A Bayesian Perspective on Extended Full Waveform Inversion¶

Tristan van Leeuwen1,2 and Yuzhao Lin3

  1. Utrecht University, the Netherlands.↩

  2. Centrum Wiskunde en Informatica, the Netherlands.↩

  3. China University of Petroleum.↩

Overview¶

  • Formulation
  • Algorithms
  • Numerical examples

Formulation¶

We use the following process and measurement models

$$d = Pu + \epsilon, \quad \epsilon \sim \mathcal{N}(0,\Sigma_m),$$$$A(m)u = q + \eta, \quad \eta \sim \mathcal{N}(0,\Sigma_p),$$

where

  • $P$ is the sampling operator
  • $A(m)$ represents the PDE with coefficients $m$
  • $q$ represents the source

This leads to a MAP estimation problem

$$\min_{m,u} \textstyle{\frac{1}{2}}\|Pu - d\|_{\Sigma_m}^2 + \textstyle{\frac{1}{2}}\|A(m)u - q\|_{\Sigma_p}^2.$$

or equivalently,

$$\min_{m,f} \textstyle{\frac{1}{2}}\|PA(m)^{-1}(q + f) - d\|_{\Sigma_m}^2 + \textstyle{\frac{1}{2}}\|f\|_{\Sigma_p}^2.$$

Various extended source arise by choosing $\Sigma_p$ appropriately1,2.

  1. Huang, Guanghui, Rami Nammour, and William W. Symes. "Volume source-based extended waveform inversion." Geophysics 83.5 (2018): R369-R387.↩

  2. Symes, William W., Huiyi Chen, and Susan E. Minkoff. "Full-waveform inversion by source extension: Why it works." SEG Technical Program Expanded Abstracts 2020. Society of Exploration Geophysicists, 2020. 765-769.↩

Advantages of these formulations include

  • Take in to account errors in the physics
  • Make inversion more robust to initialisation

Conventional FWI¶

1

  1. van Leeuwen, Tristan, and Felix J. Herrmann. "A penalty method for PDE-constrained optimization in inverse problems." Inverse Problems 32.1 (2015): 015007.↩

Extended FWI¶

A reduced formulation¶

Since, the problem is quadratic in $u$ (or $f$), we can reduce the problem to1,2

$$\min_{m} \|PA(m)^{-1}q - d\|_{\Sigma(m)}^2,$$

with

$$\Sigma(m) = \Sigma_m + P(A(m)^*\Sigma_p^{-1}A(m))^{-1}P^*.$$

  1. van Leeuwen, Tristan. "A note on extended full waveform inversion." arXiv preprint arXiv:1904.00363 (2019).↩

  2. Fang, Zhilong, et al. "Uncertainty quantification for inverse problems with weak partial-differential-equation constraints." Geophysics 83.6 (2018): R629-R647.↩

The gradient of the corresponding objective can be computed via

$$g = G(m,u_0)^*\left(w_0 - v_0\right),$$

with

  • $G(m,u) = \frac{\partial A(m)u}{\partial m},$
  • $u_0 = A(m)^{-1}q,$
  • $v_0 = A(m)^{-*}P^*\Sigma(m)^{-1}(Pu_0 - d),$
  • $w_0 = A(m)^{-1}\Sigma_p^{-1}v_0.$

Algorithms¶

The main tasks of a basic gradient-descent method are

  • Solving the forward PDE to compute the residual
  • Solving a deconvolution problem to weigh the residual
  • Solving an additional forward and adjoint PDE to compute the gradient

The deconvolution requires inverting

$$\Sigma(m) = \Sigma_m + P(A(m)^*\Sigma_p^{-1}A(m))^{-1}P^*.$$
  • want to avoid forming and storing the full matrix
  • matvecs require additional PDE solves

We are free to choose $\Sigma_p$, however, so why not choose

$$\Sigma_p^{-1} = ss^*,$$

and invert $\Sigma(m)$ using the Sherman-Morrison identity?

$$\Sigma(m)^{-1} = \Sigma_m^{-1} - \frac{\Sigma_m^{-1} zz^* \Sigma_m^{-1}}{1 + z^*\Sigma_m^{-1}z},$$

with $z = PA(m)^{-1}s.$

  • $s = \alpha \cdot q$ for computational convenience
  • $s_i = \alpha\cdot \|x_i - x_s\|_2$ to emulate source-extension annihilator
  • ...

Numerical examples¶

Summary and outlook¶

Bayesian perspective on extended waveform inversion:

  • Connections to various other extensions
  • Efficient implementations in the time-domain
  • Explore connections to data-driven approaches

Other extensions¶

  • Source extensions
  • Contrast source1
  • Adaptive Waveform Inversion2
  • ...

  1. Van Den Berg, Peter M., and Ralph E. Kleinman. "A contrast source inversion method." Inverse problems 13.6 (1997): 1607.↩

  2. Warner, Michael, and Lluís Guasch. "Adaptive waveform inversion: Theory." Geophysics 81.6 (2016): R429-R445.↩

Time-domain¶

Apply these methods in the time-domain remains challenging. Some approaches are based on solving

$$\left(A^*\!A + \rho^{-1}P^*\!P\right)u = A^*q + \rho^{-1}P^*d.$$
  • Preconditioned iterative approach1,2
  • A dual formulation3
  • Approximating the solution for large $\rho$4
  • Neural Networks5

  1. van Leeuwen, Tristan, Peter Jan van Leeuwen, and Sergiy Zhuk. "Data-Driven Modeling for Wave-Propagation." ENUMATH. 2019.↩

  2. Aghamiry, Hossein S., Ali Gholami, and Stéphane Operto. "Accurate and efficient data-assimilated wavefield reconstruction in the time domain." Geophysics 85.2 (2020): A7-A12.↩

  3. Rizzuti, Gabrio, et al. "A dual formulation for time-domain wavefield reconstruction inversion." SEG Technical Program Expanded Abstracts 2019. Society of Exploration Geophysicists, 2019. 1480-1485.↩

  4. Li, Zhen-Chun, et al. "Time-domain wavefield reconstruction inversion." Applied Geophysics 14.4 (2017): 523-528.↩

  5. Song, Chao, and Tariq Alkhalifah. "Wavefield reconstruction inversion via physics-informed neural networks." arXiv preprint arXiv:2104.06897 (2021).↩

Inverse scattering¶

  • The joint approach estimates the wavefield and parameters in an alternating fashion
  • Alternative approaches to estimating the full wavefield from partial data are available1,2
  • Can we combine these approaches?

  1. Diekmann, Leon, and Ivan Vasconcelos. "Imaging with the exact linearised Lippmann-Schwinger integral by means of redatumed in-volume wavefields." SEG Technical Program Expanded Abstracts 2020. Society of Exploration Geophysicists, 2020. 3598-3602.↩

  2. Druskin, Vladimir, Shari Moskow, and Mikhail Zaslavsky. "Lippmann-Schwinger-Lanczos algorithm for inverse scattering problems." Inverse Problems (2021).↩

Thanks for your attention¶