Closing the simulation-to-field gap in passive waveguide-invariant estimation from tonal ship noise

The waveguide invariant $\beta$ summarizes the dispersive interference structure of shallow-water acoustic propagation in a single scalar and enables single-hydrophone passive ranging from broadband striations or tonal fading. Harms, Odom, and Krolik (ICASSP 2015) introduced a maximum-likelihood estimator of $\beta$ from the range-dependent fading of the strong tonal components radiated by transiting ships of known location, but verified it only in simulated Pekeris waveguides; the move to practical passive estimation remains open along the gaps implied by these stated assumptions and documented by the subsequent literature: real-data validation with AIS-tracked ships, joint estimation under uncertain source kinematics (with robustness to uncertain source/receiver depths desirable but not required), robustness to environmental model mismatch and laterally varying $\beta$, and data-driven treatment of the noise variance. Real-data tonal and waveguide-invariant-based estimation and ranging has since been demonstrated (SWellEx-96, AIS-tracked ships off Southern California, SBCEX17 commercial ships with range errors within $\pm 4\%$ out to 62 km), so the real-data feasibility item is resolved in the literature; the 2025 state of the art nevertheless still requires the source range rate from AIS (acoustic range-rate estimation was reported unreliable on ship radiated noise), assumes range-independent propagation, and treats data-estimated statistical and noise parameters as perfectly known. The surviving open core is the conjunction: a passive tonal-based framework that jointly estimates $\beta$ and source kinematics without external track information, remains valid in range-dependent environments with laterally varying $\beta$, and propagates noise-parameter estimation uncertainty, validated on real at-sea data.

Background

In shallow-water ocean acoustics, sound propagates through a set of dispersive normal modes — each a trapped acoustic field pattern whose phase and group velocities vary with frequency — whose mutual interference produces sloped intensity fringes ("striations") in the frequency–range or frequency–time plane. The slope of these striations is summarized, to the extent the environment cooperates, by a single scalar, the waveguide invariant $\beta$: two points $(r,f)$ and $(r',f')$ on the same striation satisfy $\beta = -(r/f)(df/dr)$ along the striation, and $\beta$ is called an invariant because in many environments it is approximately independent of range and frequency. For the Pekeris waveguide — an idealized shallow-water model consisting of an isovelocity water layer over a fluid, homogeneous half-space seabed — a closed-form analysis gives $\beta \approx \cos^2\theta$, with $\theta$ the modal grazing angle, so $\beta$ is close to 1 for near-horizontal propagation and departs from unity as bottom interaction strengthens (Byun and Song, 2022), and the classic analysis of D'Spain and Kuperman (1999) extended striation analysis to environments that vary in range and azimuth. Because $\beta$ encodes the environmental dispersion, knowing it enables passive source ranging from a single hydrophone with minimal environmental knowledge, and estimating it from ships of opportunity (rather than dedicated sources) is attractive for passive sonar, navigation, and environmental characterization. Ships of opportunity can be geolocated via the Automatic Identification System (AIS), the maritime VHF ship-tracking system. Earlier passive waveguide-invariant methods processed broadband striations; Harms, Odom, and Krolik (ICASSP 2015) instead exploited the strong narrowband tonal components (machinery lines) that dominate many ship spectra: as a ship transits, the received intensity of each tonal pair fades periodically with range in a pattern governed by $\beta$, and the paper derived a maximum-likelihood (ML) estimator of $\beta$ by relating the fading of different tonals over range for ships of known location (e.g., from AIS), with a noise variance $\sigma^2$ entering the likelihood. The method was verified only on simulated Pekeris-waveguide data, and the move toward practical passive estimation remains open along four directions implied by the paper's stated assumptions (ships of known location, simulation-only verification, and a noise variance entering the likelihood) and documented by the authors' follow-ups and the 2025 state of the art: real-data validation with tonal ship data and AIS-derived kinematics; joint estimation when source kinematics (relative range and range rate, i.e., radial velocity) are uncertain rather than known, with robustness to uncertain source/receiver depths desirable but not required; robustness to environmental model mismatch, including deviations from the Pekeris idealization, laterally varying $\beta$, and complex bathymetry; and estimation or adaptation of the unknown noise variance. The subsequent literature resolved the real-data feasibility of tonal and waveguide-invariant-based estimation (single-hydrophone ML estimation of $\beta$ and range on real SWellEx-96 data by Young and Harms in 2016; $\beta$ determination from AIS-tracked ships by Verlinden et al. in 2017; the journal-scale framework of Young et al. in 2020; the broadband-plus-tonal statistical model of Jang, Hodgkiss, and Meyer in 2025 validated on Seabed Characterization Experiment 2017 (SBCEX17) recordings of transiting commercial ships), while confirming that the other three directions remain: the 2016 work requires a known track and a separate range-rate estimate and degrades when they are wrong; the 2025 state of the art obtains the range rate from AIS, assumes a range-independent environment, and treats its data-estimated statistical parameters as perfectly known. Here SWellEx-96 denotes the 1996 shallow-water experiment off San Diego with GPS-tracked sources and ships of opportunity, and KRAKEN is a widely used normal-mode propagation model used to generate simulated replicas. A standard tool for judging such estimation problems is the Cramér–Rao bound, a fundamental lower bound on the variance of any unbiased estimator, computed from the Fisher information of the statistical model.

Problem Statement

Develop, and validate on real at-sea data, a passive estimation framework based on the tonal components radiated by transiting ships — received on a single hydrophone without coherent array aperture — that closes the remaining simulation-to-field gap left by Harms, Odom, and Krolik (2015) and still open in the 2025 state of the art. The framework must simultaneously achieve: (i) joint estimation — the waveguide invariant $\beta$ and the source kinematics required by the likelihood (at minimum the relative range and range rate between source and receiver; desirably also robustness to uncertain or mismatched source and receiver depths) must be estimated jointly from the acoustic data, so that externally supplied tracks (AIS/GPS) are needed only as ground truth for evaluation rather than as estimator inputs; Cramér–Rao-type identifiability analysis (treating tonal fading and Doppler information, observation time, and signal-to-noise ratio) may support the estimator design and performance assessment but does not by itself substitute for a working joint estimator; (ii) environmental robustness — the framework must remain accurate, by the range-error metric used for validation, in shallow-water environments that deviate from the ideal range-independent Pekeris model; laterally and range-varying $\beta$ and sloped or complex bathymetry are in-scope examples of such deviations rather than a conjunctive checklist, to be treated e.g. by incorporating range-dependent waveguide-invariant formulations into the statistical model; quantifying degradation under mismatch alone, without a range-dependent treatment, does not satisfy (ii); and (iii) noise-parameter treatment — the statistical parameters of the interference-plus-noise model, including the noise variance $\sigma^2$, must be estimated or marginalized from the data, with their estimation uncertainty propagated to the $\beta$ and range estimates rather than assumed perfectly known. Validation must be against independent positional ground truth (AIS/GPS-derived ranges) on real tonal ship data from the archived at-sea recordings of SWellEx-96 (event S5) or of SBCEX17 (transiting commercial ships), with performance compared against established baselines such as the ML estimators of Young et al. (2020) and the broadband-plus-tonal statistical method of Jang et al. (2025); simulated or re-simulated data do not satisfy this real-data validation requirement.

References

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