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To my Teachers...


The goal of this thesis is to design new methods for imaging complex geologic structures of the Earth's Lithosphere. Seeing complex structures is important for both exploration and non-exploration studies of the Earth and it involves, among other things, dealing with complex wave propagation in media with large velocity contrasts.

The approach I use to achieve this goal is depth imaging using acoustic waves. This approach consists of two components : migration and migration velocity analysis. No accurate imaging is possible without accurate, robust and efficient solutions to both components.

The main technical requirements I impose on both imaging components call for the use of as much information as possible from the recorded wavefields, design of methods consistent with one-another, and accurate modeling of wave phenomena within the constraints of the available computational resources.

I address both migration and migration velocity analysis in the general framework of one-way wavefield extrapolation. In this context, both imaging components are consistent and use the entire acoustic wavefields with accurate, robust and computationally feasible techniques.

The migration state-of-the-art involves downward continuation of wavefields recorded at the Earth's surface. I introduce Riemannian wavefield extrapolation as a general framework for wavefield extrapolation. Downward continuation or extrapolation in tilted coordinates are special cases. With this technique, I overcome the steep-dip limitation of downward continuation, while retaining the main characteristics of wave-equation techniques.

Riemannian wavefield extrapolation propagates waves in semi-orthogonal coordinate systems that conform with the general direction of wave propagation. Therefore, extrapolation is done forward relative to the direction in which waves propagate, so I achieve high-angle accuracy with small-angle operators. Riemannian wavefield extrapolators can also be used for diving waves that cannot be easily handled using conventional downward continuation.

The velocity estimation state-of-the-art involves traveltime tomography from sparse reflectors picked on migrated images. I introduce wave-equation migration velocity analysis as a more accurate and robust alternative. With this technique, I overcome the instability of traveltime tomography caused by ray tracing in areas with high velocity contrasts.

I formulate wave-equation MVA with an operator based on linearization of wavefield extrapolation using the first-order Born approximation. I define the optimization objective function in the space of migrated images, in contrast with wave-equation tomography with objective function defined in the space of recorded data. Since the entire images are sensitive to migration velocities, I use image perturbations for optimization, in contrast with traveltime tomography which employs traveltime perturbations picked at selected locations. I construct image perturbations with residual migration operators by measuring flatness of angle-domain common image gathers, or by measuring spatial focusing of diffracted energy.


All of the figures in this thesis are marked with one of the three labels: [ER], [CR], and [NR]. These labels define to what degree the figure is reproducible from the data directory, source code and parameter files provided on the web version of this thesis [*].

denotes Easily Reproducible. My claim is that you can reproduce such figures from the programs, parameters, and data included in the electronic document. I assume that you have a UNIX workstation with Fortran, C, X-Window system, and the software on the webpage at your disposal. Before the publication of the electronic document, someone else in the SEP group tested my claims by destroying and rebuilding all ER figures.
denotes Conditional Reproducibility. My claim is that the commands are in place to reproduce the figure if certain resources are available. For example, you might need a large or proprietary data set. You may also need a super computer, or you might simply need a large amount of time (more than two hours) on a workstation. Before the publication of the electronic document, someone else in the SEP group tested my claims by following all rules used to generate a figure.

CR figures in adcig and storm require more than several hours of processing on a Linux computer cluster. The real datasets used in the examples are part of the SEP data library. CR figures in example use proprietary Gulf of Mexico data which cannot be distributed with the thesis. They also require processing on Linux computer clusters for one day or more.

denotes Non-Reproducible. This class of figure is considered non-reproducible. Figures in this class are scans and artists' drawings.


I am dedicating this thesis to my Teachers , by which I mean family, friends, professors and others who knowingly or unknowingly have taught me something throughout my life. I am grateful to all, since who I am and what I know is due to them to the largest degree.

My learning journey began with my immediate family, parents and grandparents, from whom I learned the values of hard work, high standards and focus on what is important. My parents, in particular, dedicated much of their energy and resources to their children's education. My current achievements are their achievements, to a large degree, and I am grateful for the education they gave me, which is their most valuable and most enduring gift.

I am grateful to the Stanford Exploration Project students, faculty and sponsors, without whose support I could not have completed this thesis. My advisers, Biondo Biondi and Jon Claerbout, helped me get started on my research and bombarded me with ideas and advice, but then allowed me to venture at will in uncharted research territory. This is what I think great advising is all about, and both deserve recognition for fulfilling their duty. I am also grateful to the SEP alumni, whose outstanding achievements turned SEP into one of the leaders in seismic imaging and who remain a model for current and future generations of students.

Of all my SEP colleagues, Sergey Fomel deserves a special mention. He was my student mentor upon my arrival at SEP many years back, officemate for some years, a frequent collaborator in many research projects ever since, and remains a close friend today. James Rickett, Antoine Guitton and Jeff Shragge have also been influential collaborators at various times of my SEP years and I am grateful for their insight and challenges to my thinking.

I have also benefited from interaction with my other SEP colleagues of older or younger generations: Morgan Brown, Marie Clapp, Daniel Rosales, Louis Vaillant, Gabriel Alvarez, Alejandro Valenciano, Nick Vlad, Bill Curry, Jesse Lomask, Guojian Shan, Brad Artman and Thomas Tisserant. We have shared way too many seminar hours, although I think that in some strange way this brought us closer, helped us understand each-other better and stimulated collaboration and research.

I would also like to thank Diane Lau who helped me in countless occasions do what was necessary to bypass seemingly impossible administrative hurdles. She made my life at Stanford much easier than it could have been. I am a grateful beneficiary of her kindness and generous support.

I am also grateful to close friends of older SEP generations, mainly Mihai Popovici, who many years ago brought me straight from San Francisco International Airport to Mitchell Building (where Jon promptly added me to the seminar list). I had no idea then that it would take me so long to make the trip back, but it was a fine ride and I enjoyed it a lot.

Finally, I would like to thank my ultimate teachers, Diana and Iulia, who with love, patience and generosity teach me every day what real life is all about. There are no words to express how happy and grateful I am to have them by my side.


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