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Full Waveform Inversion Using Student’s t Distribution: a Numerical Study for Elastic Waveform Inversion and Simultaneous-Source Method

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Abstract

Seismic full waveform inversion (FWI) has primarily been based on a least-squares optimization problem for data residuals. However, the least-squares objective function can suffer from its weakness and sensitivity to noise. There have been numerous studies to enhance the robustness of FWI by using robust objective functions, such as l 1-norm-based objective functions. However, the l 1-norm can suffer from a singularity problem when the residual wavefield is very close to zero. Recently, Student’s t distribution has been applied to acoustic FWI to give reasonable results for noisy data. Student’s t distribution has an overdispersed density function compared with the normal distribution, and is thus useful for data with outliers. In this study, we investigate the feasibility of Student’s t distribution for elastic FWI by comparing its basic properties with those of the l 2-norm and l 1-norm objective functions and by applying the three methods to noisy data. Our experiments show that the l 2-norm is sensitive to noise, whereas the l 1-norm and Student’s t distribution objective functions give relatively stable and reasonable results for noisy data. When noise patterns are complicated, i.e., due to a combination of missing traces, unexpected outliers, and random noise, FWI based on Student’s t distribution gives better results than l 1- and l 2-norm FWI. We also examine the application of simultaneous-source methods to acoustic FWI based on Student’s t distribution. Computing the expectation of the coefficients of gradient and crosstalk noise terms and plotting the signal-to-noise ratio with iteration, we were able to confirm that crosstalk noise is suppressed as the iteration progresses, even when simultaneous-source FWI is combined with Student’s t distribution. From our experiments, we conclude that FWI based on Student’s t distribution can retrieve subsurface material properties with less distortion from noise than l 1- and l 2-norm FWI, and the simultaneous-source method can be adopted to improve the computational efficiency of FWI based on Student’s t distribution.

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Acknowledgments

This work was financially supported by the Human Resources Development program (No. 20134010200510) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Ministry of Trade, Industry, and Energy, and by the “Development of Technology for CO2 Marine Geological Storage” grant funded by the Ministry of Oceans and Fisheries of Korea. We would like to thank the editor and the anonymous reviewers for their constructive comments.

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Authors and Affiliations

  1. Research Institute of Energy and Resources, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, 151-744, Korea

    Woodon Jeong & Won-Ki Kim

  2. Daesung Energy Co., Ltd., 68 Ujeongguk-ro, Jongno-gu, Seoul, 110-300, Korea

    Minji Kang

  3. Department of Energy Systems Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, 151-744, Korea

    Shinwoong Kim & Dong-Joo Min

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  1. Woodon Jeong
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Correspondence to Dong-Joo Min.

Appendix

Appendix

We compare elastic inversion results obtained by the l 2-norm, l 1-norm, and t distribution FWIs for noise-free data from the Marmousi-2 model in Figs. 19, 20, 21, and 22. The details of the FWI setups are the same as those in the examples for noisy data. In Figs. 19, 20, 21, and 22, we can note that the three objective functions give similarly good results.

Fig. 19
figure 19

Inversion results of the P-wave velocities obtained using a the l 2-norm, b the l 1-norm, and c the t distribution objective functions for noise-free data from the Marmousi-2 model

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Fig. 20
figure 20

Inversion results of the S-wave velocities obtained using a the l 2-norm, b the l 1-norm, and c the t distribution objective functions for noise-free data from the Marmousi-2 model

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Fig. 21
figure 21

Depth profiles of P-wave velocity models at distances of 2 km (left), 5 km (center), and 7 km (right) for the Marmousi-2 model. The thinnest line indicates the true velocity. The grey line represents the l 2-norm, and the black solid and dashed lines denote the l 1-norm and Student’s t distribution, respectively. The velocities are shown in km s−1

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Fig. 22
figure 22

Depth profiles of S-wave velocity models at distances of 2 km (left), 5 km (center), and 7 km (right) for the Marmousi-2 model. The thinnest line indicates the true velocity. The grey line represents the l 2-norm, and the black solid and dashed lines denote the l 1-norm and Student’s t distribution, respectively. The velocities are shown in km s−1

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Jeong, W., Kang, M., Kim, S. et al. Full Waveform Inversion Using Student’s t Distribution: a Numerical Study for Elastic Waveform Inversion and Simultaneous-Source Method. Pure Appl. Geophys. 172, 1491–1509 (2015). https://doi.org/10.1007/s00024-014-1020-7

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