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Optimization Algorithms

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Full-3D Seismic Waveform Inversion

Part of the book series: Springer Geophysics ((SPRINGERGEOPHYS))

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Abstract

Seismic inverse problems are often solved using optimization algorithms. The formulation given in Chaps. 3 and 4 provides the machinery for constructing the gradient and the approximate Hessian of the objective function that can be used in local optimization algorithms to search for a local optimal model that provides smaller misfits with observed waveforms than its neighbors. Local optimization algorithms can be classified based on whether an approximate Hessian of the objective function is explicitly constructed. The scattering-integral (SI) method is based on explicitly constructing the approximation Hessian in the Gauss-Newton algorithm or its variants. The adjoint-wavefield (AW) method is based on constructing and utilizing only the gradient of the objective function in a conjugate-gradient or quasi-Newton-type optimization algorithm. In Sect. 5.1 I will discuss the SI method and the various ingredients related to its efficient implementation. In Sect. 5.2, I will derive the AW method using the adjoint representation theorem and give a tutorial about how to compute the event kernels based on the AW method.

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Notes

  1. 1.

    Once a descent direction is given, the line search determines how far we should move along the descent direction, i.e., the optimal step length along the descent direction. The other widely used technique is known as the “trust region” method.

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Authors and Affiliations

  1. Department of Geology & Geophysics, University of Wyoming, Laramie, Wyoming, USA

    Po Chen

  2. Department of Earth Sciences, National Cheng Kung University, Tainan, Taiwan

    En-Jui Lee

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Chen, P., Lee, EJ. (2015). Optimization Algorithms. In: Full-3D Seismic Waveform Inversion. Springer Geophysics. Springer, Cham. https://doi.org/10.1007/978-3-319-16604-9_5

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