The frequency-difference and frequency-sum autoproducts are quadratic products of complex acoustic fields at two recorded frequencies that can mimic genuine fields at the difference and sum frequencies, extending coherent array processing beyond a transmitted signal's bandwidth. Ocean measurements from the COAST 2012 experiment showed that both autoproducts typically have longer spatial coherence than the genuine field, yet autoproduct-based matched-field source localization anomalously failed at frequencies of constructive autoproduct interference (around 80 and 160 Hz), where autoproduct coherence was high. The cause is structural: the quadratic construction rectifies the pressure-release sea surface into an apparent rigid boundary, shifting the Lloyd's-mirror interference pattern in a way that genuine-field-style replicas — even with a sign-flipped surface reflection coefficient — do not reproduce. The open problem is to develop replica fields for autoproduct matched-field processing that consistently incorporate the boundary-condition, amplitude, and phase modifications imposed by the quadratic construction, and to evaluate them systematically so that the autoproducts' extended coherence becomes exploitable for source localization throughout the frequency range.

Background

Matched-field processing (MFP) is an array-signal-processing technique for locating an underwater sound source: replica fields — the acoustic pressures predicted by a propagation model at each receiving array element for a grid of trial source positions — are correlated with the measured field, and the trial position that maximizes the correlation (commonly a coherent Bartlett sum over array elements) is taken as the source location (Bucker 1976; Baggeroer et al. 1993). MFP performance is limited by the spatial coherence of the measured field: once the array aperture exceeds the coherence length — the receiver separation at which the normalized field coherence falls below a chosen threshold, commonly e−1e^{-1} — additional elements no longer contribute coherently.

The frequency-difference and frequency-sum autoproducts are surrogate fields that extend such processing to frequencies outside a recording's bandwidth. For a recorded complex pressure P(rj,ω)P(\mathbf{r}_j,\omega) at receiver jj, the autoproducts are the quadratic products APΔ=P(rj,ω+)P∗(rj,ω−)AP_\Delta = P(\mathbf{r}_j,\omega_+)P^*(\mathbf{r}_j,\omega_-) and APΣ=P(rj,ω+)P(rj,ω−)AP_\Sigma = P(\mathbf{r}_j,\omega_+)P(\mathbf{r}_j,\omega_-), with ω±=ω±Δω/2\omega_\pm = \omega \pm \Delta\omega/2; averaged over the recorded signal bandwidth, these can mimic genuine acoustic fields at the difference frequency Δω\Delta\omega and the sum frequency Σω=2ω\Sigma\omega = 2\omega, respectively, because propagation through a multipath environment imprints nearly the same phase on nearby frequencies (Worthmann and Dowling 2017; see Dowling 2018 for a review). This idea underlies frequency-difference beamforming (Abadi et al. 2012) and frequency-difference MFP: Worthmann, Song and Dowling (2015, 2017) demonstrated below-band source localization at high frequencies in a shallow sound channel, and Geroski and Dowling (2019, 2021) developed long-range frequency-difference source localization in the deep ocean, matching the measured autoproduct against composite mode-ray replicas and phase-only matched-autoproduct processors. Laboratory measurements in a Lloyd's mirror environment — the direct-plus-surface-reflected-path interference pattern produced by a boundary above a source — first verified autoproduct mimicry near a pressure-release surface (Lipa et al. 2018).

Joslyn, Douglass and Dowling (2022) measured and compared the spatial coherence of the genuine field and both autoproducts using airgun signals (10–200 Hz) from the COAST 2012 experiment off the coast of Washington state, recorded on a nominally 8-km-long, 636-element seismic streamer towed at 9 m depth over roughly 1.9 km of water. Both autoproducts extended coherence to frequencies outside the signal band and generally produced longer coherence lengths than the genuine field (frequency-averaged 16.7 and 17.6 wavelengths for the difference and sum autoproducts versus 12.0 for the genuine field). A Bartlett MFP with image-theory replica weight vectors — image theory representing each boundary reflection by a mirror-image virtual source — combined a bottom-reflected path and a bottom-surface-reflected path, wj=e−iωRb,j/c/Rb,j±e−iωRs,j/c/Rs,jw_j = e^{-i\omega R_{b,j}/c}/R_{b,j} \pm e^{-i\omega R_{s,j}/c}/R_{s,j}, with the surface-reflection sign negative for in-band processing and positive for out-of-band autoproduct processing. This processor localized the source over most frequencies but showed anomalously low success fractions at 80 and 160 Hz.

The reason is a structural property of the quadratic construction. The sea surface is approximately pressure-release (a Dirichlet boundary with pressure reflection coefficient −1). Near such a surface the genuine field vanishes linearly with depth, so any quadratic product of fields at nearby frequencies varies quadratically with depth: its normal derivative vanishes at the surface, and the product's surface reflection coefficient is rectified, since R⋅R∗=∣R∣2=+1R \cdot R^* = |R|^2 = +1. The autoproducts therefore behave as if the ocean surface were a rigid (Neumann) boundary, which shifts the depths and frequencies of the Lloyd's-mirror destructive-interference nulls; this is why the autoproducts retain coherence at frequencies where the genuine field is nulled at the 9-m receiver depth. A genuine-field replica at the difference or sum frequency — even with its surface reflection coefficient sign flipped, as in the 2022 paper — captures only the crudest part of this physics. It omits, among other things, the quadratic amplitude scaling (the 2022 authors note that the 1/R→1/R21/R \to 1/R^2 change appropriate for the quadratic construction was not made, for simplicity), the phase structure of bandwidth-averaged field products, and the cross terms generated by the quadratic construction in multipath environments. The 2022 paper concludes that, at frequencies of constructive autoproduct interference, an alternate replica field scheme may be required in matched-field applications, and it explicitly leaves the systematic development and evaluation of such replicas outside its scope.

Problem Statement

Develop replica fields (weight vectors) for frequency-difference and frequency-sum autoproduct matched-field processing that are consistently derived from the quadratic autoproduct construction — incorporating the rectified rigid surface reflection condition, the resulting shifted surface-interference (Lloyd's-mirror) pattern, the quadratic amplitude scaling, and the effects of bandwidth averaging and cross terms — and evaluate them systematically, for example on the COAST 2012 towed-streamer geometry or comparable surface-reflection-dominated environments, to determine whether they restore source-localization performance at frequencies of constructive autoproduct interference (such as the 80 and 160 Hz failures identified by Joslyn, Douglass and Dowling, 2022), where genuine-field-style replicas with a simple surface-reflection sign flip fail despite high measured autoproduct coherence.

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