Direct Waveform Inversion Using Explicit Time-Space Causality Principle

I have developed a Direct Waveform Inversion (DWI) scheme to simultaneously address several existing challenges in full waveform inversion (FWI). A key ingredient in the DWI is the explicit use of the wavefield time-space causality property in the inversion, which allows us to convert the global non-linear optimization problem in the FWI, without losing information, into local linear inversions that can be readily solved. The DWI is a recursive scheme which sequentially inverts for the subsurface velocity and density structures in a shallow-to-deep fashion. Therefore, it does not need a global initial velocity model. The DWI is unconditionally convergent, and the inversion process stops when the reflection traveltime from the boundary of the inverted model is beyond the finite recording time in seismic data. The DWI must use the full seismic wavefield, including internal and free-surface multiples, and it combines seismic migration and velocity model inversion into one single process. In this dissertation, I firstly illustrate the basic idea of the DWI in 1D layered models with plane wave incidence using numerical examples. The basic idea is to build a recursive scheme by decomposing and extrapolating wavefields in each layer and applying the localized inversion to determine the properties of the next layer. I have applied the DWI scheme to different cases, including inversion for both P-wave velocity and density in the 1D stratified layered model using both plane wave and point sources, and in a 2D model with a point source. For the simultaneous inversion of velocity and density structures, I have used the angle-dependent reflections and data from multiple plane wave sources to improve the stability of the inversion. For the 2D model with point sources, I have used boundary integrals to decompose and extrapolate the spherical waves and integrate a localized reverse time migration (RTM) method to locate the layer bottoms. For the inversion of the 1D stratified layered model with a point source, I develop a new 1D scheme for the spherical wave case. Numerical examples are presented for these studies. I have also reviewed applications of the Gelfand-Levitan-Marchenko (GLM) equation in geophysics studies and discussed their relations to the DWI.