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Reduced Order Modeling for First Order Hyperbolic Systems with Application to Multiparameter Acoustic Waveform Inversion

Liliana Borcea, Josselin Garnier, Alexander V. Mamonov, Jörn Zimmerling

TL;DR

The paper develops a data-driven reduced order model (ROM) for general first-order hyperbolic systems, enabling efficient, non-iterative construction of a data-consistent ROM from sensor measurements. By projecting the time-stepping scheme onto the snapshot space and leveraging a data-interpolation relation, the ROM yields an internal-vector wave field that aligns with measurements but requires accurate medium estimates to satisfy the underlying hyperbolic equations. The authors specialize the framework to multiparameter acoustic inversion, where wave speed $c$ and density $\rho$ are recovered by minimizing a ROM-based objective, using layer-stripping and Gauss-Newton with regularized mass matrices to stabilize the inversion. Numerical results on disjoint inclusions, Marmousi-like models, and crack scenarios show the ROM approach mitigates cycle skipping and cross-talk inherent in full waveform inversion (FWI), delivering more accurate reconstructions under various noise levels. The work broadens ROM-based inversion to vectorial waves and non-coincident source-receiver geometries, with potential impact on radar, seismology, and ultrasound imaging where reliable multiparameter estimates are essential.

Abstract

Waveform inversion seeks to estimate an inaccessible heterogeneous medium from data gathered by sensors that emit probing signals and measure the generated waves. It is an inverse problem for a second order wave equation or a first order hyperbolic system, with the sensor excitation modeled as a forcing term and the heterogeneous medium described by unknown, spatially variable coefficients. The traditional ``full waveform inversion" (FWI) formulation estimates the unknown coefficients via minimization of the nonlinear, least squares data fitting objective function. For typical band-limited and high frequency data, this objective function has spurious local minima near and far from the true coefficients. Thus, FWI implemented with gradient based optimization algorithms may fail, even for good initial guesses. Recently, it was shown that it is possible to obtain a better behaved objective function for wave speed estimation, using data driven reduced order models (ROMs) that capture the propagation of pressure waves, governed by the classic second order wave equation. Here we introduce ROMs for vectorial waves, satisfying a general first order hyperbolic system. They are defined via Galerkin projection on the space spanned by the wave snapshots, evaluated on a uniform time grid with appropriately chosen time step. Our ROMs are data driven: They are computed in an efficient and non-iterative manner, from the sensor measurements, without knowledge of the medium and the snapshots. The ROM computation applies to any linear waves in lossless and non-dispersive media. For the inverse problem we focus attention on acoustic waves in a medium with unknown variable wave speed and density. We show that these can be determined via minimization of an objective function that uses a ROM based approximation of the vectorial wave field inside the inaccessible medium.

Reduced Order Modeling for First Order Hyperbolic Systems with Application to Multiparameter Acoustic Waveform Inversion

TL;DR

The paper develops a data-driven reduced order model (ROM) for general first-order hyperbolic systems, enabling efficient, non-iterative construction of a data-consistent ROM from sensor measurements. By projecting the time-stepping scheme onto the snapshot space and leveraging a data-interpolation relation, the ROM yields an internal-vector wave field that aligns with measurements but requires accurate medium estimates to satisfy the underlying hyperbolic equations. The authors specialize the framework to multiparameter acoustic inversion, where wave speed and density are recovered by minimizing a ROM-based objective, using layer-stripping and Gauss-Newton with regularized mass matrices to stabilize the inversion. Numerical results on disjoint inclusions, Marmousi-like models, and crack scenarios show the ROM approach mitigates cycle skipping and cross-talk inherent in full waveform inversion (FWI), delivering more accurate reconstructions under various noise levels. The work broadens ROM-based inversion to vectorial waves and non-coincident source-receiver geometries, with potential impact on radar, seismology, and ultrasound imaging where reliable multiparameter estimates are essential.

Abstract

Waveform inversion seeks to estimate an inaccessible heterogeneous medium from data gathered by sensors that emit probing signals and measure the generated waves. It is an inverse problem for a second order wave equation or a first order hyperbolic system, with the sensor excitation modeled as a forcing term and the heterogeneous medium described by unknown, spatially variable coefficients. The traditional ``full waveform inversion" (FWI) formulation estimates the unknown coefficients via minimization of the nonlinear, least squares data fitting objective function. For typical band-limited and high frequency data, this objective function has spurious local minima near and far from the true coefficients. Thus, FWI implemented with gradient based optimization algorithms may fail, even for good initial guesses. Recently, it was shown that it is possible to obtain a better behaved objective function for wave speed estimation, using data driven reduced order models (ROMs) that capture the propagation of pressure waves, governed by the classic second order wave equation. Here we introduce ROMs for vectorial waves, satisfying a general first order hyperbolic system. They are defined via Galerkin projection on the space spanned by the wave snapshots, evaluated on a uniform time grid with appropriately chosen time step. Our ROMs are data driven: They are computed in an efficient and non-iterative manner, from the sensor measurements, without knowledge of the medium and the snapshots. The ROM computation applies to any linear waves in lossless and non-dispersive media. For the inverse problem we focus attention on acoustic waves in a medium with unknown variable wave speed and density. We show that these can be determined via minimization of an objective function that uses a ROM based approximation of the vectorial wave field inside the inaccessible medium.
Paper Structure (20 sections, 1 theorem, 99 equations, 5 figures, 2 algorithms)

This paper contains 20 sections, 1 theorem, 99 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. ROM for first order hyperbolic systems
  3. Time stepping and the snapshots
  4. Galerkin approximation of the snapshots
  5. Galerkin projection ROM
  6. Data driven ROM
  7. Properties of the ROM
  8. Acoustic waves
  9. Derivation of the first order system \ref{['eq:LS1']}
  10. The data
  11. Multiparametric inversion
  12. The inversion algorithm
  13. Regularization of the ROM computation
  14. Numerical results
  15. Disjoint inclusions
  16. ...and 5 more sections

Key Result

Theorem 2.2

\newlabelthm.1 The ROM can be computed directly from the array response matrix eq:LS3, that defines the "data matrices" The data matrices satisfy where the second equality means that the ROM interpolates the measurements on the uniform time grid. The ROM is computed from $\{{\itbf D}_j\}_{j=0}^{n_t}$ and its propagator matrix eq:LS31 has an unreduced, upper Hessenberg block structure.

Figures (5)

  • Figure 3.1: Illustration of the acoustic setup for inverse scattering with polarized exitations ${\bm F}_{\bm \epsilon}( \bm x)$. The inversion domain is shown in darker blue and is surrounded by a homogeneous, reference medium. The sensors are placed above this domain and generate waves with $d$ different polarizations (here $d=2$).
  • Figure 4.1: Two disjoint inclusions. Top row: true velocity and density; middle row: ROM based estimates; bottom row: MFWI estimates. Source locations are yellow crosses. Axes are in km. The colorbar shows the contrast i.e., ratios of $c$ and $\rho$ with the reference values $c_o$ and $\rho_o$ of the medium near the array.
  • Figure 4.2: Two-coefficient Marmousi model. Top row: true velocity and density; middle row: ROM based estimates; bottom row: MFWI estimates. Source locations are yellow crosses. Axes are in km. The colorbar shows the contrast.
  • Figure 4.3: First crack model. Top row: true velocity and density; second row: ROM based estimates from noiseless data; third row: ROM based estimates from noisy data with $b = 10^{-2}$; bottom row: MFWI estimates from noiseless data. Source locations are yellow crosses. Axes are in km. The colorbar shows the contrast.
  • Figure 4.4: Second crack model. Top row: true velocity and density; second row: ROM based estimates from noiseless data; third row: ROM based estimates from noisy data with $b = 3 \cdot 10^{-2}$; bottom row: MFWI estimates from noiseless data. Source locations are yellow crosses. Axes are in km. The colorbar shows the contrast.

Theorems & Definitions (5)

  • Remark 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4