Engineering

On the Steadfastness of the Least-Square Reverse-Time Migration Wavefield Extrapolation via 1st-Order Riemannian Axis Finite-Difference Solver

Authors

  • HUSSEIN MUHAMMED

    Shandong Provincial Key Laboratory of Reservoir Geology, China University of Petroleum (East China)
    Author
    https://orcid.org/0000-0002-0453-3686

  • Xiaodong Sun

    Shandong Provincial Key Laboratory of Reservoir Geology, China University of Petroleum (East China), Qingdao, 266580, China
    Author

  • Liping Gao

    College of Sciences and Arts, School of Mathematics, China University of Petroleum (East China), Qingdao, 266580, China
    Author

  • AbdelHafiz Gadelmula

    Centre for Seismological Phenomena, Department of Geology, Faculty of Science, University of Khartoum, Sudan
    Author

  • Zhenchun Li

    Shandong Provincial Key Laboratory of Reservoir Geology, China University of Petroleum (East China), Qingdao, 266580, China
    Author

DOI:

https://doi.org/10.71107/dh3px571

Keywords:

Seismic waves propagation; Stability analysis; Riemannian metric tensors; Claerbout’s principle; Finite-Difference modelling.

Abstract

Exploring Earth's deep regions through pioneering Least-Squares Reverse-Time Migration (LSRTM) methods is of significant interest due to its exceptional structural clarity. This cutting-edge seismic imaging technique is time-consuming and memory-intensive, so wavefield extrapolation is proposed in the Pseudodepth domain (1st-order Riemannian coordinate system’s axis) to address these issues and prevent oversampling/aliasing when modeling deeper subsurface zones. Stabilizing the generated Riemannian wavefield involves implementing an appropriate mapping velocity and obtaining the vertical axis operator that partially converts the finite difference solver from time to frequency domains. Each Cartesian point  has a corresponding vertical-time point , allowing interpolation of the reconstructed source wavefield through a Cartesian-to-Riemannian mapping function. Our stability and convergence analysis indicates that the spatial derivatives of the 1st-order Riemannian axis can be approximated by Fourier pseudo-spectral methods and fast-Fourier transforms using a special Gaussian-like impulse function. This function generates the source term vector-matrix within the finite-difference operator. The mapping velocity, derived as a differential form of the initial input velocity model, controls the CFL conditions of the associated Riemannian-finite difference operator. Numerical, synthetic, and seismic field data examples show that this approach is more stable and efficient in extrapolating a smooth 1st-order Riemannian axis-based finite-difference wavefield while adhering to Claerbout’s principle for locating subsurface reflectors. Additionally, choosing the appropriate sampling rate for the new vertical axis is inversely related to the maximum frequency of the impulse wavelet and directly related to the minimum velocity value in the model.

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Author Biographies

  • HUSSEIN MUHAMMED, Shandong Provincial Key Laboratory of Reservoir Geology, China University of Petroleum (East China)

    B.Sc. (Hons.) degree in Geology from the University of Khartoum, Sudan (Oct. 2015) and M.Sc. degree in Geological Engineering from China University of Petroleum (East China). PhD in math at Northwestern Polytechnical University. His research interests include: Geophysical Exploration Methods, Unified Field theory, Finite-Difference Methods, Finite-Element modeling, CFDs and Numerical Analysis.

  • Xiaodong Sun, Shandong Provincial Key Laboratory of Reservoir Geology, China University of Petroleum (East China), Qingdao, 266580, China

    Shandong Provincial Key Laboratory of Reservoir Geology, China University of Petroleum (East China), Qingdao, 266580, China

  • Liping Gao, College of Sciences and Arts, School of Mathematics, China University of Petroleum (East China), Qingdao, 266580, China

    College of Sciences and Arts, School of Mathematics, China University of Petroleum (East China), Qingdao, 266580, China

  • AbdelHafiz Gadelmula, Centre for Seismological Phenomena, Department of Geology, Faculty of Science, University of Khartoum, Sudan

    Centre for Seismological Phenomena, Department of Geology, Faculty of Science, University of Khartoum, Sudan

  • Zhenchun Li, Shandong Provincial Key Laboratory of Reservoir Geology, China University of Petroleum (East China), Qingdao, 266580, China

    Shandong Provincial Key Laboratory of Reservoir Geology, China University of Petroleum (East China), Qingdao, 266580, China

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Published

2026-05-08

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Vol. 2 No. 1 (2026): Conclusions in Engineering

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Copyright (c) 2026 HUSSEIN MUHAMMED, Xiaodong Sun, Liping Gao, AbdelHafiz Gadelmula, Zhenchun Li (Author)

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How to Cite

On the Steadfastness of the Least-Square Reverse-Time Migration Wavefield Extrapolation via 1st-Order Riemannian Axis Finite-Difference Solver. (2026). Conclusions in Engineering, 2(1), 42-56. https://doi.org/10.71107/dh3px571

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