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Recovering the sources in the stochastic wave equations from multi-frequency far-field patterns

Yan Chang, Yukun Guo, Zhipeng Yang, Yue Zhao

TL;DR

This work addresses the inverse stochastic source problem for both acoustic and elastic waves, where the source is modeled as a mean component plus white-noise driven fluctuations. It introduces a non-iterative Fourier-based reconstruction that links the far-field statistics to the Fourier coefficients of the mean $g$ and the variance $\sigma^2$, using carefully chosen multi-frequency data and sparse observation directions. The method yields explicit coefficient formulas for both the deterministic and stochastic parts and applies to 2D acoustic and elastic models with a unified framework, including specialized handling of the zero-frequency modes. Numerical results on both acoustic and elastic tests demonstrate accurate reconstructions and robustness to moderate noise, highlighting the practical viability of sparse, multi-frequency far-field measurements for characterizing source statistics.

Abstract

This paper concerns the inverse source scattering problems of recovering random sources for acoustic and elastic waves. The underlying sources are assumed to be random functions driven by an additive white noise. The inversion process aims to find the essential statistical characteristics of the mean and variance from the radiated random wave field at multiple frequencies. To this end, we propose a non-iterative algorithm by approximating the mean and variance via the truncated Fourier series. Then, the Fourier coefficients can be explicitly evaluated by sparse far-field measurements, resulting in an easy-to-implement and efficient approach for the reconstruction. Demonstrations with extensive numerical results are presented to corroborate the feasibility and robustness of the proposed method.

Recovering the sources in the stochastic wave equations from multi-frequency far-field patterns

TL;DR

This work addresses the inverse stochastic source problem for both acoustic and elastic waves, where the source is modeled as a mean component plus white-noise driven fluctuations. It introduces a non-iterative Fourier-based reconstruction that links the far-field statistics to the Fourier coefficients of the mean and the variance , using carefully chosen multi-frequency data and sparse observation directions. The method yields explicit coefficient formulas for both the deterministic and stochastic parts and applies to 2D acoustic and elastic models with a unified framework, including specialized handling of the zero-frequency modes. Numerical results on both acoustic and elastic tests demonstrate accurate reconstructions and robustness to moderate noise, highlighting the practical viability of sparse, multi-frequency far-field measurements for characterizing source statistics.

Abstract

This paper concerns the inverse source scattering problems of recovering random sources for acoustic and elastic waves. The underlying sources are assumed to be random functions driven by an additive white noise. The inversion process aims to find the essential statistical characteristics of the mean and variance from the radiated random wave field at multiple frequencies. To this end, we propose a non-iterative algorithm by approximating the mean and variance via the truncated Fourier series. Then, the Fourier coefficients can be explicitly evaluated by sparse far-field measurements, resulting in an easy-to-implement and efficient approach for the reconstruction. Demonstrations with extensive numerical results are presented to corroborate the feasibility and robustness of the proposed method.

Paper Structure

This paper contains 12 sections, 4 theorems, 84 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Acoustic waves
  3. Problem formulation
  4. Reconstruct the mean function $g$
  5. Reconstruct the variance function $\sigma^2$
  6. Numerical simulation
  7. Elastic waves
  8. Problem formulation
  9. Recover the mean function
  10. Recover the variance function
  11. Numerical simulations
  12. Conclusion

Key Result

Theorem 1

For $\bm l\in \mathbb{Z}^2$, let the wavenumber $k_{\bm l}$ and the observation direction $\widehat{x}_{\bm l}$ be defined by eq: admissible_mean and eq: xhat, respectively. Then the Fourier coefficients $\{\widehat{g}_{\bm l}\}$ of $g$ in eq: g can be determined by $\mathbf{E}[u^\infty(\widehat{x}_

Figures (6)

  • Figure 1: The exact source function. (a) mean function $g$ (b) the variance function $\sigma^2.$
  • Figure 2: The reconstructed source function for $\delta = 5\%$. (a) mean function $g_N$ (b) the variance function $\sigma_N^2.$
  • Figure 3: The reconstructed source function for $\delta=10\%$. (a) the mean $g_N$ (b) the variance $\sigma_N^2.$
  • Figure 4: The exact mean functions and variance functions.
  • Figure 5: The reconstructed mean functions and variance functions for $\delta = 5\%$.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 1: Admissible wavenumber to reconstruct the mean function
  • Theorem 1
  • proof
  • Definition 2: Admissible wavenumber to reconstruct the variance function
  • Theorem 2
  • proof
  • Definition 3: Admissible angular frequencies to reconstruct the mean function $\boldsymbol{g}$
  • Theorem 3
  • proof
  • Definition 4: Admissible angular frequency to recover the variance function
  • ...and 2 more