ABSTRACT

The research project was carried out to study the estimations of interval velocity from seismic reflection data of the acoustic P waves inside the Earth using the method of time migration which is necessary for obtaining seismic images in regular Cartesian coordinates. The main goals of the research work were to develop algorithms to convert time-migration velocities to true seismic velocities, and to convert time-migrated images to depth images in regular Cartesian coordinates. The results were the established theoretical relation between the true seismic velocities and the 'time-migration velocities' using the paraxial ray tracing, the formulated appropriate inverse problem describing the relation between time-migration velocities and depth velocities, and the developed numerical algorithms to solve regularized versions of the formulated equations which can be used to recover smoothed velocity variations. The developed algorithms include the efficient time-to-depth conversion algorithms, the level set and ray tracing algorithms for transforming Dix velocities into interval velocities. These algorithms are applied to both two- and three-dimensional problems, and they are tested on a collection of both synthetic and field data, which yielded improved seismic images.

TABLE OF CONTENTS

Title Page………………………………………………………..……….         i

Declaration……………………………………………………………....         ii

Certification……………………………………………………………...        iii

Dedication…………………………………………………………….….        iv

Acknowledgement……………………………………………………….         v

Abstract…………………………………………………………………..        vi

Table of Contents………………………………………………………...        vii

List of Tables………………………………………………………….….         x

List of Figures……………………………………………………………          xi

CHAPTER ONE: INTRODUCTION

1.1 Background of Study………………………………………………...           1

1.2 Statement of the Problem…………………………………………….           2

1.3 Aims and Objectives of the Study……………………………………          3

1.4 Significance of the Study……………………………………………..          4

1.5 Operational Definition of the Terms………………………………….          5

CHAPTER TWO: LITERATURE REVIEW

2.1 Time Migration Coordinates and Image Rays…………………….….          7

2.2 Travel Time Approximation……………………………………….…          8

2.3 Emerging Wave Front in 3D Case………………………………...…..       10

2.4 Simplifications in 2D………………………………………………….       11

2.5 Interval Velocity Estimation by Dix Inversion…………………….           12

2.6 Forward Modeling of the Time-Migration Velocities……………....          14

2.6.1 Paraxial Ray Tracing………………………………………………         14

2.6.2 Relation between the Matrices K and the True Seismic Velocities in 3D………………………………………………………………………            18

2.6.3 Relation between the Dix and the True Interval Velocities in 2D…         23

2.7 Stability of the Forward and Backward (Inverse) Construction Problem………………………………………………………………….          24

2.7.1 The Inverse Problem in 2D………………………………………..          24

2.7.2 Eulerian Formulation of the Inverse Problem……………………..          29

2.7.3 The Inverse Problem in 3D…………………………………….......         30

CHAPTER THREE: MATERIALS AND METHODS/ RESEARCH METHODOLOGY

3.1 Numerical Algorithms Design in 2D…………………………………        31

3.1.1 Efficient Time-to-Depth Conversion Algorithms…………………..  31

3.1.2 Causality……………………………………………………………    38

3.1.3 Boundary Effects…………………………………………………...   39

3.2 Algorithms Producing the Seismic Velocities from the Migration Velocities…………………………………………………………………     40

 3.2.1 Ray Tracing Approach…………………………………………….   41

3.2.2 Level Set Approach…………………………………………………   44

3.3 Numerical Algorithms Design in 3D…………………………………   47

3.3.1 Ray Tracing Algorithm……………………………………………..   48

3.3.2 Re-computation of the velocity using the Found Image Rays………          54

3.3.3 Time-to-Depth Conversion Algorithm……………………………… 54

CHAPTER FOUR: RESULT ANALYSIS

4.1 Synthetic Data Examples in 2D………………………………………  58

4.2 Field Data Examples………………………………………………….   63

4.3 Synthetic Data Examples in 3D………………………………………  67

CHAPTER FIVE: DISCUSSION OF FINDINGS, CONCLUSION AND RECOMMENDATIONS

5.1 Discussion of Findings………………………………………………… 77

5.2 Conclusion……………………………………………………………..  78

5.3 Recommendations……………………………………………………… 78

Reference…………………………………………………………….……    80

Appendix A………………………………………………………………….         82

LIST OF TABLES

Table 4.1: The maximal relative errors produced by the time-to-depth conversion, the ray tracing and the level set algorithms on the data from the velocity field in example two ……………………………………………        62

LIST OF FIGURES

Fig.1.1: A Velocity Model v(x)………………………………………………           3

Fig.1.2: The Approaches and the Algorithms: left, 2D; right, 3D…………..          5

Fig.2.1: Image Rays and Time-Migration Coordinates…………………..…           8

Fig.2.2: Travel Time Approximation…………………………………….…            9

Fig.2.3: Emerging Wave Front……………………………………………...         11

Fig. 2.4: Dix Inversion……………………………………………………... 13

Fig.2.5: Paraxial Ray Tracing……………………………………………… 15

Fig.2.6: Illustration for Theorem 1………………………………………… 17

Fig. 3.1: Relation between (x, z), x₀ and T………………………………….     32

Fig.3.2: Fast Marching Method: Black, grey and white dots represent 'accepted', 'considered' and 'unknown' points respectively…………………………….         38

Fig. 4.1: (a) The Exact Velocity; (b) The Input Data v(x₀, T). (c) The Found Velocity v(x, z). (d) The Relative Error: its maximus is less than 5 percent…  59

Fig. 4.2: The Image Rays Computed for the Exact Velocity…………….   60

Fig. 4.3: (a) The Exact Velocity v(x, z). (b) The Input Data f(x₀, t) ≡ v_Dix(x₀, t). (c) The Found Velocity v(x, z)……………………………………..…              61

Fig. 4.4: (a) The Exact Velocity v(x, z). (b) The Input Data: the Dix velocity converted to depth by 'vertical stretch'. (c) The Found Interval Velocity v(x, z) and the Image Rays………………………………………………………        63

Fig. 4.5: Left: seismic image from North Sea obtained by prestack time migration using velocity continuation (Fomel, 2003). Right: the corresponding time-migration velocity……………………………………………….               64

Fig. 4.6: The Estimated Interval Velocity v(x, z) and the Image Rays Computed from It……………………………………………………………………    65

Fig. 4.7: The Smoothed Dix Velocity v_Dix(x₀, t₀) (left) Versus the Estimated Interval Velocity v(x, z) (right)………………………………………...            66

Fig.4.8: (a) The Poststack Depth-Migrated Image Obtained with the Found v(x, z). (b) The Prestack Time-Migrated Image Converted to Depth…….......................................................................................           67

Fig. 4.9: The Poststack Depth Migration Using the Dix Velocities (left) Versus the Poststack Depth Migration Using the Estimated Interval Velocities (right)……………………………………………………………………       68

Fig. 4.10: (a) The Exact Velocity, (b) The Velocity Found by the Ray Tracing Approach. (c) The Heuristic Estimate Analogous to the Dix Inversion, Converted to Depth. (d) The Image Rays Projected onto the Earth Surface…………………………………………………………………..      70

Fig. 4.11: (a) The Exact Velocity, (b) The Velocity Found by the Ray Tracing Approach. (c) The Heuristic Estimate Analogous to the Dix Inversion, Converted to Depth. (d) The Image Rays Projected onto the Earth Surface……………………………………………………………………     72

Fig. 4.12: (a) The Exact Velocity, (b) The Velocity Found by the Ray Tracing Approach. (c) The Heuristic Estimate Analogous to the Dix Inversion, Converted to Depth. (d) The Image Ray Projected onto the Earth Surface……………………………………………………………………     75