Broadband ML-approach to environmental parameter estimation in shallow ocean at low SNR
Introduction
Cost-efficient estimation of the geophysical structure of marine sediments just beneath the seafloor is an important topic of research in monitoring of coastal areas. Model-based methods that attempt to invert the acoustic field for obtaining a detailed and global estimate of the seafloor structure have been widely investigated in the past few years [16], [19], [20], [23], [36]. Still, these methods require relatively sophisticated and costly acoustic sensing devices. Regarding the experimental aspect, most studies reported so far required data measurement setups involving extremely long arrays and sound sources either towed or suspended from auxiliary ships. In an effort carried out under MAST II, it was shown that geoacoustic parameter estimation is possible using a moderate aperture horizontal linear array allowing for the use of a single ship towing both the source and the receiver [23], [24].
Usually, in matched field processing (MFP) the knowledge about the environment is exploited to estimate the location parameters of unknown sources [1], [21]. Vice versa, one can estimate environmental parameters provided the sources are known [14], [36]. However, if the sources are unknown, the estimation must be performed simultaneously for environment and source locations: one approach is known as focalization [8]. In their article, Collins and Kuperman show the feasibility of focalization via numerical simulations. They investigate both ray- and normal-mode models for a range-dependent deep ocean and a single-frequency time-harmonic source. It is not clear, however, how to generalize their cost-function-based approach to the broadband case in a statistically justified way. Focalization attempts to determine the ocean acoustic parameters only along the propagation paths between the sources and the array of hydrophones. The claim in [8] is that it is possible to obtain correct localization even when the environmental parameters do not converge to the correct ocean-acoustic parameters.
Recently, some results have been obtained for simultaneous estimation of environmental parameters, source location and receiver calibration [16]. It is shown in [16] that bottom parameter estimation as an optimization procedure for a goal-function allows direct inclusion of the source location parameters into the search space. In practice, this increases the computational burden of the estimation procedure.
From the numerical point of view, estimation of environmental parameters is a highly non-linear and non-convex optimization problem in a high-dimensional bounded domain [37]. These inverse problems are ill-posed, so some sort of regularization scheme is necessary [7], [34]. Various cost functions have been proposed in the literature, most of them for the single-source/single-frequency case [37]. The single-frequency case is often badly behaved in terms of ambiguity, which can be partly overcome in a full broadband treatment [27], [36]. This fact stems directly from the ambiguity of time-delay estimation in the purely time-harmonic case. Although the full benefit of broadband signal processing for environmental sensing is not yet clear, slight amelioration for small bandwidths between 10% and 60% of center frequency have been observed in sensitivity studies [23].
The aim of this paper is to demonstrate that the broadband conditional maximum-likelihood (ML) estimator combined with a simplistic ocean waveguide model allows to estimate source and environmental parameters simultaneously with a good accuracy. The paper is organized as follows: in the next section, we formulate the acoustic propagation model. Section 3 is devoted to the data model. In Section 4, we briefly describe the theoretical background of the conditional maximum-likelihood estimator applied to our propagation model. Thereafter, we present experimental results of real passive sonar data. Mutual correlations between estimated parameters are analyzed by Monte Carlo simulations. We conclude with some remarks concerning depth estimation.
Section snippets
Regularized propagation model
The Green's function is the acoustic response observed at location to a time-harmonic point source with frequency ω at location . In cylindrical coordinates, we havewhere ∂zz=∂2/∂z2, etc. In this paper, we consider the simplistic ocean waveguide model shown in Fig. 1. That is, we assume c=const. throughout. Further, G obeys the Sommerfeld radiation condition for r→∞, homogeneous Dirichlet boundary conditions at the
Data model
We use the above propagation model to describe the output of the horizontal array of N hydrophones. The array output is sampled after low-pass filtering and sectioned into K stretches of duration T. Each of these data stretches, in turn, is divided into K′ time batches of length T′=T/K′. Then, they are short time Fourier-transformed using a multi-window technique [29], [35] to obtain for k=0,…,K′−1 and ℓ=0,…,L′−1. The number L′ of orthonormal windows used depends on the selected
Matched-field conditional ML estimator
The asymptotic statistics of the data in frequency domain motivate the application of a conditional ML estimator for simultaneous estimation of signal, noise and environmental parameters [4]. In the broadband case, we maximize the conditional log-likelihood functionwhere is the vector of noise spectral parameters and is a vector which summarizes all non-linear parameters of the model, i.e. source locations for m=1,…,M
Application of bootstrap principle
Lower bounds on the asymptotic variances and cross correlations of estimates can be obtained from the Cramér-Rao bound (CRB) [3] which depends on local properties of around the true parameters. In the non-asymptotic case, i.e. for finite observation time, the performance of the conditional ML estimator can (at present) only be judged numerically because analytical results are missing. In a Bayesian context, the performance of the estimator could be judged from calculating a posteriori
Description of experimental data
The experimental site was located in the Bornholm Deep, to the east of Bornholm island in the Baltic Sea. The bathymetry is depicted in Fig. 2 which was generated from a high-resolution topographic grid provided by Insitut für Ostseeforschung Warnemünde (IOW) [30]. Several sonar experiments were conducted by Atlas Elektronik (Bremen) during the cruise between October 3 and 13, 1983.
A 16-element hydrophone array was towed by the surface ship Walther von Ledebur. The rear sensor was defective,
Results
We started our investigation with simulated data to obtain some insight into the SNR needed for the estimation procedure. Fig. 5 shows likelihood (8) as a function of the ocean depth d. The artificial data were generated by a Monte Carlo procedure which generated Wishart-distributed matrices for a selected ocean depth and source configuration with M=4 sources present. We selected source location parameters to be identical with the estimated source configuration at data-piece No. 43 by
Conclusion
We have estimated an environmental parameter (ocean depth) using unknown broadband sources as sources of acoustic energy. Although more investigations should be conducted with more realistic environmental models, this result implies feasibility of purely passive bathymetry and geophysical parameter estimation for low SNR. Monte Carlo simulations have shown that source range, bearing, sound speed and ocean bottom can be estimated with low variance. Source depth, however, shows considerably more
Acknowledgements
We thank Manfred Siegel, STN Atlas Elektronik, Bremen, Germany, for providing the passive sonar data; Torsten Seifert, Institut f. Ostseeforschung Warnemünde (IOW), Warnemünde, Germany, for providing the high-resolution bathymetric data; Harry Dooley, International Council for the Exploration of the Sea (ICES), Copenhagen, Denmark, for access to the helcom bmp k2 measurements, and Tobias Fassbender, Ruhr-University Bochum, for implementing the genetic algorithm.
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Supported by a research grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.