Bayesian tracking predictions in an uncertain ocean environment.

This paper develops an approach to three-dimensional source tracking in an uncertain ocean environment using a horizontal line array (HLA). The tracking algorithm combines matched-field focalization for environmental (seabed and water column) and source-bearing model parameters with the Viterbi algorithm for range-depth estimation and includes physical constraints on source velocity. The ability to track a source despite environmental uncertainty is examined using synthetic test cases for various track geometries and with varying degrees of prior information for environmental parameters. Performance is evaluated for a range of signal-to-noise ratios in terms of the probability of estimating a track within acceptable position/depth errors. The algorithm substantially outperforms tracking with poor environmental estimates and generally obtains results close to those obtained with exact environmental knowledge. The approach is also applied to measured narrowband data recorded on a bottom-moored HLA in shallow water (the Barents Sea) and shown to successfully track both a towed submerged source and a surface ship in cases where simpler tracking algorithms failed.

Shallow water is a complex sound propagation medium, which is affected by the varying spatial–temporal ocean environment. Taking this complexity into account, the classical processing techniques of source localization and environmental inversion may be improved. In this work, a joint tracking approach for the moving source and environmental parameters of the range-dependent and time-evolving environment in shallow water is presented. The tracking scheme treats both the source parameters (e.g., source depth, range, and speed) and the environmental parameters (e.g., water column sound speed profile (SSP) and sediment parameters) at the source location as unknown variables that evolve as the source moves. To counter sample impoverishment and robustly characterize the evolution of the parameters, an improved particle filter (PF), which is an extension of the standard PF, is proposed. Two examples with simulated data in a slowly changing environment and experimental data collected during the ASIAEX experiment are utilized to demonstrate the effectiveness of the joint approach. The results show that we were able to track the source and environmental parameters simultaneously, and the uncertainties were evaluated in the form of time-evolving posterior probability densities (PPDs). The performance comparison confirms that the improved PF is superior to the standard PF, as it can reduce the parameter uncertainties. The tracking capabilities of the improved PF were verified with high accuracy in real-time source localization and well-estimated rapidly varying parameters. Moreover, the influence of different particle numbers on the improved PF tracking performance is also illustrated.

When under uncertain ocean environment, the environmental model of matched field processing (MFP) is different from the real world. The performance of MFP decreases largely beacuse of the environmental mismatch. In order to improve its robustness to the environmental mismatch, a MFP with posterior probability constraints (MFP-PPC) is proposed. The algorithm derives the posterior probability density function (PDF) of the source locations from Bayesian Criterion, then the main lobe of AMFP is protected by posterior PDF, so the MFP-PPC has not only the merit of high resolution as AMFP, but also the advantage of robustness. We use experimental data to evaluate the algorithm, the results shown that in the uncertain ocean environment the MFP-PPC is robust not only to the moored source, but also to the moving source. Meanwhile, the localization and tracking is consistent with the trajectory of the moving source.

This paper considers sequential Bayesian strategies for geoacoustic inverse problems which are difficult to solve simultaneously due to computational constraints. Bayesian inference provides a powerful approach to learning problems such as this since sequential inversions of multiple data sets [with the posterior probability density (PPD) of one inversion applied as prior information in the subsequent inversion] are equivalent to simultaneous inversion of all data. However, passing PPDs forward as priors has its own challenges when the PPD is sampled numerically for nonlinear inverse problems, particularly when the model parameter space is of high dimensionality and the data information content is high. In such cases, approximations are required to efficiently carry PPD information forward to subsequent inversions. The approach developed here represents numerically sampled PPDs in terms of discretized marginal probability distributions for principal components of the parameters, which minimizes the loss of information in representing inter-parameter correlations. The sequential Bayesian approach is applied to seabed reflectivity inversion with multiple data sets representing travel-time data and frequency-domain reflection coefficient data for a series of increasing penetration depths. Data information content is quantified by accounting for potential error biases as well as data error covariances. [Work supported by the Office of Naval Research.]

The environmental parameters are usually uncertain in complex shallow ocean environment and restrict the performance of the matching model-like method. Therefore, we need a more tolerant detection method for detecting underwater targets in the uncertain shallow ocean environment. The previous mode-subspace detection method has the characteristics of both high performance and robustness. However, the robust mode-subspace detector is suitable for vertical arrays and its performance is limited by shallow ocean environment. Therefore, we propose the tolerant detection method for estimating the robust signal subspace with horizontal arrays. We estimate the robust signal subspace by bringing uncertain parameters into the observation matrix of a horizontal array. Combined with the robust signal subspace estimation, we propose a subspace detector that tolerates uncertain parameters. The results on simulation in a uncertain shallow ocean environment show that the detector we proposed has a high average detection capability and a certain tolerance for uncertain parameters.

SUMMARY The seismic waveform inversion problem is usually cast into the framework of Bayesian statistics in which prior information on the model parameters is combined with the data and physics of the forward problem to estimate the a posteriori probability density (PPD) in model space. The PPD is a function of an objective or fitness function computed from the observed and synthetic data. In general, the PPD or the fitness function is multimodal and its shape is unknown. Global optimization methods such as simulated annealing (SA) and genetic algorithms (GA) do not require that the shape of the fitness function be known. In this paper, we investigate GA to rapidly sample the most significant portion or portions of the PPD, when very little prior information is available. First, we use a simple three operator (selection, crossover and mutation) GA acting on a randomly chosen finite population of haploid binary coded models. We use plane wave transformed synthetic seismic data and a normalized cross-correlation function [E(m)] in the frequency domain as a fitness function. A moderate value of crossover probability, a low value of mutation probability, a high value of update probability and a proper population size are required to reach very close to the global maximum of the fitness function. Next, with an attempt to accelerate convergence we show that the concepts from simulated annealing can be used in stretching of the fitness function, i.e., we use exp [E(m)/T] rather than E(m) as the fitness function, where T is a control parameter analogous to temperature in simulated annealing. By a schemata analysis, we show that at low temperatures, schemata with above average fitness values are reproduced in large numbers causing a much more rapid convergence of the algorithm. A high value of temperature T assigns nearly equal selection probability to most of the schemata and thus retains diversity among the members of the population. Thus a GA with a step function type cooling schedule (very high temperature in the beginning followed by rapid cooling to a very low temperature) improves the performance dramatically: high values of the fitness function are obtained rapidly using only half as many models as would be required by a conventional GA. Similar performance could also be achieved by first using a high mutation probability and then decreasing the mutation probability to a very low value, while retaining the same low temperature throughout. We also address the problem of ‘genetic drift’ which causes the finite GAs to converge to one peak or the other when the algorithm is applied to a highly multimodal fitness function with several peaks of nearly the same height. A parallel genetic algorithm based on the concept of ‘punctuated equilibria’ is implemented to circumvent the problem. We run several GAs each with a finite subpopulation in parallel and collect many good models from each one of these runs. These are then used to grasp the most significant portion(s) of the PPD in model space. We then compute the weighted mean model and use the derived good models to estimate uncertainty in the derived model parameters.

In Bayesian inversion, the solution is characterized by its posterior probability density (PPD). A fast Gibbs sampler (FGS) has been developed to estimate the multi-dimensional integrals of the PPD, which requires solving the forward models many times and leads to intensive computation for multi-frequency or range dependent inversion cases. This paper presents an alternative approach in order to speed this estimation process based on a neighbourhood approximation Bayes (NAB) algorithm. For lower dimension geoacoustic inversion, the NAB can approximate the PPD very well. For higher dimensional problems and sensitive parameters, however, the NAB algorithm has difficulty to estimate the PPD accurately with limited model samples. According to the preliminary PPD estimation by NAB, this paper developed a multi-step inversion scheme, which adjusts the parameter search intervals flexibly, in order to improve the approximation accuracy of NAB and obtain more complete parameter uncertainties. The prominent feature of NAB is to approximate the PPD by incorporating all models for which the forward problem has been solved into the appraisal stage. Comparison of FGS and NAB for synthetic benchmark test cases indicates that NAB provides reasonable estimates of the PPD moments while requiring less computation time

The present work addresses the problem of structural damage identification built on the statistical inversion approach. Here, the damage state of the structure is continuously described by a cohesion parameter, which is spatially discretized by the finite element method. The inverse problem of damage identification is then posed as the determination of the posterior probability densities of the nodal cohesion parameters. The Markov Chain Monte Carlo method, implemented with the Metropolis–Hastings algorithm, is considered in order to approximate the posterior probabilities by drawing samples from the desired joint posterior probability density function. With this approach, prior information on the sought parameters can be used and the uncertainty concerning the known values of the material properties can be quantified in the estimation of the cohesion parameters. The assessment of the proposed approach has been performed by means of numerical simulations on a simply supported Euler–Bernoulli beam. The damage identification and assessment are performed considering time domain response data. Different damage scenarios and noise levels were addressed, demonstrating the feasibility of the proposed approach.

where jct 's are known real numbers called the observation points. It is sometimes more natural to consider the parameters a and β in the above logistic way than something like e/(l + e)'s. For example, let us cosider a random variable Y on {0, 1} with small P[Y = 1], where the value 1 stands for a serious accident which we must avoid definitely. Since we are sensitive on the value P[Y = 1], we take the measurement log P[Y = 1] instead of the value itself. In this case, the logistic parametrization is suitable. In the same reason, it is natural to assume that the prior distribution a and β is uniform, that is, the joint prior density for (α, β) is given by p(a, β) = 1 on R. Then we discuss the posterior distribution on (α, β) under a set of observations F; = v; (ΐ = l , . . . ,/ i) . By the Bayes formula, the posterior probability density, is given by

Computing position error margin for a USV due to wind and current with a trajectory model

This paper presents a derivation of the Maximum Likelihood (ML) detector for acoustic signals that have propagated through a random or uncertain ocean environment. The derivation requires probability distribution functions (pdfs) of relevant signal and noise parameters belonging to the exponential class. (A companion paper describes how available knowledge of the environment and the Maximum Entropy method can be used to calculate exponential class pdfs for signal and noise parameters.) The resultant ML detector operates on the observations to compute parameter means, referred to as conditional mean estimates (CMEs), and then correlates the CME with the observations to obtain a detection statistic. For this reason, the detector is referred to as an Estimated Ocean Detector (EOD). For Gaussian signal and noise, the EOD reduces to the weighted sum of a correlation detector (CD) and an energy detector (ED), which agrees with Van Trees' result. When the signal variance is low the weights strongly favor the CD, while for high signal variance the weights strongly favor the ED. A closed-form solution is not available for the Gaussian EOD. However, results obtained using a Monte Carlo simulation show that when the signal variance is low, EOD performance equals that of the CD, while under high signal variance, performance equals the ED. For intermediate signal variance, the simulation shows that the EOD weights the CD and ED outputs so as to perform better than either detector by itself. Finally, EOD robustness to incorrect estimates of received signal and noise parameters pdfs is investigated using the Monte Carlo simulation. The EOD is shown to be a promising method for improving passive sonar performance by directly incorporating available information about the ocean environment without sacrificing robustness to errors in the signal parameter distributions

Incomplete environmental knowledge is a factor that can adversely affect the prediction capability of physics-based signal processors to localize acoustic sources in ocean waveguides. Environmental uncertainty effectively introduces unwanted additional degrees of freedom into the signal processing scheme. Including a quantitative measure of this incomplete information into the processor can be accomplished by considering the influence of alternative environments, each weighted by their probability of occurrence. An approach is outlined to account for environmental uncertainty in matched field processing by the use of polynomial chaos basis expansions to represent the replica vectors in terms of stochastic Green’s functions. These bases are interpreted as representing the effect of introducing additional degrees of freedom due to environmental uncertainty, because the pressure field replicas are specified as random processes. In this manner, a processor correlates the measured pressure field with a set of statistically weighted propagators between a replica source location and a sensor array. [Work sponsored by ONR through NRL base funding.]

Time reversal (TR) processing is derived from the invariance of the wave equation for a lossless medium to change in the sign of the time variable. By retransmitting the TR version of the time‐dispersed received signal propagated from a probe source (PS) to a source‐receiver array, one can reacquire the transmitted pulse at the PS location (time compression and spatial focusing) when the waveguide environment is time invariant. However, if some environmental variations occur between the two transmissions, the retrofocusing signal will be defocused. This paper presents a novel method of environmental variation inversion by the focused/defocused signals measured at the PS location with environmental perturbations between the PS transmission and the SRA retransmission. Because the sound speed profile (SSP) plays a critical role in an uncertain ocean environment, inversion of the SSP represented in terms of the empirical orthogonal functions is developed and discussed in detail. Examples of the method applied to synthetic data are provided to demonstrate the method’s feasibility.

In this paper an advanced anti-jam indoor adaptive GNSS signal acquisition and tracking algorithm is considered. Initially, we were able to determine that double-dwell structure (DDS) reduces the processing time penalty caused by false alarm. Nevertheless, the double-dwell structure is still vulnerable to interference and jamming. In order to determine a suitable advanced anti-jam indoor adaptive GNSS signal acquisition and tracking algorithm for DDS we first perform the Bayesian parameter estimation; i.e., we analytically compute the posterior Bayes probability density function (pdf) and cumulative distribution function (cdf) by applying the Bayes theorem in three steps. First, we compute the complex signal distribution and complex matrix variate signal distribution. This is an original new result never published before. Second, we provide an introduction of the complex Bessel or parabolic function interference distribution due to earlier assumptions of the DDS interference distribution models. We further explain that the process that is required to produce Complex Matrix Variate Bessel Interference Distribution or Complex Matrix Variate Parabolic Function Interference Distribution is an extremely daunting task let alone the Complex Matrix Variate DDS Bayesian posterior density it would seem at the moment nearly an impossible task because it may require the computation of functions such as Kampe de Feriet function and Jack functions of matrix arguments. Third, instead of performing the second step, we relax the assumption of interference to normal distribution. In both the scalar case and complex matrix variate cases we observe that the complex matrix variate Bayesian posterior pdf or cdf is invariant of the observation data or is identical to the prior complex matrix variate signal distribution model. This is an original new and very powerful result never published before. Why this result is so powerful is because up until now we never had a complete theoretical validation of our GNSS receiver design based on either autocorrelation or cross-correlation properties since, the complex matrix variate Bayesian posterior pdf or cdf is invariant of the observation data or is identical to the prior complex matrix variate signal distribution model. Future simulation results will illustrate that an anti-jam indoor adaptive DDS stable detection structure reduces by half the average acquisition time and significantly outperform its predecessor against interference and jamming. Moreover, future simulation results will show further advancements of the Giftet Inc. MATLAB library capability to perform advanced numerical computations based on closed form expressions of the generalized Bessel function distributions and generalized parabolic cylinder function distribution models via Kampe de Feriet function and Jack functions.

The localization of an acoustic source in the oceanic waveguide is a difficult task because the oceanic environment is often poorly known. Uncertainty in the environment results in uncertainty in the source position and poor localization results. Hence, localization methods dealing with environmental uncertainty are required. In this paper, a Bayesian approach to source localization is introduced in order to improve robustness and obtain quantitative measures of localization uncertainty. The Green's function of the waveguide is considered as an uncertain random variable whose probability density accounts for environmental uncertainty. The uncertain distribution over range and depth is then obtained through the integration of the posterior probability density (PPD) over the Green's function probability density. An efficient integration technique makes the whole localization process computationally efficient. Some results are presented for a simple uncertain Green's function model to show the ability of the proposed method to give reliable PPDs.