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Simultaneous Estimation of Seabed and Its Roughness With Longitudinal Waves

Babak Maboudi Afkham, Ana Carpio

TL;DR

This work addresses the ill-posed inverse problem of seabed characterization from surface acoustic data by formulating an infinite-dimensional Bayesian framework that jointly infers the seabed profile and its roughness using a fractional differentiability parameter $s$. The forward model uses an elastic (scalar) wave equation, while the posterior is built with a Gaussian knee prior on seabed realizations via a Karhunen–Loève expansion and a surface-measurement likelihood; the posterior is shown to be well-posed. A Metropolis-within-Gibbs sampler with pCN updates for the seabed and a Metropolis-Hastings step for the regularity parameter $s$ enables uncertainty quantification and robust estimation even when the truth lies outside the prior support. Numerical experiments demonstrate accurate seabed reconstruction and plausible uncertainty bands in both known and uncertain regularity settings, including out-of-prior seabed scenarios, highlighting practical potential for large-scale seabed mapping. The approach provides a principled, discretization-invariant methodology for interpretable seabed inference with quantified uncertainty and lays groundwork for extensions to more realistic forward models and 3D geometries.

Abstract

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.

Simultaneous Estimation of Seabed and Its Roughness With Longitudinal Waves

TL;DR

This work addresses the ill-posed inverse problem of seabed characterization from surface acoustic data by formulating an infinite-dimensional Bayesian framework that jointly infers the seabed profile and its roughness using a fractional differentiability parameter . The forward model uses an elastic (scalar) wave equation, while the posterior is built with a Gaussian knee prior on seabed realizations via a Karhunen–Loève expansion and a surface-measurement likelihood; the posterior is shown to be well-posed. A Metropolis-within-Gibbs sampler with pCN updates for the seabed and a Metropolis-Hastings step for the regularity parameter enables uncertainty quantification and robust estimation even when the truth lies outside the prior support. Numerical experiments demonstrate accurate seabed reconstruction and plausible uncertainty bands in both known and uncertain regularity settings, including out-of-prior seabed scenarios, highlighting practical potential for large-scale seabed mapping. The approach provides a principled, discretization-invariant methodology for interpretable seabed inference with quantified uncertainty and lays groundwork for extensions to more realistic forward models and 3D geometries.

Abstract

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.
Paper Structure (20 sections, 8 theorems, 57 equations, 10 figures, 1 algorithm)

This paper contains 20 sections, 8 theorems, 57 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. The Forward Model: Scalar Wave Equation
  4. Goal-Oriented Inverse Problem
  5. Finite Element Discretization
  6. Bayesian Formulation of Seabed Inference
  7. Prior Modeling
  8. Example
  9. Likelihood
  10. Posterior
  11. Numerical Algorithm to Explore The Posterior
  12. Sampling from Lg
  13. Sampling from Lg
  14. Numerical Experiments and Results
  15. Numerical Details
  16. ...and 5 more sections

Key Result

Theorem 3.1

ibragimov2012gaussian Let $\eta$ be an $H^{\tau}$-valued Gaussian random function, then we can find $m \in H^{\tau}$ and trace-class, symmetric, and non-negative linear operator $\mathcal{C}:H^{\tau} \to H^{\tau}$, known as the covariance operator, such that where $\mathbb E$ denotes expectation ibragimov2012gaussian. We then define the push-forward probability measure $\mathcal{N}(m, \mathcal{C}

Figures (10)

  • Figure 1: Information on the computational domain for seabed tomography.
  • Figure 1: prior samples of $\eta$ for different regularity parameter $s$.
  • Figure 1: Diagnostic plots for the pCN and r methods. Top row: pCN results — (a) trace plot of the first 5 coefficients $\beta_j$, (b) posterior mean and highest posterior density interval for each coefficient. Bottom row: FES results — (c) trace plot of the first 5 coefficients, (d) posterior mean and highest posterior density interval.
  • Figure 2: Time snapshots of the solution to the discrete problem \ref{['eq:fully-discrete']}.
  • Figure 2: Estimated seabed and quantified uncertainty with 99% highest posterior density region. Left: pCN. Right: FES.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proof 1
  • Proposition 3.4
  • Proof 2
  • Theorem 3.5
  • Proof 3
  • ...and 7 more