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High-resolution closed-form estimation of normal mode parameters of a partially sampled water column

H. Gazzah,
Department of Electrical and Computer Eng.,
University of Sharjah, 27272 UAE
LARSyS, University of Algarve, 8005-139 Faro, Portugal

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Ref.: OCEANS MTS/IEEE, Marseille (France), June 2019

Abstract: Propagation of acoustic waves in a shallow water is modeled using the normal mode model. Reliable and fast estimation of the model parameters is of the utmost importance to many underwater observation systems. A vertical linear array is deployed that partially covers the water column, with the objective to estimate the model functions at the sensor depths, as well as the wavenumbers relative to the different modes.

For this purpose, a monochromatic source is placed at a given range from the array. It is activated successively at the same sensor depths. Pressure field measurements are collected and grouped into a first data matrix. Then, the source is moved to another location at a different range, and the experiment is repeated. A second data matrix is collected.

We design a subspace algorithm that computes the sought-after model parameters. Unlike existing subspace algorithms, the presented algorithm is not based on the decomposition of the array data matrix, and consequently, does not require full and dense coverage of the water column. This unpractical condition is, unfortunately, needed to obtain orthogonal columns that could be identified by means of a singular vector decomposition. We manage to design a subspace algorithm that does not require this condition because we do not process the brute data matrix. Instead, we introduce a non-trivial combination of the two data matrices collected as mentioned above. In this matrix, sampled modal functions appear as eigenvectors and wavenumbers appear as eigenvalues. Contrarily to singular vectors, eigenvectors do not have to be orthogonal, explaining why the proposed algorithm does not require orthogonality (of modal functions).

Not only does the resulting algorithm operate under realistic conditions, it also computes the model parameters in a closed-form, search-free and fully-automatic manner, that contrasts with many of the existing heuristic techniques. It, also, does not make any assumption about the statistical distribution of observation noise. The computation burden is that of one eigenvector decomposition and one singular vector decomposition, where the matrix size is determined by the number of array sensing elements.

By a proper theoretical development, we prove that the so-obtained estimates are exact if pressure measurements are error-free. From this point-of-view, we propose a truly high-resolution algorithm whose accuracy is not limited by the size of the sensing array, but rather by the observation noise. In the presence of noise, estimation accuracy can be improved by collecting more field measurements, over longer periods of time and/or using longer arrays.

The developed algorithm was tested with two different 300m deep waveguides: a Pekeris waveguide and a lossy waveguide with an underlying sediment layer, both excited by a source emitting at 60 Hz. Pressure field was generated using the KRAKEN propagation model and computer-generated random noise was artificially added. The estimation algorithm was run with different combinations of SNR levels, source-to-VLA separation, inter-sensor spacing, array length and depth. Good estimation performance is obtained, more difficultly for the lossy waveguide, naturally. Below, we show typical estimation performance for the modal functions expressed in terms of an averaged normalized mean square error, for the two waveguides, as function of the number of sensors and the depth of the bottom sensor.

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