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==Interpolation with prediction-error filters and training data== | ==Interpolation with prediction-error filters and training data== | ||

- | //by [[http://sepwww.stanford.edu/sep/jeff|William Cury]]// | + | //by **William Cury**// |

- | Full thesis ([[http://sepwww.stanford.edu/data/media/public/docs/sep137/sep137.pdf|PDF]]) | + | Full thesis ([[http://sepwww.stanford.edu/data/media/public/docs/sep135/sep135.pdf|PDF]]) |

- | * [[http://sepwww.stanford.edu/data/media/public/docs/sep137/chap0.pdf|Title Pages]] | + | * [[http://sepwww.stanford.edu/data/media/public/docs/sep135/chap0.pdf|Title Pages]] |

- | * Chapter 1: [[http://sepwww.stanford.edu/data/media/public/docs/sep137/chap1.pdf|Introduction]] | + | * Chapter 1: [[http://sepwww.stanford.edu/data/media/public/docs/sep135/chap1.pdf|Introduction]] |

- | * Chapter 2: [[http://sepwww.stanford.edu/data/media/public/docs/sep137/chap2.pdf|RWE: Non-orthogonal coordinate systems]] | + | * Chapter 2: [[http://sepwww.stanford.edu/data/media/public/docs/sep135/chap2.pdf|Prediction-error filters and interpolation]] |

- | * Chapter 3: [[http://sepwww.stanford.edu/data/media/public/docs/sep137/chap3.pdf|Shot-profile migration in elliptic coordinates]] | + | * Chapter 3: [[http://sepwww.stanford.edu/data/media/public/docs/sep135/chap3.pdf|Interpolation of irregularly sampled data]] |

- | * Chapter 4: [[http://sepwww.stanford.edu/data/media/public/docs/sep137/chap4.pdf|Generalized-coordinate ADCIGs]] | + | * Chapter 4: [[http://sepwww.stanford.edu/data/media/public/docs/sep135/chap4.pdf|Interpolation of near offsets using multiples]] |

- | * Chapter 5: [[http://sepwww.stanford.edu/data/media/public/docs/sep137/chap5.pdf|Delayed-shot migration in TEC coordinates]] | + | * Chapter 5: [[http://sepwww.stanford.edu/data/media/public/docs/sep135/chap5.pdf|Interpolation in the frequency-space domain with nonstationary PEFs]] |

- | * [[http://sepwww.stanford.edu/data/media/public/docs/sep137/appendix.pdf|Appendices]] | + | * Chapter 6: [[http://sepwww.stanford.edu/data/media/public/docs/sep135/chap6.pdf|Conclusions]] |

- | * [[http://sepwww.stanford.edu/data/media/public/docs/sep137/bibliography.pdf|Bibliography]] | + | * [[http://sepwww.stanford.edu/data/media/public/docs/sep135/bibliography.pdf|Bibliography]] |

**Abstract** | **Abstract** | ||

- | Wave-equation migration using one-way wavefield extrapolation operators is commonly used in industry to generate images of complex geologic structure from 3D seismic data. By design, most conventional wave-equation approaches restrict propagation to downward continuation, where wavefields are recursively extrapolated to depth on Cartesian meshes. In practice, this approach is limited in high-angle accuracy and is restricted to down-going waves, which precludes the use of some steep dip and all turning wave components important for imaging targets in such areas as steep salt body flanks. | + | A Ph.D. is a much longer endeavor than I had originally thought, and I have many people to thank for getting me through it. This list is far from exhaustive, but I would like to single out people who played particularly strong roles during my extended stay at SEP. |

- | This thesis discusses a strategy for improving wavefield extrapolation based on extending wavefield propagation to generalized coordinate system geometries that are more conformal to the wavefield propagation direction and permit imaging with turning waves. Wavefield propagation in non-Cartesian coordinates requires properly specifying the Laplacian operator in the governing Helmholtz equation. By employing differential geometry theory, I demonstrate how generalized a Riemannian wavefield extrapolation (RWE) procedure can be developed for any 3D non-orthogonal coordinate system, including those constructed by smoothing ray-based coordinate meshes formed from a suite of traced rays. I present 2D and 3D generalized RWE propagation examples illustrating the improved steep-dip propagation afforded by the coordinate transformation. | + | I would like to start by thanking my senior students and mentors Morgan Brown, Bob Clapp, and Antoine Guitton. They led me toward a research project and pro- vided many hours of conversation about it and many other topics of mutual interest in geophysics and computing. I have since had the pleasure of having Bob on my de- fense committee and Antoine as a supervisor during an internship, and learned even more from them as a result. I thank my cohort, Gabriel Alvarez, Brad Artman, An- drey Karpushin, and Nick Vlad, who I shared many experiences with, and who gave me much needed support. I also thank the many students who arrived before and after me, in particular Guojian Shan, Jesse Lomask, Daniel Rosales, Doug Gratwick, Alejandro Valenciano, Jeff Shragge, Yaxun Tang, Madhav Vyas, Pierre Jousselin, Ben Witten, Claudio Cardoso, Gboyega Ayeni, and Roland Gunther for many useful discussions about our research and the world at large. |

- | One consequence of using non-Cartesian coordinates, though, is that the corresponding 3D extrapolation operators have up to 10 non-stationary coefficients, which can lead to imposing (and limiting) computer memory constraints for realistic 3D applications. To circumvent this difficulty, I apply the generalized RWE theory to analytic coordinate systems, rather than numerically generated meshes. Analytic co- ordinates offer the advantage of having straightforward analytic dispersion relationships and easy-to-implement extrapolation operators that add little computational overhead. In particular, I demonstrate that the dispersion relationship for 2D elliptical geometry introduces only an effective velocity model stretch, permitting the use of existing high-order Cartesian extrapolators. The results of elliptical coordinate shot- profile migration tests demonstrate the improvements in steep dip reflector imaging facilitated by the coordinate system transformation approach. | + | I owe a large debt to Jon Claerbout and Biondo Biondi for establishing and main- taining the intellectual framework of SEP and for keeping us focused on interesting topics that have applicability in the real world. They have given me the viewpoint that research is the most fun when working on new ways to approach a difficult field data set. Diane Lau deserves much credit for keeping a group as large as SEP running smoothly, and I owe the sponsors of SEP much for their financial support. |

+ | Anyone reading this thesis should thank Ken Larner for taking what was a semi- intelligible mess and helping me rewrite and refine it into what you are reading today. He was extremely supportive and helpful during his brief stay at SEP. Norm Sleep also deserves much credit for his thorough review of this thesis. It is much stronger because of his help. | ||

- | I extend the analytic coordinate system approach to 3D geometries using tilted elliptical-cylindrical (TEC) meshes. I demonstrate that propagation in a TEC coordinate system is equivalent to wavefield extrapolation in elliptically anisotropic media, which is easily handled by existing industry practice. TEC coordinates also allow steep dip propagation in both the inline and cross-line directions by virtue of the associated elliptical and tilting Cartesian geometries. Observing that a TEC coordi- nate system conforms closely to the shape of a line-source impulse response, I develop an TEC-coordinate, inline-delay-source migration strategy that enables the efficient migration of individual sail-line data. I argue that this strategy is more robust than 3D plane-wave migration because of the reduced migration aperture requirements and, commonly, a lower number of total migration runs. Synthetic imaging tests on a 3D wide-azimuth data set demonstrate the imaging advantages offered by the TEC coordinate transformation, especially in the cross-line direction. Field data tests on a Gulf of Mexico data set similarly indicate the advantage of TEC coordinates. | + | Doug Schmitt and Mauricio Sacchi at the University of Alberta sparked my in- terest in geophysics, and I enjoyed their courses greatly. Bill MacDonald was my supervisor during an extended internship in Calgary. He sent me out on a seismic survey for a week, which got me thinking about interpolation as a research topic. |

+ | | ||

+ | I would like to thank my many friends at Stanford and elsewhere who helped me in my attempt to remain as well-rounded as possible. They are too numerous to mention, and are in geophysics, earth sciences, and further abroad. | ||

+ | | ||

+ | Finally, I thank my family: my father and my brother for their focus on science and education, and my mother for her continual support. |