Solving the frequency-domain acoustic VTI wave equation using physics-informed neural networks

Type
Article

Authors
Song, Chao
Alkhalifah, Tariq Ali
Waheed, Umair bin

KAUST Department
Earth Science and Engineering Program
Physical Science and Engineering (PSE) Division
Seismic Wave Analysis Group

Online Publication Date
2021-01-11

Print Publication Date
2021-02-15

Date
2021-01-11

Submitted Date
2020-08-03

Abstract
SUMMARY Frequency-domain wavefield solutions corresponding to the anisotropic acoustic wave equation can be used to describe the anisotropic nature of the Earth. To solve a frequency-domain wave equation, we often need to invert the impedance matrix. This results in a dramatic increase in computational cost as the model size increases. It is even a bigger challenge for anisotropic media, where the impedance matrix is far more complex. In addition, the conventional finite-difference method produces numerical dispersion artefacts in solving acoustic wave equations for anisotropic media. To address these issues, we use the emerging paradigm of physics-informed neural networks (PINNs) to obtain wavefield solutions for an acoustic wave equation for transversely isotropic (TI) media with a vertical axis of symmetry (VTI). PINNs utilize the concept of automatic differentiation to calculate their partial derivatives, which are free of numerical dispersion artefacts. Thus, we use the wave equation as a loss function to train a neural network to provide functional solutions to the acoustic VTI form of the wave equation. Instead of predicting the pressure wavefields directly, we solve for the scattered pressure wavefields to avoid dealing with the point-source singularity. We use the spatial coordinates as input data to the network, which outputs the real and imaginary parts of the scattered wavefields and auxiliary function. After training a deep neural network, we can evaluate the wavefield at any point in space almost instantly using this trained neural network without calculating the impedance matrix inverse. We demonstrate these features on a simple 2-D anomaly model and a 2-D layered model. Additional tests on a modified 3-D Overthrust model and a 2-D model with irregular topography further validate the effectiveness of the proposed method.

Citation
Song, C., Alkhalifah, T., & Waheed, U. B. (2021). Solving the frequency-domain acoustic VTI wave equation using physics-informed neural networks. Geophysical Journal International, 225(2), 846–859. doi:10.1093/gji/ggab010

Acknowledgements
We thank KAUST for its support and the SWAG group for the collaborative environment. This work utilized the resources of the Supercomputing Laboratory at King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia, and we are grateful for that. We thank the editor, Dr Andrew Valentine, assistant editor, Fern Storey, and two reviewers Dr Anandaroop Ray and Dr Tarje Nissen-Meyer, for their critical and helpful review of the manuscript. We thank Bin She, from the University of Electronic Science and Technology, for sharing the 3-D plotting tool. We thank Dr Fabio Crameri for releasing a perceptually uniform colour map.

Publisher
Oxford University Press (OUP)

Journal
Geophysical Journal International

DOI
10.1093/gji/ggab010

arXiv
2008.01865

Additional Links
https://academic.oup.com/gji/article/225/2/846/6081098

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