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UNIVERSITY OF CALGARY Stability and Accuracy Analysis of Space-frequency Wavefield Extrapolators for Depth Migration by Kun Liu A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF GEOLOGY AND GEOPHYSICS CALGARY, ALBERTA April, 2005 © Kun Liu 2005 UNIVERSITY OF CALGARY FACULTY OF GRADUATE STUDIES The undersigned certify that they have read, and recommend to the Faculty of Graduate Studies for acceptance, a thesis entitled "Stability and Accuracy Analysis of Space- frequency Wavefield Extrapolators for Depth Migration" submitted by Kun Liu in partial fulfilment of the requirements of the degree of Master of Science. Supervisor, Dr. J. C. Bancroft Department of Geology and Geophysics Co-supervisor, Dr. H. D. Geiger Department of Geology and Geophysics Dr. G. F. Margrave Department of Geology and Geophysics Dr. R. H. Johnston Department of Electrical and Computer Engineering _______________________ Date ii Abstract Stability and accuracy are major concerns when designing recursive wavefield extrapolators implemented in the space-frequency domain. In this thesis, a common set of tools has been developed and utilized to conduct a comprehensive stability and accuracy analysis of selected space-frequency extrapolators. The analysis shows that the Hale extrapolator is superior over a limited-aperture Rayleigh extrapolator spatially tapered by either a Hanning or a Gaussian window. A key merit of the Hale extrapolator is that it preserves as much of the phase and amplitude of the propagating wavefield as possible, yet attenuates propagating waves where the extrapolator has significant phase error. A new adaptive tapering scheme is proposed and compared with conventional tapering methods using both simple and complex models. The results show that adaptive tapering can be applied in the first extrapolation step to improve the image at shallow depth from a common-source migration. iii Acknowledgements I would like to extend my gratitude to my supervisor, Dr. John C. Bancroft, for his patient guidance and generous help throughout my entire Master study. I also thank my co-supervisor, Dr. Hugh D. Geiger, for his innovative ideas, beneficial comments and strong support of my research. I’m grateful to all the faculty members, staff and students in the CREWES Project and the department who offered tremendous help to me during the past years. The sponsors of CREWES Project are greatly appreciated for their financial support. Special thanks to Dr. Gary F. Margrave and the POTSI Project who provided the Linux Clusters, Mr. Kevin Hall and Mr. Henry Bland who helped solve computer problems, Dr. Rolf Maier who proofread my thesis, Dr. Yanpeng Mi who made Marmousi dataset available, Mr. Xinxiang Li who is always ready to help, and Dr. Helen Isaac who helped with the data processing. Finally, I would like to sincerely thank my wife, Tao Cheng, who always supports me and encourages me to overcome all difficulties. iv Dedication To my beloved wife Tao Cheng v Table of Contents Approval Page...........................................................................................................................ii Abstract....................................................................................................................................iii Acknowledgements..................................................................................................................iv Dedication..................................................................................................................................v Table of Contents.....................................................................................................................vi List of Tables..........................................................................................................................vii List of Figures and Illustrations.............................................................................................viii Chapter One: INTRODUCTION.........................................................................................1 1.1 Wavefield extrapolation techniques....................................................................1 1.2 Stability and accuracy of space-frequency extrapolators....................................3 1.3 Adaptive tapering for the first extrapolation step...............................................9 1.4 Contribution of the thesis..................................................................................10 1.5 Structure of the thesis........................................................................................11 Chapter Two: REVIEW OF WAVEFIELD EXTRAPOLATORS.....................................12 2.1 One-way wave propagation..............................................................................12 2.2 Rayleigh extrapolator........................................................................................15 2.3 Hale extrapolator...............................................................................................22 2.4 Nautiyal extrapolator........................................................................................28 2.5 Common-source migration and imaging condition..........................................30 2.6 Chapter summary..............................................................................................32 Chapter Three: STABILITY AND ACCURACY OF THREE EXTRAPOLATORS.....34 3.1 Three extrapolators in the wavenumber-frequency domain.............................34 3.2 Stability analysis...............................................................................................37 3.3 Accuracy analysis.............................................................................................42 3.4 Zero-offset impulse responses..........................................................................55 3.5 Prestack synthetic test.......................................................................................58 3.6 Marmousi synthetic test....................................................................................62 3.7 Computation costs.............................................................................................69 3.8 Chapter summary..............................................................................................71 Chapter Four: ADAPTIVE TAPERING IN WAVEFIELD EXTRAPOLATION.............73 4.1 Adaptive tapering for the first extrapolation step.............................................76 4.2 Synthetic tests...................................................................................................77 4.3 Chapter summary..............................................................................................82 Chapter Five: CONCLUSIONS AND FUTURE WORK..................................................83 REFERENCES........................................................................................................................87 vi List of Tables Table 1: Tools of stability and accuracy analysis in this thesis and their purposes..........71 Table 2: Comparison of stability, accuracy and efficiency of various extrapolators. More stars is better.............................................................................................................72 vii List of Figures and Illustrations Figure 1.1: Comparison of the images of the 2-D SEG-EAGE salt model from (a) nonrecursive Kirchhoff migration; (b) wavefield-continuation migration based on recursive wavefield extrapolation (Soubaras, 2002)...................................................2 Figure 1.2: Schematic implementation of a 2-D x-ω (space-frequency) extrapolation......3 Figure 2.1: (a) Amplitude and (b) phase spectra of the desired extrapolator in the k -ω x domain. Wavenumber and phase are normalized by the factors of 2π/∆x and 2π, respectively...............................................................................................................14 Figure 2.2: (a) Amplitude and (b) phase spectra of a 19-point Rayleigh extrapolator in the k -ω domain. The parameters are the same as those in Figure 2.1...........................20 x Figure 2.3: (a) Amplitude and (b) phase spectra of a 19-point Rayleigh extrapolator at the frequency of 31.25 Hz (blue lines). The evanescent boundaries are at k =±0.25. The x dashed red lines are the amplitude and phase spectra of the desired extrapolator....21 Figure 2.4: Amplitude spectrum of a 200-point Rayleigh extrapolator at the frequency of 31.25 Hz (blue lines). The close-up of (a) is shown in (b). The evanescent boundaries are at k =±0.25. The dashed red lines are the desired amplitude spectrum. x ...................................................................................................................................21 Figure 2.5: (a) Amplitude and (b) phase spectra of a 19-point Hanning-tapered Rayleigh extrapolator in the k -ω domain. The parameters are the same as those in Figure 2.1. x ...................................................................................................................................22 Figure 2.6: (a) Amplitude and (b) phase spectra of a 19-point Hanning-tapered Rayleigh extrapolator at the frequency of 31.25 Hz (blue lines). The evanescent boundaries are at k =±0.25. The dashed red lines are the amplitude and phase spectra of the x viii desired extrapolator...................................................................................................22 Figure 2.7: (a) Amplitude and (b) phase spectra of a 19-point Hale extrapolator in the k - x ω domain. The parameters are the same as those in Figure 2.1................................27 Figure 2.8: (a) Amplitude and (b) phase spectra of a 19-point Hale extrapolator at the frequency of 31.25 Hz (blue lines). The evanescent boundaries are at k =±0.25. The x dashed red lines are the amplitude and phase spectra of the desired extrapolator....27 Figure 2.9: Phase spectra of a 19-point Hale extrapolator (a) in the k -ω domain and (b) at x the frequency of 31.25Hz. All the values outside the evanescent boundaries are zeroed........................................................................................................................27 Figure 2.10: (a) Amplitude and (b) phase spectrum of a 19-point Nautiyal extrapolator in the k -ω domain. The parameters are the same as those in Figure 2.1......................30 x Figure 2.11: (a) Amplitude and (b) phase spectra of a 19-point Hale extrapolator at the frequency of 31.25 Hz (blue lines). The evanescent boundaries are at k =±0.25. The x dashed red lines are the amplitude and phase spectra of the desired extrapolator....30 Figure 2.12: Schematic implementation of a wavefield-continuation migration based on the wavefield extrapolation. (a) Forward extrapolate the source wavefield by modeling; (b) backward extrapolate the receiver wavefield; (c) the horizontal reflector is imaged by applying imaging condition (figure courtesy Bancroft)........31 Figure 3.1: 3-D plot of various band-limited extrapolators in the k -ω domain. Amplitude x spectra are on the left, and phase spectra are on the right. (a) The desired extrapolator (from equation (2.21)); (b) the Hanning edge-tapered Rayleigh extrapolator; (c) the Hale extrapolator; (d) the Nautiyal extrapolator. An operator length of 19 points is used by operators (b), (c) and (d) in the space domain. The ix