Parameter selection in seismic data analysis problems

Seismic imaging is an essential tool for non-invasive subsurface evaluation. It enables Earth scientists to create a picture of the planet's interior, predicting the rocks and structures that lie below. This can enable characterization of tectonic margins to better understand the deep history of the planet, delineation of aquifers to provide water, and the safe and economic exploration for commercial oil and gas accumulations for energy production.

To generate these images numerous observations of the subsurface are taken and they are transformed to a common domain where observations of the same point in the subsurface overlay. These transformations typically are linear on the observed data and usually depend on a parameter related to seismic wave propagation, like the speed at which a seismic wave travels through the subsurface, in a non-linear manner. Selecting and determining these parameters is a crucial step in the generation of seismic images. Using inaccurate parameters in the transformations involved in seismic data processing results in seismic images that are distorted, inaccurate representations of the subsurface. Because these parameters are related to seismic wave propagation, their values can provide insight into the composition of the Earth's interior, including the rocks or fluids present.

In this dissertation, I present methods for accurately determining those parameters and how they may be used to efficiently generate accurate, well resolved images of the Earth's interior. I show how dynamic time warping may be used to create an operator which efficiently corrects for the blurring and distortion present in seismic images caused by seismic anisotropy, or wave propagation speed changing with the direction of travel, while simultaneously characterizing and quantifying that anisotropy. I demonstrate how slope-decomposed seismic images may be transported along their characteristics in a process called oriented velocity continuation to efficiently generate a suite of images over a range of plausible migration velocities, and how oriented velocity continuation may be used with seismic diffraction imaging to determine migration velocity. The use of oriented velocity continuation is further expanded on to generate a framework for probabilistic diffraction imaging using a collection of weights computed from slope-decomposed images that represent the probability of a correctly imaged diffraction existing at a point in space for a given migration velocity, while simultaneously outputting the most likely migration velocity at each point in space. This method generates seismic images with significantly improved signal to noise ratios compared to conventional approaches. Finally, I formulate a variational method for picking an optimal surface representing how a parameter evolves in space from a volume representing the quality of fit for different parameter values based on iteratively minimizing a functional. I prove that minimizers for that functional exist, and that an iterative method will converge to a minimizer in an infinite dimensional setting. The method is applied using continuation, or graduated optimization, to avoid local minima and used to determine seismic velocities as a component of seismic processing workflows and perform automatic interpretation of a seismic horizon.