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Coupling Deep Learning with Full Waveform Inversion

Wen Ding, Kui Ren, Lu Zhang

TL;DR

...

Abstract

Full waveform inversion (FWI) aims to reconstruct unknown physical coefficients in wave equations using the wavefield data generated from multiple incoming sources. In this work, we propose an offline-online computational strategy for coupling classical least-squares-based computational inversion with modern learning-based approaches for FWI to achieve advantages that can not be achieved with only one of the components. \RED{In brief, we develop an offline learning strategy to construct a robust approximation of the inverse operator through weighted optimization and utilize it to design a new objective function for approximate online inversion with new datasets. The approximate online inversion then serves as a warm start for the true online inversion.} We demonstrate through numerical simulations that our coupling strategy improves the computational efficiency of FWI with reliable offline training on moderate computational resources (in terms of both the size of the training dataset and the computational cost needed).

Coupling Deep Learning with Full Waveform Inversion

TL;DR

...

Abstract

Full waveform inversion (FWI) aims to reconstruct unknown physical coefficients in wave equations using the wavefield data generated from multiple incoming sources. In this work, we propose an offline-online computational strategy for coupling classical least-squares-based computational inversion with modern learning-based approaches for FWI to achieve advantages that can not be achieved with only one of the components. \RED{In brief, we develop an offline learning strategy to construct a robust approximation of the inverse operator through weighted optimization and utilize it to design a new objective function for approximate online inversion with new datasets. The approximate online inversion then serves as a warm start for the true online inversion.} We demonstrate through numerical simulations that our coupling strategy improves the computational efficiency of FWI with reliable offline training on moderate computational resources (in terms of both the size of the training dataset and the computational cost needed).
Paper Structure (42 sections, 4 theorems, 71 equations, 28 figures, 3 tables, 2 algorithms)

This paper contains 42 sections, 4 theorems, 71 equations, 28 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Coupling learning with FWI
  3. Robust offline learning of main features
  4. Online stage (i): main feature reconstruction
  5. Online stage (ii): full least-square refinement
  6. The benefits of the coupling approach
  7. Formal understanding of the coupling
  8. Elements of network training
  9. Sampling training data.
  10. Heuristic analysis of training error distribution.
  11. Inversion with accurate training
  12. Computational simplifications
  13. Refinement to reconstruct outside of training domain
  14. Computational implementation
  15. Computational setup
  16. ...and 27 more sections

Key Result

Proposition 3.1

Let $\Omega$ be a smooth domain, and $m\in [\underline{m}, \overline{m}]$ for some $0<\underline{m}<\overline{m}<+\infty$. Assume that $m \in W^{k,\infty}(\Omega)$ and $h\in W^{k-1/2,\infty}((0, T]\times \partial\Omega)$ ($k\ge 1$). Then there is a unique solution $u$ to EQ:Wave and it satisfies th Assume further that $m\in \mathcal{C}^4(\bar{\Omega})$. Then the map: is Fréchet differentiable at

Figures (28)

  • Figure 1: The two-dimensional computational domain $\Omega=(0, L)\times(-H, 0)$ for wave propagation. Periodic boundary conditions are imposed on the left and right boundaries. In geophysical applications, sources and detectors are placed on the top boundary (left), while in medical ultrasound applications, sources (red dots) and detectors (blue triangles) can be placed on both the top and the bottom boundaries (right).
  • Figure 2: Network flow for learning the approximate inverse operator. Training objective is to select $\theta$ such that $\mathbf g=D_\theta(E_\theta(\mathbf g))$ and $m=P_\theta(E_\theta(\mathbf g))$ for every datum pair ($\mathbf g$, $m$).
  • Figure 3: Random samples of the velocity field for training of the neural networks. Top row: velocity fields generated from \ref{['EQ:Velocity Model 1']} with $M = 2$; bottom row: velocity fields generated from \ref{['EQ:Velocity Model 2']} with $M = 4$.
  • Figure 4: The left panel presents time series wave signals at the bottom surface generated from a velocity model satisfying (\ref{['EQ:Velocity Model 1']}) with $M = 2$, while the right panel shows time series wave signals at the bottom surface generated from a velocity model constructed by (\ref{['EQ:Velocity Model 2']}) with $M = 4$. From the top to the bottom are time series wave signals without noise, with $10\%$ multiplication Gaussian noise and with $10\%$ additive Gaussian noise, respectively.
  • Figure 5: Three randomly selected velocity fields from the testing dataset: $5\times 5$ coefficients Fourier model, $8\times 8$ coefficients Fourier model, $10\times 10$ coefficients Fourier model. All cases have decay rate $\alpha=0$ (column 1), the corresponding predictions by the trained neural network (column 2), the error of the prediction (column 3), and the error in the neural network prediction ($\widetilde{m}(\mathbf x)$) in the Fourier domain ($\mathfrak{m}(\mathbf k)-\widetilde{\mathfrak{m}}(\mathbf k)$) (column 4).
  • ...and 23 more figures

Theorems & Definitions (7)

  • Proposition 3.1: BaSy-CPDE96DiDoNaPaSi-SIAM02Isakov-Book06
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof