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Full Waveform Inversion and Lagrange Multipliers

Ali Gholami

TL;DR

This work presents a unified Lagrangian framework for full waveform inversion (FWI) that encompasses quadratic penalty and augmented Lagrangian (AL) formulations, recasting them in terms of both wavefields and Lagrange multipliers. By analyzing standard, penalty-based (WRI) and AL-based (IR-WRI) schemes, the authors reveal two equivalent yet complementary implementation paths: wavefield-oriented and multiplier-oriented, with multipliers interpreted as physically meaningful scattering sources. The paper further connects these optimization-based methods to scattering theory via the Lippmann–Schwinger formulation, derives a suite of iterative schemes (Gauss-Seidel, Gauss-Newton, and their split variants), and proposes refined approaches to account for high-order scattering and error terms, including memory-efficient formulations. The insights enable time-domain, physically interpretable, and computationally robust FWI methods that improve convergence and practical applicability, especially for inaccurate initial models and complex subsurface structures.

Abstract

Full-waveform inversion (FWI) is an effective method for imaging subsurface properties using sparsely recorded data. It involves solving a wave propagation problem to estimate model parameters that accurately reproduce the data. Recent trends in FWI have led to the development of extended methodologies, among which source extension methods leveraging reconstructed wavefields to solve penalty or augmented Lagrangian (AL) formulations have emerged as robust algorithms, even for inaccurate initial models. Despite their demonstrated robustness, challenges remain, such as the lack of a clear physical interpretation, difficulty in comparison, and reliance on difficult-to-compute least squares (LS) wavefields. This paper is divided into two critical parts. In the first, a novel formulation of these methods is explored within a unified Lagrangian framework. This novel perspective permits the introduction of alternative algorithms that employ LS multipliers instead of wavefields. These multiplier-oriented variants appear as regularizations of the standard FWI, are adaptable to the time domain, offer tangible physical interpretations, and foster enhanced convergence efficiency. The second part of the paper delves into understanding the underlying mechanisms of these techniques. This is achieved by solving the FWI equations using iterative linearization and inverse scattering methods. The paper provides insight into the role and significance of Lagrange multipliers in enhancing the linearization of FWI equations. It explains how different methods estimate multipliers or make approximations to increase computing efficiency. Additionally, it presents a new physical understanding of the Lagrange multiplier used in the AL method, highlighting how important it is for improving algorithm performance when compared to penalty methods.

Full Waveform Inversion and Lagrange Multipliers

TL;DR

This work presents a unified Lagrangian framework for full waveform inversion (FWI) that encompasses quadratic penalty and augmented Lagrangian (AL) formulations, recasting them in terms of both wavefields and Lagrange multipliers. By analyzing standard, penalty-based (WRI) and AL-based (IR-WRI) schemes, the authors reveal two equivalent yet complementary implementation paths: wavefield-oriented and multiplier-oriented, with multipliers interpreted as physically meaningful scattering sources. The paper further connects these optimization-based methods to scattering theory via the Lippmann–Schwinger formulation, derives a suite of iterative schemes (Gauss-Seidel, Gauss-Newton, and their split variants), and proposes refined approaches to account for high-order scattering and error terms, including memory-efficient formulations. The insights enable time-domain, physically interpretable, and computationally robust FWI methods that improve convergence and practical applicability, especially for inaccurate initial models and complex subsurface structures.

Abstract

Full-waveform inversion (FWI) is an effective method for imaging subsurface properties using sparsely recorded data. It involves solving a wave propagation problem to estimate model parameters that accurately reproduce the data. Recent trends in FWI have led to the development of extended methodologies, among which source extension methods leveraging reconstructed wavefields to solve penalty or augmented Lagrangian (AL) formulations have emerged as robust algorithms, even for inaccurate initial models. Despite their demonstrated robustness, challenges remain, such as the lack of a clear physical interpretation, difficulty in comparison, and reliance on difficult-to-compute least squares (LS) wavefields. This paper is divided into two critical parts. In the first, a novel formulation of these methods is explored within a unified Lagrangian framework. This novel perspective permits the introduction of alternative algorithms that employ LS multipliers instead of wavefields. These multiplier-oriented variants appear as regularizations of the standard FWI, are adaptable to the time domain, offer tangible physical interpretations, and foster enhanced convergence efficiency. The second part of the paper delves into understanding the underlying mechanisms of these techniques. This is achieved by solving the FWI equations using iterative linearization and inverse scattering methods. The paper provides insight into the role and significance of Lagrange multipliers in enhancing the linearization of FWI equations. It explains how different methods estimate multipliers or make approximations to increase computing efficiency. Additionally, it presents a new physical understanding of the Lagrange multiplier used in the AL method, highlighting how important it is for improving algorithm performance when compared to penalty methods.
Paper Structure (24 sections, 46 equations, 1 figure)

This paper contains 24 sections, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. NOTATION
  3. Lagrangian based full waveform inversion
  4. The Standard Lagrangian
  5. Regularization by the quadratic penalty method
  6. Wavefield-oriented penalty method
  7. Multiplier-oriented penalty method
  8. Wavefield-oriented vs. multiplier-oriented approaches
  9. Connection with WRI
  10. Regularization by the AL method
  11. Wavefield-oriented AL method
  12. Multiplier-oriented AL method
  13. Penalty vs. AL methods
  14. The basic iterations, mechanisms, and physical interpretations
  15. The Gauss-Seidel iteration
  16. ...and 9 more sections

Figures (1)

  • Figure 1: Interconnections between iterative methods for solving FWI equations. The circled numbers represent the corresponding equation numbers in the text. Each iteration consists of solving a series of equations sequentially, updating the variables denoted with $\bullet^+$ using the background values $\bullet$ and the previous values $\bullet^-$.