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WISER: multimodal variational inference for full-waveform inversion without dimensionality reduction

Ziyi Yin, Rafael Orozco, Felix J. Herrmann

TL;DR

FWI is a high-dimensional, nonlinear inverse problem with a multimodal posterior, making uncertainty quantification essential. The paper introduces WISER, a semi-amortized variational inference framework that blends offline conditional normalizing flows (CNFs) with online physics-based refinements to approximate the posterior $p( ext{velocity}| ext{data})$ without dimensionality reduction. A key idea is to use physics-informed statistics (CIGs) during CNF training and a weak, nested refinement to bridge the amortization gap, enabling accurate, full-resolution uncertainty estimates with reduced computational cost. Case studies on 2D Compass-model data show improved posterior fidelity and more continuous, accurate imaging under both in-distribution and out-of-distribution conditions, highlighting WISER’s potential for scalable, robust multi-D FWI and paving the way for 3D deployment.

Abstract

We present a semi-amortized variational inference framework designed for computationally feasible uncertainty quantification in 2D full-waveform inversion to explore the multimodal posterior distribution without dimensionality reduction. The framework is called WISER, short for full-Waveform variational Inference via Subsurface Extensions with Refinements. WISER leverages the power of generative artificial intelligence to perform approximate amortized inference that is low-cost albeit showing an amortization gap. This gap is closed through non-amortized refinements that make frugal use of acoustic wave physics. Case studies illustrate that WISER is capable of full-resolution, computationally feasible, and reliable uncertainty estimates of velocity models and imaged reflectivities.

WISER: multimodal variational inference for full-waveform inversion without dimensionality reduction

TL;DR

FWI is a high-dimensional, nonlinear inverse problem with a multimodal posterior, making uncertainty quantification essential. The paper introduces WISER, a semi-amortized variational inference framework that blends offline conditional normalizing flows (CNFs) with online physics-based refinements to approximate the posterior without dimensionality reduction. A key idea is to use physics-informed statistics (CIGs) during CNF training and a weak, nested refinement to bridge the amortization gap, enabling accurate, full-resolution uncertainty estimates with reduced computational cost. Case studies on 2D Compass-model data show improved posterior fidelity and more continuous, accurate imaging under both in-distribution and out-of-distribution conditions, highlighting WISER’s potential for scalable, robust multi-D FWI and paving the way for 3D deployment.

Abstract

We present a semi-amortized variational inference framework designed for computationally feasible uncertainty quantification in 2D full-waveform inversion to explore the multimodal posterior distribution without dimensionality reduction. The framework is called WISER, short for full-Waveform variational Inference via Subsurface Extensions with Refinements. WISER leverages the power of generative artificial intelligence to perform approximate amortized inference that is low-cost albeit showing an amortization gap. This gap is closed through non-amortized refinements that make frugal use of acoustic wave physics. Case studies illustrate that WISER is capable of full-resolution, computationally feasible, and reliable uncertainty estimates of velocity models and imaged reflectivities.
Paper Structure (17 sections, 2 equations, 2 figures, 1 algorithm)

This paper contains 17 sections, 2 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Amortized VI with WISE (lines 1---20)
  3. Dataset generation (lines 3---9)
  4. Network training (lines 11---12)
  5. Online inference (lines 14---20)
  6. Physics-based refinements (lines 22---32)
  7. Case studies
  8. Case 1: in distribution
  9. Observations
  10. Impact on imaging
  11. Case 2: out of distribution
  12. Observations
  13. Impact on imaging
  14. Discussion and conclusions
  15. Acknowledgement
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Comparison between WISE and WISER for an in-distribution case. (a) Unseen ground-truth velocity model. (b) Estimated velocity models from WISE. The conditional mean estimate (CM) is shown in the center. For posterior samples, horizontal traces at $Z = 2.7\,\mathrm{km}$ and vertical traces at $X = 3.6\,\mathrm{km}$ are displayed on the top and on the right, respectively. (d) Imaged reflectivity samples from WISE. (f) Zoom-in views of (d) overlaying on the CM of WISE. (c)(e)(g) are the counterparts from WISER, showcasing significant improvements.
  • Figure 2: OOD case study. (a) Curves for velocity-value perturbations; (b) histograms of values at the depth of $0.5\,\mathrm{km}$ and $2.8\,\mathrm{km}$ in the original velocity model (Figure \ref{['fig-true']}), perturbed velocity model (Figure \ref{['fig-v-ood']}), posterior samples of WISE, and WISER, shown in red, blue, yellow and green color, respectively. (c)---(i) Comparison between WISE and WISER. The ordering remains the same as in Figure \ref{['fig-wise-vs-wiser']}. (j) FWI result starting with a posterior sample from WISE. (k) A posterior sample from WISER.