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Recovering a (1+1)-dimensional wave equation from a single white noise boundary measurement

Emilia L. K. Blåsten, Tapio Helin, Antti Kujanpää, Lauri Oksanen, Jesse Railo

TL;DR

The paper addresses recovering a first-order coefficient in a (1+1)-dimensional wave equation on the half-line from passive boundary data where the Neumann input is Gaussian white noise and the Dirichlet trace is measured at the boundary. It introduces a correlation-imaging framework that converts stochastic boundary data into a deterministic Neumann-to-Dirichlet map, leveraging microlocal and energy-estimate tools to establish convergence of a correlation operator to the time-reversed ND map. A central contribution is proving that a single realization of white-noise boundary excitation suffices to distinguish different coefficients almost surely, yielding a uniqueness-type result in this passive setting. The findings have potential applications in acoustic sensing of pipe cross-sections and vocal-tract geometry, enabling imaging with minimal active probing by exploiting ambient fluctuations. The approach integrates stochastic analysis with inverse problems and microlocal theory to extend passive imaging methods to a one-point measurement regime.

Abstract

We consider the following inverse problem: Suppose a $(1+1)$-dimensional wave equation on $\mathbb{R}_+$ with zero initial conditions is excited with a Neumann boundary data modelled as a white noise process. Given also the Dirichlet data at the same point, determine the unknown first order coefficient function of the system. We first establish that direct problem is well-posed. The inverse problem is then solved by showing that correlations of the boundary data determine the Neumann-to-Dirichlet operator in the sense of distributions, which is known to uniquely identify the coefficient. This approach has applications in acoustic measurements of internal cross-sections of fluid pipes such as pressurised water supply pipes and vocal tract shape determination.

Recovering a (1+1)-dimensional wave equation from a single white noise boundary measurement

TL;DR

The paper addresses recovering a first-order coefficient in a (1+1)-dimensional wave equation on the half-line from passive boundary data where the Neumann input is Gaussian white noise and the Dirichlet trace is measured at the boundary. It introduces a correlation-imaging framework that converts stochastic boundary data into a deterministic Neumann-to-Dirichlet map, leveraging microlocal and energy-estimate tools to establish convergence of a correlation operator to the time-reversed ND map. A central contribution is proving that a single realization of white-noise boundary excitation suffices to distinguish different coefficients almost surely, yielding a uniqueness-type result in this passive setting. The findings have potential applications in acoustic sensing of pipe cross-sections and vocal-tract geometry, enabling imaging with minimal active probing by exploiting ambient fluctuations. The approach integrates stochastic analysis with inverse problems and microlocal theory to extend passive imaging methods to a one-point measurement regime.

Abstract

We consider the following inverse problem: Suppose a -dimensional wave equation on with zero initial conditions is excited with a Neumann boundary data modelled as a white noise process. Given also the Dirichlet data at the same point, determine the unknown first order coefficient function of the system. We first establish that direct problem is well-posed. The inverse problem is then solved by showing that correlations of the boundary data determine the Neumann-to-Dirichlet operator in the sense of distributions, which is known to uniquely identify the coefficient. This approach has applications in acoustic measurements of internal cross-sections of fluid pipes such as pressurised water supply pipes and vocal tract shape determination.

Paper Structure

This paper contains 10 sections, 15 theorems, 158 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Test functions and distributions
  4. The wave front set of a distribution
  5. The space D’_Σ(Rn )
  6. Gaussian white noise
  7. The initial-boundary value problem
  8. Derivation of Theorem \ref{['themain']}
  9. Energy estimates
  10. Sondhi and Gopinath 1971

Key Result

Theorem 1.3

Let the cut-off function $\chi \in C^\infty( \mathbb{R})$ be supported in $(0,\infty)$ and satisfy $\chi-1 \in C^\infty_c(0,\infty)$. Assume that $A_\nu \in C^\infty(\mathbb{R})$, $\nu = 1,2$ are admissible and denote by ${\Lambda}_{\nu}$ the Neumann-to-Dirichlet map (Definition DN-def) correspondin

Figures (3)

  • Figure 1: A visualization of the sets $(-\infty, \delta) \times \mathbb{R}$ and $W$. The light gray area represents the half space $(-\infty, \delta) \times \mathbb{R}$. The wedge $W$ is the darker shape on the right.
  • Figure 2: A visualization of $X$ in relation to $W$ and $(-\infty, \delta)\times \mathbb{R}$. The set $X$ is shown in dark gray. The medium and light gray areas represent $W$ and $(-\infty,\delta) \times \mathbb{R}$, respectively.
  • Figure 3: A visualization of $V$ in relation to the sets $W$ and $(-\infty, \delta ) \times \mathbb{R}$. The wedge $V$ is shown in dark gray. The medium and light gray areas represent $W$ and $(-\infty,\delta) \times \mathbb{R}$, respectively.

Theorems & Definitions (35)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Proposition 3.1: Existence of solutions
  • Lemma 3.2
  • proof : Proof of Proposition \ref{['exist-pro']}
  • proof : Proof of Lemma \ref{['weaker-exist']}
  • ...and 25 more