IEEE Workshop on Underwater Acoustic Signal Processing, October 1999
Title: Design of environmentally robust nonuniform arrays for shallow water model filtering
Authors: John R. Buck, Tsung-Jieh Shiao

The pressure field in a shallow water channel is often characterized by a finite, discrete set of propagating normal modes. Mode filtering is defined as the estimation of the amplitudes of these normal modes from the observed pressure samples obtained at a vertical hydrophone array. Many researchers have proposed the use of model amplitudes, or coefficients, for underwater source localization or remote sensing of oceanographic features. The success of any of these techniques relies on the ability to obtain accurate estimates of the mode amplitudes.

The Cramer-Rao Bound (CRLB) gives the smallest possible variance of any unbiased estimator, and [Buck, et al. JASA, April 1998[ derives this bound for the variance of the mode filter. The resulting expression depends on two factors: the shapes of the normal modes at the array location, and the depths of the hydrophones in the array. The former is beyond our control, but is determined by the ocean  environment when and where the array is deployed. Consequently, the hydrophone locations are the only factor which we can control that affect the CRLB, the fundamental limit on the performance of the mode filter. We want to choose the array positions to give us the best performance guarantee possible given the uncertainty about the environmental factors. The difficulty lies in designing an appropriate array for the range of frequencies of interest in the notoriously variable environmental conditions of coastal waters.

The goal of our research is to develop algorithms to design hydrophone arrays whose mode filtering performance is robust to environmental variations. Specifically, we wish to design nonuniformly spaced vertical arrays whose worst case performance evaluated over a given range of environmental variations is as favorable as possible. This strategy addresses the realistic deployment scenario in which the ocean environment is not known in advance, and may vary substantially over the course of the deployment.

Conventional wisdom holds that uniformly-spaced arrays provide the most flexibility for multiple or extended deployments. However, in the preliminary results of the environments with known water column over both rigid and unknown bottoms, we find this is not always the case. In the former example, we assumed the water sound speed (1550 m/s) and depth (15m) were known. To limit the size of the problem, we chose the frequency (100 Hz) such that the waveguide supports only two trapped modes, and attempt to design the best two hydrophone vertical array to estimate these modes amplitudes. The total CRLB for the mode filter performance for the optimal array (-20 dB) is roughly 3 dB better (smaller) than the bound for the uniform array (-17dB). For the latter example, we have a 19m water column with known sound speed (1490 m/s) and density (1 g/cm³). The frequency of the propagating sound is known to be 100 Hz, and the bottom attenuation is known to be 0.5 db/λ. The uncertainty is in the bottom sound speed which can be anywhere in the range 1650-5250 m/s, and the bottom density which may range from 1.9-2.7 g/cm³. This set of environments support two propagating modes, and the total CRLB for the mode filter performance for the optimal array is nearly 8 dB lower than the uniform array.

With only two hydrophones in above two classes of environments, it's possible to visualize the total error variance as a surface over a two-dimensional plane, and each point in the plane corresponds to a different array. An important feature of thee two error surfaces is that they are convex, and thus unimodal. This suggests that steepest descent algorithm would be an efficient technique for finding the optimal array. As expected, the algorithm proceeds directly downhill to the same optimal array found through exhaustive search.

We will then move to a broader range of ocean environments and hope to combine some of these cases to provide accurate models of the uncertainty faced in realistic deployments. We will also study each case to determine if its performance surface is convex, and if it is possible to apply efficient optimization algorithms to design the arrays. Besides, quasi-exhaustive search techniques will be explored to verify the optimality of our results when the number of hydrophones precludes an exhaustive search to confirm the global optimum.

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