Array coherence loss stops being discarded as noise: Topophone2 inverts array covariance into a slowness energy distribution
Synopsis
Array processing has long treated inter-sensor coherence loss as a nuisance, and conventional beamforming assumes a far-field point source with incoherent noise between sensors. Topophone2 models the array covariance as a mixture of energy distributions over slowness, describing signal and environmental noise with skew-normal and angular-harmonic distributions that have analytical characteristic functions, then computes the posterior by variational Bayesian inference. On a synthetic abyssal T-phase, adding a fourth hydrophone raises the information gain on the angular-harmonic noise coefficients by 142%, while the gain on signal parameters and SNR rises by less than 8%.
Interpretation
Inter-sensor coherence loss becomes information to be inverted rather than a nuisance to discard, yielding a posterior probability density over the energy distribution in slowness instead of a single best-fitting model. Conventional and adaptive beamforming assume a far-field point source, and detectors such as the F-detector further assume noise is incoherent between sensors; both assumptions fail for spatially distributed sources, multipathing and coherent environmental noise. On the synthetic abyssal T-phase, all true signal parameters lie within the 90% posterior credible intervals, SNR is recovered within its error bound of the true 10 dB, and the posterior probability that SNR exceeds 9.5 dB is 90%.
The inversion splits the wavefield into components, each with energy incoherently distributed over slowness according to a normalised density, and the cross-spectral density between a sensor pair is that density's characteristic function. The skew-normal distribution is parameterised by a location vector, scale matrix and shape vector, and its analytical characteristic function is known; the zero-shape limit recovers the coherency of a normal wavefield and the zero-scale limit recovers a single plane wave. The synthetic case uses five frequencies within a quarter-octave band centred on 5 Hz, synthesising a noise-only and a signal-plus-noise sample-covariance matrix at each, averaged over 96 and 16 realisations respectively, and inverting the ten matrices jointly.
The posterior feeds experimental design, model comparison and hypothesis testing, not only parameter estimation. Information gain is the KL divergence of the posterior from the prior, evaluated for parameter subsets, so array designs and noise levels can be compared. Relative to the baseline, adding a fourth hydrophone at the centre of the triplet raises information gain by less than 8% for signal parameters and SNR but by 142% for the angular-harmonic noise coefficients; raising environmental noise energy by 10 dB lowers information gain on SNR, mean signal slowness and slowness beamwidth by 15%, 31% and 56%.
Perspective
The demonstrated scope is one synthetic abyssal T-phase hydrophone-triplet case and one observed IS37 thermospheric infrasound return; the former is synthesised with a Munk sound-speed profile and a patch of incoherent seafloor sources, the latter is the recording of an approximately 20 t surface chemical explosion at Hukkakero on 19 August 2016. The method is provided as the open-source Python package Topophone2.
The assumptions behind the complex Wishart likelihood are rarely satisfied exactly, and most violations make the posterior over-confident rather than biased; unmodelled differences between sensors, such as timing offsets, gain mismatches, reversed polarity and unequal self-noise, instead bias the posterior, so data quality must be verified before inversion. The forward problem is restricted to incoherently distributed wavefields, and the authors propose partially coherent wavefields and multicomponent vector sensors as natural extensions.