Stability and Accuracy Analysis of Space-frequency Wavefield Extrapolators for Depth Migration (MSc Thesis)

105 Pages·2005·4.65 MB·English

by  Liu K.

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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.
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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.
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Dedication

To my beloved wife Tao Cheng

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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
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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

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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
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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
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