Convergence Acceleration in Scattering Series and Seismic Waveform Inversion Using Nonlinear Shanks Transformation



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An iterative solution process is fundamental in seismic inversion, such as in full-waveform inversions and inverse scattering methods. However, the convergence process could be slow or even divergent depending on the initial model used in the iteration. We propose to apply Shanks transformation (ST for short) to accelerate the convergence of the iterative solution. ST is a local nonlinear transformation, which transforms a series locally into another series with improved convergence property. ST separates the smooth background trend called the secular term versus the oscillatory transient term and then accelerates the convergence of the secular term. Because the transformation is local, we do not need to know all the terms in the original series and this is very important in the numerical application of ST. I propose to apply the ST in the context of both the forward Born series and the inverse scattering series (ISS). I test the performance of the ST in accelerating the convergence using several numerical examples, including three examples of forward modeling using the Born series and two examples of velocity inversion based on ISS. We observe that ST is very helpful in accelerating the convergence and it can achieve convergence even for a weakly divergent scattering series. As such, it provides us a useful technique to invert for a large-contrast medium perturbation in seismic inversion.The method developed in this dissertation can also be used in other fields such as in electromagnetics, quantum mechanics, and possibility medical imaging.



Shanks transformation