Measurement of autoproduct fields in a Lloyd's mirror environment

Coherent reflection recovery in scattering from the ocean surface using the frequency-difference autoproducta)

The coherence of rough sea-surface-scattered acoustic fields decreases with increasing frequency. The frequency-difference autoproduct, a quadratic product of acoustic fields at nearby frequencies, mimics a genuine field at the difference frequency. In rough-surface scattering, the autoproduct's lower effective frequency decreases the apparent surface roughness, restoring coherent reflection. Herein, the recovery of coherent reflection in sea surface scattering via the frequency-difference autoproduct is examined for data collected off the coast of New Jersey during the Shallow Water '06 (SW06) experiment. An acoustic source at depth 40 m and receiver at depth 24.3 m and range 200 m interrogated 160 independent realizations of the ocean surface. The root mean square surface height h was 0.167 m, and broadcast frequencies were 14-20 kHz, so that 2.5 ≤kh cos θ≤ 3.7 for acoustic wavenumber k and incidence angle θ. Measured autoproducts, constructed from scattered constituent fields, show significant coherent reflection at sufficiently low difference frequencies. Theoretical results, using the Kirchhoff approximation and a non-analytic surface autocorrelation function, agree with experimental findings. The match is improved using a numerical strategy, exploiting the relationship between autoproduct-based coherence recovery, the ocean-surface autocorrelation function, and the ocean-surface height spectrum. Error bars computed from Monte Carlo scattering simulations support the validity of the measured coherence recovery.

High-resolution frequency-difference beamforming for a short linear array

Conventional beamforming (CBF) is a commonly employed approach for detecting and estimating the direction-of-arrival (DOA) of acoustic signals in underwater environments. However, CBF becomes ambiguous due to spatial aliasing when the received signal's half wavelength is smaller than the array spacing. Frequency-difference beamforming (FDB) allows for processing data in the lower frequency Δf without encountering spatial aliasing by utilizing the product of array data at frequency f with its complex conjugate at frequency f+Δf. However, lower frequency results in a wider mainlobe, which can lead to poorer DOA performance for short arrays. In this paper, a fourth-order cumulants FDB method and a conjugate augmented FDB method are proposed to extend an M-element uniform linear array to 2M-1 and 4M-3 elements. The proposed methods generate narrower beams and lower sidelobe levels than the original FDB for short arrays with large spacing. And by setting the signal subspace dimension reasonably, the proposed methods can improve the weak target detection ability under strong interference compared with FDB. Last, we verify the excellent performance of the proposed methods through simulations and experimental data.

Recovery of coherent reflection from rough-surface scattered acoustic fields via the frequency-difference autoproduct

The acoustic field reflected from a random rough surface loses coherence with the incident field in the Kirchhoff approximation as kh cos θ increases, where k is the incident field wavenumber, h is the root mean square roughness height, and θ is the incidence angle. Thus, for fixed rough-surface properties and incidence angle, a reflected field at lower wavenumber should retain more coherence. Recent results suggest that the frequency-difference autoproduct formed from complex acoustic field amplitudes at two nearby frequencies can recover acoustic information at the difference of those frequencies even when the difference frequency is below the recorded field's bandwidth. Herein analytical, computational, and experimental results are presented for the extent to which the frequency-difference autoproduct recovers coherence from randomly rough-surface-scattered constituent fields that have lost coherence. The analytical results, developed from the Kirchhoff approximation and formal ensemble averaging over randomly rough surfaces with Gaussian height distributions and Gaussian correlation functions, indicate that the coherence of the rough-surface-reflected frequency-difference autoproduct depends on the surface correlation length and Δkh cos θ, where Δk is the difference of the autoproduct's constituent field wavenumbers. These results compare favorably with Monte Carlo simulations of rough surface scattering, and with laboratory experiments involving long surface correlation lengths where 1  ≤kh cos θ≤ 3.

Measurements of the correlation of the frequency-difference autoproduct with acoustic and predicted-autoproduct fields in the deep ocean

Frequency-domain spatial-correlation analysis of recorded acoustic fields is typically limited to the bandwidth of the recordings. A previous study [Lipa, Worthmann, and Dowling (2018) J. Acoust. Soc. Am. 143(4), 2419-2427] suggests that limiting such analysis to in-band frequencies is not strictly necessary in a Lloyd's mirror environment. In particular, below-band field information can be retrieved from the frequency-difference autoproduct, a quadratic product of measured complex pressure-field amplitudes from two nearby frequencies. The frequency-difference autoproduct is a surrogate field that mimics a genuine acoustic field at the difference frequency. Here, spatial-correlation analysis is extended to deep-ocean acoustic fields measured during the PhilSea10 experiment. The frequency-difference autoproduct, at difference frequencies from 0.0625 to 15 Hz, is determined from hundreds of Philippine Sea recordings of 60 or 100 Hz bandwidth signals with center frequencies from 172.5 to 275 Hz broadcast to a vertical receiving array 129-450 km away. The measured autoproducts are cross correlated along the array with predicted acoustic fields and with predicted autoproduct fields at corresponding below-band frequencies. Stable measured cross correlations as high as 80%-90% are found at the low end of the investigated difference-frequency range, with consistent correlation loss due to mismatch at the higher below-band frequencies.

Frequency-difference autoproduct cross-term analysis and cancellation for improved ambiguity surface robustness

Frequency-differencing, or autoproduct processing, techniques are one area of research that has been found to increase the robustness of acoustic array signal processing algorithms to environmental uncertainty. Previous studies have shown that frequency differencing techniques are able to mitigate problems associated with environmental mismatch in source localization techniques. While this method has demonstrated increased robustness compared to conventional methods, many of the metrics, such as ambiguity surface peak values and dynamic range, are lower than would typically be expected for the observed level of robustness. These previous studies have suggested that such metrics are reduced by the inherent nonlinearity of the frequency-differencing method. In this study, simulations of simple multi-path environments are used to analyze this nonlinearity and signal processing techniques are proposed to mitigate the effects of this problem. These methods are used to improve source localization metrics, particularly ambiguity surface peak value and dynamic range, in two experimental environments: a small laboratory water tank and in a deep ocean (Philippine Sea) environment. The performance of these techniques demonstrates that many source localization metrics can be improved for frequency-differencing methods, which suggests that frequency-differencing methods may be as robust as previous studies have shown.

The effects of refraction and caustics on autoproducts

Quadratic products of complex amplitudes from acoustic fields with nonzero bandwidth, denoted "autoproducts," can mimic acoustic fields at frequencies lower or higher than the bandwidth of the original field. While this mimicry has been found to be very promising for a variety of signal processing applications, its theoretical extent has, thus far, only been considered under the most elementary ray approximation. In this study, the combined effects of refraction and diffraction are considered in environments where refraction causes neighboring rays to cross and form caustics. Acoustic fields on and near caustics are not well-predicted by elementary ray-acoustic theory. Furthermore, caustics introduce frequency dependence to the nearby acoustic field and a phase shift on the acoustic waves that passes through them. The effects these caustics have on autoproducts is assessed here using two simple, range-independent waveguides with index of refraction (n) profiles that are n²-quadratic and n²-linear. It is found that in multipath regions where rays have passed through differing numbers of caustics, the ability of autoproducts to mimic out-of-band fields is substantially hindered.

Autoproducts in and near acoustic shadow zones created by barriers

The autoproducts are nonlinear mathematical constructs developed from acoustic fields with non-zero bandwidth. When averaged through the field's bandwidth, the autoproducts may mimic a genuine acoustic field at frequencies that are lower or higher than the original field's bandwidth. The resulting opportunity to extend signal processing to user-selectable below- or above-band frequencies is intriguing for many signal processing algorithms. Based on prior work, the limitations of the autoproducts' mimicry of out-of-band fields are understood when the in-band acoustic field is well-represented by ray acoustics. Thus, the focus in this study is on autoproducts in acoustic shadow zones behind barriers containing only diffracted acoustic fields where a sum of ray-path contributions is not an adequate field description. Diffraction is expected to be a detriment to autoproduct techniques due to its sensitivity to frequency. Two ideal shadow-zone environments with exact analytic Helmholtz-equation solutions are considered: Sommerfeld's half-plane problem, also known as knife-edge diffraction, and Mie scattering from a sphere with ka = 40, where k is the wavenumber and a is the sphere's radius. With the exception of the shadow regions, autoproducts experience only mild degradation in field-mimicry performance when compared to what the ray-based theory would predict.

Long-range frequency-difference source localization in the Philippine Sea

Matched field processing (MFP) refers to a variety of source localization schemes for known complicated environments and involves matching measured and calculated (replica) fields to identify source locations. MFP may fail for several reasons, most notably when the calculated fields are insufficiently accurate. This error commonly prevents MFP-based long-range (>100 km) source localization in the deep ocean (from 5 to 6 km depth) for signal frequencies of hundreds of Hz, even when extensive high-signal-to-noise ratio field measurements are available. Recently, below-band MFP utilizing the frequency-difference autoproduct [Worthmann, Song, and Dowling (2015). J. Acoust. Soc. Am, 138(6), 3549-3562] achieved some shallow-ocean localization success at a 3 km source-to-array range with signal frequencies in the tens of kHz. The performance of this technique, when extended to matching the measured frequency-difference autoproduct with a composite mode-ray replica, is described here for deep ocean source localization. The ocean propagation data come from the PhilSea10 experiment and involve source-to-array ranges from 129 to 379 km and nominal 100-Hz-bandwidth signals having center frequencies from 250 to 275 Hz. Based on an incoherent average of five signal samples, the frequency-difference technique was 90%-100% successful at four different source-to-array ranges using single-digit-Hz difference frequencies.

Frequency-difference beamforming in the presence of strong random scattering

Frequency-difference beamforming [Abadi, Song, and Dowling (2012). J. Acoust. Soc. Am. 132, 3018-3029] is a nonlinear, out-of-band signal processing technique used to beamform non-zero bandwidth signals at below-band frequencies. This is accomplished with the frequency-difference autoproduct AP(Δω)=P(ω₂)P^*(ω₁), a quadratic product of complex field amplitudes that mimics a genuine field at the difference frequency, Δω=ω₂-ω₁. For frequency-difference beamforming, AP(Δω) replaces the in-band complex field in the conventional beamforming algorithm. Here, the near-field performance of frequency-difference beamforming is evaluated in the presence of 1 to 30 high-contrast spherical scatterers with radius a placed between, and in the plane defined by the source and a 12-element linear receiving array with element spacing d. Based on the center frequency wave number, k, of the 150-200 kHz frequency sweep source signal, the scatterers are large, ka ≈ 15; the array is sparse, kd = 37; and the average source-to-receiver distance is up to 4.3 mean-free-path lengths. Beamforming results from simulations and experiments show that in-band beamforming loses peak-to-sidelobe ratio and fails to reliably locate the source as the scatterer count increases. Using the same signals, frequency-difference beamforming with difference frequencies from 5 to 25 kHz localizes sources reliably with higher peak-to-side-lobe ratios, though with reduced resolution.

Frequency-sum beamforming for passive cavitation imaging

Beamforming includes a variety of spatial filtering techniques that may be used for determining sound source locations from near-field sensor array recordings. For this scenario, beamforming resolution depends on the acoustic frequency, array geometry, and target location. Random scattering in the medium between the source and the array may degrade beamforming resolution with higher frequencies being more susceptible to degradation. The performance of frequency-sum (FS) beamforming for reducing such sensitivity to mild scattering while increasing resolution is reported here. FS beamforming was used with a data-dependent [minimum variance (MV)] or data-independent (delay-and-sum, DAS) weight vector to produce higher frequency information from lower frequency signal components via a quadratic product of complex signal amplitudes. The current findings and comparisons are based on simulations and passive cavitation imaging experiments using 3 MHz and 6 MHz emissions recorded by a 128-element linear array. FS beamforming results are compared to conventional DAS and MV beamforming using four metrics: point spread function (PSF) size, axial and lateral contrast, and computation time. FS beamforming produces a smaller PSF than conventional DAS beamforming with less computation time than MV beamforming in free space and mild scattering environments. However, it may fail when multiple unknown sound sources are present.