Robust Tomographic Full Waveform Inversion using Adaptive Amplitude Estimation

I examine the basis of slow convergence of tomographic full waveform inversion (TFWI) and discover that the reason behind it is the unbalanced effects of amplitudes and phase in the design of the regularization term. This imbalance results in a strong reliance of the kinematic updates on the amplitude fitting, slowing down the convergence. To mitigate the problem I propose two modifications to the tomographic inversion. First, by modifying the regularization term to focus more on the phase information, and second, simultaneously updating the source function for modeling. The adjustments reduce the gradient artifacts and allow for explicit control over the amplitudes and phases of the residuals. Tomographic full waveform inversion (Symes, 2008; Sun and Symes, 2012; Biondi and Almomin, 2012) is an innovative inversion technique that preserves all the advantages and benefits of full waveform inversion (FWI) while at the same time bypassing its strict initial model requirement and cycle-skipping challenges. To reach this objective, TFWI alters FWI by merging its classical form with a modified form of wave-equation migration-velocity analysis (WEMVA). This combination displays itself as an extension of the velocity model through virtual axes (Biondi and Almomin, 2013). The modeling operator is able to match the observed data by extending the velocity model with the proper axis, no matter what the accuracy of the initial model is, by using kinematic information from the extended axis with disregard to the occurrence of cycle skipping. The inversion is set up to extract all the essential information from the virtual axes and smoothly fold them back into their original, nonextended form of the model. The kinematic and dynamic information of the data were successfully inverted with exceptional robustness and precision. Even though cycle-skipping is not an issue with TFWI, this method creates its own challenges, which are; its high computational cost and the big number of iterations that it needs (Almomin and Biondi, 2013). The conventional FWI uses only a single frequency per iteration to match the phase (Pratt, 1999; Shin and Ha, 2008). Not using amplitudes reduces the accuracy of the solution because it prevents the simultaneous inversion of scales. Modifying the gradient calculation is another method that is used to reduce some "kinematic" artifacts (Fei and Williamson, 2010; Shen and Symes, 2015). These methods are appropriate for image-domain velocity analysis methods, such as WEMVA. The explicit calculations of the nonlinear modeling operator and residuals in the data space prevents it from being applied in TFWI. Two adjustments to TFWI are proposed to reduce the slow convergence and allow for more control of the ratio between amplitude and phase. These adjustments are consistent in the framework of TFWI and allow for an accurate calculation of the gradient in the data space. The adjustments were tested and resulted in a reduction in the kinematic artifacts in the gradient.