Nicholas C. Makris
Naval Research Laboratory
Washington D.C. 20375
Popular version of paper 1pAO4
Presented Monday Afternoon, 13 May 1996
Acoustical Society of America, Indianapolis, IN
Embargoed until 13 May 1996
The purpose of this research is to provide a means of determining whether the information contained in remote acoustic measurements, which suffer statistical fluctuation, is sufficient to accurately estimate the oceanographic structure of the water column in the littoral zone, or to be used in the more traditional localization of submerged objects. A method by which such estimates are often made is known as matched field inversion. The concept is to rapidly probe the ocean over wide areas by transmitting sound either from a known source, which may be controlled or may also be a source of opportunity such as a marine mammal or crashing surf, through the water column to a receiver. In the case of matched field tomography, the supposition is that the unique temperature, density and salinity structure of the water column, through which the sound propagates, will lead to an equally unique sound field structure on the array of hydrophones comprising the receiver. These unique oceanographic properties of the water column can then be inverted for given the measured acoustic data, our knowledge of the physical processes by which sound propagates through such a spatially varying medium, and the known source characteristics. The term "matched field" specifically refers to the method by which the estimate is arrived at. This method is to find the medium that leads to an acoustic field structure, determined by propagation modeling, that best matches that measured. In matched field processing, the location of a submerged object is estimated in a similar way. Here the assumption is that the object radiates sound that has a unique spatial structure on the receiving array for every unique position the object may occupy in the water column.
However, all of the matched field techniques just described are ideally based upon the assumption that the relationship between the acoustic measurements and the parameters to be estimated is entirely deterministic. But this is very seldom the case in reality. For example, natural disturbances such as passing surface and internal gravity waves often place shallow water waveguides in such a state of flux that a signal, deterministic when transmitted from a source, becomes fully randomized after propagating only several channel depths away in range to a receiver. When not properly accounted for, such randomization can severely degrade the accuracy of a matched field inversion, which presumably is for parameters that remained fixed during the measurement process.
The general performance of a matched field inversion is examined from the perspectives of statistical estimation and information theory for the worst case scenario of an environment that may cause random fluctuations in the received acoustic signal during a given transmission or across a set of transmissions. Specifically, a quantitative measure of the minimum mean-square error that can be attained in an unbiased parameter estimate is sought. According to classical estimation and information theory, the inverse of this lower bound on mean-square error is proportional to the amount of information that can be extracted about the parameter in question from the given measurements. Optimal methods of estimating the parameters are unbiased and attain the lower bound on error, and therefore extract all the information possible about the desired parameter from the measurements. However, computation of the lower bound on error, which applies to any unbiased estimate, is itself useful because it quantifies the intrinsic accuracy of an experiment. For example, once the error bound is known for a given experimental geometry and set of acoustic sources and receivers, it is possible to determine how much averaging of independent samples is necessary to reduce the error of the parameter estimate to fall within a tolerable threshold. Specific results along these lines indicate that acoustic tomography for oceanographic parameters of the water column is practical given current acoustic technology, but is impractical for the localization of submerged objects using acoustic sources of opportunity such as the noise of crashing waves on the sea surface.
Finally, it is interesting that, after propagation through the fluctuating ocean, the received sound field has been so fully randomized that it typically obeys the same kind of Gaussian field statistics that are commonly observed in the related disciplines of optics, radar, and medical ultrasound. This commonality follows from the central limit theorem which applies in all these applications. It has recently been shown, that given such fully randomized Gaussian fields, the optimal way to find a pattern in an intensity image is to take the log transform of the intensity measurements and correlate these with the expected value of the log transform of the hypothetical pattern. This is particularly appealing because it is common practice in many engineering applications, including ocean acoustics, to first plot intensity data in decibel units, which are a log transform of measured intensity units, and then look for patterns in the data. It is also appealing because the optical and acoustic fields received by the human eye and ear often undergo fully randomized Gaussian fluctuations. The optimality of looking for patterns in the logarithmic domain may then explain the logarithmic response of human auditory and visual perception to intensity stimulus known as the Weber-Fechner laws.
1. N. C. Makris, "A foundation for logarithmic measures of fluctuating intensity in pattern recognition," Optics Letters 20, 2012-2014 (1995).
2. N. C. Makris, "Parameter resolution bounds that depend on sample size," J. Acoust. Soc. Am. 99, May 1996.
3. N. C. Makris, F. Ingenito and W. A. Kuperman, "Detection of a submerged object insonified by surface noise in an ocean waveguide," J. Acoust. Soc. Am. 96, 1703-1724 (1994).
4. N. C. Makris, S.P. Heckel, J.S. Perkins and J. Catipovic, "Optimizing experimental design for shallow water sound speed inversion," in Full Field Inversion Methods in Ocean and Seismic Acoustics, edited by O. Diachok, Kluwer, Dordrecht (1994).
5. N. C. Makris, "Optimal pattern recognition in signal-dependent noise as a possible basis for the Weber-Fechner Laws," J. Acoust. Soc. Am. 98, 2907 ABSTRACT (Nov. 1995).