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Regularized Laplace–Fourier-Domain Full Waveform Inversion Using a Weighted l 2 Objective Function

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Abstract

Full waveform inversion (FWI) can be applied to obtain an accurate velocity model that contains important geophysical and geological information. FWI suffers from the local minimum problem when the starting model is not sufficiently close to the true model. Therefore, an accurate macroscale velocity model is essential for successful FWI, and Laplace–Fourier-domain FWI is appropriate for obtaining such a velocity model. However, conventional Laplace–Fourier-domain FWI remains an ill-posed and ill-conditioned problem, meaning that small errors in the data can result in large differences in the inverted model. This approach also suffers from certain limitations related to the logarithmic objective function. To overcome the limitations of conventional Laplace–Fourier-domain FWI, we introduce a weighted l 2 objective function, instead of the logarithmic objective function, as the data-domain objective function, and we also introduce two different model-domain regularizations: first-order Tikhonov regularization and prior model regularization. The weighting matrix for the data-domain objective function is constructed to suitably enhance the far-offset information. Tikhonov regularization smoothes the gradient, and prior model regularization allows reliable prior information to be taken into account. Two hyperparameters are obtained through trial and error and used to control the trade-off and achieve an appropriate balance between the data-domain and model-domain gradients. The application of the proposed regularizations facilitates finding a unique solution via FWI, and the weighted l 2 objective function ensures a more reasonable residual, thereby improving the stability of the gradient calculation. Numerical tests performed using the Marmousi synthetic dataset show that the use of the weighted l 2 objective function and the model-domain regularizations significantly improves the Laplace–Fourier-domain FWI. Because the Laplace–Fourier-domain FWI is improved, the frequency-domain FWI, in which the Laplace–Fourier-domain FWI result is used as the starting model, yields inversion result much closer to the true velocity.

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Acknowledgments

We would like to thank STATOIL for supporting this study. This work was also supported by the Energy Efficiency&Resources Core Technology Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea (Nos. 20132510100060 and 20152520100740).

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Authors and Affiliations

  1. Department of Energy Systems Engineering, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul, 08826, Republic of Korea

    Hyunggu Jun, Jungmin Kwon & Changsoo Shin

  2. Statoil, 2107 CityWest Blvd, Houston, TX, 77042, USA

    Hongbo Zhou & Mike Cogan

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  1. Hyunggu Jun
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  2. Jungmin Kwon
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  3. Changsoo Shin
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  4. Hongbo Zhou
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  5. Mike Cogan
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Correspondence to Jungmin Kwon.

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Jun, H., Kwon, J., Shin, C. et al. Regularized Laplace–Fourier-Domain Full Waveform Inversion Using a Weighted l 2 Objective Function. Pure Appl. Geophys. 174, 955–980 (2017). https://doi.org/10.1007/s00024-016-1398-5

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  • DOI: https://doi.org/10.1007/s00024-016-1398-5

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