INCORPORATING PERFORMANCE PREDICTION UNCERTAINTY INTO DETECTION AND TRACKING. 21 February 2006

Transcription

1 INCORPORATING PERFORMANCE PREDICTION UNCERTAINTY INTO DETECTION AND TRACKING 21 February 2006 Lawrence D. Stone, Bryan R. Osborn Metron Inc Freedom Drive, Suite 800 Reston, VA Robert T. Miyamoto, Chris Eggen, Marc Stewart, Andrew A. Ganse Appied Physics Laboratory University of Washington 1013 NE 40 th Street Seatte, WA Brian R. La Cour Appied Research Laboratories The University of Texas at Austin P.O. Box 8029 Austin, TX Danie N. Fox Rite Soutions 88 Siva Lane, Suite 220 East Middetown, RI POC: Lawrence D. Stone, x440 (Ph) (Fax) Short Tite: Performance Prediction Uncertainty and Tracking Distribution authorized to DoD and DoD contractors ony; Critica Technoogy; June 2004; other requests for this document sha be referred to NAVSEA, IWS5A1B at Washington Navy Yard, Washington, DC. 1

2 ABSTRACT As part of the Uncertainty Directed Research Initiative of the Office of Nava Research, the authors have deveoped methods to characterize and quantify uncertainty in acoustic environmenta predictions. We have deveoped methods for refecting this uncertainty in tactica systems such as senor performance prediction, search pan recommendation, and track and detect systems that rey on environmenta inputs. Uncertainty in performance prediction can resut from mode error and uncertainty in environmenta information such as the bottom composition, sound speed profie, and background interna waves. In this paper, we characterize and quantify uncertainty in environmenta predictions for the components of the sonar equation for mutistatic active detection, and incorporate this uncertainty into a Bayesian track-before-detect system caed the Likeihood Ratio Tracker (LRT). We present an exampe that appies LRT to mutistatic active detection and tracking. For this exampe, we show that by incorporating environmenta uncertainty into the LRT state space, we can make use of performance prediction whie maintaining robustness to prediction errors. 2

3 I. INTRODUCTION For many years the U.S. Navy has supported a vigorous program of coecting environmenta data and deveoping modes of the ocean environment. A major goa of this program is to provide environmenta predictions that wi aow the Navy to estimate detection performance for active and passive sonar systems used to detect submarines. These predictions are usefu in optimizing sensor empoyment, panning searches, and performing tracking and detection functions. However, many times the uncertainties in these predictions prevent their effective use. In 2001 the Office of Nava Research inaugurated the Capturing Uncertainty Directed Research Initiative (DRI). The goa of this DRI is to characterize and quantify the uncertainty in acoustic environmenta predictions and to deveoped methods for refecting this uncertainty in tactica systems such as sensor performance prediction systems that rey on environmenta inputs. In this paper we describe the process that we have deveoped for quantifying uncertainty in environmenta predictions. We then show how this uncertainty can be incorporated into a detection and tracking system in a way that aows us to use performance predictions whie maintaining robustness to prediction errors. Importance of Probem The United States Navy is charged with the task of ensuring support for joint operations in support of United States nationa interests. A major component of supporting operations in foreign ands is to provide safe passage and areas of operation for ships at sea. Many foreign governments have turned to submarines as an effective weapon to defend their nationa interests. Highy capabe submarines are easiy obtained and can operate effectivey beneath the ocean. This new generation of submarine threat is a serious chaenge to U.S. supremacy at sea. The 3

4 quiet diese submarine, operating with modern technoogies, can defeat most current United States ASW systems. The extremey effective passive sonar systems, deveoped for the cod war, are not proving effective against these quieter, smaer diese submarines. Active acoustic systems can provide a more reiabe detection capabiity but make surface ships and submarines vunerabe to counter-detection. Air-depoyed active mutistatic sonobuoy systems, such as the Extended Echo Ranging (EER) system, provide active capabiities that can counter quiet diese submarines, but they have had serious probems being introduced into the feet. An active sonar mutistatic system, such as EER, must dea with two major issues: fase targets and ocaization. Fase targets, arising from surface ships, bottom features, wrecks, and pipeines, are ubiquitous, consuming vauabe operator time to vaidate each target-ike echo and requiring more aircraft to vaidate contacts. Poor estimation of a target s position and path resuts in a arge area of uncertainty. This consumes vauabe resources (i.e., sonobuoy and aircraft time) to ocate the suspected target and provide a definitive passive or non-acoustic cassification for aunching weapons. Therefore, faiure to reduce fase aarms and improve ocaization substantiay degrades our operationa capabiity Dependence on Environment Whie a system, such as EER, has been designed for singe ping detection and ocaization, this hasn t proven to be a reiabe capabiity. The Likeihood Ratio Tracker (LRT) described beow is a Bayesian track-before-detect system that has been appied to EER. It is capabe of integrating severa beow threshod responses over sensors and time to determine a detection and improve the detection and tracking performance of EER systems. LRT requires sensor performance predictions in order to cacuate the ikeihood functions used by the tracker. These performance predictions require good estimates of the environment. If these are correct, then they can enhance tracker performance. If the estimates are incorrect, they can degrade the 4

5 performance of the tracker. By designing LRT to account for the uncertainty in performance prediction generated by uncertainty in the environmenta predictions, we can obtain improved tracker performance whie maintaining robustness to prediction error. Motivation for Approach Whie significant gains can be reaized through the appication of LRT to EER, providing accurate sensor performance prediction estimates is difficut. Sensor performance predictions are transated to LRT by using acoustic modes to estimate signa excess (SE) on a target. Signa excess is the decibe eve of a target s echo over a detection threshod above the background acoustic interference. Whie the actua acoustic interference (i.e., reverberation or noise) coud be measured from the sensor system itsef, sonobuoys are not caibrated and used in such a fashion. Rather, a components of the SE are cacuated from acoustic modes and their inputs. However, there are imitations on the accuracy of the acoustic modeing and their inputs. It is, in fact, impossibe to reproduce nature precisey. Definition of Uncertainty We account for two types of uncertainty in this work, namey short term and mode uncertainty. The short term uncertainty is caused by tempora fuctuations in the environment that occur over a period of minutes. These may be caused by interna waves for exampe. Mode uncertainty is caused by our ack of knowedge of sowy varying or constant environmenta parameters. For, exampe, we may have uncertain knowedge of the bottom type. This wi produce uncertainty in predictions of propagations oss and signa excess. However, the bottom type wi not change during the time that a buoy fied is depoyed. This ong term uncertainty is caed mode uncertainty. It can aso refect errors in our acoustic modes. We mode both types of uncertainty through the use of probabiity distributions. We iustrate this by giving an exampe of short term and ong term uncertainty modes for signa excess prediction. 5

6 Uncertainty Modes We suppose there are a finite number of environmenta modes { E E E },,, I that represent the environmenta uncertainty in the region of interest. One of these modes is the correct mode for the area, but we are not sure which one it is. For each mode we can compute an expected signa excess 1 2 SE i (in db) for a specified source and receiver pair and target state (position and veocity). The actua signa excess (given mode SE = SE i + ξ i E i is correct) is given by where ξ is a random variabe with a specified distribution. For exampe, the distribution coud be norma with mean 0 and specified standard deviation or it coud have some other distribution such as a Rayeigh distribution. We assume that the vaue of ξ at time t is independent of the vaue at time t + δ where δ is the time between pings in a mutistatic active situation. This represents the short term uncertainty in signa excess. The ong term or mode uncertainty is represented by a discrete probabiity distribution on the finite set { E E E },,, I of environmenta modes. This distribution is defined by 1 2 i { E } p = Pr is the correct mode for i= 1,, I i and I pi = 1. i= 1 Environmenta Uncertainty Environmenta uncertainty arises from both incompete knowedge of the environment and incompete physics in the acoustic modeing. Whie modes are not perfect, in genera, most errors arise because of a ack of environmenta characterization. As an exampe, the East China Sea (ECS) is an area that has one acoustic bottom oss versus bottom grazing ange function in the standard Navy database 1 for acoustic frequencies between 1000 and 10,000 Hz. The bottom oss function is a defaut vaue for a the word s oceans ess than 200 meters and was meant to 6

7 support a deep water sonar that bounced energy off the bottom to make a target contact. Direct measurements of the surficia sediments and acoustic measurements of propagation oss indicate that the standard Navy bottom is seriousy in error. Moreover, the temperature and sainity from which sound speed is computed has substantia changes in time and space due to externa mixing (winds) and pressure (tides), thus making an accurate estimate of the sound speed profie extremey difficut. Rather, an approach must be adopted that recognizes that there are inherent uncertainties in the estimation of the predicted acoustic environment that ed to the acoustic detections that are actuay seen. Target Strength Uncertainty Predictions of signa excess rey not ony on environmenta modeing but on modeing of the target as we. This adds an additiona and important source of uncertainty for any active sonar processor which attempts to use modeed signa excess eves to compare with measured contact ampitudes. An obvious source of uncertainty is the choice of target submarine cass to be hypothesized. Given the target identity, however, uncertainties wi nevertheess arise from deficiencies in the target mode. Much as in the case of environmenta predictions, this may be due either to insufficient knowedge about the physica structure of the submarine in question or to intrinsic deficiencies in the scattering mode. Finay, variabiity in time, say, from ping to ping, may arise due to changes in the target s kinematic state. In the LRT this kinematic state is the target s position and veocity, which are part of the hypothesized state variabes of the tracker. Given this state, then, a prediction of target strength may be made based soey on the reative positions of the source, target, and receiver. The remaining variabiity may be modeed stochasticay and arises physicay from acoustic propagation effects couped with an unknown target depth. 7

8 II. CHARACTERIZING ENVIRONMENTAL UNCERTAINTIES Temperature Custering Traditiona approaches to identifying representative sound speed profies use historica (cimatoogica) data sets grouped into months or seasons as we as geographica areas and then either average by depth or fit a representative curve. The Navy s officia database is the Generaized Dynamic Environmenta Mode (GDEM) based on many years of data coections from different systems over most of the word. From that a variance about the nomina sound speed versus depth can provide an estimate of the uncertainty. However, the vertica structure of the water is not retained. To support tracking, it is important to retain the vertica structure of the water coumn that resuts in an acoustic fied that governs detections. A new approach to identifying possibe acoustic conditions is needed. Raw historica profies in an area can show considerabe variabiity, but upon coser examination it is often the case that the profies consist of a sma number of modes, each of which consists of a famiy of simiary appearing profies with a much smaer variabiity. In Fig. 1, the historica profies in the East China Sea area are shown, aong with a custering which separates them into three internay simiar subsets, each of which has significanty ess variabiity than the origina set. This aows us to represent the uncertainty in an area more accuratey. For exampe, if we know that the water in the area on a given day is in mode 2, then we know from this anaysis of historica profies what the variabiity (uncertainty) wi be. This, in turn, aows us to make a more accurate assessment of how this variabiity transates into uncertainty in the detection probabiity, for exampe. 8

9 Figure1. Mean profies for each of the four ECS sound speed custers. The red ines in the custer pots are historica measured sound speed profies from the ECS taken over severa years during the months of Juy to August. The green ine within the custer pot represents the average for each depth. The bue ine represents a measured profie that is cosest by Eucidean distance to the average profie. Finay, a three average profies are shown. It may be possibe to determine the particuar mode from remote sensing. The surface temperature and dynamic height of each profie can be reated to a custer. Sateite measurements of surface temperature and height (via atimetry) woud indicate the most ikey custer, which woud then aow us to estimate not ony the profie shape but its uncertainty. In custer anaysis, a series of 'attributes' is assigned to each datum, and the agorithm attempts to subdivide the dataset into subsets with simiar attributes. In the case of profie data, we start with measurements of temperature (and possiby sainity) at arbitrary depths. This data is preprocessed to interpoate the measurements to fixed, standard depth eves. If measured 9

10 sainities are not avaiabe, they are assigned using the database of historica T-S reationships avaiabe in the Moduar Ocean Data Assimiation System (MODAS). The temperature and sainity measurements are then converted to sound speed, and additiona attributes (such as the depth of the mixed ayer, the sonic ayer depth, and the near-surface vertica sound speed gradient) are computed. There are three basic steps to custering as appied here 2. Custer anaysis uses simiarity measures to group observations together. There are two parameters that must be defined in custer anaysis. First, a metric must be defined that quantifies profies that are simiar. This cacuation is caed the resembance coefficient. A traditiona metric is the Eucidian distance between two vectors and is used for our anaysis. Specificay, the metric is the sum of the squared sound speed differences at the depths of 0.0, 2.5, 7.5, 12.5, 17.5, 25.0, 32.5, 40.0, 50.0, 62.5, 75.0, meters. These specific depths are chosen since they are standard depths in oceanographic databases. Second, the custering method defines how the pairs of observations are grouped together based on their resembance coefficients. Each custer is defined as containing a observations whose resembance coefficients are equa to within some toerance. The custering method starts by defining a toerance such that each observation is in its own custer. By repeatedy reaxing the toerance, groups of simiar observations fa into fewer and fewer custers unti finay a the observations are in a singe custer. Ward s Minimum Variance method 3 is used to custer the sound speed data from the East China Sea. Data for the custer anaysis on Sound Veocity Profie (SVP) data are cacuated from temperature profies taken during a Navy exercise, Ship Anti-submarine warfare Readiness and Effectiveness Measuring (SHAREM) 134, in the East China Sea. For this initia evauation, data from a tweve months were processed from standard nationa ocean data archives. A the data 10

11 was from regions of reativey shaow water. Ony data with quaity controed, compete temperature profies were used. Lasty, once the custering has been performed, the custering toerance is fixed to determine the simiar observations. This choice defines the number of significant custers and the observations that make up each custer. Rather than setting the toerance a priori, the data can be viewed at different toerances and the custers reviewed for the best groupings. For the ECS the four custers were chosen to be representative. These custers are shown in Fig. 1 aong with the average profie for each custer. Whie there aren t dramatic differences in the ECS mean profies, these are the ikey states for acoustic propagation based on historica and in situ data. The impact of these profies on the acoustic fieds, incuding the impact of the bottom composition, wi be discussed further in the section on propagation of uncertainty. Bottom Characterization Simiar to the sound speed, the bottom composition needs to be characterized into possibe conditions based on sources of error. These sources are the foowing: variations in measured grain size, variations in thickness of the sediment ayer, and statistica uncertainty in the regressions reating grainsize to geoacoustic parameters such as sediment density, attenuation, scattering strength, and sound speed. The variabiity and uncertainty about means in these three sources propagate through the bottom mode to create variabiity in the resuting bottom refection oss (BL) and the bottom scattering strength (BSS) curve pairs. Reaistic test distributions for each of these sources were modeed as Gaussian distributions derived from either the ASIA Experiment (ASIAEX) 13 measured data (for sediment thickness and measured grain sizes) or from regression statistics cacuated from gobay measured data (grain-size to geoacoustic-parameter reations). 11

12 The bottom acoustic mode used in this anaysis is a 2-ayer anaytic one deveoped by Darre Jackson 11 of APL-UW. From 13 parameters of bottom properties such as density and sound speed in the two ayers, as we as frequency and sound speed at the base of the water coumn, the mode produces a curve pair of BL and BSS across the specified grazing anges. A Monte Caro simuation with 1000 runs of the bottom acoustic mode was set up to produce 1000 couped BL and BSS curve pairs as shown in Fig. 2. Each run consisted of five sub-runs at frequencies from 150Hz to 1000Hz, for which the resuts were averaged together. The resuting dataset of 1000 couped BL and BSS curves serve as a PDF of oss and scattering for an area, which can be samped for use in some broader ocean acoustic simuation. The compicated correations between both the BL and BSS curves within a pair, as we as between each curves vaues at different anges, made deriving some theoretica, continuous, joint PDF very daunting. However, by keeping a arge enough dataset of BL and BSS curves, we inherenty hande those compicated correations. (For exampe, N = 1000 gives a margin of error of 1/ N which is equa to approximatey 3% about the mean BL or BSS vaue at a given ange.) 12

13 Figure 2. Bottom figures: a possibe bottom oss and backscatter curves from 1000 reaizations of possibe geoacoustic bottoms for ECS. Top figures: coor coded histogram of the most ikey bottom oss and backscatter curves with white ine overay of the most ikey condition To simpify the cacuation for the LRT simuation, four bottom types were seected from the distribution of possibe sediment conditions. The four bottom types were seected by first averaging the bottom refection oss and the bottom scattering strength for each reaization over the range of grazing anges from 5-15 degrees which are the most important for shaow water acoustics. The median of the averaged curves was computed for the 1000 reaizations. The median was used to separate the curves into high and ow oss and scattering sets. Within the 500 ow bottom refection oss data set the median average bottom scattering strength was used as a threshod to further separate 500 reaizations into a ow bottom refection oss data set, a ow bottom scattering strength dataset, and a ow bottom refection oss with high bottom scattering strength subsets. Likewise, within the 500 high-bl data set, the median average BSS was used 13

14 as a threshod to further separate 500 reaizations into a high bottom refection oss with ow bottom scattering strength, and a (330) high bottom refection oss with a high bottom backscatter strength subsets. For each subset a most ikey bottom was found by computing a db average and standard deviation of bottom refection oss and scattering strength over a the reaizations in the subset. Note that these are averages and standard deviations over a reaization of data previousy averaged over grazing ange intervas 5-15 deg. From each subset average we computed the root mean squared difference between the ange averaged bottom refection oss and scattering strength for vectors in the subset and the means over the subset and then divide the average by the corresponding standard deviations to normaize the differences and sum the resuting normaized two curves. We ordered the difference sums to determine the highest and owest differences and seected the smoothest curves. The fina seection was made by human judgment. The four possibe bottom types are shown in Fig

15 Figure 3. Resuting 4 possibe bottom compositions that range from ow bottom refection oss with ow bottom backscattering strength (back) to high bottom refection oss with high bottom backscatter strength (green). III. PROPAGATION OF UNCERTAINTY The three possibe sound speed conditions and the four possibe bottom types are now considered the possibe acoustic environments, and the Comprehensive Acoustic Sonar Simuation (CASS) 4 suite of software is used to cacuate reverberation and transmission oss. In a of the modeing here, the propagation mode was the Gaussian Ray Bunde (GRAB) approved as a standard Navy mode. The resuting tweve different acoustic environments represent the uncertainty in the acoustic environments to be used by the tracker. In order to provide suitabe inputs for an active mutistatic simuation, the reverberation needed to be cacuated for bistatic geometries (separated source and receivers). For this simpe 15

16 case we kept a constant bottom depth of 100 meters and the cimatoogica average wind speed of 8.8 meters/second. The bottom composition was assumed to be constant over the simuation area. The impuse source eve was 245 db for a 0.1 second puse for a transmitter depoyed at 30 meters depth. The receivers were aso considered to be omni-directiona at 10 meters depth. The frequency of a acoustic cacuations was 750 Hz and was assumed to be constant over a band of 500 Hz. The transmission oss cacuation was modified from the conventiona assumption of a continuous source to recognize that a short impusive transmission woud spread the energy in time due to mutipaths. Therefore, the transmission oss cacuation is the peak incoherent addition of a of the propagation paths to a fixed point in range and depth for a target depth of 50 meters. Discussion of Acoustic Mode Outputs Since active mutistatic systems have spatiay separated sources and receivers, reverberation eves are cacuated for discrete spacings in increments of 2.5 km. The resuting set of 12 reverberation curves is shown in Fig. 4. The peak bistatic reverberation eves drop off as the receiver separation increases; however, they tend to converge at ater times. It is cear from the pots that the uncertainty in bottom refection oss dominates the reverberation for ECS because of the many bottom interactions. A sma uncertainty in bottom oss over many bounces is much more important than an uncertainty in the bottom backscatter strength or sound speed. 16

17 Figure 4. Monostatic and bistatic reverberation eves from CASS acoustic mode using the four possibe bottom compositions and the three possibe sound speed profies A set of active acoustic TL cacuations is shown in Fig. 5. These TL cacuations show that the bottom refection oss is the dominant factor in transmission oss over the range of possibe sound speed conditions for this simuation. It is interesting to note that the high refection oss bottom has a arger change in TL due to uncertainty in the sound speed. This is because there is a greater difference in bottom oss as the principa grazing anges to the bottom change due to the change in the sound speed profie. The ight bue ines bracketing the dark bue ine are one standard deviation above and beow the mean transmission oss for SSP1. These are representative of the TL uncertainty. Since the mean TL curves for SSP2 and SSP3 fa outside of the TL uncertainty, it appears that our custering provides significanty different environments. 17

18 Figure 5. Transmission oss curves for the four possibe bottom compositions and the three possibe sound speed profies as dark bue, red and green. The ight bue curves are the mean pus or minus one standard deviation of the transmission oss resuting from the uncertainty in the sound speed for SSP1 from the custering anaysis The acoustic mode cacuations provide the basis for the signa excess cacuation. A of the system and environmenta components are incuded. In order for the signa excess to be cacuated a that remains is the target strength needed to cacuate the target echo eve. IV. TARGET STRENGTH UNCERTAINTIES AND FLUCTUATIONS Target strength (TS) is the component of the sonar equation which represents the response of a scatterer (i.e., a target submarine) to ensonification by an active source. Target strength is traditionay defined as the ratio of the intensity of the equivaent scattered pane wave to that of the incident pane wave 5. Athough the energy from such an interaction is aways 18

19 dispersed and attenuated, the target strength vaue itsef is often over unity due to the fact that the incident intensity is measured at the acoustic scattering center, whereas the scattered intensity is measured with respect to a nonzero reference distance (usuay 1 m) from this center. Defined as such, target strength is an inherenty far-fied, incoherent concept. As a far-fied quantity, target strength is in genera a function of the source and receiver directions reative to the target. In addition, the target response to ensonification may be frequency dependent. Since target depth is generay not known or difficut to estimate, target strength predictions are often based on the assumption that both source and receiver are ocated in the horizonta pane of the target. At the mid-frequencies ranges typicay used for active antisubmarine warfare (ASW), scattering is argey specuar or ray-ike and frequency dependence is sight. Composite Target Modeing A standard approach to computing target strength for compex objects is to treat the submarine as a composite of severa more basic constituent objects, often referred to as primitives. Let usr (,, f ) denote the tota compex scattering function such that TS( SRf,, ) = 20og usrf (,, ) 10 is the target strength, in decibes, for a source at S ensonifying a target at frequency f and a receiver at R. Assuming inearity of the target response, the compex scattering function is a coherent sum of the individua compex scattering functions for each of the constituents. Thus, we may write j n (,, ) (,, ) ( S, R, f u S R f = g S R f e δ ), n n 19

20 where g ( S, R, f ) is the compex scattering function for the n th n constituent, situated at the origin, and δ ( SRf,, ) gives the phase deay due to the actua position of the object reative to the target origin. n To incorporate the broadband response of an active sonar transmission, one may integrate the product of the compex, frequency-dependent scattering function and transmit waveform spectrum over the frequency band to obtain an effective broadband target strength. Thus, the broadband target strength is given by TS( SR, ) = 20 og usr (, ), 10 where usr (, ) = usr (,, f) ρ( f) df, and ρ ( f ) is the spectra density of the transmitted waveform. In the high frequency regime where specuar refections dominate, R.E. Ke 6 showed in a cassic resut that a reationship exists between monostatic and bistatic radar cross-sections (RCS). This resut transates directy in terms of target strength predictions and may be described as foows. Let u ( θ, f) denote the monostatic scattering function at frequency f M and ange θ, say, counter-cockwise from the target bow in horizonta pane. Let θ S and θ R denote the anges of the source and receiver. The bistatic theorem states that the bistatic scattering function, u ( θ, θ, f) may be written in terms of the monostatic scattering function as foows 7 B S R { } u ( θ, θ, f) = u ( θ + θ ) / 2, f sec[( θ θ ) / 2]. B S R M S R S R This reationship is vaid when the source and receiver directions are cose to one another; i.e., when the bistatic ange, θs θr, is sma. Discrepancies between the true bistatic 20

21 target strength and that predicted by the bistatic theorem may be attributed to wave effects, such as diffraction or the excitation of eastic waves, which become important at ower frequencies. Target Strength Uncertainty As discussed earier, target strength is not a competey deterministic quantity, even when one is given the in-pane source and receiver directions reative to the target. A common approach to incorporating this additiona eement of uncertainty is to add to the scattering function a zero-mean random variabe representing the short term, tempora variabiity of the measured target strength. We sha denote this random variabe by Z( θ, θ, f ), recognizing that it genera it may depend upon both the source/receiver directions as we as the frequency of ensonification. S R A standard assumption is that Z( θ, θ, f ) is a compex Gaussian random variabe with a S mean of zero and a standard deviation σθ (, θ S R, f ). The resuting magnitude of the tota R scattering function, X = u( θ, θ, f) + Z( θ, θ, f), now has a Rician distribution. Let f ( x ) S R S R be the probabiity density function (PDF) for X. Then X x x + u 2 x u fx ( x) = exp I for x 0 2 σ σ σ, where I () 0 is the zero-order modified Besse function of the first kind, σ= σ ( θ, θ, ), and u= u( θ, θ, f). S R S R f Written in terms of decibes, the target strength PDF is og10 y/20 y/20 fy( y) = 10 fx(10 ), 20 where Y = 20og 10( X). Sampe pots of this distribution are shown in Fig. 6. It may be shown that the expectation of Y is given by 21

22 10 EY = u + Γ u σ og ( ) 20og 10( ) (0, / ), where Γ ( az, ) is the incompete gamma function. For arge SNR vaues, this expression is 2 2 approximatey equa to 10og ( u + σ ), the expectation of 10 2 the standard deviation are on the order of 3 db (i.e., σ = 2 ). 2 X in decibes. Typica vaues for Figure 6: Pot of the target strength probabiity density function on a decibe scae. The three curves correspond to three different vaues of 20og 10 u with σ = 1. Note that for sma signa-to-noise ratio (SNR), the distribution is highy skewed due to the noninear decibe scae. At arger SNR vaues the distribution becomes more Gaussian in shape. Generic Diese Eectric Submarine The target mode used for the resuts discussed in this paper is the Bistatic Acoustic Simpe Integrated Structure (BASIS) Target Strength Mode deveoped by D. Drumheer at NRL-DC 8. BASIS modes targets as composites of simpe geometric primitives whose individua responses are added coherenty. The target strength predictions are fundamentay monostatic in that the bistatic theorem is used to determine bistatic target strength. Forward scattering is computed using Babinet s principe and each primitive s aspect-dependent cross section. For the resuts described in this paper, a mode of a generic diese eectric submarine, 56 22

23 meters in ength, provided in BASIS was used. The detais of this mode are provided in the above reference. Based on comparisons with finite eement cacuations, this mode is expected to be vaid for frequencies between 250 Hz and 2000 Hz. For the present appication, we considered the broadband target strength for a fatspectrum waveform with a bandwidth of 500 Hz centered at 750 Hz. As was noted in the previous section, this frequency band is appropriate for an impusive EER source. The broadband target strength was computed by coherenty averaging the narrowband scattering function over a fat spectrum with frequency band Hz. In Fig 7, the resuts of the monostatic target strength cacuations are shown and compared with predictions for 100 Hz and 0 Hz (narrowband) bandwidths. The vaues potted are the expectation vaues for a constant σ of 3 db. It may be noted that the variabiity of the target strength as a function of aspect ange tends to diminish as the bandwidth increases. This may be understood as a consequence of the coherent frequency averaging, whereby the detaied target responses at each frequency tends to be suppressed as they interfere with one another. The broad secondary peak appearing just above 90 o is due to the fat pane of the wedge used to mode the sai. A more reaistic, curved sai woud suppress this peak. The uncertainty in the target strength predictions is iustrated in Fig. 8. The figure shows the vaue of the target strength PDF for each aspect. For regions far off beam aspect (i.e., 90 o ), where the target strength is ow, the PDF is broad is heaviy skewed toward ower vaues. By contrast, near the peaks in the monostatic target strength, the PDF is sharpy peaked and neary symmetric. Finay, the broadband bistatic target strength, as computed from the bistatic theorem, is shown in Fig. 9. The monostatic prediction can be seen aong the horizonta ine at zero bistatic ange. The two ridges at bistatic aspect anges 90 o and 270 o correspond to specuar refections, where the ange of incidence from the source equas the ange of refection to the receiver. The 23

24 broad ridge at bistatic ange 180 o corresponds to forward scattering, where the source, target, and receiver are coinear. Figure 7. Pot of monostatic target strength for a generic diese eectric submarine modeed in BASIS. The vaues are the expected target strength for a constant standard deviation of 3 db. The three overays show the broadband target strength predictions for a puse centered at 750 Hz with bandwidths of 0 (bue), 100 (red), and 500 (back) Hz. Due to the port-starboard symmetry of the target, ony the target strength from bow to stern aspect is shown. 24

25 Figure 8. Pot of the broadband monostatic target strength PDF as a function of aspect, with each aspect corresponding to a separate PDF. The coor indicates the vaue of the PDF for a given target strength and aspect, with bue indicating a ow vaue and red indicating a high vaue. Figure 9. Pot of broadband bistatic target strength for generic diese eectric submarine mode in BASIS. The coor scae indicates the target strength in decibes. The horizonta axis is the bisector ange between the source and receiver directions. The vertica axis is the anguar separation between these two directions. 25

26 For purposes of mutistatic detection, it is important to note that most detections wi tend to occur when the target is oriented with respect to the source and receiver such that a near specuar refection is provided. It is cear from the figure that most combinations of source and receiver directions wi not provide a specuar return, which creates a probem for bistatic detection. The forward scattering case, whie providing a significant target strength vaue, may be difficut to utiize, as it may be difficut to distinguish from the direct bast. In the next section we wi discuss a Bayesian approach to mutistatic tracking which takes into account these subteties by incorporating target strength predictions directy into the observation ikeihood function. V. LIKELIHOOD RATIO TRACKER This section gives a brief description of the Likeihood Ratio Tracker (LRT) and discusses how we incorporated environmenta uncertainty into LRT. Likeihood ratio (detection and) tracking is based on an extension of the singe target tracking methodoogy to the case where there is either one or no target present. It is a Bayesian form of track-before-detect. LRT is designed for ow Signa-to-Noise-Ratio (SNR) situations where severa sensor responses must be integrated over time whie the target is moving in order to ca a detection. It is aso appropriate for high fase aarm situations such as those that arise in mutistatic active systems that are operated with ow contact threshods. LRT s abiity to integrate beow-normadetection-threshod data or even unthreshoded data over space and time to detect weak targets aows it to produce improved detection performance without excessive fase aarm rates. Description of LRT For LRT we identify a region R of interest. We et S be the target state space for a target in the region R. Typicay, S wi the position and veocity of the target. If no target exists in R, we sha formay refer to this state as the nu state and designate it by the symbo φ. We augment the target state space S with this nu state to make S + = S φ. The augmented 26

27 state space S + incudes not ony a of the states within R but the discrete nu state φ as we. We sha assume there is a prior probabiity (density) function p 0 defined on S +. Since we are assuming that there is at most one target in the region R, we may write p0( φ) + p0( s) ds= 1. s S Let X ( t ) be the (unknown) target state at time t. We start the probem at time 0 and are interested in estimating X ( t ) for t 0. The prior information about the target is represented by a Markov process { X( t); t 0}. This process is specified by a prior probabiity distribution on the state of the target at time 0 and a Markov motion mode that describes the target s motion through the state space S. There are one or more sensors that produce a set of K observations or measurements Y () t = ( Y1,, Y K ) obtained in the time interva [0, t]. The observations are received at the discrete (possiby random) times ( t1,, t K ) where t0 = 0 t1... tk t. From this information we can cacuate the posterior state probabiity (density) in the standard Bayesian fashion 8. (, ()) = { () = ()} p ts Y t Pr Xt s Y t for s S. This is the posterior distribution on target state that is the standard output of a Bayesian tracker. We can aso cacuate the probabiity of receiving the sensor responses Y () t given no target is present. This is (, φ ( )) = { ( ) = φ ( )} p t Y t Pr X t Y t. We define the ratio of the posterior state probabiity (density) pts (, ( t) ) state probabiity pt (, φ Y ( t) ) as the target ikeihood ratio (density) ( ts, ( t) ) (, Y( t) ) (, φ Y( t) ) Y to the posterior nu Λ Y ; that is, Λ pts ( ts, Y( t) ) = for s S pt. (1) 27

28 At each time t, ( t, () t ) Λ Y is a ikeihood ratio surface defined over the state space S. Peaks in this surface indicate ikey target ocations. When the peak crosses a specified threshod, a detection is decared and the region near the peak is converted into a probabiity distribution for target state. This can be used to initiate a track on the target. Target Motion Target motion is modeed by a Markov process with transition function { } q ( s s ) = Pr X( t ) = s X( t ) = s for k 1. k k k 1 k k k 1 k 1 Between the observation times, target motion is accounted for by appying this transition function to the target state distribution at time tk 1 to produce the projected target state distribution at time t k as foows: (, Y( )) = ( φ) (, φ Y( )) p t s t q s p t t k k 1 k k 1 k 1 ( Y ) + q ( s s ) p t, s ( t ) ds for s S. k k 1 k 1 k 1 k 1 k 1 (2) In the case of LRT, the state space incudes φ, the target not present state, as we as the norma kinematic states. Thus we must specify the transition probabiities for eaving the region R as we as entering R from the state φ. Thus we may compute (, φ Y( )) = ( φ φ) (, φ Y( )) p tk tk 1 qk p tk 1 tk 1. (3) q ( φ s) p t, s ( t ) ds. + S ( Y ) k k 1 k 1 From (2) and (3) we can compute the motion updated ikeihood ratio surface ( k, Y( k 1 )) (, φ Y( )) p t s t Λ ( tk, s Y( tk 1) ) = for s S. (4) p t t k k 1 28

29 The ikeihood ratio density has the same dimensions as the state probabiity density. Furthermore, from the ikeihood ratio density one may easiy recover the state probabiity density as we as the probabiity of the nu state. Likeihood Functions The sensor response or measurement Yk ikeihood functions. Specificay, = y is incorporated into LRT through k { } L ( y s) = Pr Y = y X( t ) = s for s S +. k k k k k Note that the data or observation Yk = yk is known but the target state s is not. Thus the ikeihood function L ( y ) is function of target state. Likeihood functions aow us to convert k k sensor responses or measurements of amost any kind into a common currency of information on the state space. In LRT, we use the measurement ikeihood ratio. It is the ratio of the ikeihood of receiving the measurement given a target is present to the ikeihood of receiving the measurement given no target is present. Specificay, we define the measurement ikeihood ratio for Yk = y to be k Lk( yk s) L k( yk s) = for s S. L ( y φ) k k Likeihood ratio detection and tracking recursion assume that We can write a simpified recursion for ikeihood ratio detection and tracking if we (, φ Y( )) = ( φ φ) (, φ Y( )) + ( φ ) (, Y( )) p t t q p t t q s p t s t ds k k 1 k k 1 k 1 k k 1 k 1 = pt ( φ Y t ), ( ). k 1 k 1 S (5) 29

30 This assumption says that under the motion mode, the amount of probabiity mass that moves out of the region R is equa to the amount that moves in for a given time period. Under this assumption we obtain the foowing ikeihood ratio detection and tracking recursion. Simpified Likeihood Ratio Detection and Tracking Recursion Initiaize Likeihood Ratio p0() s Λ ( 0, s Y (0)) = for s S (6) p ( φ) 0 For k 1 and s S Motion Update (, ( )) ( φ) ( ) (, ( )) Λ t s Y t = q s + q s s Λ t s Y t ds (7) k k 1 k k k 1 k 1 k 1 k 1 k 1 Measurement Likeihood Ratio S Lk( yk s) L k( yk s) = for s S (8) L ( y φ) Information Update k k ( Y ) ( ) ( Y ) Λ t, s ( t ) = L y s Λ t, s ( t ) for s S. (9) k k k k k k 1 Extension of LRT state space Ordinariy LRT uses the target s kinematic variabes for the tracker state space. For exampe, the state is often taken to be the target s 2 (or 3) dimensiona position and veocity. For the anaysis in this paper, we assume there are a finite number of possibe environments with a prior distribution on which environment is the correct one. We extend the kinematic state space (2 dimensiona position and veocity) to incude environment. As sensor responses are obtained, LRT produces a joint estimate of target kinematic state and environment. 30

31 Numerica impementation of LRT The impementation of LRT used to produce the resuts in the exampes beow empoys a discrete grid in position, veocity, and environment. The methodoogy is simiar to the gridded version of Nodestar described in Chapter 3 of Stone, Barow, and Corwin 9. Measurement Likeihood Ratio Function The exampes presented in this paper invove muti-static active (MSA) sonar. There are a set of 10 to 30 buoy pairs. Each pair consists of a receiver buoy and a source buoy with two exposive charges. These buoy pairs are distributed over an area where a target submarine may be present. The charges on the source buoys are detonated sequentiay, one every few minutes. Between detonations, the hydrophones in the receiver buoys isten for echoes of the shockwave as it scatters off objects, potentiay incuding a moving target submarine. The time between the reception of the direct bast and an echo produces an eipse of possibe ocations for the echo producer. By accumuating a number of these echoes from the target, it is possibe to ca a detection and deveop a track on the target, i.e., a distribution on the target s position and veocity. The signa processing agorithms associated with these buoys are presumed to process the acoustic time-series at the receiver hydrophones and ca detections. These agorithms produce a set of time vaues for each hydrophone where the signa or matched-fiter output exceeds some threshod. Each of these detections comes from one of three things: random stochastic fuctuations in the noise signa (fase aarms), cutter echo, or a target echo. These are the measurements used by LRT. In some situations the Signa-to-Noise Ratio (SNR) at the receiving buoy is high enough that a bearing is aso obtained. However, for the exampes presented beow, we assume that no bearing information is avaiabe. 31

32 Known Probabiity of Detection To simpify our presentation, we first introduce the measurement ikeihood ratio function for MSA sonar in the case where the detection probabiity is known. For a singe ping, et τ = { τ,, τ } be the set of echo times detected at buoy. The measurement ikeihood ratio for 1 τ is n L ( τ s) Pr Pr { Obtaining τ at buoy target in state s} { Obtaining τ at buoy no target present} n 1 f( τi s) = Pd () s P s i= 1 λ + w( τ ) i ( 1 d ()) (10) where s= target state ( x, v), position and veocity f ( τ s) = Pr{ receiving an echo at buoy at time τ target detected in state s} w( τ) = Pr{ receiving an echo at time τ echo generated by fase aarm} The number of fase aarms per ping per buoy is Poisson distributed with mean λ P s = Pr buoy detects a target in state s on a singe ping. d () { } The derivation of (10) is given in the appendix of Stone and Osborn 10. To form the composite measurement ikeihood ratio function for a singe ping, we mutipy the measurement ikeihood ratio function for each of the buoys as foows. Let τ = ( τ1,, τ ). Then LC N r ( s) L( s) = = 1 N r τ τ (11) where N r is the number of receiver buoys. This measurement ikeihood ratio is mutipied into the motion updated cumuative ikeihood ratio to obtain the posterior cumuative ikeihood ratio in the information update step in (9). This process is repeated for each ping. 32

33 The measurement density function f ( τ s) represents the eipse information from the time of detection. Let m x s be the position of source m and x r be the position of the receiver buoy. Then the measurement error density function for the arriva time τ from source m is m 1 ( c (, ) (, )) 2 sτ d xs x d x x r f ( τ s= ( x, v) ) = exp 2 2 2πσ cs2σ t t (12) where d( x1, x 2) is the distance between positions x 1 and x 2, and c s is the sound speed. We use σ t = 1 second. This form of the measurement error density function can aso be used to account for uncertainty in buoy position to first order, as we have shown esewhere. The fase aarm density function w is uniform over a 90 second interva after the arriva of the direct bast. The P ( ) d s factor is where we require a performance prediction mode for the sensors, and this is where environmenta information is incorporated into the LRT agorithm. To compute P ( ) d s for a given source m, we begin by computing the mean signa excess SE in db at the receiver buoy as foows. SE () s = SL TL () s + TS () s TL () s ( RL () s + AN ) DT (13) m m m m m where SLm = source eve at source m TLm ( s) = transmission oss from source m to state s TSm( s) = target strength for a signa from source m refected from a target in state s to sensor TL ( s) = transmission oss from state s to sensor RLm( s) = reverberation at sensor from source m at the time an echo arrives from state s AN = ambient noise eve DT = detection threshod. The vaue of SE m given by equation (13) represents the mean signa excess eve. The actua vaue of signa excess SE is a random variabe with fuctuations about this mean that are normay distributed with mean 0 and standard deviation σ SE generay taken to be between 5 to 33

34 10 db. Detection occurs when SE > 0. Having cacuated SE () s from (13), we can compute P ( ) d s as foows 2 Pd () s = η( x, SE( s), σse) dx 0 m where 2 η(, μ, σ ) is the density function for a norma distribution with mean μ and variance 2 σ. Incorporating Prediction Uncertainty The procedure for computing the ikeihood function described in the previous section assumes we know the terms in equation (13) with certainty. When there is uncertainty in these terms, we must modify our procedure. We account for this uncertainty by extending the LRT state space to incude an environmenta-uncertainty dimension where each vaue in this dimension represents one possibe environment, E. The ikeihood function then becomes n 1 f( τi s) L( τ se, ) = Pd ( se, ) ( 1 Pd ( se, )), i= 1 λ + (14) w( τi) where we now have a ikeihood function that is defined not ony over the target s kinematic state space, but aso over the extra environmenta dimension. The vaues of E { E1, E2, } represent a possibe environments. For the exampes that we consider beow, each environment consists essentiay of a reverberation and transmission oss characterization. Furthermore, as we are using range independent environments for this work, we have RL RL( d, t) and TL TL( d ): reverberation is a function of the distance between source and receiver and time i i since bast whie transmission oss is a function of distance ony. Thus, E { RL, TL}. The Appied Research Laboratory, University of Texas, and the Appied Physics Laboratory at University of Washington have computed these functions for us from more basic environmenta parameters such as bottom type and sound speed profies appropriate for a specific ocean area. i 34

35 VI. EXAMPLE This section provides an exampe of using LRT with environmenta prediction uncertainty incorporated by the method described above. For the exampe, we consider the East China Sea region described above and use the tweve environmenta modes seected to span the uncertainty in the East China Sea s bottom type and sound speed profie (SSP). Four representative bottom types are used with three representative SSPs to arrive at tweve distinct environments. Tabe 1 ists these environments aong with their probabiities of representing the correct environment. Each of these tweve environments gives rise to its own mean reverberation and transmission oss estimates. The three SSPs are abeed 1, 2, 3. The bottom refection oss (BL) and the bottom scattering strength (BSS) are abeed as H or L for high or ow. Thus the environment designated 1/H:H has SSP 1, high BL, and high BSS. Env # Designation SSP BL BSS Probabiity 1 1 / H:H 1 H H / / H:L 1 H L / / L:H 1 L H / / L:L 1 L L / / H:H 2 H H / / H:L 2 H L / / L:H 2 L H / / L:L 2 L L / / H:H 3 H H / / H:L 3 H L / / L:H 3 L H / / L:L 3 L L / 3.0 Tabe 1. Summary of the tweve environmenta modes. Each of the tweve environments is a combination of an SSP, BL, and BSS. Each SSP has an equa probabiity of occurrence. The BL/BSS combinations of H/H and L/L have a probabiity of whie the H/L and L/H combinations have a probabiity of.191. In what foows, we appy the ikeihood ratio tracker described in the previous sections to a set of simuated EER detection data. We show that the tracker successfuy detects the target 35