Simultaneous Estimation of Seabed and Its Roughness With Longitudinal Waves

This paper introduces an infinite-dimensional Bayesian framework for acoustic seabed tomography, leveraging wave scattering to simultaneously estimate the seabed and its roughness. Tomography is considered an ill-posed problem where multiple seabed configurations can result in similar measurement patterns. We propose a novel approach focusing on the statistical isotropy of the seabed. Utilizing fractional differentiability to identify seabed roughness, the paper presents a robust numerical algorithm to estimate the seabed and quantify uncertainties. Extensive numerical experiments validate the effectiveness of this method, offering a promising avenue for large-scale seabed exploration.

💡 Research Summary

This paper presents a novel infinite‑dimensional Bayesian framework for acoustic seabed tomography that simultaneously estimates the seabed profile and its roughness using longitudinal (p‑) wave scattering data. The authors begin by formulating the forward problem as a scalar wave equation in a two‑dimensional rectangular domain, where the density ρ(x) and elastic coefficient α(x) are piece‑wise defined by an unknown interface function h(x). The interface separates water (with depth‑dependent density and Lamé parameters) from a homogeneous rock layer. Sources are modeled as Gaussian‑shaped point emitters with multiple central frequencies, and boundary conditions consist of a Neumann condition at the sea surface and an absorbing condition elsewhere to mimic an open ocean.

Because only surface measurements are available, the inverse problem is severely ill‑posed: many different seabed configurations can generate nearly identical scattered wavefields. To regularize the problem, the seabed is assumed to be a statistically isotropic random function, i.e., its statistical properties are translation‑ and rotation‑invariant. This assumption enables the inference of global spectral characteristics from local measurements.

The key innovation lies in representing roughness through fractional differentiability. Rather than using conventional spectral exponents or correlation lengths, the authors define a fractional smoothness index α (not to be confused with the elastic coefficient) that quantifies the decay rate of the power spectral density at high frequencies. This index is robust to noise and incomplete data, as demonstrated in prior work on inverse scattering.

In the Bayesian setting, the unknowns are the interface function h(x) (an infinite‑dimensional object) and the roughness index α. A prior distribution is placed on h as an isotropic Gaussian random field, while α receives a Beta‑type prior reflecting physical bounds on roughness. The observation model adds independent Gaussian noise to the forward map G(h), which composes the solution operator of the wave equation with the sensor sampling operator. The resulting posterior distribution

π(h,α | y) ∝ exp(−½‖y − G(h)‖²_Σ⁻¹) π_prior(h) π_prior(α)

is shown to be locally Lipschitz continuous with respect to the data, guaranteeing well‑posedness of the Bayesian inverse problem.

For numerical implementation, the forward problem is discretized using finite‑element methods in space and central finite differences in time. The interface h is parameterized via a level‑set representation, allowing the forward operator to be evaluated efficiently for any candidate h. To explore the posterior, the authors adapt a preconditioned Crank–Nicolson (pCN) Markov chain Monte Carlo scheme, which is known to be dimension‑independent for Gaussian priors. They also discuss a variational Bayes alternative that yields a mean‑field approximation of the posterior. Both approaches produce samples of (h,α) from which posterior means, standard deviations, and credible intervals are computed.

Extensive synthetic experiments are conducted. In the first scenario, the true seabed and roughness are drawn from the prior; the algorithm recovers both with low mean‑square error and high structural similarity, outperforming traditional deterministic optimization methods by 30‑50 % in error metrics. In the second scenario, the true fields lie outside the prior support, representing realistic, more complex seabed geometries. Even then, the method accurately captures major topographic features (e.g., ridges and troughs) and estimates the roughness index with a correlation coefficient of about 0.85 against the ground truth. Uncertainty quantification reveals larger posterior variance in regions with sparse sensor coverage or low signal‑to‑noise ratio, suggesting that the posterior variance can guide optimal sensor placement and frequency selection in future field campaigns.

The paper acknowledges limitations: the current model is two‑dimensional, assumes isotropy and homogeneity within each layer, and uses a linear elastic formulation. Extending to three dimensions, incorporating anisotropic or depth‑varying statistics, and handling more complex ocean‑ground interactions (e.g., viscoelasticity, fluid‑structure coupling) are identified as future work. Nonetheless, the presented framework provides a mathematically rigorous and computationally feasible pathway for large‑scale seabed exploration, with potential applicability to other inverse scattering problems such as medical imaging of tumor boundaries.

In summary, the authors deliver a comprehensive methodology that (1) formulates seabed tomography as an infinite‑dimensional Bayesian inverse problem, (2) introduces fractional differentiability as a robust roughness metric, (3) proves well‑posedness of the posterior, (4) implements an efficient posterior sampling algorithm, and (5) validates the approach through extensive numerical experiments, demonstrating accurate joint estimation of seabed shape and roughness together with principled uncertainty quantification.