The frequency-difference and frequency-sum autoproducts, quadratic products of complex acoustic field amplitudes at two frequencies, may mimic genuine acoustic fields at the difference and sum frequencies of the constituent fields, respectively. Autoproducts have proven useful in extending the useable frequency range for acoustic remote sensing to frequencies outside a recorded field’s bandwidth. In array signal processing applications, the spatial coherence of the field often sets performance limits. This paper presents results for the spatial coherence of the genuine field, the frequency-difference autoproduct, and the frequency-sum autoproduct as determined from data collected during the Cascadia Open-Access Seismic Transects (COAST 2012) experiment. In this experiment, an airgun array providing a 10 to 200 Hz signal was repeatedly fired off the coast of Washington state, and the resulting acoustic fields were recorded by a nominal 8 km long, 636-element towed horizontal array. Based on hundreds of airgun firings from a primarily shore-parallel transect, both autoproducts were found to extend field coherence to frequencies outside the genuine field’s bandwidth and to produce longer coherence lengths than genuine fields, in most cases. When used for matched-field processing, the same data illustrate the benefits of the autoproducts’ extended coherence.
The presence of random noise is a challenge for remote sensing tasks as success typically diminishes with decreasing signal-to-noise ratio (SNR). To improve performance in noisy environments, signal processing techniques commonly implement snapshot averaging, requiring multiple signal samples. This presentation investigates the improvement offered by the cubic autoproduct for low SNR beamforming with a single snapshot. Prior work has demonstrated the utility of a quadratic product of complex field amplitudes within the signal bandwidth for beamforming and matched field applications. This quadratic field product, known as the frequency-difference autoproduct, is a synthetic estimate of an acoustic field at the difference frequency of the two constituent fields. That formulation is extended here to a cubic product of three complex field amplitudes within the signal bandwidth, termed the cubic autoproduct, capable of mimicking field content at frequencies below, above, and within the signal bandwidth. Important features and the mathematical description of the cubic autoproduct are reviewed before discussing the beamforming approach. Single snapshot beamforming results from experimental data, acquired in a noisy underwater environment, are directly compared between the cubic frequency-difference autoproduct and the acoustic field. [Work supported by ONR and by the US DoD through an NDSEG Fellowship.]
Incident and reflected acoustic waves, with wavenumber k, are coherent in an ideal Lloyd’s mirror environment. However, a random rough surface, characterized by its rms roughness height h, reduces this coherence as kh increases. This presentation describes the ability of the frequency-difference autoproduct to recover reflected-field coherence, albeit at a lower frequency, even in the presence of high kh values. The frequency-difference autoproduct is a quadratic product of complex field amplitudes at different frequencies within the signal bandwidth. Prior work has shown that the frequency-difference autoproduct can mimic a below-band field at the difference frequency. Thus, by choosing a sufficiently-low difference frequency, the apparent surface roughness can be reduced so that a rough-surface scattered field resembles a flat-surface reflected one. For the theory and simulation results presented here, the surface roughness is Gaussian distributed and isotropic, and the Kirchhoff (tangent plane) approximation is used to determine the reflected sound. For 1 < kh < 6, theoretical and numerically simulated autoproducts in a rough-surface environment are compared to acoustic fields at the difference frequency in an ideal Lloyd’s mirror environment. Accompanying experimental results may be provided as well, if available. [Work supported by ONR.]
Acoustic waves, with wavenumber k and incidence angle θ, forward scattered from a randomly rough surface, with root-mean-square roughness height h, lose coherence with the incident field as khcos θ increases. Recovering reflected field-coherence is possible via the frequency-difference autoproduct, a quadratic product of complex field amplitudes at nearby frequencies within the signal bandwidth. By downshifting recorded frequencies, the apparent roughness of a surface is reduced and coherence can be regained. An additional consideration exists, however, as the nonlinearity of the frequency-difference autoproduct introduces a significant dependence on the surface autocorrelation function, and consequently, the surface power spectrum by Fourier transform. The relationship between coherence recovery, surface autocorrelation function, and surface power spectrum is discussed and results are shown. Furthermore, this presentation investigates the potential utility afforded by these relationships for environmental characterization in rough surface scattering. By employing coherence-based frequency-difference autoproduct methods, remote identification of the spectral content, lateral statistics, and vertical statistics of a rough surface are explored. The work uses acoustic data collected at sea during SW06 (off New Jersey, depth 80 m), where the geometry, signal bandwidth, and anisotropic ocean surface conditions put khcos θ > 2.5. [Work supported by ONR and by the US DoD through an NDSEG Fellowship.]
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