Lincoln Laboratory ASAP-2003 Worshop Array Shape Tracing Using Active Sonar Reverberation Vijay Varadarajan and Jeffrey Kroli Due University Department of Electrical and Computer Engineering Durham, NC 27708 Supported by the ONR 321US 6.1 Program 1
Array Shape Tracing from Active Sonar Clutter OBJECTIVE: To estimate and trac the shape of a towed distorted-linear array using reverberation from active sonar pings as a distributed source of opportunity. BACKGROUND: Array shape estimation using heading and depth sensors has been previously developed, e.g. Gray et al.(1993), by applying Kalman filtering with a state equation derived from spatio-temporal discretization of the simplified Paidoussis equation (water-pulley model) for array motion. Acoustic sources of opportunity have been employed for array shape estimation by fitting transverse element displacements to measuring inter-element phase differences e.g. Owsley (1980). For mid-frequency active sonar arrays the above methods may be precluded since it may be infeasible to instrument the array with a sufficient number of heading sensors and strong point sources of opportunity may not always be available in the presence of strong reverberation. Clutter has previously been used for element gain and phase calibration of a uniform linear airborne radar array (Robey, Fuhrmann, and Krisch, 1994) but not for array shape estimation. We propose constrained maximum lielihood array shape estimation from clutter (ASEC) which uses an array shape-dependent model of spatially-distributed, Doppler-spread reverberation. Array shape parameters are traced within and across pings by using ML ASEC heading estimates as input to a Kalman filter whose state equation incorporates a dynamical model for array motion. 2
Motivation for Array Shape Estimation Detection Detection of of slowly-moving slowly-moving targets targets with with Doppler-sensitive Doppler-sensitive waveforms waveforms is is limited limited by by beamformer beamformer sidelobes. sidelobes. Uncompensated Uncompensated array array distortion distortion causes causes increased increased clutter clutter ran ran and andtarget masing. masing. Simulation Simulation example example angle-doppler angle-doppler spectrum spectrum from from perfectly perfectly compensated compensated array array (left) (left) vs. vs. uncompensated uncompensated distorted distortedarray (right). (right). Note Note target target mased mased by by reverberation reverberation due due to to beamformer beamformer sidelobes sidelobes of of uncompensated uncompensated array array Target 3
Array Shape Estimation from Clutter Concept Constrained Constrained ML ML ASEC ASEC assimilates assimilates clutter clutter data data with with a a few few heading heading sensor sensor outputs outputs to to obtain obtain array array shape shape parameters parameters from from each each sub-pulse sub-pulse length length space-time space-time snapshot. snapshot. Kalman Kalman tracer tracer filters filters sub-cpi sub-cpi shape shape coefficients coefficients using using a a model model for for array array dynamics. dynamics. Example SWAC-3 Reverberation Return Sub-CPI space-time clutter snapshots r Heading sensors g r r 1 Constrained ML ASEC Tow-cable Excitation u µ Kalman Tracer with Array Dynamics estimated heading vector Filtered array shape 4
Reverberation Received at a Distorted Array th The clutter data from the n sensor located at ( x, y ) at time t mt for a distorted array moving with velocity v rnm, l l (, ) l e a n n m r along the x direction is modeled as: 2 2va j (sin cos x cos cos y ) j2 sin cos mt l n l n l r where, is the complex Gaussian scatter amplitude from clutter at azimuth,, and multipath elevation,. l Sensor coordinates can be expressed in terms of heading and inter-element spacing n as x d cos, y d sin, 0 n N 1, x,y 0,0. n i n i i 1 i 1 n 0 0 d 5
Space-Time Reverberation Model The space-time data snapshot at time t j max consisting of clutter from all azimuths,, 1 i N and elevation angles,1 j N, can be written as i 1 N 1 T where µ, Vµ v ( 1, 1, µ ) v ( N, N, µ ) is the clutter MN 1 2 steering matrix, represents unnown scattering, and noise e has covariance I. r V µ e th i N j V µ v i j µ b ij a i j µ i j The ( 1) column of is (,, ),, which represents the return from a single clutter patch at location,. e MN Array shape parameters µ are assumed constant over the sub-cpi. Array shape parameters µ estimated by fitting the low ran ( MN) clutter subspace of V µ to the observed received space-time snapshot, r. 6
Clutter Eigenvalues for a Distorted Linear Array Eigenvalues Eigenvaluesof of the the asymptotic asymptotic space-time space-time covariance covariance matrix matrix for for a a modestly modestly distorted distorted array array versus versus ideal ideal uniform uniform linear linear array array for for N=32 N=32 and and M=5. M=5. Observe Observe that that ran ran inflation inflation with with 00 db dbbaclobes baclobesagrees agrees with with 2-D 2-D Brennan s Brennan s rule rule [Varadarajan [Varadarajanand andkroli, 2002]. 2002]. 0-50 Eigen Values (db) -100-150 -200-250 1-D Brennan's rule 2-D Brennan's rule -300 Linear Array Distorted Array (No-Multipath) Distorted Array (Multipath) -350 0 20 40 60 80 100 120 140 160 Eigen Value Index 7
Maximum Lielihood Array Shape Estimation The N -1 array shape headings can be parameterized by a low dimensional subspace using N 1 ( L N ) arbitrary but nown shape basis as µ =. The lielihood function for for the model reduces to minimizing the projection: ˆ argmin f ( ) P ( ) r 2 where P ( ) I V ( ) V ( ) H 1 orthogonal complement of the clutter subspace. H V ( ) V ( ) is the projection matrix onto the 1 0.8 0.6 75 70 65 Evaluation of log( f( )) (right) for L 2 demonstrates left-right ambiguity in shape estimate corresponding to mirrored solutions about the array axis. 2 0.4 0.2 0-0.2-0.4-0.6-0.8-1 -1-0.5 0 0.5 1 1 60 55 50 45 40 35 30 8
Constrained ML ASEC Algorithm The left-right shape ambiguity can be resolved with nowledge of the position or heading of a single off-axis sensor which can be expressed as a linear constraint on the shape basis coefficients as T c g. The ML ASEC estimate can be obtained iteratively using a gradient projection approach (e.g. Frost (1972)): j 1 j j ˆ ˆ c P c f' ˆ j T 1 th where c P c ccc ( ) g is the projection of the current solution (j iteration) onto the constraint j subspace, Pc I-Pc, f( ) -1 f' H H H = =-2 r P ( ) V i( ) V ( ) V ( ) V ( ) r, 0< 1. i i j ˆ V ( ) The matrix V ( ) can be computed analytically by assuming the temporal component of the i i steering matrix is independent of array shape distortion over a sub-cpi, and an analytic form for V ( ) obtained by judicious sampling of the clutter wavenumber spectrum. The ran of V ( ) is assumed constant and can be chosen so that it does not change over the set of possible array distortions. 9
Incorporating Array Dynamics into ASEC A Kalman filter can be used to trac the array shape coefficients across multiple sub-cpi intervals during a ping. The state vector µ can be related to µ using the water-pulley model: 1 µ Fµ u v 1 1 H 1 N 2 2 v1 and represents white state noise with v I where F (1 ) I L is the state transition matrix, u 0 is the tow-cable driving term, 1. It can be shown that the displacement velocity along the array is determined by 2. An observation equation can be defined using the MLE z µ ˆ from each sub-cpi: z µ v 2 2 where v2 is the measurement noise with covariance v I. 2 The predicted MMSE array shape is used to initialize the next ASEC MLE search. 10
Array Dynamical Model Validation Using TB-29 Data Real Real heading heading sensor sensor data data from from a a TB-29 TB-29 array array (left) (left) vs. vs. time time (sec), (sec), courtesy courtesy of of Bruce Bruce Newhall Newhall at at APL/JHU, APL/JHU, indicates indicates water water pulley pulley model model valid valid for for mild mild maneuvers. maneuvers. Transverse Transverse distortion distortion over over CPI CPI (right) (right) validates validates assumption assumption of of rigid rigid short-time short-time motion. motion. 11
TB-29 Heading Sensor Eigenbasis Empirical Empirical orthogonal orthogonal basis basis vectors vectors (left) (left) for for headings headings derived derived from from 66 heading heading sensor sensor TB-29 TB-29 data data (interpolated (interpolated to to 11 11 sensors) sensors) demonstrate demonstrate characteristic characteristic behavior. behavior. TB-29 TB-29 heading heading sensor sensor eigen-spectrum eigen-spectrum (right) (right) indicates indicates most most of of the the variation variation captured captured by by less less than than 44 modes. modes. 12
ASEC versus Heading Only Shape Estimation Illustrative Illustrative simulation simulation comparison comparison of of ASEC ASEC versus versus headings headings only only tracing tracing of of a a mild mild maneuver maneuver as as seen seen in in the the middle middle of of the the array array based based on on TB-29 TB-29 heading heading data data (left). (left). Simulation Simulation assumes assumes ASEC ASEC with with 1.2 1.2 s. s. moving moving window window sub-cpi, sub-cpi, 44 EOF EOF basis, basis, spacetimtime snapshot snapshot dimension dimension 144 144 and and clutter clutter ran ran of of 55. 55. space- Heading Heading (left) (left) and and error error (right) (right) for for ASEC ASEC and and 22 heading-sensor heading-sensor tracing tracing as as in in MFTA. MFTA. 13
RMS Heading Error along the Array verus Time Simulated Simulated RMSE RMSE for for 2-sensor 2-sensor headings-only headings-only tracing tracing error error (left) (left) and and ASEC ASEC (right) (right) over over 40 40 second second ping ping along along the the array, array, ensemble ensemble averaged averaged over over initial initial condition condition perturbations perturbations and and clutter clutter realizations. realizations. Observe Observe propagation propagation of of error error down down the the array array with with time time using using water-pulley water-pulley model model (left). (left). Error Error largest largest in in the the middle middle of of the the array array since since headings headingsnown at at the the end end and and at at tow-point tow-point offset offset from from first first sensor. sensor. 14
Angle-Doppler Spectrum for Distorted Array Impact Impact of of ASEC ASEC tracing tracing on on sonar sonar performance performance illustrated illustrated by by conventional conventional beamforming-doppler beamforming-doppler spectra spectra outputs outputs simulated simulated for for continuous continuous array array motion. motion. Time-evolving Time-evolving simulated simulated array array shapes shapes based based on on scaled scaled TB-29 TB-29 motion motion model model (left). (left). Angle-Doppler Angle-Doppler spectrum spectrum with with 10 10 db db target target for for perfect perfect array array compensation compensation (right). (right). Target 15
ASEC vs. Headings-only Tracing Angle-Doppler Spectra Spectrum Spectrum for for conventional conventional array array tracing tracing for for WP WP model model with with 22 heading heading sensors sensors (left). (left). Spectrum Spectrum for for ASEC ASEC tracing tracing with with 44 basis basis functions, functions, 1.2 1.2 s. s. sub-cpi sub-cpi sliding sliding window, window, and and CNR CNR = 30 30 db db (right). (right). Target Target visible visible at at zero-doppler zero-doppler and and 45 45 degree degree bearing. bearing. Target 16
ASEC Summary and Future Wor Distortion of nominally linear arrays will often result in a substantial increase in spatial sidelobe levels which can mas slowly-moving targets. ML estimation of array shape is facilitated by fitting a reduced-ran reverberation model to a sliding window of sub-cpi space-time snapshots over the extent of each sonar return. Shape ambiguities can be resolved by efficiently maximizing lielihood subject to a constraint which incorporates measurements from at least a single heading sensor. Constrained ML ASEC heading estimates are used as inputs to a Kalman tracer incorporating an array dynamical model and driven by a tow-cable heading sensor output. Simulations using array motion scaled from real TB-29 heading data suggests that ASEC tracing can facilitate improved array compensation compared to headings-only tracing. Future wor will include ASEC performance evaluation with real 53C/MFTA data. 17