Electronic Thesis and Dissertation Repository


Doctor of Philosophy




Pratt, R. Gerhard


Seismic full waveform inversion (FWI) is a non-linear problem. The Born approximation provides a way to linearize FWI and obtain a gradient in a computationally efficient manner. However, this linearization is only valid if the background velocity is sufficiently known, which often is not possible in practice. There have been various attempts at solving problems associated with the non-linearity of FWI by separating the problems of background and scatterer inversion. Most of the methods, however either depend on the availability of low frequencies and large offsets in the data, or separate the spatial scales completely, which removes the scattered information from the gradient. A complete separation of scales can fail to solve the problem of false local minima. Constrained scale separation methods have also been proposed, however these either require extra computational cost or a priori information about the reflectivity. Cycle-skipping in FWI is an offset dependent phenomenon; a differential semblance approach has been used to take this offset dependance into account. However differentiating the residuals with offset creates a preferred weighting on large offset arrivals, which generally correspond to longer path lengths. In this thesis, I propose scaled-Sobolev methods, which can be applied with negligible extra computational cost per iteration. To this end, I will define a scaled Sobolev inner product (SSIP) to take the scaled derivatives of a function into account when defining a norm, and use it to derive scaled-Sobolev pre-conditioners (SSP) for model and data domain pre-conditioning. The model domain SSP provides a constrained scale separation. The offset dependance of cycle-skipping is taken into account by a scaled-Sobolev objective function (SSO). I apply the scaled-Sobolev methods in both model and data domains using 2D synthetic examples within the acoustic approximation. Finally, I apply the scaled-Sobolev methods to the ocean bottom wide-angle velocity experiment (OBWAVE). The OBWAVE inversion results show that the scaled-Sobolev methods managed to correct some large traveltime errors and suppress the artifacts in the gradient, thereby mitigating the non-linearity in the FWI. The results revealed deeper structures interpreted as the Moho discontinuity and showed good agreement with previous studies for the shallow structures.