and

eigenvectors, vi. As the primary concern is for the uncer-tainty in the radar transmitter’s position, consider the reducedtwo by two covariance matrix, P2x2 (which corresponds tothe position covariance), and let its eigenvalues be λ1 > λ2.The confidence ellipse can be defined by its semi-major axis,SMA, its semi-minor axis, SMI , and its orientation angle,

φ. These are defined by the following.

SMA =√

2λ1σ2F 1−αp,n−p (25)

SMI =√

2λ2σ2F 1−αp,n−p (26)

Here, σ2 represents the calculated error variance, F rep-resents the percentage point of the Fisher distribution, αrepresents the confidence level, p is the number of degreesof freedom and n is the number of data points. The Fisherdistribution is itself a ratio of Gamma function and generallyhas numerical values available in standard math tables. Theorientation of the ellipse can defined as the angle of the firsteigenvector (i.e. the direction of the semi-major axis) withrespect to the first coordinate axis. This is given by thefollowing.

φ = arctan

(

v12v11

)

(27)

Here, v1l represents the lth component of the first eigenvector.This method has the advantage of being well suited to a WLSapproach as the covariance is already calculated. However, itmakes the assumption that the confidence ellipse is centeredupon the solution position, which is not necessarily the case.Additionally, the ellipse is derived from the covariance of thelinearized system, which may not yield the correct result in anonlinear system.

Thus, a nonlinear method for developing the confidence re-gion is needed. References [21] and [22] both discuss severalmethods for developing a confidence region in the context ofPSO. A general confidence region can be expressed by theformula for the likelihood method which is as follows.

J(x)− J(xo) ≤ σ2pF 1−αp,n−p (28)

This equation, which was first formalized in Reference [23],states the confidence region is the locus of points x in state-space such that the difference between the cost function atthat point and the cost function at the optimal point x0 iswithin a constant, which itself is a function of the errorvariance, degrees of freedom and the associated percentagepoint of the Fisher distribution.

This definition of a confidence region lends itself well to thePSO process. A key feature of PSO is the evaluation ofthe cost functional at a large sampling of points. Thus, abyproduct of this is the ability to evaluate whether a pointsatisfies Equation 28, and thereby create the locus of pointsthat represents the confidence region.

This process is implemented in several steps. First, withinthe PSO algorithm a check can be added for each particleas to whether it satisfies Equation 28. If it is satisfied,the point and value of the cost function are stored for postprocessing. After the PSO has run to completion, a checkmust be run to ensure each point that has previously satisfiedEquation 28 still indeed satisfies the equation (as the valueof the optimal cost has no doubt been updated throughouteach iteration). The remaining set of points represents thenonlinear confidence region. A set of points, however, isdifficult to express by a few meaningful quantities. Thus,it was decided that the minimum volume containing ellipse(MVCE) should be fit to these points so that the size andorientation of region may be expressed in identical terms asin the linearized confidence region. In order to make thisprocess more efficient, the convex hull of this set of points

5

may be found before calculating the MVCE (since the MVCEof a set of points is identical to the MVCE of its convex hull).The convex hull is defined as the minimum subset of pointssuch that all points are contained within them.

This process has a few benefits derived from the fact that theconfidence region is explicitly linked to the evaluation of thecost functional. First, this method is able to identify whena cost minimum represented by the ghost point is within theconfidence region of the true minimum. This can be a qualitymeasure by the end user to determine whether sufficient datahas been taken to definitively locate the radar transmitter.An additional benefit of this method is that the ellipse is notassumed to be centered upon the optimal solution. This factcan aid in identifying modeling problems by more readilydistinguishing when the true solution does not lie within theconfidence region in excess of what is statistically predicted.

6. TESTING RESULTS

An exercise was conducted where a fixed receiver and amoving, ship-based, mobile receiver were collecting datawhile in range of three separate radar transmitters to testthese methods. These transmitters had varied characteristics.These receivers collected a synchronized time tag, their ownposition as well as radar data. The relative geometries ofthe radar transmitters and the receivers are shown in Figure3 below.

As can be seen, the fixed radar transmitters were installedin a coastal location. The ship-board receiver traveled alongthe coast so that the effects of relative geometries couldbe observed. Figure 3 also shows how the data have beendivided into subsets, each set ranging from 10 to 30 minutesof collection time.

Both the WLS method and the PSO algorithm were imple-mented on the dataset. The estimated radar position wascompared against the known position for each data segment.The results of this analysis are summarized in Table 1 below.As an additional point of comparison, data for a line of bear-ing (LOB) geolocation process was simulated. The techniqueuses the angle of arrival (AOA) of the signal and essentiallyuses triangulation to determine the radar transmitter’s loca-tion. References [2] and [5] contain detailed discussion ofthis method. This was done so that an appropriate comparisoncan be made, as the LOB technique also does not require pre-cision time synchronization and may be used with distributedreceivers. Both the WLS and LOB results are taken fromReference [9] to serve as a point of comparison against thePSO results.

As can be seen the SearchLight algorithm provides geolo-cation results with much higher accuracy than the line ofbearing method. The positions calculated via the SearchLightalgorithm are on average twenty times more accurate.

Furthermore, it is clear that the PSO generally provides moreaccurate results than the WLS method. The average missdistance is almost 150 meters lower for the PSO method.In addition to the more accurate results, the PSO methodrequires much less data preprocessing than the WLS method.The WLS requires not only an initial guess for the radartransmitter’s position and scan rate, but it also requires moredata checking to ensure that it will not converge to the ghostpoint. The PSO method, on the other hand, requires only a

2WLS and LOB results are from [9]

Table 1. Data Analysis Results Summary of Miss Distancesin Meters2

Dataset PSO Method WLS Method LOB Method

F1 11.0 9.3 926.0

F2 38.5 53.7 1296.4

F3 199.2 398.2 6667.2

F4 80.4 196.3 2592.8

F5 160.1 368.5 7408.0

F6 72.8 561.2 14630.8

F7 667.0 822.3 11112.0

L1 72.1 20.4 1296.4

L2 45.3 131.5 9074.8

N1 136.5 785.2 1852.0

N2 732.3 724.1 1481.6

N3 550.9 492.6 2963.2

Min 11.0 9.3 926.0

Max 732.3 822.3 14630.8

Avg 230.5 380.3 5108.4

rough search space definition and is able to uniformly processdata with little intervention. And since the WLS methodrequires additional preprocessing, the run times for bothmethods are comparable, even though a PSO is traditionallymore computationally intensive.

Next, the confidence region of these solutions can be calcu-lated from both the linearized and nonlinear methods. Theseregions can be constructed for a given confidence level (e.g.95%) and represent where the estimated solution may lie ifmore or varied data is used to compute the solution. It shouldbe noted that this does not necessarily mean that the truesolution will lie within the confidence region at that givenprobability.

Confidence ellipses were calculated at a 99.9% confidencelevel in both the linearized and nonlinear method as describedabove for each dataset. It was also noted whether the confi-dence region contained the true solution or not. These resultsare presented in Table 2 below.

It can be seen from those results that nonlinear methodprovides confidence regions that are much smaller, but onlycontain the solution for half of the cases while the linearizedsolution provides confidence regions that are much larger butcontain the true solution for all datasets. At this chosenconfidence level, it would be expected that each confidenceregion would approximately always contain the true solutionif the system is correctly modeled. This indicates that systemmodeling can be improved. Indeed, upon examination of acontour plot of the error residuals, it can be seen that thetrue solution is not located within a minimum error residuallocation.

A plot of the results for dataset F1 are shown below in Figure4. As can be seen, the estimated radar transmitter location iswithin 11 meters of the true transmitter location. The 99.9%nonlinear confidence ellipse is plotted as well for a point ofreference. Figure 5 shows a plot of the results for the F3dataset. These results show the case where the true solutionis not contained within the confidence ellipse. It should be

6

Figure 3. Plot Showing Relative Positions of Radar Transmitters Versus Receivers Over Collection Period

Table 2. Results Summary for Confidence Region Analysis

Nonlinear Method Linearized Method

Dataset SMA [m] SMI [m] Contained SMA [m] SMI [m] Contained

F1 332.10 163.34 In 288.81 139.45 In

F2 93.09 20.36 Out 274.02 57.37 In

F3 1286.39 28.64 Out 7380.09 80.43 In

F4 313.16 22.11 Out 1660.67 80.14 In

F5 570.94 28.49 Out 4470.12 99.37 In

F6 2635.31 45.32 Out 21014.90 138.75 In

F7 5259.61 126.09 In 50190.45 315.56 In

L1 1872.81 202.33 In 3584.14 367.26 In

L2 1749.42 116.66 In 7644.55 216.75 In

N1 19217.66 1988.07 In 24307.59 345.76 In

N2 349.64 155.37 Out 1200.12 508.43 In

N3 7769.74 6073.10 In 30136.18 72.13 In

noted that the confidence ellipse extends beyond the boundsof the plot and is indeed elliptical. It can be seen that the truetransmitter location does not correspond to a location withminimal error, suggesting that modeling errors are present.

7. CONCLUSIONS

A novel method for geolocation of a circularly scanning radartransmitter based on observing times between detection ofa distributed set of receivers was introduced. This methodhas the benefits of minimum complexity and minimal timingrequirements. This, in turn, enables the technology to be usedon a variety of platforms in a multitude of configurationswithout the need for precise timing calibration or high data

7

283.99 283.992 283.994 283.996Longitude

36.92

36.922

36.924

36.926

36.928

36.93

36.932La

titud

eConfidence Ellipse

True Location

Estimated Location

Figure 4. Plot Showing Nonlinear Analysis Results forDataset F1 with Error Contours

283.95 283.96 283.97Longitude

36.925

36.93

Latit

ude

Confidence Ellipse

True Location

Estimated Location

Figure 5. Plot Showing Nonlinear Analysis Results forDataset F3 with Error Contours

transfer rates. Both linear and nonlinear methods wereexplored for solving this system. Using a particle swarmoptimizer enabled precise geolocation, while being tolerantof the nonlinearities of the solution space. An analysis ofthe confidence region associated with the solution showedthat the uncertainty in the position of solution for the radartransmitter is relatively small; however, it also suggests thatmodeling improvements may be made.

With this in mind, future work includes incorporating ascan rate drift term in the estimation process. This requiresrelatively low effort for implementation in the PSO and willallow a more accurate result given a non-constant scan rate.In addition, incorporating angle of arrival (AOA) informationwithin the estimation process should increase the accuracyof the results while maintaining minimal complexity. Uncer-tainty in AOA measurements are generally orthogonal to theuncertainty in the DCTOA measurement. This should allowfor increased accuracy.

Ultimately, the technology shows enormous potential in avariety of fields. The future of some fields, such as maritime

domain awareness, are dependent upon the use of a networkof disparate sensors. This method will enable these sensorsto perform precision radar geolocation without the need forexpensive precision timing calibration and high data transferrates.

APPENDIX

The partial derivatives of the nonlinear measurement givenin Equation 1 may be calculated as follows. Consider thefollowing system.

x =

[

rω

]

(29)

y =1

ωcos−1

(

(r1 − r) · (r2 − r)

|r1 − r| |r2 − r|

)

(30)

(31)

Now assume that positions are restricted to the local tangentplane and let ρi = ri − r. So then, the following are true.

x =

[

xyω

]

(32)

y =1

ωcos−1

(

(ρ1) · (ρ2)

|ρ1| |ρ2|

)

(33)

(34)

So then the partial derivatives of h with respect to the firstposition state, x is as follows.

∂h

∂x= ∓

1

ω√

1− γ2

∂u

∂x(35)

γ =ρ1 · ρ2|ρ1| |ρ2|

(36)

∂u

∂x=

−ρ1(1)− ρ2(1)

|ρ1| |ρ2|(37)

+ ρ1(1)ρ1 · ρ2

|ρ1|3 |ρ2|

(38)

+ ρ2(1)ρ1 · ρ2

|ρ1| |ρ2|3

(39)

Here, ρ(i) represents the ith component of ρ vector. Thepartial derivative of h with respect to the second positionstate, y, is as follows

∂h

∂y= ∓

1

ω√

1− γ2

∂u

∂y(40)

∂u

∂y=

−ρ1(2)− ρ2(2)

|ρ1| |ρ2|(41)

+ ρ1(2)ρ1 · ρ2

|ρ1|3 |ρ2|

(42)

+ ρ2(2)ρ1 · ρ2

|ρ1| |ρ2|3

(43)

Finally, the partial derivative of h with respect to the scanrate, ω, is as follows.

∂h

∂ω= ∓

1

ω2arccos

(

ρ1 · ρ2|ρ1| |ρ2|

)

(44)

8

The sign ambiguity on these terms is related to the ambiguitypresented in Equation 4. The partial derivatives are negativefor the case where the radar has swept through θ or 2π + θ,while the partial derivatives are positive for the case wherethe radar has swept out 2π − θ.

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BIOGRAPHY[

John G. Warner focuses his expertiseto solve problems relating to astrody-namics and nonlinear estimation. Hejoined the US Naval Research Labora-tory last year as a Karle’s Fellow freshfrom receiving MS and BS degrees inaerospace engineering from the Univer-sity of Illinois at Urbana-Champaign.While at NRL he has lead efforts to im-prove atmospheric modeling within orbit

determination software and has applied nonlinear optimiza-tion techniques to solve a range of problems.

Jay W. Middour has more than twenty-five years experience doing space sys-tems research and development. He ispresently the head of the Space SystemsTechnology Branch at the Naval Re-search Laboratory in Washington, DC,and is the former head of the NRLAstrodynamics Office. Mr. Middourleads a broad range of investigations andprojects including Radio Frequency and

Optical Space Payloads, Satellite Laser Ranging, Signal Pro-cessing, Calibration, and Space Navigation Systems.

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