theory of seismic imaging

7/27/2019 theory of seismic imaging

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Theory of seismic imaging: lecture notes

Robert J. Ferguson, [email protected]


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Contents

1 Background 1

1.1 Planewaves and the Fourier transform . . . . . . . . . . . . . 1

1.1.1 Orthogonality and the function . . . . . . . . . . . . 21.1.2 Complex exponentials . . . . . . . . . . . . . . . . . . 11

1.1.3 Fourier transforms . . . . . . . . . . . . . . . . . . . . 121.1.4 Discrete Fourier transforms and the FFT . . . . . . . 13

1.1.5 Amplitude and phase . . . . . . . . . . . . . . . . . . 13

1.1.6 Phase shift . . . . . . . . . . . . . . . . . . . . . . . . 141.1.7 Snells Law . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.8 Taylor series . . . . . . . . . . . . . . . . . . . . . . . 16

1.1.9 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . 17

1.1.10 Vector calculus . . . . . . . . . . . . . . . . . . . . . . 181.1.11 Numerical derivatives . . . . . . . . . . . . . . . . . . 19

1.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Overview: seismic reflection 25

2.1 Goals in seismic imaging . . . . . . . . . . . . . . . . . . . . . 25

2.2 Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Noise reduction . . . . . . . . . . . . . . . . . . . . . . 30

2.3.2 Trace edit . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.3 Trace balance . . . . . . . . . . . . . . . . . . . . . . . 312.3.4 Spherical divergence . . . . . . . . . . . . . . . . . . . 31

2.3.5 Band-pass . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.6 Deconvolution . . . . . . . . . . . . . . . . . . . . . . 322.3.7 Model building . . . . . . . . . . . . . . . . . . . . . . 322.3.8 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4 Rays and waves . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.5 Two kinds of waves . . . . . . . . . . . . . . . . . . . . . . . . 56

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iv CONTENTS

2.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3 A model: seismic imaging 63

3.1 Point diffractors and reflections . . . . . . . . . . . . . . . . . 63

3.2 A model of seismic imaging . . . . . . . . . . . . . . . . . . . 64

3.3 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 A model: seismic reflection 69

4.1 A model of reflection . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Planewaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4 Scalar wave equation . . . . . . . . . . . . . . . . . . . . . . . 71

4.5 Fluid medium assumption . . . . . . . . . . . . . . . . . . . . 71

4.6 R in a fluid medium . . . . . . . . . . . . . . . . . . . . . . . 72

4.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Computational reflection and extrapolation 75

5.1 R as a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 W+ and W as a matrices . . . . . . . . . . . . . . . . . . . 78

5.3 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Some simplifications 83

6.1 W

RW+ in a constant velocity medium . . . . . . . . . . . 836.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 The exploding reflector model 89

7.1 Secondary sources . . . . . . . . . . . . . . . . . . . . . . . . 90

7.2 Half offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.3 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8 Phase shift migration 97

8.1 Inversion of U = [W]

2 SS . . . . . . . . . . . . . . . . . . 98

8.2 Imaging condition . . . . . . . . . . . . . . . . . . . . . . . . 98

8.3 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

8.4 Depth-variable velocity c (z) . . . . . . . . . . . . . . . . . . . 99

8.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

8.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104


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CONTENTS v

9 Stolt migration 105

9.1 Migration by Fourier transform . . . . . . . . . . . . . . . . . 1059.2 Stolt stretch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.3 Apply stretch . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 10

9.4 Migrate the stretched data . . . . . . . . . . . . . . . . . . . . 114

9.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

10 Kirchhoff migration 117

10.1 Domain change kh xh . . . . . . . . . . . . . . . . . . . . . 11710.2 Method of stationary phase . . . . . . . . . . . . . . . . . . . 118

10.3 Laplaces formula . . . . . . . . . . . . . . . . . . . . . . . . . 121

10.4 Determine stationary points k0 and solve . . . . . . . . . . . 122

10.5 Apply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10.6 The xh xh integral . . . . . . . . . . . . . . . . . . . . . . . 1 2310.7 Variable velocity . . . . . . . . . . . . . . . . . . . . . . . . . 124

10.8 Q uestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 27

11 x migration 12911.1 Space-variable . . . . . . . . . . . . . . . . . . . . . . . . . 129

11.2 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . 130

11.3 E rror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

11.4 Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

11.5 x migration . . . . . . . . . . . . . . . . . . . . . . . . . . 131

12 SS, GS, and PSPI 137

12.1 Split-step Fourier . . . . . . . . . . . . . . . . . . . . . . . . . 137

12.2 Generalized screens . . . . . . . . . . . . . . . . . . . . . . . . 139

12.3 PSPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

12.4 O th e r op e r ator s . . . . . . . . . . . . . . . . . . . . . . . . . . 143

13 RTM in point form 147

13.1 In tr od u c tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

13.2 Explicit difference approximation . . . . . . . . . . . . . . . . 14713.3 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . 149

13.4 Energy absorbing boundaries . . . . . . . . . . . . . . . . . . 150

A Appendix: Phase shift code 151


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vi CONTENTS

A Appendix: Normal Move Out (NMO) 155

A.1 cRMS: hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . 1 55A.2 Small offset assumption . . . . . . . . . . . . . . . . . . . . . 155A.3 Hyperboloid reflections . . . . . . . . . . . . . . . . . . . . . . 157A.4 RMS velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 57A.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 58

B Appendix: The scalar wave equation 161B.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . 161B.2 Conservation of force . . . . . . . . . . . . . . . . . . . . . . . 163B.3 Linearized wave equations . . . . . . . . . . . . . . . . . . . . 166B.4 Scalar wave equation . . . . . . . . . . . . . . . . . . . . . . . 169

C Appendix: NMO and SWE Questions 171


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List of Figures

1.1 A common-shot record for a simple reflector at z = 1000mand = 3000m/s. . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 A single planewave from the planewave decomposition of thedata in Figure 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 All planewaves associated with a single frequency from theplanewave decomposition of the data in Figure 1.2 are summed. 5

1.4 All planewaves associated with a large number of frequenciesfrom the data in Figure 1.2 are summed. . . . . . . . . . . . . 6

1.5 A common-shot record for a simple reflector at z = 800m and = 3000m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 A single planewave from the planewave decomposition of thedata in Figure 1.6. . . . . . . . . . . . . . . . . . . . . . . . . 8



1.9 Sinusoids cos and sin . . . . . . . . . . . . . . . . . . . . . 12

1.10 Sinusoids cos and sin are advanced. . . . . . . . . . . . . . 14

1.11 Sinusoids cos and sin are delayed. . . . . . . . . . . . . . . 15

1.12 A reflection raypath in a homogeneous medium. . . . . . . . . 16

1.13 A function f (a) and its derivative f (b). . . . . . . . . . . . 20

2.1 A Gibson c

Explorer (top) and a Fender c

Jazzmaster (bottom). 27

2.2 Phase from a Gibson cExplorer combined with amplitudefrom a Fender cJazzmaster produces a rather fuzzy Gibson cExplorer. 28

2.3 Small land 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Large marine 2D . . . . . . . . . . . . . . . . . . . . . . . . . 29

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viii LIST OF FIGURES

2.5 Geometry of a 2D seismic line. a) Common shot. b) Com-

mon mid point (CMP). Acquisition parameters are: 6 shotswith shot interval xs = 80m, and 12 receivers with receiverinterval xr = 80m. . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Unprocessed (raw) seismic gather. The gather (a) and the spectrum(b) are dominated by a few high amplitude traces. . . . . . . . . . 33

2.7 Bad traces are zeroed. The gather (a) looks better, and some re-flections are identifiable. The spectrum (b) is still dominated by a

few high amplitude traces. . . . . . . . . . . . . . . . . . . . . . 34

2.8 Traces are corrected for differences in source/receiver responses bydivision of each trace by its maximum value. . . . . . . . . . . . 35

2.9 Spherical spreading applied to achieve a visual balance betweennear/far and shallow/deep reflections. . . . . . . . . . . . . . . . 36

2.10 Band-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.11 Processed shot (no filter) (a) and its auto correlation (b). On(b), non-random features are found around zero lag (t = 0) (in this

example, between 0 and 90 seconds). . . . . . . . . . . . . . . 382.12 Filtered shot with source effects and multiples partially re-

moved (a), and the resulting amplitude spectrum (b). . . . . 39

2.13 A (z) velocity model. . . . . . . . . . . . . . . . . . . . . . . 41

2.14 A synthetic data gather that corresponds to the velocity model

in Figure 2.13. . . . . . . . . . . . . . . . . . . . . . . . . . . 422.15 The synthetic data gather of Figure 2.14 stretched t t2 and

x x2. Hyperbolas are now linear events with slopes equalto 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.16 Synthetic data with NMO applied. a) Linear events corre-spond to flattened Hyperbolas. A top mute is applied toremove direct arrivals. b) Random noise (top mute applied).c) The random noise in a) is added to the data in b). . . . . 45

2.17 The data in Figure 2.16 are stacked over t = 1s for all x. Thenoise free data stack to a value of -4.75. The noise stacks to

a value close to zero. The noisy data stacks to a value of -1.75. 462.18 The data in Figure 2.16 are stacked over x for all t. a) Trace

1 from Figure 2.16c. b) The stack of all traces in Figure2.16c reflection events are now more distance and noise isreduced. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


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LIST OF FIGURES ix

2.19 Spectrum comparison between the traces in Figure 2.18. a)

Spectrum of trace 1 from Figure 2.16b similar amplitudefor all frequencies. b) Spectrum of trace 1 from Figure 2.18a.This spectrum is a sum of the noise in (a) and the data in (d)below. c) Spectrum of trace 1 from Figure 2.18b. Stackinghas reduced noise in the spectrum. d) Spectrum of trace 1from Figure 2.16a. This noise free spectrum is similar to thespectrum of the stacked data. . . . . . . . . . . . . . . . . . 47

2.20 Normal moveout (NMO) applied. The data within the NMO cor-rected gather (a) are stacked into a single trace (b). The resulting

NMO-velocity function (c). . . . . . . . . . . . . . . . . . . . . 48

2.21 Comparison of time and depth migration. a) Seismic-data.

b) Erroneous velocity model. c) Time migration - insensitiveto the velocity model. d) Depth migration - sensitive to thevelocity model. . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.22 Time migration of a zero-offset section. A zero-offset section (a) isa gather of stacked traces. b) The time-migrated section. . . . . . 51

2.23 Prestack-depth migration. . . . . . . . . . . . . . . . . . . . . . 52

2.24 Laser pointer with fish-eye lens. A magnet, a paper clip, anda ball bearing. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.25 Poissons Spot Augustin Fresnel, Simeon Poisson, Do-minique Arago (1818). This projection is achieved with theequipment pictured in Figure 2.24. . . . . . . . . . . . . . . 55

2.26 A source of rays / waves and a reflecting disk in plan view. Aray model indicates a shadow on the far side of the disk. Thewave model indicates a shadow with a bright spot on the farside of the disk. . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.27 a) A salt body embedded in a linear velocity medium. b)Seismic data associated with the model in (a). c) A ray-basedimage. d) A wave-based image. . . . . . . . . . . . . . . . . 57

2.28 A zoom of Figure 2.27. a) A small target lies below the saltbody at 1.6 km depth between 8 and 9 km. b) Seismic dataassociated with the small target in (a) found between 1.6 and

1.7 seconds and between 8 and 9 km. c) A ray-based image -the target is not visible. d) A wave-based image - the targetis visible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.29 A time snapshot of Numerical model of reflection, refraction,and critical refraction. . . . . . . . . . . . . . . . . . . . . . 59


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x LIST OF FIGURES

2.30 A time snapshot of controlled reflection, and transmission.

Only one set of reflected and transmitted waves due to alimited set of reflectors (red dots) are indicated. . . . . . . . 60

2.31 a) Structural model. b) Seismic data. c) Fourier migration.d) Reverse time migration. . . . . . . . . . . . . . . . . . . . 61

3.1 Downgoing wavefield: W+ PS. . . . . . . . . . . . . . . . . . 65

3.2 Diffracted wavefield: R W+ PS. . . . . . . . . . . . . . . . . . 65

3.3 Upgoing wavefield: PU = W R W+ PS. . . . . . . . . . . . . 66

3.4 Recorded wavefield: PR = PU|z=0. . . . . . . . . . . . . . . . 66

4.1 A boundary between elastic media. T and I indicate Transmissionand Incident sides of the interface. . . . . . . . . . . . . . . 70

4.2 A boundary between fluids. . . . . . . . . . . . . . . . . . . . 71

5.1 Some ray paths between a source and a diffractor. . . . . . . 76

5.2 kz as a function of p. (a) For constant , kz is symmetric forall p and . (b) For constant , kz is symmetric for all p and. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3 R as a function of p. (a) R is symmetric with p. (b) R as aconvolution matrix (the real part), where pD are the down-going ray coordinates, and pU are the upgoing rayparameters. 77

5.4 A diffractor sends energy in all directions. . . . . . . . . . . . 77

5.5 The real parts of the components of W+. (a) The t

Fourier transform matrix. (b) The x kx Fourier transformmatrix. (c) Extrapolation operator eikzz. . . . . . . . . . . 80

6.1 Monochromatic (a single ) extrapolation down (W+), re-flection (R), and extrapolation up (W) as matrices. . . . . 84

6.2 Monochromatic down reflect up. . . . . . . . . . . . . . 846.3 R and ei kz z are symmetric so ei kz z R ei kz z = ei 2 kz zR. 85

6.4 By assuming horizontal reflectors, (R) is diagonalized. . . . . 86

7.1 PU (x, 0) = I F F T

R (kx, z) S (kx, 0) eizkz(kx) eizkz(kx)

is equivalent to PU (x, 0) = I F F T SS (kx, 0) e

i2zkz(kx). . . 947.2 A four trace, common source gather. The black circle rep-

resents the source location, and circles represent receiver lo-cations. Trace spacing is x, and half-offset spacing is xh.Rays correspond to reflections from the horizontal reflectorat depth z. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


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LIST OF FIGURES xi

7.3 The same gather as in Figure 7.2, but annotated from the

perspective of secondary sources. Black circles represent sec-ondary sources spaced xh = x/2 apart, and rays corre-spond to direct propagation from the reflector to the receiversat half velocity c = /2. . . . . . . . . . . . . . . . . . . . . 95

7.4 A four trace CMP that corresponds to the right most reflec-tion point in Figures 7.2 and 7.3. Four common source gatherscontribute a single trace to this gather, and trace spacing inthis gather is the shot interval xs. . . . . . . . . . . . . . . 95

7.5 NMO correction and stack of the CMP gather in Figure 7.4.Trace labels s1, s2, , s4 indicate the correspondance be-tween each NMO corrected trace and the sources. NMOeliminates the horizontal distance travelled by the secondarysource, and so the rays are now vertical (they are zero offset).The trace labeled xh is the stacked trace located at distancexh from the survey origin. . . . . . . . . . . . . . . . . . . . 96

8.1 Models of density, velocity, and the corresponding reflectivity.a) Density . b) Velocity = 2 c. c) Reflectivity rj . Note, inthis Figure, CDP is equivalent to CMP. . . . . . . . . . . . . 101

8.2 Synthetic data. a) Zero offset. b) Migrated. c) Filtered r forcomparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.3 Snapshots of data during post stack imaging. a) The wave-field at z = 210m. The values along the top row correspond to

t = 0, and these are written to z = 210m in the output space.b) The wavefield at z = 790m. Data from the top row arewritten to z = 790m in the output space c) The wavefield atz = 1460m. Data from the top row are written to z = 1460min the output space d) The wavefield at z = 2010m. Datafrom the top row are written to z = 2010m in the output space 103

9.1 Calculations leading to . a) Interval velocity (cint) vs. depth.b) Interval velocity and RMS velocity (RMS) vs. time. c)2RMS t. d) vs time. . . . . . . . . . . . . . . . . . . . . 110

9.2 Input wavefield (zero offset). Velocity variation is = 3000 ms +z

5

m

s , where 0 z 4000m. . . . . . . . . . . . . . . . . . . . 1119.3 Input wavefield after Stolt stretch where ref = 3500

ms . Diffrac-

tions steeper. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

9.4 Input wavefield generated for ref = 3500ms . The stretched

section of Figure 9.3 attempts to mimic this section. . . . . . 113


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xii LIST OF FIGURES

9.5 Stolt migration of the stretched data in Figure 9.3. The depth

axis is converted to time for comparison with the input data.The point diffractors have focused quite well with some spu-rious energy present. . . . . . . . . . . . . . . . . . . . . . . 114

9.6 A true (z) migration of the un-stretched data in Figure 9.4.The depth axis is converted to time for comparison with theinput data. Focusing is better than in Figure 9.5 and spuriousenergy is reduced. . . . . . . . . . . . . . . . . . . . . . . . . 115

10.1 Behaviour of g ei[0] about stationary point k0 = 0.a) g is sinusoidal. b) kh e

i[0] = 0 at k0 = 0. c) g ei[0]

becomes sinusoidal as . The envelope of g ei[0]oscillates about 0 and is symmetric around k

0= 0, integra-

tion to compute ei0H tends to decrease the value of thefunction away from k0 = 0. As , ei0H is nonzeroonly at k0 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 119

10.2 Behaviour ofg ei[0]. a) g. b) ei[0]. c) g ei[0].Integration to compute ei0H 0 as . . . . . . 120

10.3 45o phase rotation of a Ricker wavelet. . . . . . . . . . . . . . 124

10.4 Input data with (xh xh, z) annotated for a few diffractors.Here, c = 3000/2 and all scattering is due to contrasts. . . . 125

10.5 Migrated from Figure 10.4. . . . . . . . . . . . . . . . . . . 126

10.6 Migrated data from Figure 10.4. . . . . . . . . . . . . . . . . 126

11.1 Three term cos and cos 1. a) cos . Divergent beyond 20degrees. b) cos 1. Divergent beyond 30 degrees. . . . . . 132

11.2 SEG/EAGE salt model. . . . . . . . . . . . . . . . . . . . . . 135

11.3 Exploding reflector data. . . . . . . . . . . . . . . . . . . . . . 135

11.4 Finite difference migration (n = 4, 65 degree). Note dip limiton left side of salt. . . . . . . . . . . . . . . . . . . . . . . . . 136

12.1 Zero offset migration of the SEG salt model by Split-stepFourier [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

12.2 GS migration (n = 4). . . . . . . . . . . . . . . . . . . . . . . 140

12.3 Constant velocity and frequency (k = /v in the figure) is

used to compute kz associated with the x operator. . . . 14212.4 Vertical wavenumber for a linear-varying velocity (increasing

from left to right) associated with the GS operator. . . . . . . 142

12.5 PSPI migration (n = 4). . . . . . . . . . . . . . . . . . . . . . 144

12.6 Variable v as in sediment - salt transition. . . . . . . . . . . . 144


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LIST OF FIGURES xiii

12.7 Variable v as across dipping sediments. . . . . . . . . . . . . . 145

A.1 A refracted raypath from a source to a reflector (a) and (b)two-way traveltime indicated on the corresponding seismictrace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

A.2 A refracted raypath (red line) and straight raypath (blue line)from a source to a reflector (a), and (b) two-way traveltimesindicated on the corresponding seismic trace. Straight raytraveltime computed with RMS is a good approxima-tion to the true refracted-time compared to straight ray trav-eltime computed with average velocity = mean. . . . . . . 159

B.1 Mass flux through infinitessimal volume dV = h dS. Velocity

of mass transport is vector v, and n is a unit vector normalto infinitessimal surface-area dS. . . . . . . . . . . . . . . . . 162


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xiv LIST OF FIGURES


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List of Tables

3.1 The assumptions of homogeneity and angle-of-incidence areoften relaxed when methods are implemented with varying

degrees of success. . . . . . . . . . . . . . . . . . . . . . . . . 67

12.1 Table of GS variables. . . . . . . . . . . . . . . . . . . . . . . 140

xv


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Chapter 1

Background

A number of mathematical concepts are helpful to students of seismic imag-ing. To develop a wave equation to govern the propagation of seismic waves,concepts from vector calculus are important. Because planewave decompo-sition of seismic wavefields is central to nearly all imaging methods, studentsmust be familiar with integrals associated with spatial and temporal Fouriertransforms. Following on to the Fourier integrals, for numerical implemen-tation, the relationship between continuous integration and discrete inte-gration is important. To understand discrete integration, simple conceptsfrom matrix analysis will help understanding and program development, aswill the fundamental theorem of calculus and numerical implementation ofderivatives in the time and space domain and in the Fourier domain. Wewill find that seismic imaging in its purest form is very costly to imple-ment in 3D for large datasets. To improve cost / accuracy performance, wewill employ Taylor series expansions of wavefield extrapolation to providediscrete series that may be truncated.

1.1 Planewaves and the Fourier transform

Planewave decomposition is central to most seismic imaging methods. Theyare a solution to the scalar wave equation, and from there, seismic reflectioncoefficients, the notion of phase velocity, and phase velocity in anisotropic

media are all developed. Any signal can be decomposed into simple func-tions, and these simple functions can be used for simple physical descrip-tions.

Figures 1.1 through 1.8 demonstrate one of the utilities of planewaves:wave propagation. As in Figure 1.1, a synthetic seismic experiment re-

1


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2 CHAPTER 1. BACKGROUND

sults in a band-limited shot record. Planewave decomposition of the shot

record results in a number of very simple datasets. Figure 1.2 give a singleplanewave from this shot record. Sum up all dips for a single frequency toget a shot-record-looking result (Figure 1.3). Then, sum up all dips andfrequencies to get the shot-record back (Figure 1.4). Figure 1.4 shows thesum of most of the frequencies from the lowest to a fraction of the largestfrequency. Summing from the lowest to the highest returns the wavefield.

A similar experiment as the one above - for a different reflector - resultsin another band-limited shot record (Figure 1.5). For the new experiment,the identical planewave as is shown in Figure 1.2 can be seen in Figure 1.6but shifted slightly. The sum all dips for a single frequency is also shifted(Figure 1.7). Then, the sum of all dips and all frequencies gets the shot-record back - shifted (Figure 1.8). So, shifts applied to the dataset can bedone to each planewave one-at-a-time.

To understand the planewave, we first begin with a very general de-composition of a one-dimensional function f into an infinite number of eignfunctions and eign values. Any value of f is the sum of an infinite num-ber of scalars (eign values F) and orthogonal functions (eign functions G)according to:

f(m) =

F (n) G (m, n) dn, (1.1)

where n 1 is an input coordinate, and m 1 is an output coordinate.

1.1.1 Orthogonality and the functionOrthogonal G means that G cant be constructed from any linear combina-tion of G, for example

G (m, n) =

F

n, n

G

m, n

dn, (1.2)

is valid only for F(n, n) = (n n). That is, under integration, (n n)has the action

G (m, n) =

n n G m, n dn, (1.3)where n 1 is a dummy variable1.

1By dummy variable, we mean an integration variable that does not appear as an inputor an output coordinate.


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1.1. PLANEWAVES AND THE FOURIER TRANSFORM 3

0 500 1000 1500 2000 2500

1500

1000

500

0

500

1000

1500

2000

2500

Distance (m)

Depth(m)

source

receiver

Time(s)

Distance (m)

All frequencies, all angles

0 500 1000 1500 2000 2500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 1.1: A common-shot record for a simple reflector at z = 1000m and = 3000m/s.


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Figure 1.2: A single planewave from the planewave decomposition of thedata in Figure 1.1 .


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Figure 1.3: All planewaves associated with a single frequency from theplanewave decomposition of the data in Figure 1.2 are summed.


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Figure 1.4: All planewaves associated with a large number of frequenciesfrom the data in Figure 1.2 are summed.


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0 500 1000 1500 2000 2500

1500

1000

500

0

500

1000

1500

2000

2500

Distance (m)

Depth(m)

source

receiver

Time(s)

Distance (m)

All frequencies, all angles

0 500 1000 1500 2000 2500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 1.5: A common-shot record for a simple reflector at z = 800m and = 3000m/s.


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Figure 1.6: A single planewave from the planewave decomposition of thedata in Figure 1.6.


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1.1.2 Complex exponentials

A good choice for G are complex exponentials of the form

G (m, n) = ei n m = cos (n m) + i sin(n m), (1.4)

because they are orthogonal, and because they are solutions for the equationthat governs wave propagation (see Appendix B).

Figure 1.9 shows one cycle each of a sine wave and a cosine wave. Inte-grated over an infinite number of cycles, values for sine cancel so the sinesums to 0. Cosine sums to 2. Also, complex exponentials have nice prop-erties under integration in that

ei n mdm = 2 (n) . (1.5)

To see orthogonality, substitute equation 1.4 for G in equation 1.2

ei n m =

F

n, n

ei nm dn, (1.6)

and recognize that integration of equation 1.6 over m will give the result

2 (n) =

F

n, n

ei nm dn dm. (1.7)

So that equation 1.7 balances, F(n, n) (n n), and so equation 1.6becomes

ei n m =

n n ei nm dn = ei n m, (1.8)and ein m is orthogonal. Write f, now, in terms of equation 1.4 as

f(m) =

F(n) ei n m dn. (1.9)

F is often called a spectrum, and F(n) are the spectral elements. To seewhat they are, multiply both sides of equation 1.9 by ein

m and integrateover m

f(m) e

i nm

dm = F(n) e

im [nn]

dndm

=

F(n) 2

n n dn= 2 F

n

(1.10)


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0 50 100 150 200 250 300 350

1

0.5

0

0.5

1

(degrees)

ei

900

cos

sin

Figure 1.9: Sinusoids cos and sin .

due to the action of . Dummy variables n can be renamed, so F is givenby

F (n) =1

2

f(m) ei n m dm. (1.11)

1.1.3 Fourier transforms

Equations 1.9 and 1.11 are general forms of the forward and inverse Fouriertransform.

Planewave decomposition is multi-dimensional for seismic data P(x,y ,t),and it consists of three stages (the infinite limits of integration are omittedhere for brevity) according to

P(x,y ,) =

P(x,y ,t) ei t dt, (1.12)

that applies the t transform,

P(kx, y , ) =1

2 P(x,y ,) eikx x dx, (1.13)

transforms x kx, and

(kx, ky, ) =1

2

P(kx, y , ) e

iky y dx, (1.14)


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transforms y ky.Spectrum (kx, ky, ) is a planewave (see Figures 1.1 through 1.8).Written as a single operation, planewave decomposition of a seismic

recording is

(kx, ky, ) =1

(2 )2

P(x,y,t) ei[ky y+kx x t] dxdy dt. (1.15)

Note, the choice of signs in equation 1.15 is made in advance of solving awave equation - it results in the extremely useful dispersion relation.

1.1.4 Discrete Fourier transforms and the FFT

For numerical implementation, Fourier transforms have finite rather thaninfinite bounds. Finite bounds means that Fourier wrap-around occurs un-less we pad data sufficiently. For example, the apparent reflection from thesides of Figure 1.5 are the result of Fourier wrap-around - a remedy is tohave a much longer and wider data grid.

Numerically, the t transform in equation 1.15 is often implementedas

fm =N

n=1

Fn ei 2 (m1) (n1)/N, (1.16)

where m is the mth output sample, N is the number of input samples, and

1 n N. The x kx and y ky transforms in equation 1.15 are thenimplemented numerically as

Fn =1

N

Mm=1

fm ei 2 (m1) (n1)/M. (1.17)

The sign choice in 1.15 is the reason for the sign difference between equations1.16 and 1.9 and between equations 1.17 and 1.11.

Computationally, equations 1.16 and 1.17 cost N log2 N floatingpoint operations where N is the number of samples.

1.1.5 Amplitude and phase

Planewave can be written in terms of amplitude and phase according to

(kx, ky, ) = A (kx, ky, ) ei (kx,ky,), (1.18)


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0 50 100 150 200 250 300 350

1

0.5

0

0.5

1

(degrees)

e(i + )

900

cos

sin

Figure 1.10: Sinusoids cos and sin are advanced.

whereA (kx, ky, ) = | (kx, ky, )| , (1.19)

and

(kx, ky, ) = tan1

{ (kx, ky, )}{ (kx, ky, )}

, (1.20)

where {a + i b} = a, and {a + i b} = b.

1.1.6 Phase shiftThe difference between Figures 1.1 through 1.4 and 1.5 through 1.8 is the re-sult of time-shifting the individual planewaves. To see this, compare Figures1.3 and 1.7 - the planewaves are identical except that one is time shifted rel-ative to the other. The sinusoids are Figures 1.10 and 1.11 are time-shiftedversion of Figure 1.9. A positive phase change () shifts the sinusoids leftand a negative change shifts them right.

Most imaging methods have time shifts at their core - in 3D instead of1D.

1.1.7 Snells Law

Snells Law provides a relationship between planewaves and rays, and it isvery useful for the mapping of physical intuition from the physical worldto the planewave domain. Figure 1.12 shows a reflected ray (dashed line)where the ray is normal to an advancing planewave. Angle is the angle


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0 50 100 150 200 250 300 350

1

0.5

0

0.5

1

(degrees)

e(i )

900

cos

sin

Figure 1.11: Sinusoids cos and sin are delayed.

between the ray and the normal to the reflecting boundary, and the sine ofthis angle is given by

sin =

x 2 + z2

x, (1.21)

where

x 2 + z2 is distance travelled along the ray. If both travel timet along the ray and wave velocity are known, then

x 2 + z2 = t, (1.22)and equation 1.21 simplifies to

sin

=

t

x. (1.23)

For a 2D planewave (equation 1.15 with ky = 0), = kx x t remainsconstant so that planewaves are in phase). This stationary requirementleads to

0 = =

xx +

tt = kxx t. (1.24)

Therefore, using equation 1.23 and 1.24

sin

=

t

x=

kx

= p, (1.25)

where p is rayparameter.


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0 50 100 150 200 250 300 350 400 450 500

0

50

100

xx

z

Depth(m)

Distance (m)

Figure 1.12: A reflection raypath in a homogeneous medium.

1.1.8 Taylor series

In the development of a number of concepts in imaging, we will employTaylor series according to:

f(t + t) = f(t) +

j=1

1

j !

jt f(t)

tj , (1.26)

where j ! = 1 2 ... j = jk=1 k is the factorial function, and

jt f(t) = j

j tf(t) . (1.27)

To linearize equations, series expansion is important in particular the ex-pansions

1 + a 1 + 1

2a , (1.28)

and1

1 + a 1 1

2a + , (1.29)

for

1 < a

1, as well as

ea 1 + a + a2

2 !+ , (1.30)

where < a < .


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1.1.9 Matrix algebra

Matrix representations of numerical calculations are very useful in imaging,in particular matrix - vector multiplication

a = B c (1.31)

where indicates vector transpose, a = [a1 a2 aM], c = [c1 c2 cN],and, for M and N positive integers, B is the following M N matrix:

B =

B11 B12 B1NB21 B22 B2N

.... . .

...BM1 BM2 BMN

. (1.32)

For example, when a = [a1 a2 a3 a4] and c = [c1 c2 c3], equation 1.34 is

a1a2a3a4

=

B11 B12 B13B21 B22 B23B31 B32 B33B41 B42 B43

c1c2

c3

=

B11 c1 + B12 c2 + B13 c3B21 c1 + B22 c2 + B23 c3B31 c1 + B32 c2 + B33 c3B41 c1 + B42 c2 + B43 c3

. (1.33)

As a code snippet, equation 1.33 is

a = zeros(4,1)

F o r j = 1 - > 4

F o r k = 1 - > 3

a(j) = a(j) + B(j,k) * c(k)

end

end

Similarly, matrix - matrix multiplication is given by

A = B C, (1.34)

where, for M N matrix B, C is an N P matrix, and A is an M P


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matrix. For M = 4, N = 3, and P = 2, equation 1.34 is

a11 a12a21 a22a31 a32a41 a42

=

(B11 c11 + B12 c21 + B13 c31) (B11 c12 + B12 c22 + B13 c32)(B21 c11 + B22 c21 + B23 c31) (B21 c12 + B22 c22 + B23 c32)(B31 c11 + B32 c21 + B33 c31) (B31 c12 + B32 c22 + B33 c32)(B41 c11 + B42 c21 + B43 c31) (B41 c12 + B42 c22 + B43 c32)

.

(1.35)

where C = c11 c12c21 c22

c31 c32

.As a code snippet, equation 1.35 is implemented as

A = zeros(4,2)

F o r h = 1 - > 2

F o r j = 1 - > 4

F o r k = 1 - > 3

A(j,h) = A(j,h) + B(j,k) * C(k,h)

end

end

end

1.1.10 Vector calculus

The concepts of div, grad, and curl will be important - especially in the

derivation of a wave equation. For a vector v =

v1 i + v2j + v3 k

, where

each vj vj (x,y,z), define div, , as

s = v = x

v1 +

yv2 +

zv3 (1.36)

where s is a scalar.

For scalar s s (x,y ,z), define grad, , asw = s =

xs i +

ysj +

zs k, (1.37)

where w is a vector.


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For vectors v and w, define curl, as

z = v = x y zv1 v2 v3

=

v3y

v2z

i

v3x

v1z

j +

v2x

v1y

k,

(1.38)

where z is a vector.

1.1.11 Numerical derivatives

From the fundamental theorem of calculus we have for the slope of a function

f(t) d

dtf(t) f(t)

t=

f(t + t) f(t)(t + t) t (1.39)

for small t.Numerically, equation 1.39 can be computed as a matrix vector operation

according to

f1f2...

fN1

=

1 1 0 0 00 1 1 0 0...

. . ....

0 0 0 1

f1f2...

fN

t, (1.40)

or, with more accuracy2, as [4, Section 4.3]

f1f2...

fN1

=

0 1 0 0 01 0 1 0 0

.... . .

...0 0 0 1 0

f1f2...

fN

2 t. (1.41)

Higher derivatives can be multiple applications of the derivative matrixor, for example, the 2nd order finite difference operator can be written ex-plicitly as

f1

f2...

fN1

=

2 1 0 0 01 2 1 0 0...

. . ....

0 0 0 1 2

f1

f2...

fN

t2. (1.42)2Better approximations exist though they are more costly.


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0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

f(t)

a)

0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

1.5

1

0.5

0

0.5

1

Time (s)

b)

f

f

Figure 1.13: A function f (a) and its derivative f (b).


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Higher derivatives also have explicit forms.

Figure 1.13 shows a function f(t) and its derivativesd

dt f(t) = f

(t) andf.Differentiation has a representation in the Fourier domain. For example,

f (t) =1

2

d

dt

F() ei td =

1

2

(i ) F() ei t d (1.43)

Higher order derivatives are computed simply as

fj (t) =1

2

(i )j F() ei t d (1.44)

where j indicates the jth derivative.


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1.2 Questions

1. For a 1D FFT, cost N log2N. What is the cost of a 2D FFT?M N[log2 M + log2 N] = M N [log2 M N].

2. What is the cost of a 3D FFT?P M N [log2 M + log2 N + log2 P] = P M N [log2 M N P].

3. (kx, ky, ) = a (kx, ky, ) + i b (kx, ky, ). Compute the amplitudeand phase of .

4. Mathematically, how is (t) advanced .01 seconds? (t + t) = IF T

() ei t

, where t = .01

5. Mathematically, how is (t) reversed .01 seconds? (t + t) = IF T

() ei t

, where t = .01

6. Compute rayparameter p for 90 90 in 30 degree increments ina medium where = 3000m/s.p = sin

2 , 3 , 6 , 0, 6 , 3 , 2 , 3000 m/s.7. Expand

2 k2x to first order in kx.

1 12

kx

2

8. For B = 1 1 12 2 2

3 3 3

and c = 222

, what is B c?B c = [06 12 18]T

9. For B =

1 1 12 2 23 3 34 4 4

, what is B c?

B c = [06 1218 24]T

10. For v = x i + yj + z k, compute v, ( v), and v. v =

vxx +

vyy +

vzz = 1 + 1 + 1 = 3,

( v) = vx i + vy j + vz k = 0 i + 0j + 0 k,and v =

vzy vyz

i +

vzx vxz

j +

vyx vxz

k = 0 i +

0j + 0 k.


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1.2. QUESTIONS 23

11. Compute the numerical derivative of [2 3 4]. Assume that t = 1.321

43

1

= [11] from equation 1.40,or0+3

22+4

2

= [3/2 1] from equation 1.41.


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Chapter 2

Overview: seismic reflection

This chapter is an introduction to the broad topic of reflection seismology.A number of concepts are introduced in a brief, pictorial form to try toprovide the student with a wide view of this field. The concepts of seismicacquisition geometry and data processing are presented, for example, toensure that students understand that digital seismic data are acquired on aspace time grid, and that they must be processed prior to imaging. Imagingitself is summarized at a conceptual level so that students become acquaintedwith the variety of seismic imaging approaches, for example ray-based verseswave-based, and so that they have the expectation that imaging accuracyand efficiency must be considered within the context of geologic complexity.

2.1 Goals in seismic imaging

Roughly speaking, there are three goals in seismic imaging. The first, andsimplest, is the determination of geologic structure. Structure is representedin the data as a change in the seismic wavefield in time due to some materialcontrast - a reflection event. Reflection events, if strong enough, are easilyidentified in data of high quality, and their arrival time, plus some knowledgeof seismic velocity, together allow identification of the spatial origin of thereflection. Geologic structure is mappable through this process. Technically,geologic structure is related to the phase (section 1.1.5) component of the

seismic wavefield.More difficult is the characterization lithology and fluids. Here, seismic

amplitude (section 1.1.5) is analyzed and compared with model amplitudesfor lithologic contrasts of interests. If a sandstone-beneath-shale scenariois the target, for example, a theoretical seismic amplitude is computed and

25


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26 CHAPTER 2. OVERVIEW: SEISMIC REFLECTION

then compared to the seismic amplitude that corresponds to a structure of

interest. Correspondence of the amplitudes allows identification of both theshale and the sand.A difficulty arises here in the preservation of amplitude fidelity. Unlike

seismic traveltime (phase), amplitude is much more ambiguous due mainlyto differences in shot and geophone coupling, shot strength, and these varia-tions seem to register most strongly in the amplitude spectrum rather thanin the phase spectrum.

Additionally, seismic noise sources plus model estimation and imagingerrors seem to distort amplitudes much more than phases.

In general, phase is much more robust than amplitude when a visualimage is considered as Figures 2.1 and 2.2 demonstrate. In Figure 2.1,two guitars manufactured by competing manufacturers - a Gibson c

(upper)

and a Fender c(lower) are presented. The Figures are digital images, andthey have unique amplitude and phase spectra (equations 1.19 and 1.20respectively). Importance of phase relative to amplitude is demonstratedthrough construction of a new guitar - the phase of the Gibson cand theamplitude of the Fender care combined. The result, given in Figure 2.2,is clearly a Gibson c, and the effect of the Fender camplitude is a simpleblurring of the image.

The third goal in imaging, and often the most difficult to obtain withsignificant accuracy, is the determination of a reservoir volume. Here, theattributes of amplitude and phase are used to construct a target volume. Asoutlined above, though seismic phase may image robustly a target structure,

amplitude errors may distort the spatial extent of reservoir quality rocks.Because the economic desirability has, at its root, an estimate of recover-able reserves, large errors in reservoir volume may lead to erroneous drillingdecisions.


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2.2. ACQUISITION 27

Figure 2.1: A Gibson cExplorer (top) and a Fender cJazzmaster (bottom).

2.2 Acquisition

Seismic data are acquired on land and in marine environments. Land data(data acquired on land) often suffer from irregular source and receiver lo-

cations (irregular geometry) plus variations in topography. This is demon-strated in Figure 2.3 - acquisition lines are crooked, and the acquisitionsurface is not flat. These inconsistencies diminish the fidelity of seismicdata if they are not accommodated in processing later on.

Marine data also suffer from irregular geometry plus a systematic restric-tion to 2D as in Figure 2.4. Irregular geometry is most often associated withtides and currents that cause drift in the acquisition system. The restric-tion to 2D is the result of acquisition with flexible hydrophone cables thatare towed behind ships. The 2D restriction is overcome somewhat throughmultiple streamers.

Ocean bottom seismometer (OBS), ocean bottom cable (OBC), and ma-

rine vertical seismic cable are acquisition variants that have their own ad-vantages and disadvantages.

For all forms of acquisition, the shot record is the basic unit of dataas in Figure 2.5a. Six source points (shots) are illustrated in Figure 2.5a.The recording system depicted in this Figure consists of 12 receiver stations


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Figure 2.2: Phase from a Gibson cExplorer combined with amplitude froma Fender cJazzmaster produces a rather fuzzy Gibson cExplorer.

(geophones). To cover a large spatial distance, the receivers are moved ina line along with a central source. Sources are either pre-drilled explosivecharges, or vibrational sources mounted on trucks. The source is excited,and the receivers record the vibrations. A common source gather for thisillustration is the collection of all traces along the xr axis in Figure 2.5aassociated with the source point xs.

The common midpoint (CMP) gather is convenient reorganization ofseismic data. The main application of the CMP is for seismic velocity anal-ysis under the assumption of horizontal reflectors (see Appendix A for ex-ample), and often seismic data are imaged in this domain. A CMP is thecollection of all traces along the xs axis in Figure 2.5b associated with the

CMP point xcmp.Traces in a CMP have a common midpoint (surface expression of the

point of reflection from all reflectors) as in Figure 2.5b, but different sources.The distance xcmp from the survey origin to a midpoint is given by

xcmp = xs +xs xr

2, (2.1)

where xs and xr are source and receiver positions along the line.

A useful measure of acquisition quality is the fold of the data. Fold isgiven by

fold = Nrr

2s, (2.2)

and with shot and receiver intervals (in units of distance) r and s re-spectively. Nr is the number of traces in a common-shot gather.


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2.2. ACQUISITION 29

40 30 20 10 0 10 20 30

40

30

20

10

0

10

20

21

1

1

1

0

0

00

1

1

1

2

2

Distance (m)

Distance(m)

ShotReciever

N

Figure 2.3: Small land 3D

Figure 2.4: Large marine 2D


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400 200 0 200 400 600

200

0

200

400

600

xr(m)

xs

(m)

a)400 200 0 200 400 600

200

0

200

400

600

xcmp

(m)

xs

(m)

b)

GeophoneShot

CMPShot

Figure 2.5: Geometry of a 2D seismic line. a) Common shot. b) Commonmid point (CMP). Acquisition parameters are: 6 shots with shot intervalxs = 80m, and 12 receivers with receiver interval xr = 80m.

2.3 Processing

In reflection imaging, we are mostly interested in reflected body-waves (P-wave and S-wave), so seismic processing is often just a set of procedures toseparate body-waves from non body-waves (noise).

Energy for the body-waves arrives in phase (the constituent frequenciesarrive with peaks aligned) along the Snell-raypath between their source atthe shop point and a receiver at the surface. The Snell ray path ensuresthat we can see the energy on a seismic trace.

Body waves reflect from density () and velocity contrasts ( for P-

waves, and for S-waves). These contrasts define a reflector. As body-wavespropagate to a reflector, they are converted from downgoing to upgoing (theyare reflected), and their relative energy changes according to , , and across the reflector. If variation in and is known roughly (a modelhas been constructed), then body-waves can be focused into a seismic image.

2.3.1 Noise reduction

Figures 2.6 through 2.12 demonstrate a few of the common noise reductionprocedures used in seismic data processing. Note, a number of proceduresthat scale the data are often applied to seismic data. This is usually done to

enhance reflections for later procedures such as velocity analysis. Thoughthe visual fidelity of the data is often improved by these procedures, theyare often base on qualitative rather than physical reasoning. Care must betaken, then, to evaluate the effect of scaling on the absolute amplitude ofdata when true amplitude fidelity is desired.


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2.3. PROCESSING 31

2.3.2 Trace edit

Seismic traces (recordings that correspond to a single shot-receiver pair)that are obviously erroneous are edited. As shown in Figure 2.6, visualcomparison of traces within a single CMP (Figure 2.6a) indicates a num-ber of traces that do not correlate with the majority, and this observationis generally supported by visual comparison of amplitude spectra (Figure2.6b). Erroneous traces edited (they are set to zero) as in Figure 2.7.

2.3.3 Trace balance

Each trace in a seismic gather corresponds to a unique receiver, as in acommon source gather (Figure 2.5a), or to different sources and receivers as

in a CMP gather (Figure 2.5b). Source strength and coupling, and receiverresponse and coupling differ, and these differences contribute non-geologicalamplitude variation to trace amplitudes. As a rough correction, traces arebalanced as in Figure 2.8, where each trace is normalized to its maximumvalue as a crude correction for source and receiver variation (much moresophisticated corrections can be applied). Note, because this is a marineexample, strong surface waves are not present, but land data require surfacewave removal prior to scaling.

2.3.4 Spherical divergence

Due to spherical spreading (finite source energy spreads out on the advanc-

ing wavefront), traces near the source register more energy that and tracesfarther from the source. Similarly, within a single trace, reflectors visibleat early times register greater energy than those that arrive later. As anapproximate correction, trace energy is amplified exponentially according toarrival time as in Figure 2.9a.

Note, because much of basic seismic processing and modelling assumesthat the seismic trace is one dimensional, and that seismic reflection ampli-tude is proportional to seismic reflectivity, amplitude decay due to sphericalspreading is removed. Once processing is complete, however, this correctionis often removed prior to imaging.

2.3.5 Band-pass

In the amplitude spectrum of Figure 2.9b, coherent amplitudes appear be-tween about 10 Hz and 40 Hz with noise outside of that range. The data,then, are band-pass filtered (as in Figure 2.10b) so that only the coherent


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part of the spectrum remains. In Figure 2.10a, the filtered trace gather

appears significantly less noisy.The sources of the noise that has been removed falls into one of two cat-egories: random and non-random. Random noise results from a number ofconditions that include wind and waves, electrical noise, and spurious equip-ment behaviour. In general, any non-predictable source of non-reflectionenergy is random noise.a Non random noise is due to the recording of anynon-reflected waves such as air blast, headwaves, and surface waves, and /or P-wave leakage onto S-wave data, and S-wave leakage onto P-wave data.Fortunately, non-random noise often separates from reflections because ofdifferences in apparent velocity.

2.3.6 DeconvolutionThe ideal seismic source is a perfect impulse restricted to the source locationat the instant of excitation. In the frequency domain, the ideal source hasa flat amplitude spectrum and a zero phase spectrum. Unfortunately, therenumerous physical effects that, during acquisition, distort the source wave-field. Such effects include ghosting and ringing in the water column and inthe near surface, combustible sources that burn rather than explode, non-elastic surface conditions on land. Further, strong reflectors add multiplereflections to the source.

Superposition of the above effects results in a source that has a non-flatfrequency spectrum and, under the most hopeful circumstance, a minimum

phase spectrum. In the case of a source with a minimum phase spectrum,a source amplitude for each trace may be deduced through auto correla-tion of the trace (Figure 2.11b), and the phase of the source is then derivedfrom the amplitude using the Hilbert transform. An inverse filter is thencomputed from the source estimate and applied to the trace in a processcalled deconvolution. Under ideal conditions, the resulting trace representsan ideal source experiment. Figures 2.12a and b are the result of deconvo-lution applied to the data in Figure 2.11a. Reflection events on 2.12a aresharpened relative to those on Figure 2.11a, and the amplitude spectrum inFigure 2.12b is much flatter through the pass band (10 - 40 Hz) than thespectrum in Figure 2.11b.

2.3.7 Model building

When traces in a gather are ordered according to increasing source - receiverdistance (offset), seismic reflections appear as hyperboloids on trace gathers


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2.3. PROCESSING 33

Time(s)

Distance (m)

a) Raw

500 1000 1500 2000 2500

0

0.5

1

1.5

2

2.5

3

Frequency(Hz)

Distance (m)

b) Amplitude spectrum

500 1000 1500 2000 2500

0

10

20

30

40

50

60

70

80

Figure 2.6: Unprocessed (raw) seismic gather. The gather (a) and the spectrum(b) are dominated by a few high amplitude traces.


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Time(s)

Distance (m)

a) Edit bad traces

500 1000 1500 2000 2500

0

0.5

1

1.5

2

2.5

3

Frequency(Hz)

Distance (m)


500 1000 1500 2000 2500

0

10

20

30

40

50

60

70

80

Figure 2.7: Bad traces are zeroed. The gather (a) looks better, and some reflectionsare identifiable. The spectrum (b) is still dominated by a few high amplitude traces.


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2.3. PROCESSING 35

Time(s)

Distance (m)

a) Trace balance on

500 1000 1500 2000 2500

0

0.5

1

1.5

2

2.5

3

Frequency(Hz)

Distance (m)


500 1000 1500 2000 2500

0

10

20

30

40

50

60

70

80

Figure 2.8: Traces are corrected for differences in source/receiver responses bydivision of each trace by its maximum value.


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Time(s)

Distance (m)

a) Spherical divergence

500 1000 1500 2000 2500

0

0.5

1

1.5

2

2.5

3

Frequency(Hz)

Distance (m)


500 1000 1500 2000 2500

0

10

20

30

40

50

60

70

80

Figure 2.9: Spherical spreading applied to achieve a visual balance betweennear/far and shallow/deep reflections.


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2.3. PROCESSING 37

Time(s)

Distance (m)

a) Bandpass filter

500 1000 1500 2000 2500

0

0.5

1

1.5

2

2.5

3

Frequency(Hz)

Distance (m)


500 1000 1500 2000 2500

0

10

20

30

40

50

60

70

80

Figure 2.10: Band-pass filter


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Time(s)

Distance (m)

a) Processed no filter

500 1000 1500 2000 2500

0

0.5

1

1.5

2

2.5

3

Time(s)

Distance (m)

b) Auto correlation

500 1000 1500 2000 2500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2.11: Processed shot (no filter) (a) and its auto correlation (b). On (b),non-random features are found around zero lag (t = 0) (in this example, between 0and 90 seconds).


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2.3. PROCESSING 39

Time(s)

Distance (m)

a) Bandpass filter

500 1000 1500 2000 2500

0

0.5

1

1.5

2

2.5

3

Frequency(Hz)

Distance (m)


500 1000 1500 2000 2500

0

10

20

30

40

50

60

70

80

Figure 2.12: Filtered shot with source effects and multiples partially removed(a), and the resulting amplitude spectrum (b).


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(Figure 2.12a for example). The shape of each hyperboloid is a record of

seismic travel time that is due to the distance travelled by the seismic source.The distance travelled is due, in turn, to velocity variation as well as thedepth to the reflector and the source-receiver distance.

To demonstrate, the velocity model in Figure 2.13 is used to generate thesynthetic data gather in Figure 2.14. Hyperbolic events in these data corre-spond to reflections, and because the velocity model is (z), the apexes ofeach hyperbola are found at zero distance on the offset axis. From AppendixA, equation A.8, we have that for the Nth reflection hyperbola, traveltimetn, and source / receiver offset xn are related according to

t2n t0n

2=

xnn

2

, (2.3)

where t0n is travel time to the apex of the nth hyperbola (zero-offset trav-

eltime). Equation 2.3 is the equation of a hyperbola where t2n and x2n are

linearly related through 2n, where is a root mean square (RMS) velocity(Appendix A, equation A.12) associated with the nth reflector. This linearrelationship is illustrated in Figure 2.15. Here, the data of Figure 2.14 arestretched from t to t2 and from x to x2 through interpolation. Hypoerbolicevents appear linear in the t2 x2 domain and, according to equation 2.3,the reciprocal of each nth slope is equal to 2n. Through the fitting of curvesto reflection data, linear curves as in Figure 2.15 and hyperbolic curves asin Figure 2.14, an RMS velocity function is derived.

This RM S function can then be used to flatten reflection hyperbolas,first as a test for the goodness of RM S as in Figure 2.16. Here, RMS isused to compute at time correction t for each t and x pair according to

tn =

(t0n)

2 +

xnn

2 t0n +

1

2

xnn

2 1t0n

, (2.4)

and

tn =1

2

xnn

2 1t0n

, (2.5)

where tn = tn t0n.Rather than flatten just reflection events, each input sample in the data

is shifted in time according to equation 2.5, and this is seen in Figure 2.16.Reflection events are flat as in Figure 2.16a, and here non-reflection eventshave been muted with a top mute. The flatness of every event indicates twopoints: 1) the reflectors in the subsurface are probably horizontal, and 2)


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2.3. PROCESSING 41

Depth(m)

Distance (m)

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

x 104

0

500

1000

1500

2000

2500

3000

3500

4000

1600 1800 2000 2200 2400 2600 2800 3000 3200 3400

receivers

shots

Figure 2.13: A (z) velocity model.


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Time(s)

Offset (m)1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 2.14: A synthetic data gather that corresponds to the velocity modelin Figure 2.13.


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2.3. PROCESSING 43

Timesquared(s2)

Offset squared (m2)

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

x 108

0

5

10

15

20

25

Figure 2.15: The synthetic data gather of Figure 2.14 stretched t t2 andx x2. Hyperbolas are now linear events with slopes equal to 2.


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n is probably correct for all 1 n N, and for all tn = n t where N isthe number of time samples, and t is the sample interval.A further benefit to flattening reflection events occurs in the presenceof noise. Noise in the form of random plus non-random noise is presentin all measurements including seismic measurements. In Figure 2.16b, forexample, random noise is illustrated and it is added to the flattened reflectiondata of Figure 2.16a, and the result is plotted in Figure 2.16c. The noisy data(Figure 2.16c) contain reflection events that are now difficult to interpret.For example, the reflection event at 1s on the noise-free data is ambiguouson the noise data.

All three images can be summed along the row (numerically) that cor-responds to 1s. The noise free data (Figure 2.16a) sums to a negative valueof about -4.75. The noise (Figure 2.16b) sums to a value close to zero (thenoise is approximately random), and the noise data sums to a value of about-1.75. Clearly, then, summing or stacking over the x dimension causes pos-itive reinforcement of data at the expense of random noise. Stacking ofthe noisy data in Figure 2.16c is demonstrated in Figures 2.18 and 2.19.In Figure 2.18, two traces are compared. The first trace is the left mosttrace from the noisy data (Figure 2.16c), and the second trace is a stackover x of the data gather in Figure 2.16c. The stacked trace has more co-herent reflection energy present with less noise when compared to the noisytrace1. A comparison of the spectra of the noisy trace (Figure 2.18 a) andthe stacked trace (Figure 2.18 b) is even more compelling. As can be seenin Figure 2.19, the spectrum of the noisy trace (Figure 2.19b) is the sum

of the spectra of the noise (Figure 2.19a) and the data (Figure 2.19d). Thespectrum of the stacked trace, however, due to the stacking out of the noiseappears more like that of the noise free data. The velocity function ,rather than vary according to reflection event, now varies with time so thatthe time correction t for each t and x datapoint becomes

t (t) =1

2

x

RM S (t)

2 1t

, (2.6)

where t is now time along the time axis, and RMS represents velocity varia-tion below the central location of the analysis gather. A model for the entiresurvey is then achieved though analysis of numerous gathers throughout the

survey, and with spatial interpolation of their RM S functions.A more realistic example of trace stacking is given in Figure 2.20. Here,The processed gather of Figure 2.12 is analyzed, and continuous velocity

1This stacked trace is usually assigned a spatial position in the survey that correspondsto the spatial location of the centre of the gather.


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2.3. PROCESSING 45

Time(s)

Distance (m)

Data

0 200 400 600 800

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time(s)

Distance (m)

Random noise

0 200 400 600 800

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time(s)

Distance (m)

Data + Noise

0 200 400 600 800

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 2.16: Synthetic data with NMO applied. a) Linear events correspondto flattened Hyperbolas. A top mute is applied to remove direct arrivals. b)Random noise (top mute applied). c) The random noise in a) is added tothe data in b).


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0 0.5 1 1.5 2 2.5 3 3.5 4

4.5

4

3.5

3

2.5

2

1.5

1

0.5

Data

NoiseData+Noise

Figure 2.17: The data in Figure 2.16 are stacked over t = 1s for all x. Thenoise free data stack to a value of -4.75. The noise stacks to a value close tozero. The noisy data stacks to a value of -1.75.

0.1 0 0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time(s)

Amplitude

Tr1

2 0 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time(s)

Amplitude

Stack

Figure 2.18: The data in Figure 2.16 are stacked over x for all t. a) Trace1 from Figure 2.16c. b) The stack of all traces in Figure 2.16c reflectionevents are now more distance and noise is reduced.


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2.3. PROCESSING 47

0 20 40 60 80 100 1200

1

2

Amplitude

Frequency (Hz)

Noise

0 20 40 60 80 100 1200

2

4

Amplitude

Frequency (Hz)

Tr1

(Data + Noise)

0 20 40 60 80 100 1200

50

100

Amplitude

Frequency (Hz)

Stack

0 20 40 60 80 100 1200

2

4

Amplitude

Frequency (Hz)

Tr1

(Data)

Figure 2.19: Spectrum comparison between the traces in Figure 2.18. a)Spectrum of trace 1 from Figure 2.16b similar amplitude for all frequen-cies. b) Spectrum of trace 1 from Figure 2.18a. This spectrum is a sum ofthe noise in (a) and the data in (d) below. c) Spectrum of trace 1 from Fig-ure 2.18b. Stacking has reduced noise in the spectrum. d) Spectrum of trace1 from Figure 2.16a. This noise free spectrum is similar to the spectrum ofthe stacked data.


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Time(s)

Offset (m)

NMO ona)

2500200015001000500

0

0.5

1

1.5

2

2.5

3

0.2 0 0.2

0

0.5

1

1.5

2

2.5

3

Time(s)

Amplitude

Stacked traceb)

1000 1500 2000 2500

0

0.5

1

1.5

2

2.5

3

Time(s)

Velocity (m/s)

c)

Figure 2.20: Normal moveout (NMO) applied. The data within the NMO cor-rected gather (a) are stacked into a single trace (b). The resulting NMO-velocityfunction (c).

function for the gather is deduced (Figure 2.20c). The gather is NMO cor-rected (Figure 2.20a) and stacked into a single trace (Figure 2.20b). Note,though the vertical axis in Figure 2.20c is in units of time, the velocity func-tion itself can be used to convert to depth as in Appendix A, equations A.7,A.8, and A.13.

Note, there are numerous synonyms for the velocity field such as moveout-velocity, stacking-velocity, RMS-velocity, NMO-velocity, as well as stacking-velocity.

2.3.8 Imaging

Conventional seismic imaging combines the mathematics of planewave de-composition and scalar waves, with a velocity model deduced from the seis-


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2.3. PROCESSING 49

mic data, such that the seismic data itself is used to construct an image.

Interchangeable with seismic imaging, seismic migration (we will see thatseismic data are migrated into the image) has two major output domains- time and depth. In time migration, the output image retains a time axissuch that the input data are comparable directly to the output image. Theadvantage of retention of a time axis rather than a depth axis resides withthe interpretation of the data through the processing sequence. The endproduct, the image, is often not drastically different than the output of thefinal stage of processing. If all has gone well, the image will appear focused,and all of the major reflection evens will remain in familiar positions. Timemigration tends to be rather insensitive to the velocity model as Figure 2.21.Here, seismic data (Figure 2.21a) are time migrated according to a seriouslyerroneous velocity model (toys embedded in a dipping, liner background,Figure 2.21b). The image that results (Figure 2.21c) is a horizontal reflector- in fact the input reflector is simply shifted by one second.

In depth migration, the image has a depth axis that is comparable di-rectly to subsurface geology rather than to the input data. The advantageof depth migration is quite obvious - drillers drill in space rather than time.A possible disadvantage of depth migration, in contrast to time migration,is its sensitivity to the velocity model as shown in Figure 2.21d. Depth mi-gration of the data with the erroneous model results in a strongly distortedwavefield - this can be considered an advantage, however, when the validity

of a model is in question. In this example, time migration provided no in-formation regarding the model, where depth migration indicates a possibleproblem. A more realistic comparison is provided in Figures 2.22 and 2.23.Here, a synthetic seismic dataset (Figure 2.22a) is time migrated with theexact velocity model (the model that from which the data are synthesized),where the model has been converted from depth to time. The resulting im-age (Figure 2.22b) is similar to the input, but with improved focusing ofdiffractions.

Prestack depth migration of the same data with the exact model is givenin Figure 2.23. This imaging procedure is applied to each common-source

gather at a time, and the results are aligned in space and summed. Figure2.23a shows the prestack depth migration of one gather, and Figure 2.23bshow the sum of a large number of migrated gathers. The depth migratedimage is superior to the time migrated image, but a very high price is paidcomputationally.


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Time(s)

Distance (km)

a)

0 10 20

0

0.5

1

1.5

2

2.5

3

0.2

0.1

0

0.1

0.2

D

epth(km)

Distance (km)

b) m/s

0 10 20

0

0.5

1

1.5

2

2.5

3 1000

2000

3000

4000

5000

6000

Time(s)

Distance (km)

c)

0 10 20

0

0.5

1

1.5

2

2.5

3

0.2

0.1

0

0.1

0.2

D

epth(km)

Distance (km)

d)

0 10 20

0

0.5

1

1.5

2

2.5

3

0.2

0.1

0

0.1

0.2

Figure 2.21: Comparison of time and depth migration. a) Seismic-data. b)Erroneous velocity model. c) Time migration - insensitive to the velocitymodel. d) Depth migration - sensitive to the velocity model.


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2.3. PROCESSING 51

Time(s)

Offset (m)

Stacked tracesa)

0 1000 2000 3000 4000 5000

0

0.5

1

1.5

2

2.5

Time(s)

Offset (m)

Time migrationb)

0 1000 2000 3000 4000 5000

0

0.5

1

1.5

2

2.5

Figure 2.22: Time migration of a zero-offset section. A zero-offset section (a) is agather of stacked traces. b) The time-migrated section.


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Depth(m)

Offset (m)

a)

4000 5000 6000 7000 8000

0

500

1000

1500

2000

2500

3000

Depth(m)

Distance (m)

b)

4000 5000 6000 7000 8000

0

500

1000

1500

2000

2500

3000

Figure 2.23: Prestack-depth migration.


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2.4. RAYS AND WAVES 53

2.4 Rays and waves

Rays are an a common approximation for wave propagation, and they arecentral to many imaging methods, in particular, to Kirchhoff migration.Ray can be drawn in space-time coordinates, so they have great intuitiveattraction, however, in complex media, they have significant limitations.

Figures 2.24 through 2.26 illustrate the major limitation of the ray model.A laser pointer, a magnet, a paper clip, and some ball bearings are depictedin Figure 2.24. The paper clip is bent and placed on the magnet such that aball bearing is held in place on the end of the paper clip. A lens is attachedto the laser pointer such that the laser diffuses into a large spot, where thediameter of the spot is much larger than the diameter of the perched bearing.When the laser is directed a the ball bearing, the bearing reflects all laserlight such that the bearing casts a shadow as in Figure 2.25. A number ofcharacteristics of this shadow (Figure 2.25) are, perhaps, unexpected. First,the cast by the bearing, as well as that of the paperclip, is not defined sharplyat the edges. Then, and perhaps least expected, there is a central spot. Thisspot is known as the Poissons spot. This spot was postulated by SimeonPoisson to refute the new assertion by Dominique Arago that light is a wavewith wave-like behaviour, rather than a particle with ray-like behaviour. In1818, before a large audience, Dominique Arago demonstrated that, ratherthan refute the wave-theory of light, the Poissons spot indeed exists2.

The spot in Figure 2.25 exists because waves bend around obstacles.The reason for this bending is shown as a model in Figure 2.26. A light

source is positioned at the bottom of this Figure, and reflecting disk isplaced a distance above the source where the normal to the plane of the distis parallel with the axis of the light source. Rays indicate the direction inwhich the source on the near side of the disk emits light. Light that travelstowards the disk is reflected as indicated. Light that grazes the edge of thedisk is modelled twice in Figure 2.26 - once as rays, and again as waves.Grazing light modelled as a ray continues in a straight trajectory. A screenplaced on far side of the disk will be illuminated beyond only a circle whosdiameter is determined by the grazing rays. The resulting shadow with asharp boundary.

Grazing light modelled as a wave excites a Huygenss diffraction all

around the disk edge. That diffraction propagates as a wave in all direc-tions, and it interferes constructively and destructively on both sides of thedisk. The wave shadow on a screen placed on the far side will have a weakly

2Today, light is known to have both ray- and wave-like behaviour.


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Figure 2.24: Laser pointer with fish-eye lens. A magnet, a paper clip, anda ball bearing.

defined boundary (rather than a sharp boundary) with a central node ofconstructive interference.

This experiment is an analogue for seismic imaging. In a subsurfacewhere strong reflection boundaries exist, and wave based seismic imaging isoften preferable. Salt in the Gulf of Mexico and the basalts of the NorthSea are example of common, strong reflection boundaries. A numericalcomparison is presented in Figures 2.27 and 2.28. In Figure 2.27a, a small,circular target is embedded in dipping medium whos velocity increases withdepth. This target is visible in the zoomed image in Figure 2.28a bat 2.9km depth between 8 and 9 km. Above the small target is a large reflector- a salt body - whos velocity is much larger than the surrounding medium.

Seismic data acquired over such a medium are shown in Figure 2.27b. Thewater bottom and the top of the strong reflector are obvious features in thedata, as well as surface multiples. The diffraction associated with the targetis just visible in Figure 2.28b between 1.6 and 1.7 seconds and 8 and 9 km.

Two imaging algorithms, one ray-based and one wave-based, are usedto image the data. Both algorithms are provided by the same vendor, so itis assumed that they are implemented at comparable levels of development,and both are provided with the exact velocity model for imaging.

The ray-based algorithm fails to provide an image of the target as shownin Figures 2.27c and 2.28c. Though the target in this experiment is knownto lie between 8 and 9 km at 2.9 km depth, it cannot be discerned in the

ray-based image.In contrast, the wave-based algorithm images the target, and this is

particularly apparent in Figure 2.28d. Though surrounded by noise dueto multiple reflections associated with the salt, there is a clear image of acircular target that corresponds to its known location.


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2.4. RAYS AND WAVES 55

Figure 2.25: Poissons Spot Augustin Fresnel, Simeon Poisson, DominiqueArago (1818). This projection is achieved with the equipment pictured in

Figure 2.24.


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Figure 2.26: A source of rays / waves and a reflecting disk in plan view. Aray model indicates a shadow on the far side of the disk. The wave modelindicates a shadow with a bright spot on the far side of the disk.

2.5 Two kinds of waves

Conservative relations for m, f, and s result in a wave equation (see Ap-

pendix B)

2 1c2

= 0 (2.7)

for use in propagation of waves through acoustic media with velocity c.A similar wave equation governs anisotropic media approximately. A fairlystandard approach from mathematical physics is to solve partial differentialequations like the wave equation numerically.

Figure 2.29 shows an example of wave propagation through numericalsolution to the wave equation. Note that all propagating modes are presentin this model including direct waves, reflections, transmitted waves, head

waves, and multiples. A specific example of a transmitted wave is indicatedby the downgoing arrow in red (at 6000 m depth, and 12000 m distance).The black rays indicate a coordinate system with one axis parallel to a majorreflector, and the reflection direction for the reflected part of this wave isindicated by the upgoing arrow in red.


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2.5. TWO KINDS OF WAVES 57

D

epth(km)

Distance (km)

a)Velocity (m/s)

0 10 20

0

0.5

1

1.5

2

2.5

3 1500

2000

2500

3000

3500

4000

4500

Time(s)

Distance (km)

b)

0 10 20

0

0.5

1

1.5

2

2.5

3 0.5

0

0.5

D

epth(km)

Distance (km)

c)

0 10 20

0

0.5

1

1.5

2

2.5

3 0.5

0

0.5

D

epth(km)

Distance (km)

d)

0 10 20

0

0.5

1

1.5

2

2.5

3 0.5

0

0.5

Figure 2.27: a) A salt body embedded in a linear velocity medium. b)Seismic data associated with the model in (a). c) A ray-based image. d) Awave-based image.


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Depth(km)

Distance (km)

a)Velocity (m/s)

4 6 8 10

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Time(s)

Distance (km)

b)

4 6 8 10

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

Depth(km)

Distance (km)

c)

4 6 8 10

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Depth(km)

Distance (km)

d)

4 6 8 10

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Figure 2.28: A zoom of Figure 2.27. a) A small target lies below the saltbody at 1.6 km depth between 8 and 9 km. b) Seismic data associated withthe small target in (a) found between 1.6 and 1.7 seconds and between 8and 9 km. c) A ray-based image - the target is

theory of seismic imaging

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