Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

ABSTRACT We propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution i...

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Vydáno v:Geophysical Prospecting Ročník 69; číslo 1; s. 70 - 84
Hlavní autoři: Araujo, Edvaldo S., Pestana, Reynam C.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Houten Wiley Subscription Services, Inc 01.01.2021
Témata:
2D FD modelling
Acoustic
Acoustic waves
Acoustics
Anisotropy
Boundary conditions
Chebyshev approximation
Computing time
Elastic waves
Evolution
Exact solutions
Exponential functions
Formulas (mathematics)
Isotropic media
Mathematical models
Modelling
Operators (mathematics)
Perfectly matched layers
Series expansion
Taylor series
Wave equations
ISSN:0016-8025, 1365-2478
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Shrnutí:ABSTRACT We propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time‐steps than traditional finite‐difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the first‐order linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post‐stack migration results are also shown using the proposed method.
Bibliografie:Corrections added on 11 November 2020 after first online publication: updates have been made to paragraph eight of the Theory section, paragraph five of the Numerical Examples section and the Acknowledgements in this version.
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ISSN:0016-8025
1365-2478
DOI:10.1111/1365-2478.13033