Wave-equation migration velocity analysis with time-lag imaging1

Tongning Yang and Paul Sava2

Center for Wave Phenomena, Colorado School of Mines3

4

(September 30, 2010)5

Running head: WEMVA with time-lag CIGs6

ABSTRACT

Wave-equation migration velocity analysis (WEMVA) is a technique designed to extract velocity7

information from migrated images. The velocity updating is done by optimizing the coherence8

of images migrated with the known background velocity model. Its capacity for handling multi-9

pathing makes it appropriate in complex subsurface regions characterized by strong velocity vari-10

ation. WEMVA operates by establishing a linear relation between a slowness perturbation and a11

corresponding image perturbation. The linear relationship is derived from conventional extrapola-12

tion operators and it inherits the main properties of frequency-domain wavefield extrapolation. A13

key step in implementing WEMVA is to design an appropriate procedure for constructing an image14

perturbation relative to an image reference that represents difference between current image and a15

true, or more correct image of the subsurface geology. The target of the inversion is to minimize16

such an image perturbation. Using time-shift common-image gathers, one can characterize the im-17

perfections of migrated images by defining the focusing error as the shift of the focus of reflections18

along the time-shift axis. Under the linear approximation, the focusing error can be transformed19

into an image perturbation by multiplying it with an image derivative taken relative to the time-lag20

parameter. As the focusing error is caused by the incorrect velocity model, the resulting image21

1

perturbation can be considered as a mapping of the velocity model error in the image space. The in-22

version is then performed to produce velocity updates. Such an approach for constructing the image23

perturbation is computationally efficient and simple to implement. Synthetic examples demonstrate24

the successful application of our method to a layers model and a subsalt velocity update problem.25

2

26

INTRODUCTION

In regions characterized by complex subsurface structure, prestack wave-equation depth migration,27

i.e. one-way wave-equation migration (WEM) or reverse-time migration (RTM), is a powerful tool28

for accurately imaging the earth’s interior (Gray et al., 2001; Etgen et al., 2009). However, the29

quality of the final image greatly depends on the accuracy of the velocity model. Thus, a key30

challenge for seismic imaging in complex geology is an accurate determination of the velocity31

model in the area under investigation (Symes, 2008; Woodward et al., 2008; Virieux and Operto,32

2009).33

Migration velocity analysis (MVA) generally refers to tomographic methods implemented in the34

image domain. These techniques are based on the principle that the quality of the seismic image is35

optimized when the data are migrated with the correct velocity model. MVA updates the velocity36

model by optimizing the properties measuring the coherence of migrated images. As a result, the37

quality of both the image and velocity model is improved by MVA.38

Typically, the input for MVA is represented by various types of common-image gathers (CIGs),39

e.g. shot-domain CIG (Al-Yahya, 1989), surface-offset CIG (Mulder and ten Kroode, 2002), subsurface-40

offset CIG (Rickett and Sava, 2002), time-shift CIG (Sava and Fomel, 2006), angle-domain CIG41

(Sava and Fomel, 2003; Biondi and Symes, 2004), or space- and time-lag extended-image gathers42

(Yang and Sava, 2010; Sava and Vasconcelos, 2010). Different kinds of CIGs determine whether43

semblance or focusing analysis should be used to measure the coherence of the image.44

In practice, there are many possible approaches for implementing MVA with different CIGs.45

However, all such realizations share the common element that they need a carrier of information46

3

to connect the input CIGs to the output velocity model. Thus, one can categorize MVA techniques47

into ray-based and wavefield-based methods in terms of the information carrier they use. Ray-based48

methods, which are often described as traveltime MVA, refer to techniques using wide-band rays as49

the information carrier (Bishop et al., 1985; Al-Yahya, 1989; Stork and Clayton, 1991; Woodward,50

1992; Stork, 1992; Liu and Bleistein, 1995; Jiao et al., 2002). Wavefield-based MVA methods refer51

to techniques using band-limited wavefields as the information carrier (Biondi and Sava, 1999; Mul-52

der and ten Kroode, 2002; Shen and Calandra, 2005; Soubaras and Gratacos, 2007; Xie and Yang,53

2008). Generally speaking, ray-based methods have the advantage over wavefield-based methods54

in that they involve simple implementation and efficient computation. In contrast, wavefield-based55

methods are capable of handling complicated wave propagation phenomena which always occur in56

complex subsurface regions. Therefore, they are more robust and consistent with the wavefield-57

based migration techniques used in such regions. Furthermore, the resolution of wavefield-based58

methods is higher than that of ray-based methods because they employ fewer approximations to59

wave propagation.60

Different from traveltime MVA, wavefield-based MVA requires an image residual as the input,61

which is equivalent to the data misfit defined in the image domain. Wave-equation MVA (WEMVA)62

(Sava and Biondi, 2004a,b) and differential semblance optimization (WEDSO) (Shen et al., 2003;63

Shen and Symes, 2008) are two common approaches for wavefield-based MVA. The image resid-64

ual for WEMVA and WEDSO are obtained either by constructing a linearized image perturbation65

(Sava et al., 2005) or by applying a penalty operator to offset- or angle-domain CIGs (Symes and66

Carazzone, 1991; Shen and Calandra, 2005), respectively.67

Focusing analysis is a commonly used method for refining the velocity model (Faye and Jeannot,68

1986). It evaluates the coherence of migrated images by measuring the focusing of reflections.69

The focusing information can be extracted from time-shift CIG (Sava and Fomel, 2006), and it is70

4

quantified as the shift of the focus of reflections along the time-shift axis from the origin. Such71

a shift is often defined as focusing error and indicates the existence of the velocity model error72

(MacKay and Abma, 1992; Nemeth, 1995). Therefore, the focusing error can be used for velocity73

model optimization. Wang et al. (2005) propose a tomographic approach using focusing analysis74

for re-datuming data sets. The approach is a ray-based method and thus may become unstable when75

the multi-pathing problem exists due to the complex geology. Higginbotham and Brown (2008)76

and Brown et al. (2008) also propose a method to convert this focusing error into velocity updates77

using an analytic formula. This approach is based on 1D assumptions and the measured focusing78

error is transformed into vertical updates only. Hence, the method become less accurate in models79

with complex subsurface environments. Nevertheless, (Wang et al., 2008) and Wang et al. (2009)80

illustrate that focusing analysis can effectively improve the image quality and be used for updating81

the velocity model in subsalt areas.82

For the implementation of WEMVA, one important component is the construction of an image83

perturbation which is linked directly to a velocity perturbation. To construct the image perturbation,84

one first needs to measure the coherence of the image. Sava and Biondi (2004b) discuss two types85

of measurement for the imperfections of migrated images, i.e. focusing analysis (MacKay and86

Abma, 1992; Lafond and Levander, 1993) and moveout analysis (Yilmaz and Chambers, 1984;87

Biondi and Sava, 1999). The most common approach for constructing the image perturbation is88

to compare a reference image with its improved version. However, such an image-comparison89

approach has at least two drawbacks. First, the improved version of the image is always obtained90

by re-migration with one or more models, which is often computationally expensive. Second, if91

the reference image is incorrectly constructed, the difference between two images might exceed92

the small perturbation assumption. This can lead to cycle skipping which changes the convergence93

properties of the inversion. An alternative to this approach, discussed in Sava and Biondi (2004a),94

5

uses prestack Stolt residual migration (Stolt, 1996; Sava, 2003) to construct a linearized image95

perturbation. This alternative approach avoids the cycle skipping problem, but suffers from the96

approximation embedded in the underlying Stolt migration.97

In this paper, we propose a new methodology of constructing a linearized image perturbation98

for WEMVA based on the time-shift imaging condition and focusing analysis. We demonstrate that99

such an image perturbation can be easily computed by a simple multiplication of the image deriva-100

tive constructed based on a reference image and the measured focusing error. We also demonstrate101

that this type of image perturbation is fully consistent with the linearization embedded in WEMVA.102

Finally, we illustrate the feasibility of our approach by testing the method on simple and complex103

synthetic data sets.104

THEORY

Under the single scattering approximation, seismic migration consists of two steps: wavefield re-105

construction followed by the application of an imaging condition. We commonly discuss about a106

“source” wavefield, originating at the seismic source and propagating in the medium prior to any107

interaction with the reflectors, and a “receiver” wavefield, originating at discontinuities and prop-108

agating in the medium to the receivers (Berkhout, 1982; Claerbout, 1985). The two wavefields109

are kinematically equivalent at discontinuities of material properties. Any mismatch between the110

wavefields indicates inaccurate wavefield reconstruction typically assumed to be due to inaccurate111

velocity. The source and receiver wavefields us and ur are four-dimensional objects as functions of112

position x = (x, y, z) and frequency ω,113

us = us (x, ω) , (1)

ur = ur (x, ω) . (2)

6

An imaging condition is designed to extract from these extrapolated wavefields the locations where114

reflections occur in subsurface. A conventional imaging condition (Claerbout, 1985) forms an image115

as the zero cross-correlation lag between the source and receiver wavefields:116

r (x) =∑ω

us (x, ω)ur (x, ω) , (3)

where r is the image of subsurface and overline represents complex conjugation. An extended117

imaging condition (Sava and Fomel, 2006) extracts the image by cross-correlation between the118

wavefields shifted by the time-lag τ :119

r (x, τ) =∑ω

us (x, ω)ur (x, ω) e2iωτ . (4)

Other possible extended imaging conditions include space-lag extension (Rickett and Sava, 2002)120

or space- and time-lag extensions (Sava and Vasconcelos, 2010), but we do not discuss these types121

of imaging conditions here.122

In the process of seismic migration, if the source and receiver wavefields are extrapolated with123

the correct velocity model, the result of cross-correlation between reconstructed wavefields, i.e. the124

migrated image, is maximized at zero time-lags. In other words, if the velocity used for migration125

is correct, reflections in an image are focused at zero offset and zero time. If the source and receiver126

wavefields are extrapolated with the incorrect velocity model, the result of cross-correlation is not127

maximized at zero lags. As a consequence, reflections in an image are focused at nonzero time but128

zero offset. This indicates the existence of error for downward continuation of the reconstructed129

wavefields. In such a situation, we can apply focusing analysis and extract the information about130

the velocity model. A commonly used approach for focusing analysis is to measure the focusing131

error in either depth domain or time domain. The depth-domain focusing error is defined as the132

7

depth difference between focusing depth df and migration depth dm (MacKay and Abma, 1992)133

∆z = df − dm . (5)

134

df =d

ρ, (6)

and135

dm = dρ . (7)

Here d is the true depth of the reflection point, ρ = VmV is the ratio between migration and true136

velocities. The time-domain focusing error is defined as the time-shift being applied to the recon-137

structed wavefields to achieve focusing of reflections. Yang and Sava (2010) quantitatively analyze138

the influence of the velocity model error on focusing property of reflections, and derive the formula139

connecting depth-domain and time-domain focusing error:140

∆τ =df − dmVm

, (8)

where Vm represents the migration velocity, Notice that the formulae for df and dm are derived141

under the assumptions of constant velocity, small offset angle and horizontal reflector. As a conse-142

quence, the analytic formula in equation 8 is an approximation of the true focusing error, and has143

limited applicability in practice.144

To use WEMVA for velocity optimization, one needs to start from linearization of the problem145

since an image is usually a nonlinear function of the velocity model. Furthermore, a nonlinear146

optimization problem is more difficult to solve than a linear optimization problem. We represent the147

8

true slowness as the sum of a background slowness sb and a slowness perturbation ∆s:148

s (x) = sb (x) + ∆s (x) . (9)

Likewise, the image can also be characterized as a sum of a background image rb and an image149

perturbation ∆r:150

r (x) = rb (x) + ∆r (x) . (10)

The perturbation ∆s and ∆r are related by the wave-equation tomographic operator L derived151

from the linearization of one-way wave-equation migration operator, as demonstrated by Sava and152

Biondi (2004a). A brief summary of derivation and construction of L can be found in appendix A.153

Consequently, we can establish a linear relationship between the image perturbation and slowness154

perturbation:155

∆r = L∆s . (11)

Sava and Vlad (2008) further illustrates the implementation of operator L for different imaging156

configurations, i.e. zero-offset, survey-sinking, and shot-record cases.157

The focusing information of ∆τ is available on the extended images constructed with equa-158

tion 4. In other words, the focus of reflections can be evaluated by locating the maximum energy of159

reflection events in time-shift CIGs. If the focus is located at zero time shift, the velocity model is160

correct. Otherwise, the distance of the focus from origin is measured along the time-shift axis and161

defined as the focusing error. Then, an improved image r̃ can be obtained by choosing the image162

from the time-shift CIG at the τ value corresponding to the focusing error:163

r̃ (x) = r (x, τ)|τ=∆τ , (12)

9

and an image perturbation can be constructed by a direct subtraction:164

∆r (x) = r̃ (x)− rb (x) . (13)

Although such an approach is straightforward, the constructed image perturbation might be phase-165

shifted too much with respect to the background image rb and thus violates the Born approximation166

required by the tomographic operator L. To overcome the challenge, Sava and Biondi (2004a)167

propose to construct a linearized image perturbation as168

∆r (x) ≈ K′∣∣ρ=1

[rb]∆ρ , (14)

where K is the prestack Stolt residual migration operator (Stolt, 1996; Sava, 2003), the ′ sign repre-169

sents derivation relative to the velocity ratio ρ, and ∆ρ = ρ− 1. Similarly, we can also construct an170

image perturbation by a linearization of the image relative to the time-shift parameter τ :171

∆r (x) ≈ ∂r (x, τ)∂τ

∣∣∣∣τ=0

∆τ , (15)

where the image derivative with respect to time-shift τ is172

∂r (x, τ)∂τ

=∑ω

(2iω)us (x, ω)ur (x, ω) e2iωτ . (16)

Comparing two different approaches for constructing the image perturbation in equation 14 and173

equation 16, they share a similar form of the formulae. The differences lie in how to construct the174

image derivative and how to measure the quantity relating to the velocity model error. The lin-175

earization of both the approaches is consistent with the characteristics of the tomographic operator176

L.177

10

One can follow the procedure outlined below to construct the image perturbations for WEMVA:178

• Migrate the image and output time-shift CIGs according to equation 4;179

• Measure ∆τ on time-shift CIG panels by picking the focus, i.e. maximum energy, for all180

reflections;181

• Construct the image derivative according to equation 16;182

• Construct the linearized image perturbation according to equation 15.183

After we construct the image perturbation ∆r, we can solve for the slowness perturbation ∆s184

by minimizing the objective function,185

J (∆s) =12‖∆r − L∆s‖2 . (17)

Such an optimization problem can be solved iteratively using conjugate-gradient-based methods.186

As most practical inverse problems are ill-posed, additional constraints must be imposed during the187

inversion to obtain a stable and convergent result. This can be done by adding a model regularization188

term in the objective function equation 17, leading to the modified objective function:189

J (∆s) =12‖∆r − L∆s‖2 + α2‖A∆s‖2 , (18)

where α is a scalar to control the relative weights between data residual and model norm and A is a190

regularization operator (Fomel, 2007). How to chose A and α is outside the scope of discussion in191

this paper.192

11

EXAMPLES

We illustrate our methodology with two synthetic examples. The first example consists of four193

dipping layers with different thickness and dipping angle. The velocities of the four layers are 1.5,194

1.6, 1.7, and 1.8 km/s, respectively. The background model is constant with velocity of 1.5 km/s.195

Figure 1(a) and 1(b) show the true and background velocity models, respectively. Figure 2(a) and196

2(b) are the image migrated with the true and background velocity model, respectively. Comparing197

these two images, one can observe that the second and third reflectors in Figure 2(b) are positioned198

incorrectly due to the error of the velocity model.199

Figure 3(a) - 3(c) shows three time-shift CIGs migrated with the background model. The gathers200

are chosen at x = 1.2, x = 1.8, and x = 2.3 km, respectively. The picking is done by applying201

an automatic picker to the envelope of the original time-shift CIGs. The solid line in the figure202

plots the picked focusing error, while the dashed line represents zero time shift. In this case, as203

the velocity errors for the second and third layers increase with depth, the focusing error increases204

with depth as well. Therefore, bigger shifts can be observed for deeper events. Figure 4(a) shows205

the focusing error as a function of position and depth. This figure depicts the focusing error picked206

from the time-shift CIGs constructed at every horizontal location x. Since the layers have different207

thickness, the associated velocity errors change as a function of horizontal and vertical directions.208

Therefore, a spatially varying focusing error of the subsurface is observed.209

We construct the image perturbation using the procedure discussed in the preceding sections.210

Next, we perform the inversion based on the objective function in equation 18. The inverse problem211

is solved using the conjugate-gradient method. For this simple model, two nonlinear iterations are212

enough to reconstruct the velocity model.213

Figure 1(c) shows the updated model after the inversion. The velocity for the second and third214

12

layer are correctly inverted. Here, the velocity update occurs in the region above the fourth layer.215

This is due to the fact that no reflection exists in the bottom part of the model, and the input image216

perturbation carries no velocity information for the bottom layer. Thus, the inversion does not217

compute velocity updates in the area. Figure 2(c) shows the image migrated with the updated218

model. The reflections are positioned in the correct depth, as compared with Figure 2(a).219

Figure 3(d) - 3(f) shows time-shift CIGs obtained by migration with the updated model. The220

gathers are chosen at the same positions as the gathers shown in Figure 3(a) - 3(c), i.e. at x = 1.2,221

x = 1.8, and x = 2.3 km, respectively. The focus of all the reflections are located at zero time-222

shift τ , as demonstrated by the fact that the solid and dashed are overlapped with each other in the223

gather. Figure 4(b) is the focusing error panel of the subsurface. The value of the focusing error is224

zero almost everywhere. This means that the inversion has achieved the target of minimizing the225

focusing error and confirms the successful reconstruction of the velocity model in this example.226

We also apply our methodology to a subsalt velocity update example. The target area is chosen227

from the subsalt portion of the Sigsbee 2A model (Paffenholz et al., 2002), ranging from x =228

9.5 km to x = 18.5 km, and z = 5.0 km to z = 9.3 km. For this example, the main goals are229

to image correctly the fault located between x = 14.0 km to x = 16.0 km, and z = 6.0 km to230

z = 9.0 km, to increase the focusing for the deeper diffractors, and to positioning correctly the231

bottom flat reflector. We assume correct knowledge of the velocity above the salt and known salt232

geometry. The background model for the target zone is obtained by scaling the true model with233

a constant factor 0.1. The true and background velocity model are depicted in Figure 5(a) and234

5(b), respectively. Figure 6(a) and 6(b) show the images migrated with the true and background235

model, respectively. Due to the error of the background velocity model, the fault is obscured by236

nearby sediment reflections, the deeper diffractions are defocused, and the bottom flat reflector is237

positioned far away from its correct depth.238

13

Figure 7(a) - 7(c) show several time-shift CIGs migrated with the background model at x =239

13.0, x = 14.8, and x = 16.6 km, respectively. The picked focusing error is directly overlain240

on the gathers, although the picking is done on the envelope of the original time-shift CIGs just241

as the previous example. Here, we notice that the picks do not start at zero on the top of the242

gathers, although the associated focusing error should be zero. Such an error is mainly caused by243

the truncation of reflections in the gathers. Figure 8(a) shows the focusing error in the target area.244

Notice that variation exists for the picked focusing error in lateral and vertical directions. Part of245

such a variation is caused by the uneven illumination of the subsalt area, which leads to difficulties246

for the automatic picker.247

Next, we use the same procedure to construct the image perturbation as introduced in the pre-248

ceding section. Then we run the inversion to obtain the velocity update by minimizing the objective249

function in equation 18. Figure 5(c) shows the updated model after three nonlinear iterations. The250

inversion has correctly reconstructed the velocity model in the subsalt area. Figure 6(c) shows the251

image migrated with the updated model. The fault located between x = 14.0 km to x = 16.0 km,252

and z = 6.0 km to z = 9.0 km is delineated and clearly visible in the image. The bottom reflector253

is positioned at the correct depth, and become more coherent. The deeper diffractions are also well254

focused. These improvements in the image imply a correct update of the velocity model.255

Figure 7(d) - 7(f) plot several time-shift CIGs migrated with the background model at x = 13.0,256

x = 14.8, and x = 16.6 km, respectively. The focusing error picks are overlain on the gathers.257

Most reflections are focused at zero time shift, which means the focusing error is reduced thanks to258

the more accurate velocity model after the inversion. Figure 8(b) plot the focusing error in the target259

area after velocity update and remigration. The focusing error is reduced after the inversion, which260

indicates the successful optimization of the velocity model.261

14

To further confirm that the velocity model is correctly inverted, we also compute angle-domain262

CIGs with the background and updated models and use the flatness of reflection events in the gathers263

as the criterion to evaluate the result of inversion. The angle-domain CIGs are chosen at the same264

locations at the time-shift CIGs. Figure 9(a) - 9(c) plot the gathers corresponding to the background265

velocity model, while Figure 9(d) - 9(f) show the gathers for the updated velocity model. In com-266

parison, the reflection events in the gathers obtained with the updated model are more flat than the267

events in the gathers obtained with the background model. This demonstrates the improvement of268

the quality of the velocity model after the optimization. However, we notice that the curvature of the269

residual moveout in the gathers obtained with the background model is not easily distinguishable.270

This can cause difficulties for curvature picking and thus degrades the velocity estimation methods271

relying on flatting the residual moveout. In such a situation, minimizing the focusing error, as in the272

case of our approach, may provide a reliable alternative for velocity optimization.273

DISCUSSION

The construction of the linearized image perturbation using time-shift imaging condition and focus-274

ing analysis is a good replacement for the approach using Stolt residual migration. Compared with275

the procedure based on Stolt residual migration, our method has the main advantage of a low com-276

putational cost, since the computation of the linearized image perturbation adds just a trivial cost277

to the construction of time-shift CIGs. More importantly, no expensive re-migration scan and angle278

decomposition are required in this procedure. Furthermore, the method based on time-shift CIGs is279

computationally more attractive in 3D application as only one additional dimension is required for280

constructing CIGs, while two additional dimensions are needed for constructing lag-domain CIGs281

(Shen and Symes, 2008).282

15

The inversion scheme described in the paper is a wave-equation-based tomographic approach.283

Although the information is extracted by applying focusing analysis to time-shift CIGs, the process284

of velocity updating does not rely on any analytic formulae as in the case for conventional depth-285

focusing analysis. Therefore, the technique discussed in this paper has applicability to models with286

arbitrary lateral velocity variations.287

Finally, we emphasize that picking the focusing error is extremely important for our approach288

as the focusing error determines the direction and magnitude of the velocity update. Therefore, the289

process of the picking should be carefully implemented.290

CONCLUSIONS

We develop a new method for implementing WEMVA based on time-lag imaging and focusing291

analysis. The objective of the velocity optimization is to minimize the focusing error measured292

from time-shift CIGs. The methodology relies on constructing linearized image perturbations by293

applying focusing analysis to time-shift CIGs. The focusing error is defined as the shift of the fo-294

cus for reflections along the time-shift axis and provides a measurement for the accuracy of the295

velocity model used in migration. We use this information in conjunction with image derivatives296

relative to the time-shift parameter to compute linearized image perturbations. The image pertur-297

bation obtained by this approach is consistent with the Born approximation used for the WEMVA298

operators. Compared with the conventional approach for constructing linearized image perturbation299

using Stolt residual migration, our approach is efficient, since the main cost is just the construction300

of the time-shift CIG and is much lower than the cost of re-migration scans and angle decomposi-301

tion. In addition, our method is accurate, since it does not make use of Stolt-like procedures which302

incorporate strong assumptions about the smoothness of the background model. Thus, our approach303

16

is suitable for the areas with complex subsurface structure.304

The synthetic examples demonstrate the validity of the linearized image perturbation constructed305

using our new method. The inversion with such an image perturbation as the input can render sat-306

isfactory updates for the velocity model. The success of the velocity optimization is confirmed by307

the fact that the focusing error is reduced after the inversion.308

309

ACKNOWLEDGMENTS

We acknowledge the support of the sponsors of the Center for Wave Phenomena at Colorado School310

of Mines. This work is also partially supported by a research grant from Statoil. The reproducible311

numeric examples in this paper use the Madagascar open-source software package freely available312

from http://www.reproducibility.org.313

17

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APPENDIX A

Wave-equation tomographic operator406

The construction of wave-equation tomographic operator L used in equation 11 starts from sepa-407

rating the total slowness s into a known background sb and a slowness perturbation ∆s, as shown408

in equation 9. Next, we linearize the depth wavenumber kz relative to the background slowness sb409

using a truncated Taylor expansion series expansion410

kz ≈ kzb+∂kz∂s

∣∣∣∣sb

∆s (x) , (A-1)

where the depth wavenumber in the background medium characterized by slowness sb (x) is411

kzb=√

[2ωsb (x)]2 − |kx|2 . (A-2)

A wavefield perturbation ∆u (x) caused at depth z + ∆z by a slowness perturbation ∆s (x) at412

depth z is obtained by subtraction of the wavefields extrapolated from z to z + ∆z in the true and413

background models:414

∆uz+∆z (x) = e−ikz∆zuz (x)− e−ikz0∆zuz (x)

= e−ikz0∆z

[e−i ∂kz

∂s |sb∆s(x)∆z − 1

]uz (x) . (A-3)

Here, ∆u (x) and u (x) correspond to a given depth level z and frequency ω. A similar relation can415

be applied at all depths and all frequencies.416

Equation A-3 establishes a non-linear relation between the wavefield perturbation ∆u (x) and417

the slowness perturbation ∆s (x). Given the complexity and cost of numeric optimization based on418

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non-linear relations between model and wavefield parameters, it is desirable to simplify this relation419

to a linear relation between model and data parameters. Assuming small slowness perturbations, i.e.420

small phase perturbations, the exponential function e±i ∂kz

∂s |sb∆s(x)∆z

can be linearized using the421

approximation eiφ ≈ 1 + iφ which is valid for small values of the phase φ. Therefore, the wavefield422

perturbation ∆u (x) at depth z can be written as423

∆u (x) ≈ −i ∂kz∂s

∣∣∣∣sb

∆z u (x) ∆s (x)

≈ −i∆z 2ωu (x) ∆s (x)√1−

[|kx|

2ωsb(x)

]2. (A-4)

In the case of shot-record migration, we have both the source and receiver wavefields. Thus, the424

wavefield perturbations for both the source and receiver can be computed by a direct generalization425

of equation A-4.426

∆us (x) ≈ +i∂kz∂s

∣∣∣∣sb

∆z us (x) ∆s (x)

≈ +i∆zωus (x) ∆s (x)√

1−[|kx|

ωsb(x)

]2, (A-5)

and427

∆ur (x) ≈ −i ∂kz∂s

∣∣∣∣sb

∆z ur (x) ∆s (x)

≈ −i∆z ωur (x) ∆s (x)√1−

[|kx|

ωsb(x)

]2. (A-6)

The image perturbation at depth z is obtained from the source and receiver scattered wavefields428

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using the relation429

∆r (x) =∑ω

(us (x, ω)∆ur (x, ω) + ∆us (x, ω)ur (x, ω)

), (A-7)

which corresponds to the frequency-domain zero-lag cross-correlation of the source and receiver430

wavefields.431

The tomographic operator L involves computing the wavefield perturbations using equation A-5432

and equation A-6 and the image perturbation using equation A-7. Therefore, given a slowness433

perturbation ∆s, the relating image perturbation is constructed via the operator L. A more detailed434

implementation of the operator L and its adjoint can be found in Sava and Vlad (2008).435

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LIST OF FIGURES

1 Velocity profiles of the layers model. (a) The true velocity model. (b) The background436

velocity model with a constant velocity of the first layer 1.5 km/s. (c) The updated velocity model.437

2 Images migrated with (a) the true velocity model, (b) the background velocity model, and438

(c) the updated velocity model.439

3 Time-shift CIGs migrated with the background model (a) at x = 1.2 km, (b) at x =440

1.8 km, and (c) at x = 2.3 km. Time-shift CIGs migrated with the updated model at (d) x = 1.2 km,441

(e) at x = 1.8 km, and (f) at x = 2.3 km. The overlain solid lines are picked focusing error. The442

dash line represents zero time shift.443

4 (a) Focusing error panel corresponding to the background model. (b) Focusing error panel444

corresponding to the updated model.445

5 Velocity profiles of Sigsbee 2A model. (a) The true velocity model, (b) the background446

velocity model, and (c) the updated model.447

6 Images migrated with (a) the true velocity model, (b) the background velocity model, and448

(c) the updated model.449

7 Time-shift CIGs migrated with the background model, (a) at x = 13.0 km, (b) at x =450

14.8 km, and (c) at x = 16.6 km. Time-shift CIGs migrated with the updated model, (d) at x =451

13.0 km, (e) at x = 14.8 km, and (f) at x = 16.6 km. The overlain solid lines are picked focusing452

error. The dash line represents zero time shift.453

8 (a) Focusing error panel corresponding to the background model. (b) Focusing error panel454

corresponding to the updated model.455

9 Angle-domain CIGs migrated with the background model, (a) at x = 13.0 km, (b) at456

x = 14.8 km, and (c) at x = 16.6 km. Angle-domain CIGs migrated with the updated model, (d) at457

x = 13.0 km, (e) at x = 14.8 km, and (f) at x = 16.6 km.458

25

459

26

(a)

(b)

(c)

Figure 1: Velocity profiles of the layers model. (a) The true velocity model. (b) The backgroundvelocity model with a constant velocity of the first layer 1.5 km/s. (c) The updated velocity model.–

27

(a)

(b)

(c)

Figure 2: Images migrated with (a) the true velocity model, (b) the background velocity model, and(c) the updated velocity model.–

28

(a) (b) (c)

(d) (e) (f)

Figure 3: Time-shift CIGs migrated with the background model (a) at x = 1.2 km, (b) at x =1.8 km, and (c) at x = 2.3 km. Time-shift CIGs migrated with the updated model at (d) x = 1.2 km,(e) at x = 1.8 km, and (f) at x = 2.3 km. The overlain solid lines are picked focusing error. Thedash line represents zero time shift.–

29

(a)

(b)

Figure 4: (a) Focusing error panel corresponding to the background model. (b) Focusing error panelcorresponding to the updated model.–

30

(a)

(b)

(c)

Figure 5: Velocity profiles of Sigsbee 2A model. (a) The true velocity model, (b) the backgroundvelocity model, and (c) the updated model.–

31

(a)

(b)

(c)

Figure 6: Images migrated with (a) the true velocity model, (b) the background velocity model, and(c) the updated model.–

32

(a) (b) (c)

(d) (e) (f)

Figure 7: Time-shift CIGs migrated with the background model, (a) at x = 13.0 km, (b) at x =14.8 km, and (c) at x = 16.6 km. Time-shift CIGs migrated with the updated model, (d) at x =13.0 km, (e) at x = 14.8 km, and (f) at x = 16.6 km. The overlain solid lines are picked focusingerror. The dash line represents zero time shift.–

33

(a)

(b)

Figure 8: (a) Focusing error panel corresponding to the background model. (b) Focusing error panelcorresponding to the updated model.–

34

(a) (b) (c)

(d) (e) (f)

Figure 9: Angle-domain CIGs migrated with the background model, (a) at x = 13.0 km, (b) atx = 14.8 km, and (c) at x = 16.6 km. Angle-domain CIGs migrated with the updated model, (d) atx = 13.0 km, (e) at x = 14.8 km, and (f) at x = 16.6 km.–

35