This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.IEEE JOURNAL OF OCEANIC ENGINEERING 1Peer-Reviewed Technical CommunicationAn Overview of Sequential Bayesian Filtering in Ocean AcousticsCaglar Yardim, Member, IEEE, Zoi-Heleni Michalopoulou, Senior Member, IEEE, and Peter GerstoftAbstract—Sequential filtering provides a suitable frameworkfor estimating and updating the unknown parameters of a systemas data become available. The foundations of sequential Bayesianfiltering with emphasis on practical issues are first reviewedcovering both Kalman and particle filter approaches. Filteringis demonstrated to be a powerful estimation tool, employingprediction from previous estimates and updates stemming fromphysical and statistical models that relate acoustic measurementsto the unknown parameters. Ocean acoustic applications are thenreviewed focusing on source tracking, estimation of environmentalparameters evolving in time or space, and frequency tracking.Spatial arrival time tracking is illustrated with 2006 ShallowWater Experiment data.Index Terms—Acoustic signal processing, acoustic tracking, en-semble Kalman filter, extended Kalman filter (EKF), ocean acous-tics, particle filter (PF), sequential importance resampling (SIR),sequential Monte Carlo methods, unscented Kalman filter (UKF).I. INTRODUCTIONAcommon feature of inverse problems in ocean acousticsis that underlying physical parameters are estimated frommeasured acoustic data. Examples include source localization[1]–[4], geoacoustic inversion [5]–[9], and marine mammalsignal processing [10]. In a Bayesian framework, prior knowl-edge and acoustic models are combined with a likelihoodfunction to provide posterior probability density functions(pdfs) of parameters of interest. This formulation was first pro-posed in source localization [11], [12]. Geoacoustic inversionwas subsequently approached in a similar fashion, estimating,in addition to source location, ocean environment parame-ters and their uncertainties [13]–[15]. Often, such parametersevolve in time or space, with acoustic data arriving online atconsecutive steps. Information on parameter value evolutionManuscript received September 02, 2010; revised November 21, 2010; ac-cepted December 03, 2010. This work was supported by the U.S. Office ofNaval Research Code 32, under Grants N00014-09-1-0313, N00015-05-1-0264,N00014-05-1-0262, and N00014-10-1-0073.Associate Editor: R. Chapman.C. Yardim and P. Gerstoft are with the Marine Physical Laboratory, ScrippsInstitution of Oceanography, University of California San Diego, La Jolla, CA92093-0238 USA (e-mail: [email protected]; [email protected]).Z.-H. Michalopoulou is with the Department of Mathematical Sciences,New Jersey Institute of Technology, Newark, NJ 071028 USA (e-mail:[email protected]).Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JOE.2010.2098810and uncertainty at preceding steps can be invaluable for thedetermination of future estimates but is frequently ignored.Depending on the data and the problem, Bayesian approachescan be used to address inversion or tracking problems (a briefcomparison is given in Appendix I). Sequential Bayesianfiltering, tying together information on parameter evolution, afunction relating acoustic field measurements to the unknownquantities, and a statistical description of the random perturba-tions in the field measurements, offers a suitable framework forthe solution of such problems.When functions relating 1) parameters between consecutivesteps and 2) data and parameters are linear and noise is additiveand Gaussian, the Kalman filter (KF) is the optimal estimator interms of minimizing mean squared error (MSE). The KF prop-agates expectations and covariances of the unknown parame-ters from step to step, fully characterizing posterior pdfs. Someearly examples used in source tracking in the ocean with KFsare given in [16] and [17]. Nonlinear functions require varia-tions or generalization of the standard filter. Implementation inocean acoustic problems of a straightforward extension, the ex-tended Kalman filter (EKF), which linearizes mildly nonlinearfunctions, has been pioneered in a series of papers [18]–[27].More recently, unscented Kalman filters (UKFs) have achievedbetter root-mean-square (RMS) error and convergence perfor-mance than EKFs by selecting deterministic points called sigmapoints to represent parameter pdfs [28], [29]. For large param-eter vectors, the ensemble KF (EnKF) is efficient [30]. Highlynonlinear systems and complex noise processes require numer-ical methods for the computation of posterior pdfs. These ap-proaches are termed particle filters (PFs) and have been encoun-tered in ocean acoustic applications in [29] and [31]–[33].In this tutorial paper, we briefly review the foundations ofsequential filters, starting from the well-known KF and itsvariants and proceeding with PFs. These techniques formu-late sequential estimation using physical relations betweenunknown parameters and measurements embedded in noiseenvironments of a diverse nature. We examine and comparefilters and present examples, illustrating practical challengesand solutions. A more detailed presentation of these methods,including theoretical derivations, can be found in many excel-lent signal processing papers [34]–[39] and texts on sequentialMonte Carlo methods [40]–[43].Once basic principles and methods of sequential filteringare discussed, the focus is shifted to an overview of sequentialfiltering implemented for parameter estimation of dynamicalsystems in ocean acoustics. We present applications in target0364-9059/$26.00 © 2011 IEEEThis article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.2 IEEE JOURNAL OF OCEANIC ENGINEERINGtracking, wave estimation, geoacoustic inversion, frequency,and arrival time tracking.The paper is organized as follows. First, a general backgroundabout the state-space formulation for the estimation of evolvingparameters in dynamical systems is given in Section II togetherwith the basics of Bayesian filtering. The KF framework isintroduced in Section III together with its extensions such asthe EKF, the UKF, and the EnKFs. The PFs are presented inSection IV. The filter equations are