One-way full-waveform inversion using frequency-domain model extension - SEP 198 (2026)

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Rustam Akhmadiev Thesis

Table of contents

• Chapter 1: Introduction
  • 1.1 Proposed approach
  • 1.2 Contributions
  • 1.3 Thesis outline
     
• Chapter 2: Theory
  • 2.1 One-way wave extrapolation operators
    • 2.1.1 Nonlinear forward modeling operator
    • 2.1.2 Forward of the Jacobian operator
    • 2.1.3 Adjoint of the Jacobian operator
  • 2.2 Frequency-extended one-way wave equation
     
• Chapter 3: Inversion methods using frequency-extended models
  • 3.1 The nonlinear inversion (FWIX and OFWI)
    • 3.1.1 Regularization and preconditioning
    • 3.1.2 Instantaneous phase objective function
  • 3.2 The variable projection approach (FWIFE)
  • 3.3 Constrained optimization using the proximal gradient method and ADMM
    • 3.3.1 Proximal Gradient Method
    • 3.3.2 Alternating Direction Method of Multipliers (ADMM)
       
• Chapter 4: Numerical examples and applications
  • 4.1 One-way full-waveform inversion (OFWI)
    • 4.1.1 OFWI with a semicircular ultrasound array
    • 4.1.2 Constrained optimization and regularization
  • 4.2 The variable projection approach (FWIFE)
  • 4.3 The nonlinear extended inversion (FWIX)
    • 4.3.1 Slowness-only parametrization
    • 4.3.2 Slowness-impedance vs. slowness-density
       
• Chapter 5: 3D field data application
  • 5.1 Geological setting of the Campos basin and Marimbá field
  • 5.2 OBC data processing
  • 5.3 Results of FWIX
     
• Chapter 6: Model extension and renormalization group theory
  • 6.1 Method of multiple scales
  • 6.2 How extended modeling resums the scattering series
    • 6.2.1 Method of multiple scales applied to one-way wave equation
    • 6.2.2 Extended imaging explained by RG equation
    • 6.2.3 Continuous formulation6.3 Discussion
       
• Chapter 7: Conclusion

Abstract

This thesis studies a practical question in seismic imaging: how to make full-waveform inversion more robust when low frequencies are missing, illumination is incomplete, and starting models are uncertain. The central idea is to combine a computationally efficient one-way wave-equation engine with frequency-domain model extension, so the inversion can absorb phase mismatch while still converging toward physically meaningful velocity models.

I develop a consistent operator framework for one-way forward modeling, linearization, and adjoint-state gradients, then build inversion workflows on top of it. Two complementary formulations are used: FWIME, a variable-projection approach that estimates an extended linear variable in an inner least-squares step while updating a physical background model in the outer loop, and FWIX, a fully nonlinear extended inversion that removes the inner loop but requires stronger parameterization and preconditioning choices to preserve scale separation.

Numerical experiments on synthetic seismic and medical-ultrasound settings show the same recurring behavior: extension improves robustness to cycle skipping and poor initial models, while constraints and regularization are required to recover stable, interpretable physical models. In particular, proximal methods and ADMM can be used as practical tools for incorporating sparsity, total variation, and shaping-style priors. Multi-parameter parameterizations (slowness–density or slowness–impedance), spline-grid preconditioning, and continuation in frequency/scale are key ingredients for separating smooth background updates from high-wavenumber reflectivity.

A 3D field-data application on the Marimba OBC survey demonstrates that the workflow is operational beyond controlled synthetic tests: the recovered models are approximately frequency consistent, produce improved image focusing, and support velocity-model building in a reflection dominated setting.

Finally, I connect the extension framework to renormalization-group ideas, interpreting extended images as coarse-grained scattering potentials and providing a conceptual bridge between perturbation theory and practical extended inversion.

Reproducibility and source codes

https://github.com/arustamm/pywem
https://github.com/arustamm/pyfwix 
https://github.com/arustamm/pysep3d
https://github.com/arustamm/pythonSolver

Defense

Rustam Akhmadiev Defense PPT