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Integrating Physics of the Problem into Data-Driven Methods to Enhance Elastic Full-Waveform Inversion with Uncertainty Quantification

Vahid Negahdari, Seyed Reza Moghadasi, Mohammad Reza Razvan

TL;DR

This paper proposes methods for the solution of time-harmonic FWI to enhance accuracy compared to pure data-driven and physics-based approaches and introduces a probabilistic deep learning method based on the physics of the problem that enables us to explore the uncertainties of the solution.

Abstract

Full-Waveform Inversion (FWI) is a nonlinear iterative seismic imaging technique that, by reducing the misfit between recorded and predicted seismic waveforms, can produce detailed estimates of subsurface geophysical properties. Nevertheless, the strong nonlinearity of FWI can trap the optimization in local minima. This issue arises due to factors such as improper initial values, the absence of low frequencies in the measurements, noise, and other related considerations. To address this challenge and with the advent of advanced machine-learning techniques, data-driven methods, such as deep learning, have attracted significantly increasing attention in the geophysical community. Furthermore, the elastic wave equation should be included in FWI to represent elastic effects accurately. The intersection of data-driven techniques and elastic scattering theories presents opportunities and challenges. In this paper, by using the knowledge of elastic scattering (physics of the problem) and integrating it with machine learning techniques, we propose methods for the solution of time-harmonic FWI to enhance accuracy compared to pure data-driven and physics-based approaches. Moreover, to address uncertainty quantification, by modifying the structure of the Variational Autoencoder, we introduce a probabilistic deep learning method based on the physics of the problem that enables us to explore the uncertainties of the solution. According to the limited availability of datasets in this field and to assess the performance and accuracy of the proposed methods, we create a comprehensive dataset close to reality and conduct a comparative analysis of the presented approaches to it.

Integrating Physics of the Problem into Data-Driven Methods to Enhance Elastic Full-Waveform Inversion with Uncertainty Quantification

TL;DR

This paper proposes methods for the solution of time-harmonic FWI to enhance accuracy compared to pure data-driven and physics-based approaches and introduces a probabilistic deep learning method based on the physics of the problem that enables us to explore the uncertainties of the solution.

Abstract

Full-Waveform Inversion (FWI) is a nonlinear iterative seismic imaging technique that, by reducing the misfit between recorded and predicted seismic waveforms, can produce detailed estimates of subsurface geophysical properties. Nevertheless, the strong nonlinearity of FWI can trap the optimization in local minima. This issue arises due to factors such as improper initial values, the absence of low frequencies in the measurements, noise, and other related considerations. To address this challenge and with the advent of advanced machine-learning techniques, data-driven methods, such as deep learning, have attracted significantly increasing attention in the geophysical community. Furthermore, the elastic wave equation should be included in FWI to represent elastic effects accurately. The intersection of data-driven techniques and elastic scattering theories presents opportunities and challenges. In this paper, by using the knowledge of elastic scattering (physics of the problem) and integrating it with machine learning techniques, we propose methods for the solution of time-harmonic FWI to enhance accuracy compared to pure data-driven and physics-based approaches. Moreover, to address uncertainty quantification, by modifying the structure of the Variational Autoencoder, we introduce a probabilistic deep learning method based on the physics of the problem that enables us to explore the uncertainties of the solution. According to the limited availability of datasets in this field and to assess the performance and accuracy of the proposed methods, we create a comprehensive dataset close to reality and conduct a comparative analysis of the presented approaches to it.
Paper Structure (20 sections, 1 theorem, 42 equations, 29 figures, 3 tables, 4 algorithms)

This paper contains 20 sections, 1 theorem, 42 equations, 29 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Governing equations and main results
  3. First method (Direct Deep Learning Inversion)
  4. Second method
  5. Least Square
  6. Inverse Convolution
  7. Linear-to-Nonlinear
  8. Third method (New Variational Autoencoder)
  9. New-VAE structue
  10. Fourth method ( Physics-Based )
  11. Gradient computation
  12. The Gauss-Newton method for approximation Hessian
  13. Truncated Newton Method (with Gauss Approximation)
  14. Dataset
  15. Generating random field
  16. ...and 5 more sections

Key Result

Theorem 2.1

(The Lippmann Schwinger integral equation) Given a real valued function $\rho \in C^{1,\gamma} (\mathbb{R}^2) (0<\gamma<1)$ with supp $(1-\rho)\subseteq \Omega$. If there exists a $\boldsymbol{u}^{sc} \in C^2(\mathbb{R}^2)$ that satisfies the boundary condition sommer and $\boldsymbol{u} = \boldsym conversely, if there exists a function $\boldsymbol{u} \in C^0(\mathbb{R}^2)$ that satisfies the in

Figures (29)

  • Figure 1: The above network is specifically configured to take the real components of surface displacement fields (for two separate horizontal and vertical sections, each of size $51$) corresponding to $k=9$ incident waves. The network's objective is to produce an output that represents the density, denoted as $\overline{\rho}$, with a total size of $N=n^2=2601$. We train this network using the dataset we made in section \ref{['Dataset']} and the $L_2$ loss function $\|\rho-\overline{\rho}\|^2_2$ between $\overline{\rho}$ and orginal $\rho$.
  • Figure 2: The above network is designed to take the real part of surface displacement fields (for $k=9$ incident waves) from two horizontal and vertical segments, each with a size of n = 51. This input is directed to achieve an estimation of the real part of field $U$ (called $\overline{ U}$) in whole area $\Omega$ with a size of $k * 2 n^2$ as an output (because for each incident wave, we have a horizontal and vertical part for displacement field). We train this network with the dataset we made in section \ref{['Dataset']}. The loss function for this network is defined as $||U - \overline{ U}||^2_2$, representing the $L_2$ norm between $\overline{ U}$ and the original $U$. We also train another network with the same structure to estimate the imaginary part of $U$.
  • Figure 3: The above network is designed to take the real part of displacement fields in the whole area $\Omega$ (for $k=9$ incident waves) from two horizontal and vertical segments, each with a size of $n=51$. This input is directed to achieve an estimation of the real part of field $\hat{\rho} U$ (called $\overline{\rho U}$) in whole area $\Omega$ with a size of $k * 2 n^2$ as an output (because for each incident wave, we have a horizontal and vertical part for displacement field). We train this network with the dataset we made in section \ref{['Dataset']} and estimations of displacement fields acquired from the beginning of this section. The loss function for this network is defined as $||\hat{\rho} U - \overline{\rho U}||^2_2$, representing the $L_2$ norm between $\overline{\rho U}$ and the original $\hat{\rho} U$. We also train another network with the same structure to estimate the imaginary part of $\hat{\rho} U$.
  • Figure 4: The above network is designed to take all $4k$ density estimates obtained in this method (4 is representative of 4 categories of real, imaginary, horizontal, and vertical components of estimation, and k = 9 is related to the number of incident waves) and produce a unique density as output, denoted as $\overline{\rho}$. We train this network with the dataset we made in section \ref{['Dataset']} and estimations of densities that were acquired from the previous network (Figure \ref{['rhou_Cnn']}). The loss function for this network is defined as $||\rho - \overline{\rho}||^2_2$, representing the $L_2$ norm between $\overline{\rho}$ and the original density $\rho$.
  • Figure 5: Structure and relation between variables in New-VAE
  • ...and 24 more figures

Theorems & Definitions (3)

  • Theorem 2.1
  • Remark 2.2
  • Remark 6.1