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A Unified Framework for Forward and Inverse Problems in Subsurface Imaging using Latent Space Translations

Naveen Gupta, Medha Sawhney, Arka Daw, Youzuo Lin, Anuj Karpatne

TL;DR

This work unifies forward and inverse problems in subsurface imaging within the Generalized Forward-Inverse (GFI) framework, leveraging latent space translations between velocity maps $v$ and seismic waveforms $p$. It introduces two architectures, Latent U-Net and Invertible X-Net, to perform domain translations in latent spaces and jointly solve both tasks. Empirical results on the OpenFWI suite show state-of-the-art performance for both forward and inverse problems, with strong zero-shot generalization to real-world-like datasets such as Marmousi and Overthrust. The findings offer practical guidance on latent-space size, architectural complexity, and the value of joint training, suggesting future directions toward dataset-agnostic and more generalizable subsurface imaging models.

Abstract

In subsurface imaging, learning the mapping from velocity maps to seismic waveforms (forward problem) and waveforms to velocity (inverse problem) is important for several applications. While traditional techniques for solving forward and inverse problems are computationally prohibitive, there is a growing interest in leveraging recent advances in deep learning to learn the mapping between velocity maps and seismic waveform images directly from data. Despite the variety of architectures explored in previous works, several open questions still remain unanswered such as the effect of latent space sizes, the importance of manifold learning, the complexity of translation models, and the value of jointly solving forward and inverse problems. We propose a unified framework to systematically characterize prior research in this area termed the Generalized Forward-Inverse (GFI) framework, building on the assumption of manifolds and latent space translations. We show that GFI encompasses previous works in deep learning for subsurface imaging, which can be viewed as specific instantiations of GFI. We also propose two new model architectures within the framework of GFI: Latent U-Net and Invertible X-Net, leveraging the power of U-Nets for domain translation and the ability of IU-Nets to simultaneously learn forward and inverse translations, respectively. We show that our proposed models achieve state-of-the-art (SOTA) performance for forward and inverse problems on a wide range of synthetic datasets, and also investigate their zero-shot effectiveness on two real-world-like datasets. Our code is available at https://github.com/KGML-lab/Generalized-Forward-Inverse-Framework-for-DL4SI

A Unified Framework for Forward and Inverse Problems in Subsurface Imaging using Latent Space Translations

TL;DR

This work unifies forward and inverse problems in subsurface imaging within the Generalized Forward-Inverse (GFI) framework, leveraging latent space translations between velocity maps and seismic waveforms . It introduces two architectures, Latent U-Net and Invertible X-Net, to perform domain translations in latent spaces and jointly solve both tasks. Empirical results on the OpenFWI suite show state-of-the-art performance for both forward and inverse problems, with strong zero-shot generalization to real-world-like datasets such as Marmousi and Overthrust. The findings offer practical guidance on latent-space size, architectural complexity, and the value of joint training, suggesting future directions toward dataset-agnostic and more generalizable subsurface imaging models.

Abstract

In subsurface imaging, learning the mapping from velocity maps to seismic waveforms (forward problem) and waveforms to velocity (inverse problem) is important for several applications. While traditional techniques for solving forward and inverse problems are computationally prohibitive, there is a growing interest in leveraging recent advances in deep learning to learn the mapping between velocity maps and seismic waveform images directly from data. Despite the variety of architectures explored in previous works, several open questions still remain unanswered such as the effect of latent space sizes, the importance of manifold learning, the complexity of translation models, and the value of jointly solving forward and inverse problems. We propose a unified framework to systematically characterize prior research in this area termed the Generalized Forward-Inverse (GFI) framework, building on the assumption of manifolds and latent space translations. We show that GFI encompasses previous works in deep learning for subsurface imaging, which can be viewed as specific instantiations of GFI. We also propose two new model architectures within the framework of GFI: Latent U-Net and Invertible X-Net, leveraging the power of U-Nets for domain translation and the ability of IU-Nets to simultaneously learn forward and inverse translations, respectively. We show that our proposed models achieve state-of-the-art (SOTA) performance for forward and inverse problems on a wide range of synthetic datasets, and also investigate their zero-shot effectiveness on two real-world-like datasets. Our code is available at https://github.com/KGML-lab/Generalized-Forward-Inverse-Framework-for-DL4SI

Paper Structure

This paper contains 45 sections, 2 theorems, 15 equations, 34 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Deep Learning for Subsurface Imaging (DL4SI)
  4. Domain Translation Methods
  5. Proposed Methodologies
  6. Generalized Forward Inverse Framework
  7. Prior Works in DL4SI as Special Cases of GFI
  8. Proposed Model: Latent U-Net
  9. Proposed Model: Invertible X-Net
  10. Experimental Setup
  11. Datasets
  12. Model Architectures
  13. Results and Discussions
  14. Key Insights on Open Questions in DL4SI
  15. What is the Effect of Latent Space Sizes on Translation Performance?
  16. ...and 30 more sections

Key Result

Lemma A.1

Let $f: \mathcal{V} \rightarrow \mathcal{P}$ be an arbitrary forward operator mapping velocity maps $\mathcal{V}$ to seismic waveforms $\mathcal{P}$. Let $E_v: \mathcal{V} \rightarrow \tilde{\mathcal{V}}$ and $D_v: \tilde{\mathcal{V}} \rightarrow \mathcal{V}$ denote the encoder and decoder for the v

Figures (34)

  • Figure 1: A unified framework for solving forward and inverse problems in subsurface imaging.
  • Figure 2: Schematic representation of proposed Latent U-Net and Invertible X-Net model.
  • Figure 3: Comparison of Latent U-Nets (Small and Large), Invertible X-Net, Invertible X-Net (Cycle) with different baseline methods across different OpenFWI datasets.
  • Figure 4: Visualization of model predictions for inverse and forward problems across different OpenFWI datasets: CVB, CFB, STA.
  • Figure 5: Effect of latent dimensions and skip connections (across latent dimensions) on the performance of Latent U-Net (Large) (a and b) and Invertible X-Net (c). 'A' and 'B' show 'low' and 'high' complexity scenarios for each data group.
  • ...and 29 more figures

Theorems & Definitions (4)

  • Lemma A.1: Forward Latent Space Translation Assumption
  • proof
  • Lemma A.2: Inverse Latent Space Translation Assumption
  • proof