msc thesis: elastic wavefield decomposition at the ocean ... · master of science thesis for the...

Master of Science in Applied Geophysics

Research Thesis

Elastic Wavefield Decomposition atthe Ocean Bottom

Including near-ocean bottom velocity estimation

A.M.A. Alfaraj

August 13, 2014

Elastic Wavefield Decomposition atthe Ocean Bottom

Including near-ocean bottom velocity estimation

Master of Science Thesis

for the degree of Master of Science in Applied Geophysics at

Delft University of Technology

ETH Zurich

RWTH Aachen University

by

A.M.A. Alfaraj

August 13, 2014

Department of Geoscience & Engineering · Delft University of TechnologyDepartment of Earth Sciences · ETH ZurichFaculty of Georesources and Material Engineering · RWTH Aachen University


Copyright c© 2014 by IDEA League Joint Master’s in Applied Geophysics:


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IDEA LEAGUEJOINT MASTER’S IN APPLIED GEOPHYSICS

Delft University of Technology, The NetherlandsETH Zurich, SwitzerlandRWTH Aachen, Germany

Dated: August 13, 2014

Supervisor(s):Dr.ir. D.J. Verschuur

Prof.dr.ir. C.P.A. Wapenaar

Committee Members:Prof.dr.ir. C.P.A. Wapenaar

Niels Grobbe, MSc.

Dr. Marina Hruska

Abstract

Multi-component ocean-bottom seismic data offers the opportunity to have access to seismicP-waves and S-waves, (mode converted waves). The initial pre-processing step to retrievereliable P- and S-wave sections is elastic decomposition of the multi-component measurements.This wavefield decomposition at the ocean bottom requires properties of the ocean floor inorder to get reliable one-way wavefields. We consider the situation that via the JMI process,(Joint Migration Inversion), an estimation for the upgoing P- and S-waves at a certain depthlevel has been obtained. We develop an inversion scheme that estimates the P- and S-wavevelocities in the remaining part of the subsurface, between the depth level below the waterbottom and the bottom itself. In addition, we apply a composition of the resulting up-goingwavefields at the bottom into the measured quantities by the 4-C receivers. Thus, a combinedtomography and decomposition problem is solved. In this way, the unknown properties ofthe ocean bottom as well as the velocities in the near bottom layer are estimated basedon comparing the predicted data with the true measurements. In the thesis, results of thismethodology are shown under the assumption of a horizontally stratified medium. The resultsindicate that it is possible to develop an inversion scheme that estimates the near-bottomvelocity model in combination with the optimum composition operators.

August 13, 2014

vi Abstract

August 13, 2014

Acknowledgements

I would like to express my gratitude to Saudi Aramco for sponsoring my MSc. studies. Inparticular, I would like to thank Dr. Kelamis, who advised me to join TU Delft and theIDEA League Program. It has been almost two years since I started the program, and I canconfidently say that it was the right choice.

I am grateful to my direct supervisor Dr. Eric Verschuur for providing me with the thesisproposal and the unique opportunity to work with him. Eric was always available to answermy questions and concerns, no matter how busy he was. Whenever I meet with Eric, I getinspired with his ideas. His guidance and support took me through the successful applicationof this thesis. It was truly a pleasure to be under Eric’s supervision. Thank you very muchEric.

My principal supervisor, Prof. Kees Wapenaar was behind my thesis topic choice. It was inhis course when I first learned about decomposition. Many thanks to Kees for providing theelastic modeling code that I used in this thesis. I also would like to thank Dr. Jan Thorbeckefor his assistance with the finite difference modelling code.

The DELPHI students, thank you for the nice ideas and discussions. Special thanks toAbdulrahman Alshuhail, who contributed to my understanding of JMI and provided me withnice suggestions. It was nice to spend most of my time in Delft with you. Gabriel LopezAngarita, thank you for your assistance. I also would like to thank Niels Grobbe for provingthe thesis template.

My beloved Wife, thank you very much for your love and support. Even though you did notwant me to be away from you during the past two years, you have always encouraged me tokeep going and become successful. My parents, I am grateful for everything you have doneto me. Without you, I would not have reached to this position.

Delft University of Technology A.M.A. Alfaraj

August 13, 2014

viii Acknowledgements

August 13, 2014

Table of Contents

Abstract v

Acknowledgements vii

1 Introduction 1

1-1 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1-2 Parameters estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1-3 Notations and transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Acoustic and Elastic Wavefield Decomposition 5

2-1 Acoustic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2-1-1 Two-way wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2-1-2 One-way wavefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2-2 Elastic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2-2-1 Two-way wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2-2-2 One-way wavefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2-3 Wavefield decomposition at the ocean bottom . . . . . . . . . . . . . . . . . . . 12

2-3-1 Acoustic decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2-3-2 Elastic decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2-4 Synthetic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2-4-1 Wavefield decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2-5 Decomposition with erroneous velocities . . . . . . . . . . . . . . . . . . . . . . 19

3 Near-bottom velocity estimation 23

3-1 Wavefield tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3-1-1 Velocity perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . 23

3-1-2 Tomographic Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

August 13, 2014

x Table of Contents

3-1-3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3-2 Wavefield composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3-2-1 Composition/decomposition Inversion . . . . . . . . . . . . . . . . . . . 32

3-2-2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Conclusions 35

Bibliography 37

August 13, 2014

List of Figures

2-1 Logs used for creating synthetic data. . . . . . . . . . . . . . . . . . . . . . . . 15

2-2 Synthetic ocean bottom data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2-3 Decomposition of the total wavefield just above the ocean bottom to (a) up-goingacoustic pressure and (b) down-going acoustic pressure. . . . . . . . . . . . . . . 17

2-4 Decomposition levels, just above the ocean bottom (z1 − ε) and just below theocean bottom (z1 + ε). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2-5 Decomposition of the total wavefield just below the ocean bottom to (a) down-going P-waves, (b) up-going P-waves, (c) down-going S-waves and (d) up-going-S-waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2-6 Elastic decomposition with 100 m/s error in velocities to (a) down-going P-waves,(b) up-going P-waves, (c) down-going S-waves and (d) up-going-S-waves. . . . . 20

2-7 Elastic decomposition with (a-b) 200 m/s and (c-d) 400 m/s error in the velocities. 21

3-1 Illustration of the depth levels to update. . . . . . . . . . . . . . . . . . . . . . . 26

3-2 Magnified view of the up-going measurements (blue), predicted (red) and residual(black), before applying wavefield tomography. Note that the up-going P- andS-wave residuals are computed using only one trace (p = 0.2622× 10−3 s/m), forthe sake of comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3-3 The measurements (blue), predicted up-going wavefields (red) and residual (black),after applying wavefield tomography. Note how well the predicted up-going wave-fields match the measured up-going wavefields. The trace we used to show theresidual corresponds to (p = 0.2622× 10−3 s/m). . . . . . . . . . . . . . . . . . 30

3-4 The true (blue), estimated (red) and initial (green) velocity models for (a) P-waveand (b) S-wave. Note that these are also magnified velocity models to show theupdate levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3-5 Plot of Jp (blue) and Js (red), versus the number of iterations. . . . . . . . . . . 31

3-6 The total measurements (blue), predicted data (red) and residual (black), afterapplying composition. Note the good match as well as the consistency of theevents in the measured and predicted data. The trace chosen to compute theresidual corresponds to (p = 0.266× 10−3 s/m). . . . . . . . . . . . . . . . . . 34

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xii List of Figures

4-1 Up-going measurements (blue), predicted wavefields (red) and residual (black).(a) and (b) correspond to data before applying wavefield tomography, while (c)and (d) correspond to data after applying wavefield tomography. The differencemight not be noticeable as it is in Figure 3-2 and Figure 3-3. . . . . . . . . . . . 40

4-2 True (blue), estimated (red) and initial (green) velocity profiles. . . . . . . . . . 41

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List of Tables

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xiv List of Tables

August 13, 2014

Chapter 1

Introduction

1-1 Decomposition

In marine seismic data acquisition, there are two types of medium to consider. The firstmedium is the ocean which only supports compressional waves (P)-waves. The second one isa solid medium (the Earth’s subsurface), in which longitudinal (P)-waves and shear (S)-wavescan propagate. Air guns towed behind vessels are the common sources used to generateP-waves. The energy sent by the sources propagates in the water into the subsurface andgets transmitted and reflected. Wave conversion from P to S-waves or vise versa may occurat any point along the propagation path in the Earth’s subsurface.

Seismic receivers can be chosen to be streamers towed behind the vessels at the oceansurface. Hence, the receivers only record the P-waves. In shallow water areas, vessels withstreamers cannot access. Another choice is to place the receivers at the ocean bottom, viaocean bottom cables (OBC) or ocean bottom nodes (OBN). Commonly multi-componentreceivers are used: 3 component-geophones and a hydrophone. This choice gives theopportunity to record P- waves and S-waves which are generated via wave conversion. Dueto low S-wave velocities, S-wave data carry high resolution information. Moreover, thereceivers can be left at the ocean bottom for a long period allowing for accurate time lapsemeasurements. Therefore, reservoirs can be well monitored with high repeatability, and costreduction of receivers deployment. A good example is the monitoring of the Valhall Field[van Gestel et al., 2008]. Another advantage of ocean bottom technology over streamers is thepossibility of making various acquisition designs with different source and receiver geometries.

S-wave seismic acquisition, processing and interpretation is getting more popular with thedemand of finding unconventional reservoirs. They offer more attributes such as shear-waveinterval attenuation for reservoir charactarization [Shekar and Tsvankin, 2011] and shear-wave AVO which allowed detection of fractures in the Vacuum Field [DeVault et al., 2002].Together, PP and PS data can enhance the interpretation and lead to direct access toelastic rock properties [Garotta et al., 2002]. Sometimes the S-wave seismic section has an

August 13, 2014

2 Introduction

advantage over the P-wave seismic section. A well know example is the presence of a gascloud in the subsurfaces which attenuates the P-waves while S-waves propagate throughwithout attenuation. [Cafarelli et al., 2000] and [Mancini et al., 2005] showed field dataexamples where mode converted waves imaged the target below the gas clouds, while P-wavesfailed to do so.

Since it is not possible yet to record pure P- and S-waves, these two types of wavesget recorded by the 4-C receivers. In order to have pure P- and S-wave records, it is requiredto perform proper elastic wavefield decomposition. The reliability of any further processingof the seismic data depends on whether the decomposition results are reliable or not. Theterms elastic or acoustic wavefield decomposition are used to refer to the medium in whichthe decomposition is performed and the outcome of the decomposition process. Whileacoustic decomposition provides up- and down-going acoustic pressure, elastic decompositiondelivers up- and down-going S-waves.

There exist different methods for decomposition. Data can be decomposed based onpolarity, as P- and S-waves propagate with different polarities. Another decompositionmethod and the most known in the industry is PZ summation, where the scaled verticalgeophone and hydrophone are summed together to get the decomposed up- and down-goingP-waves. This method is relatively cheap, but it is only valid for close to normal angle ofincidence. Wave equation based decomposition, both acoustic and elastic, is a more appro-priate way of honoring the physics of wave propagation. The theory of this method has beenexplained by different authors including [Ursin, 1983] and [Wapenaar and Berkhout, 1989].It is commonly derived in the wavenumber/rayparameter-frequency domains. This is thedecomposition approach that we describe and apply in this thesis.

1-2 Parameters estimation

Acoustic wavefield decomposition only requires the ocean properties. On the other hand,elastic wavefield decomposition requires the properties of the first layer beneath the receivers.Therefore, it is important to know the properties of the near-ocean bottom. One of thecurrent practice methods for estimating the near-bottom parameters is via refractiontomography surveys, which are based on first breaks picking. For large data sets, thismethod is time consuming. [Schalkwijk, 2001] and [Schalkwijk et al., 2003] suggested anadaptive wavefield decomposition scheme, with which the near-ocean bottom parameters canbe estimated, and applied it to field data as well. This approach allows good decompositionresults. However, it is based on different criteria that requires direct arrivals picking andevents identification.

Another near-ocean bottom problem is the complex behavior of the S-wave velocity.This leads to distortion of the seismic image and requires redatuming the seismic data toa certain depth level below the weathering layer. This involves near-ocean bottom velocityestimation as well. It is achieved in practice very often by model building of the near-bottomwith the analysis of refraction surveys.

August 13, 2014

1-3 Notations and transformations 3

In this thesis, we apply a waveform tomography inversion method to retrieve thenear-ocean bottom properties. The waveform tomography can be regarded a subset ofthe Joint Migration Inversion method established by [Berkhout, 2012] and performed by[Staal and Verschuur, 2013] and [Staal and Verschuur, 2014] on acoustic data. The methodis data-driven, capable of utilizing the total wavefield including all multiples and requiresno user interaction. In this thesis we will consider the situation that via the JMI process(or another process), an estimation for the upgoing P-waves and S-waves at a certain depthlevel has been obtained. We develop an inversion scheme that estimates the velocities in theremaining part of the subsurface, between the depth level below the water bottom and thebottom itself, and in addition apply a composition of the resulting upgoing wavefields at thebottom into the measured quantities by the 4-C receivers. Thus, a combined tomographyand decomposition problem is solved. In this way, the unknown properties at the bottomas well as the velocities in the-near bottom layer will be estimated based on comparing thepredicted with the true measurements. This leads to having a detailed ocean floor modelthat provides exact decomposition results. Moreover, statics correction becomes possible toachieve.

Note that in practice, the JMI method cannot provide the perfect upgoing P-wavesand S-waves if the near-bottom model is not yet available to large accuracy. However, thismeans that the JMI method including the described near-bottom estimation need to berun iteratively until all modelling results are in accordance to the measurements. But, forthe sake of this thesis we assume perfect knowledge of the upgoing wavefields at the certaindepth level below the water bottom.

1-3 Notations and transformations

We list in this section the transforms, conventions, symbols and definitions that are used inthis thesis.

We define the forward and inverse temporal Fourier transforms as

U(ω) =

∫ +∞

−∞e−jωt u(t) dt (1-1a)

u(t) =1

2π

∫ +∞

−∞ejωt U(ω) dω , (1-1b)

respectively, where ω denotes the angular frequency. For a real function u(t), the inverseFourier transform become

u(t) =1

πRe

∫ +∞

0ejωt U(ω) dω . (1-1c)

We define the horizontal 1D forward and inverse spatial Fourier transforms as

U(kx, ω) =

∫ +∞

−∞ejkxx U(x, ω) dx (1-2a)

U(x, ω) =1

2π

∫ +∞

−∞e−jkxx U(kx, ω) dkx , (1-2b)

August 13, 2014

4 Introduction

respectively, where kx denotes the horizontal wavenumber.

Based on the definitions of the temporal and spatial Fourier transforms, differentia-tion with respect to time in the time domain is equivalent to multiplication with (jω) in thefrequency domain, (∂t → jω). Also, Differentiation with respect to the 1D space-frequencydomain is equivalent to multiplication with (−jkx) in the wavenumber-frequency domain,(∂x → −jkx).

We define the horizontal 2D forward and inverse Fourier transforms as

U(kx, ky, ω) =

∫ +∞

−∞

∫ +∞

−∞ej(kxx+kyy) U(x, y, ω) dxdy (1-3a)

U(x, y, ω) =1

4π2

∫ +∞

−∞

∫ +∞

−∞e−j(kxx+kyy) U(kx, ky, ω) dkxdky , (1-3b)

respectively. Similarly, (∂l → −jkl), where (l = x, y).

We define the temporal forward and inverse Radon transform, respectively as

u(p, τ) =

∫ +∞

−∞u(x, τ + px)dx (1-4a)

u(x, t) =1

2π∂t

∫ +∞

−∞H[u(p, t− px)]dp , (1-4b)

where H denotes the Hilbert transform, (p =kxω

) is the horizontal slownes, and (τ = t− px)

is the time intercept.

The 1D Radon transform can also be defined in the frequency domain:

U(p, τ) =

∫ +∞

−∞e(jωpx)U(x, ω)dx (1-5a)

U(x, ω) =ω

2π

∫ +∞

−∞e(−jωpx)U(ωp, ω)dp. (1-5b)

The definition of the Kronecker delta is given by

δij =

{1 , for i = j,

0 , for i 6= j.(1-6)

They symbol (∗) indicates complex conjugate.

August 13, 2014

Chapter 2

Acoustic and Elastic WavefieldDecomposition

In this chapter, we review the theory of acoustic and elastic wavefield decomposition followingthe work of [Wapenaar and Berkhout, 1989] and [Schalkwijk, 2001]. First, we explain acousticwavefield decomposition in the wavenumber-frequency domain. Taking similar approach tothe acoustic decomposition, we then explain the theory of elastic wavefield decomposition in anisotropic medium. Finally, we apply the theory to multi-component ocean-bottom syntheticdata to get the up- and down-going acoustic pressure from the acoustic decomposition. Whilethe output of the elastic decomposition is the up- and down-going P- and S-waves.

2-1 Acoustic case

To reach to an expression for the decomposed one-way wavefields, we need to establish arelationship between the recorded two-way wavefields and the desired one-way wavefields.We Consider a horizontally layered medium where medium parameters vary with respect todepth only. The compressibility κ = κ(z), and the density ρ = ρ(z).

2-1-1 Two-way wave equation

The starting point is deriving the two-way wave equation from the linearized equation ofmotion:

∂kp+ ρ ∂tvk = fk (2-1)

and the linearized stress-strain relationship:

∂kvk + κ ∂tp = q . (2-2)

For a source-free situation, we write equations (2-1) and (2-2) in the space-frequency domain:

August 13, 2014

6 Acoustic and Elastic Wavefield Decomposition

∇P (x, y, z, ω) + jωρ(z) V(x, y, z, ω) = 0 (2-3)

∇.V(x, y, z, ω) + jωκ(z) P (x, y, z, ω) = 0 , (2-4)

and in the wavenumber-frequency domain −jkx P (kx, ky, z, ω)

−jky P (kx, ky, z, ω)

∂zP (kx, ky, z, ω)

= −jωρ(z)

Vx(kx, ky, z, ω)

Vy(kx, ky, z, ω)

Vz(kx, ky, z, ω)

, (2-5)

−jkx Vx(kx, ky, z, ω)−jky Vy(kx, ky, z, ω)+∂zVz(kx, ky, z, ω) = −jωκ(z)P (kx, ky, z, ω). (2-6)

To reduce the notations, we omit kx, ky and ω from the notations. Taking into account thefollowing relationships and considering only homogenous propagating waves,

k2z(z) =

ω2

c(z)2− k2

x − k2y , for k2

x + k2y ≤ k2

z(z) (2-7a)

c(z)2 =1

κ(z)ρ(z). (2-7b)

We can reach to a first-order two-way wave equation by eliminating Vx(z) and Vy(z)

∂z

(P (z)

Vz(z)

)=

0 −jωρ(z)k2z(z)

jωρ(z)0

( P (z)

Vz(z)

). (2-8)

In compact notations, we can written it as

∂zQ(z) = A(z)Q(z) . (2-9)

where Q contains the vertical component of the velocity and the acoustic pressure, whichare continious in a horizontally layered medium. Therefore, Q is continous. The verticalderivative represents variations of the wavefields in the vertical direction.

2-1-2 One-way wavefields

To go from the two-way wavefields representation to one-way wavefields, we perform eigen-value decomposition

A(z) = L(z)Λ(z)L−1(z) , (2-10)

August 13, 2014

2-1 Acoustic case 7

where L(z) and Λ(z) contain the eigenvectors and the eigenvalues, respectively, which we findby solving the characteristic equation

det[A− λI] = 0 (2-11a)

Λ(z) =

(−jkz(z) 0

0 jkz(z)

)(2-11b)

L(z) =

1 1kz(z)

ωρ(z)

−kz(z)ωρ(z)

(2-11c)

L−1(z) =

1/2ωρ(z)

2kz(z)

1/2−ωρ(z)

2kz(z)

. (2-11d)

Substituting equation (2-10) in the two-way wave equation results in

∂zQ(z) = L(z)Λ(z)L−1(z)Q(z) . (2-12)

Next, we define a vector D(z) that contains the one-way up- and down-going wavefields suchthat

Q(z) = L(z)D(z) (2-13a)

D(z) = L−1(z)Q(z) (2-13b)

and substitute it in equation (2-12) to get

∂z[L(z)D(z)] = L(z)Λ(z)D(z) (2-14a)

D(z) ∂zL(z) + L(z) ∂z D(z) = L(z)Λ(z)D(z) (2-14b)

L−1(z) D(z)∂zL(z) + ∂zD(z) = Λ(z)D(z) , (2-14c)

which we write as∂zD(z) = B(z)D(z) , (2-15)

withB(z) = Λ(z)− L−1(z) ∂zL(z) . (2-16)

For a homogenous medium,(∂zL(z) = 0

), which reduces equation (2-16) to

∂zD(z) = Λ(z)D(z) . (2-17)

In this case, the up- and down-going wavefields are decoupled. By defining D(z) to representthe one-way acoustic pressure, equation (2-17) becomes

∂z

(P+(z)

P−(z)

)=

(−jkz(z) 0

0 jkz(z)

)(P+(z)

P−(z)

). (2-18)

August 13, 2014


2-2 Elastic case

We follow similar approach to the acoustic case to describe the theory of elastic wavefield de-composition. We also consider elastic horizontally layered medium where medium parametersvary with respect to depth only, the density ρ = ρ(z) and the stiffness tensor Cijkl = Cijkl(z).

2-2-1 Two-way wave equation

We derive the elastic two-way wave equation from the linearized equation of motion

ρ ∂tvi − ∂jτij = fi (2-19)

and the linearized stress-strain relationship:

∂tτij − Cijkl ∂lvk = −∂tσij . (2-20)

The stiffness tensor given by [Auld, 1973] for an isotropic medium reads

Cijkl = λδijδkl + µ(δikδjl + δilδjk) . (2-21)

Substituting this equation in the stress-strain equation results in

∂tτij − [λδij∂kvk + µ(∂jvi + ∂ivj)] = −∂tσij . (2-22)

In the wavenumber-frequency domain and for a source-free situation, equation (2-19) becomes

jωρ(z)

Vx(z)

Vy(z)

Vz(z)

=

−jkxTxx(z) − jkyTxy(z) ∂zTxz(z)−jkxTyx(z) − jkyTyy(z) ∂zTyz(z)−jkxTzx(z) − jkyTzy(z) ∂zTzz(z)

(2-23)

and the components of equation (2-22) become

jωTxx = −[λ+ 2µ] jkx Vx − λjky Vy + λ ∂z Vz (2-24a)

jωTyy = −λjkx Vx − [λ+ 2µ] jky Vy + λ∂z Vz (2-24b)

jωTzz = −λjkx Vx − λjky Vy + [λ+ 2µ] ∂z Vz (2-24c)

jωTxz = jωTzx = µ[∂zVx − jkxVz] (2-24d)

jωTyz = jωTzy = µ[∂zVy − jkyVz] (2-24e)

jωTxy = jωTyx = µ[−jkyVx − jkxVy] . (2-24f)

Note that the z-dependency is still there, but we dropped it to reduce the notations. We

eliminate ~Tx and ~Ty and obtain expressions for ∂z ~Tz and ∂z ~V to reach to a first order two-way wave equation

∂zQp(z) = Ap(z)Qp(z) . (2-25)

August 13, 2014

2-2 Elastic case 9

where

Qp(z) =

Vx(z)

Vy(z)

−Tzz(z)−Txz(z)−Tyz(z)Vz(z)

(2-26a)

Ap(z) =

(0 A12(z)

A21(z) 0

)(2-26b)

A12 =

−j ωµ 0 jkx0 −j ωµ jkzjkx jky −jωρ

(2-26c)

A12 =

−jωρ−1jω [α1k

2x + µk2

y] − 1jωα2kxky jkx

λλ+2µ

− 1jωα2kxky −jωρ− 1

jω [µk2x + α1k

2y] jky

λλ+2µ

jkxλ

λ+2µ jkyλ

λ+2µ −j ωλ+2µ

(2-26d)

α1 = 4µ

(λ+ µ

λ+ 2µ

)(2-26e)

α2 = µ

(3λ+ 2µ

λ+ 2µ

). (2-26f)

Note the arrangement of Qp and Ap such that it is possible to apply eigenvalue decompositionsimilar to the acoustic case. The original form of Q is

Q(z) =

(−Tz(z)

V(z)

). (2-27)

2-2-2 One-way wavefields

To have a representation of the one-way wavefields, we apply eigenvalue decomposition

Ap(z) = Lp(z)Λ(z)Lp(z)−1 . (2-28)

Solving the characteristic equation

det[Ap − λI] = 0 (2-29)

gives the following eigenvalues

Λ(z) =

−jkz,p(z) 0 0 0 0 00 −jkz,s(z) 0 0 0 00 0 −jkz,s(z) 0 0 00 0 0 jkz,p(z) 0 00 0 0 0 jkz,s(z) 00 0 0 0 0 jkz,s(z)

, (2-30a)

August 13, 2014


wherekz,p =

√k2p − k2

x − k2y (2-30b)

kz,s =√k2s − k2

x − k2y (2-30c)

k2p =

ω2

c2p

=ρω2

λ+ 2µ(2-30d)

k2s =

ω2

c2s

=ρω2

µ, (2-30e)

and the following eigenvectors

Lp(z) =

(Lp1(z) Lp1(z)

Lp2(z) −Lp2(z)

), (2-31a)

where

Lp1(z) =

`1p η1 + η2 η′1 + η′2kykx`1p −kx

kyη1 −kx

kyη′1

µ

ωkx[k2s − 2k2

x − 2k2y]`1p] −

2kxµ

ωη2 −2kxµ

ωη′2

(2-31b)

and

Lp2(z) =

`2p ξ1 + ξ2 ξ′1 + ξ′2kykx`2p −kx

kyξ1 −kx

kyξ′1

ω

2kxµ− kxω

µ[k2s − 2k2

x − 2k2y]ξ2 − kxω

µ[k2s − 2k2

x − 2k2y]ξ′2

. (2-31c)

The parameters (`1p, η1, η2, η′1, η′1) and (`2p, ξ1, ξ2, ξ′1, ξ′1) are normalization parameters whichvary depending on the desired one-way wavefields choice. We chose the desired wavefields torepresent the P- and S-wave potentials, φ and ~ψ, respectively. The definition for the so calledLame parameters with respect to velocity in an isotropic medium is

∂t~v =−1

ρ[∇φ+∇× ~ψ ] , (2-32)

which implies

−ρ∂t~vp = ∇φ (2-33a)

−ρ∂t~vs = ∇~ψ , (2-33b)

where~ψ = (ψx , ψy , ψz)

T . (2-33c)

Note that ψx, ψy and ψz denote the S-wave potentials polarized in the y− z, x− z and x− yplanes, respectively. The eigenvectors for the desired one-way wavefields become

Lp1(z) =µ

ω2ρ

ωkxµ

−ωkxkyµkz,s

−ω(k2s − k2

x)

µkz,sωkyµ

ω(k2s − k2

y)

µkz,s

ωkxkyµkz,s

k2s − 2k2

x − 2k2y −2kykz,s 2kxkz,s

(2-34a)

August 13, 2014

2-2 Elastic case 11

and

Lp2(z) =µ

ω2ρ

2kxkz,p −2kxky −(k2

s − 2k2x)

2kykz,p (k2s − 2k2

y) 2kxkyωkz,pµ

−ωkyµ

ωkxµ

. (2-34b)

Analogous to the acoustic case, define a vector Dp that contains a permutation of the one-wayup- and down-going wavefields such that

Qp(z) = Lp(z)Dp(z) (2-35)

Vx(z)

Vy(z)

−Tzz(z)−Txz(z)−Tyz(z)Vz(z)

=

(Lp1(z) Lp1(z)

Lp2(z) −Lp2(z)

)

Φ+(z)

Ψ+x (z)

Ψ+y (z)

Φ−(z)

−Ψ−x (z)

−Ψ−y (z)

. (2-36)

From the arrangement of equation (2-36), it is possible to write Lp in the following notation

Lp(z) =

(L+,p

1 (z) L−,p1 (z)

L+,p2 (z) −L−,p2 (z)

), (2-37)

such that [L+,p1 = Lp1] and [L−,p1 = Lp1(−kz,s)] form the down-going wavefield composition sub

matrices and [L+,p2 = Lp2] and [L−,p2 = Lp2(−kz, p) form the up-going wavefield composition sub

matrices. The vertical wavenumber changes to negative for the up-going wavefield compositionsub matrices to match the sign convention, the positive vertical wavenumber corresponds tothe down-going wavefield and the negative vertical wavenumber corresponds to the up-goingwavefield.

We return back to use the original form of Q and arrange the matrices accordingly

Q(z) =

(−Tz(z)

V(z)

)=

(L+

1 (z)

L+2 (z)

)D+(z) +

(L−1 (z)

L−2 (z)

)D−(z) =

(L+

1 (z) L−1 (z)

L+2 (z) L−2 (z)

)(D+(z)

D−(z)

)=

(L+

1 (z) L−1 (z)

L+2 (z) L−2 (z)

)

Φ+(z)

Ψ+x (z)

Ψ+y (z)

Φ−(z)

Ψ−x (z)

Ψ−y (z)

= L(z)D(z) , (2-38)

where

L±1 (z) =µ

ω2ρ

±2kxkz,p −2kxky −(k2s − 2k2

x)

±2kykz,p k2s − 2k2

y 2kxkyk2s − 2k2

x − 2k2y ∓2kykz,s ±2kxkz,s

(2-39a)

August 13, 2014


and

L±2 (z) =1

ωρ

kx ∓kxky

kz,s∓k

2s − k2

x

kz,s

ky ±k2s − k2

y

kz,s±kxkykz,s

±kz,p −ky kx

. (2-39b)

The decomposition equation reads

D(z) = L−1(z)Q(z) = N(z)Q(z) , (2-40)

where

N±1 (z) =1

2

± kxkz,p

± kykz,p

1

0 1 ∓ kykz,s

−1 0 ± kxkz,s

(2-41a)

and

N±2 (z) =µ

2ω

2kx 2ky ±

k2s − 2k2

x − 2k2y

kz,p

∓kxkykz,s

±k2s − k2

x − 2k2y

kz,s−2ky

∓k2s − k2

x − k2y

kz,s±kxkykz,s

2kx

. (2-41b)

The derivation details of the composition/decomposition operators can also be found in[Kennett, 1983]. Following the same steps as in the acoustic case, we reach to the one-waywave equation

∂zD(z) = B(z)D(z) , (2-42)

withB(z) = Λ(z)− N(z) ∂zL(z) . (2-43)

For a homogenous medium(∂zL(z) = 0

)and

∂zD(z) = Λ(z)D(z) . (2-44)

In this case, the up- and down-going wavefields are again decoupled.

2-3 Wavefield decomposition at the ocean bottom

At the ocean bottom, (z = z1) the following boundary conditions apply:

1. −τzz(z1) = p(z1) ;

2. τxz(z1) = τyz(z1) = 0.

August 13, 2014

2-3 Wavefield decomposition at the ocean bottom 13

2-3-1 Acoustic decomposition

Equation (2-13) defines the relationship between the two-way wavefields in Q(z) and the one-way wavefields in D(z). They are related by the composition matrix L(z), which composesthe total wavfields from the one-way wavefields.(

P (z1)

V (z1)

)=

1 1kz,0(z1)

ωρ0(z1)

−kz,0(z1)

ωρ0(z1)

( P+(z1)

P−(z1)

). (2-45)

Using the decomposition matrix N(z), we can decompose the total wavefield to the one-waywavefields

(P+(z1)

P−(z1)

)=

1/2ωρ0(z1)

2kz,0(z1)

1/2−ωρ0(z1)

2kz,0(z1)

( P (z1)

Vz(z1)

), (2-46)

and in compact notation this reads

P±(z1) =1

2P (z1)± ωρ0(z1)

2kz,0(z1)Vz(z1) . (2-47)

This equation shows that by scaling the frequency-wavenumber transformed acoustic pressureand vertical velocity component measurements, it is possible to decompose the total wave-field to up- and down-going wavefields just above the ocean bottom (z1 − ε, ε → 0). Sincedecomposition is done just above the ocean bottom, it only requires the properties of theocean (ρ0(z1), kz,0(z1)).

2-3-2 Elastic decomposition

For the elastic decomposition, we consider waves that are polarized in the (x−z) plane whereky = 0. We also neglect SH waves and take into account only P-SV waves. Hence, thecomposition and decomposition equations become

−Txz(z1)

−Tzz(z1)

Vx(z1)

Vz(z1)

=

(L+

1,p−sv(z1) L−1,p−sv(z1)

L+2,p−sv(z1) L−2,p−sv(z1)

)Φ+(z1)

Ψ+y (z1)

Φ−(z1)

Ψ−y (z1)

(2-48)

and Φ+(z1)

Ψ+y (z1)

Φ−(z1)

Ψ−y (z1)

=

(N+

1,p−sv(z1) N+2,p−sv(z1)

N−1,p−sv(z) N−2,p−sv(z1)

)−Txz(z1)

−Tzz(z1)

Vx(z1)

Vz(z1)

, (2-49)

where

L±1,p−sv(z1) =µ

ω2ρ

[±2kxkz,p −(k2

s − 2k2x)

k2s − 2k2

x ±2kxkz,s

](2-50a)

August 13, 2014


L±2,p−sv(z1) =1

ωρ

[kx ∓kz,s±kz,p kx

](2-50b)

and

N±1,p−sv(z1) =1

2

±kxkz,p

1

−1 ± kxkz,s

(2-51a)

N±2,p−sv(z1) =µ

2ω

2kx ±k2s − 2k2

x

kz,p

∓k2s − k2

x

kz,s2kx

. (2-51b)

Using the OBC measured data (p(z1),vx(z1),vz(z1)), and taking into account the boundaryconditions, the total wavefield can be decomposed into up- and down-going P- and S-waves,just below the ocean bottom. Hence, the decomposition and composition matrices depend onthe properties of the layer just below the receivers.

2-4 Synthetic data

We show synthetic data to demonstrate the wavefield decomposition theory. The receiversmeasure the acoustic pressure (p), vertical particle velocity (vz) and the horizontal particlevelocity (vx). They are placed at the ocean bottom (z1 = 500 m). The source is a dipole P-wave zero-phase wavelet positioned at the water surface (z = 0). The medium below the oceanbottom consists of a gradient mimicked by multi thin layers to resemble the earth’s subsurfacewhere velocities and density often increase gradually close the sea bottom. Below that, thereis a horizontally layered medium with sharp boundaries. We performed 1.5 D elastic modelingon the provided density, P and S velocity logs, Figure 2-1. We used the reflectivity method[Muller, 1985] to compute the synthetic data in the wavenumber-frequency domain. Thesynthetic data includes the primaries as well as the surface and internal multiples, Figure 2-2.

2-4-1 Wavefield decomposition

Applying equation (2-47) to the acoustic pressure and the vertical component of the particlevelocity gives the up- and down-going acoustic pressure just above the ocean bottom,Figure 2-3. (p+) does not contain the up-going primary reflections or any source-sidemultiples. On the other hand, (p−) contains the up-going wavefield, the direct arrivaland any receiver-side multiple. This is because decomposition is performed just above theocean bottom. Any down-going energy reflected from the ocean bottom is mapped into theup-going wavefield component, Figure 2-4.

Using the three components of the synthetic data, we apply equation (2-49) to decomposethe data into up- and down-going P- and S-waves, Figure 2-5. Since the decomposition levelis just below the ocean bottom, the up-going wavefields are free from the direct arrival andany receiver-side multiples.

August 13, 2014

2-4 Synthetic data 15

0

200

400

600

800

1000

1200

1400

dept

h (m

)

500 1000 1500 2000 2500 3000density (kg/m3)

(a) Density

0

200

400

600

800

1000

1200

1400

dept

h (m

)

0 500 1000 1500velocity (m/s)

(b) S-wave velocity

0

200

400

600

800

1000

1200

1400

dept

h (m

)

1000 1500 2000 2500 3000 3500velocity (m/s)

(c) P-wave velocity

Figure 2-1: Logs used for creating synthetic data.

August 13, 2014


0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(a) p

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(b) vx

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(c) vz

Figure 2-2: Synthetic ocean bottom data.

August 13, 2014

2-4 Synthetic data 17

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(a) p+

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(b) p−

Figure 2-3: Decomposition of the total wavefield just above the ocean bottom to (a) up-goingacoustic pressure and (b) down-going acoustic pressure.

z1-‐ε z1 z1+ε

z0

Figure 2-4: Decomposition levels, just above the ocean bottom (z1−ε) and just below the oceanbottom (z1 + ε).

August 13, 2014


0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(a) ϕ+

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(b) ϕ−

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(c) ψ+

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(d) ψ−

Figure 2-5: Decomposition of the total wavefield just below the ocean bottom to (a) down-goingP-waves, (b) up-going P-waves, (c) down-going S-waves and (d) up-going-S-waves.

August 13, 2014

2-5 Decomposition with erroneous velocities 19

2-5 Decomposition with erroneous velocities

In order to get the true one-way wavefields based on the theory of elastic wavefield decompo-sition, the ocean floor parameters need to be exact. To test the sensitivity of decompositionto the ocean floor parameters, we apply decomposition to the synthetic data with erroneousvelocities. Instead of using cp(1) = 1550m/s and cs(1) = 100m/s, we first test decompositionwith velocities 100 m/s higher than the true ones, see Figure 2-6. In this case, the up-goingP- and S-waves start to contain some of the down-going energy of the direct arrival. Thisis reasonable since the amplitude of the direct arrival is relatively low at the apex comparedwith the other events. Increasing the error to 200 m/s leads to boosting the direct down-going arrival in the up-going sections, see Figure 2-7a and Figure 2-7b. Further increasingthe velocity error to 400 m/s, see Figure 2-7c and Figure 3-6d, leads to interchanging of theup- and down-going events and almost no difference between the up-going wavefields and thetotal measurements. From here, we conclude the importance of a reasonable estimate of thenear-bottom velocities, such that reliable one-way wavefields are obtained.

August 13, 2014


0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(a) ϕ+

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(b) ϕ−

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(c) ψ+

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(d) ψ−

Figure 2-6: Elastic decomposition with 100 m/s error in velocities to (a) down-going P-waves,(b) up-going P-waves, (c) down-going S-waves and (d) up-going-S-waves.

August 13, 2014

2-5 Decomposition with erroneous velocities 21

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(a) ϕ−

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(b) ψ−

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(c) ϕ−

0

0.5

1.0

1.5

2.0

2.5

time

(s)

-2000 -1000 0 1000 2000offset (m)

(d) ψ−

Figure 2-7: Elastic decomposition with (a-b) 200 m/s and (c-d) 400 m/s error in the velocities.

August 13, 2014


August 13, 2014

Chapter 3

Near-bottom velocity estimation

Since elastic wavefield decomposition at the ocean bottom requires properties of the oceanfloor, an accurate near-ocean bottom model is essential. In this chapter, we perform wave-field tomography to build P- and S-wave velocity models of the near-ocean bottom layers.[Berkhout, 2012] proposed the Joint Migration Inversion scheme, which aims at estimatingthe velocity and reflectivity. Total wavefield tomography is the part of JMI that estimatesthe velocities. This approach does not require first arrivals picking as in travel time to-mography, but rather finds a distribution of velocities and reflections such that the two-way modeled wavefield matches the observed data. The methodology was implemented by[Staal and Verschuur, 2013] and [Staal and Verschuur, 2014] on acoustic data considering thetotal wavefield. Like stated in the introduction, we assume that via the JMI method (oranother approach), we have obtained the upgoing P- and S- wavefields at a known depth inthe vicinity of the water bottom (e.g. few 100 meters below the bottom) and our task is tofind the velocities of the layers between this depth and the bottom. With those estimated ve-locities, we compose the up-going P- and S-waves to form the modeled vertical and horizontalvelocity components which can then be matched with the measurements. Since compositionrequires the down-going P- and S-waves as well, we express them in terms of the acousticpressure measurement. To retrieve the velocities of the first layer below the ocean bottom, weapply another inversion loop that minimizes the mismatch between the measured data andthe modeled data that we compose with the retrieved velocities from the tomography part.For both inversion loops, we apply the gradient descent method. All the derivations are inthe rayparameter - frequency domain, and are based on a 1.5D model.

3-1 Wavefield tomography

3-1-1 Velocity perturbation theory

If the wavefield propagates between two levels, zn and zm, in two different velocity models,the wavefields experience two different extrapolations. Assume that the velocity models form

August 13, 2014

24 Near-bottom velocity estimation

the true and background velocity models with their corresponding extrapolation operatorsW− and W−0 , respectively. The expression of the true extrapolation operator in terms of thebackground extrapolation operator reads

W−(p, zm, zn, ω) = W−0 (p, zm, zn, ω) + ∆W−(p, zm, zn, ω). (3-1)

These extrapolation operators are phase phase shift operators expressed as

W−(p, zm, zn, ω) = e−jωq∆z , (3-2)

where q is dependent on considering P- or S-wave propogation:

qp =

√1

c2p

− p2 , qs =

√1

c2s

− p2 . (3-3)

For the rest of the derivations, we drop the rayparameter p and ω from the equations fornotations reduction. The subscript p is not to be confused with the rayparameter p, theearlier denotes P-wave.Define β to be the P- and S-wave velocity contrasts at each depth level

βp(z) = 1− cp0(z)2

cp(z)2, βs(z) = 1− cs0(z)2

cs(z)2. (3-4)

Perform Taylor series expansion for W at β = 0 and ignore higher order terms:

W−(zm, zn) ≈ W−0 (zm, zn) +∂W0

−(zm, zn)

∂ββ(z) + ... (3-5a)

∆W−(zm, zn) ≈ ∂W0−

(zm, zn)

∂ββ(z) (3-5b)

∆W−(zm, zn) ≈ G−0 (zm, zn) β(z), (3-5c)

with

G−0 (zm, zn) = jω∆zq2o

2qe−jωq∆z (3-6)

q =√q2

0(1− β(z))− p2 and q0 =1

c2. (3-7)

A stabilization parameter ε is necessary for 90◦ angle of propagation, where G0 becomesundefined, (q = 0).

G−0 (zm, zn) ≈ jω∆z

2q2o

q∗

qq∗ + εe−jωq∆z . (3-8)

From expressions (3-1) and (3-5c), it can be seen that it is possible to retrieve the trueextrapolation operator from the known background extrapolation operator and the linearizeddifference extrapolation operator. Note that β = 0 and W = W0 is only valid when thebackground model matches the true model, c0 = c. This implies that the linearization of thedifference extrapolation operator requires a good initial model that is close enough to the truemodel.

August 13, 2014

3-1 Wavefield tomography 25

Define Q−p (zn) and Q−s (zn) to be the upgoing wavefields at a certain depth level. These

upgoing wavefields are the assumed-known Φ−(zN+1) and Ψ−(zN+1), forward extrapolatedwith W−0 (zn, zN+1). This is just an assumption to reduce the wavefield tomography problemto a subset of a larger scheme, in which the up-going wavefields are actually estimated.The cumulative effect of the velocity perturbations from all the depth levels on the upgoingwavefields recorded at the ocean bottom (z1) are given by

∆Φ(z1) =

N∑n=2

G−p0(z1, zn)βp(zn)Q−p (zn) (3-9a)

∆Ψ(z1) =N∑n=2

G−s0(z1, zn)βs(zn)Q−s (zn) , (3-9b)

where

G−p0(z1, zn) = W−p0(z1, zn−1)G−p0(zn−1, zn) (3-9c)

G−s0(z1, zn) = W−s0(z1, zn−1)G−s0(zn−1, zn) (3-9d)

and

Q−p (zn) = W−p0(zn, zN+1)Φ−(zN+1) (3-9e)

Q−s (zn) = W−s0(zn, zN+1)Ψ−(zN+1) . (3-9f)

Note that Ψ = Ψy is the S-wave potential polarized in the (x − z) plane. An illustration ofthe depth levels to update is shown in Figure 3-1. Note that we assumed correct velocitybetween depth level zN + 1 and zN . The tomography takes place from depth levels zN to z1

Assume that there is a velocity contrast β between the background and the true velocitymodels, then equation (3-9) can predict the difference between the two wavefields propagatingin each model. This estimated residual might be different from the actual residual in thecase that the true wavefield experience multiple reflections that are not experienced by theestimated wavefield. This is because the estimated wavefield takes into account only thetransmission effects.

3-1-2 Tomographic Inversion

In order to find an estimated velocity model of the near ocean bottom, we implement the gra-dient descent method. The method minimizes the difference between the measured wavefieldand the modeled wavefield in the estimated model in a least-squares sense. The wavefieldsrepresent the decomposed P- and S-wave potentials which we use to retrieve the P- and S-wavevelocity profiles, independently.

The two objective functions are

Jp =

p2∑p1

ω2∑ω1

||Φ−obs(z1)− Φ−mod(z1)||2 =

p2∑p1

ω2∑ω1

||∆Φ−(z1)||2 (3-10a)

Js =

p2∑p1

ω2∑ω1

||Ψ−obs(z1)− Ψ−mod(z1)||2 =

p2∑p1

ω2∑ω1

||∆Ψ−(z1)||2 . (3-10b)

August 13, 2014


z1

zN

zN+1

z0

ψ- φ-

z2

Figure 3-1: Illustration of the depth levels to update.

Substitute equations (3-9a) and (3-9b) in the objective functions and take the derivative withrespect to β to get the following gradients

∇βp(zn) =

p2∑p1

2[G−p0(zn, z1)]∗ ∆Φ−(z1)[Q−p (zn)]∗ (3-11a)

∇βs(zn) =

p2∑p1

2[G−s0(zn, z1)]∗ ∆Ψ−(z1) [Q−s (zn)]∗ . (3-11b)

Note that ∇β(zn) contains the velocity contrasts based on wavefields that propagate witha range of angles, p values. The angle-dependent contributions at a certain depth level arecombined to result in one ∇β(zn) describing the velocity variations as a function of depth.

Using the computed gradients, we predict the wavefield perturbations for each p value

∆Φ−∇β(z1) =N∑n=2

G−p0(z1, zn)∇βp(zn)Q−p (zn) (3-12a)

∆Ψ−∇β(z1) =N∑n=2

G−s0(z1, zn)∇βs(zn)Q−s (zn) . (3-12b)

Now, we can update the velocity contrasts β

βp(z)new = βp(z)

old + αp∇βp(z) (3-13a)

βs(z)new = βs(z)

old + αs∇βs(z) , (3-13b)

by matching the predicted wavefield updates, ∆Φ−∇β and ∆Ψ−∇β with the observed data mis-

matches, ∆Φ− and ∆Ψ− and by finding the corresponding scale factors αp and αs. In fact,αp and αs are calculated by solving

August 13, 2014

3-1 Wavefield tomography 27

min (||∆Φ− αp∆Φ∇βp ||2) (3-14)

min (||∆Ψ− αs∆Ψ∇βs ||2) , (3-15)

after which with the new β’s, the update of the velocities can be represented as:

cp(z)new =

cp(z)old√

1− αp∇βp(z)(3-16a)

cs(z)new =

cs(z)old√

1− αs∇βs(z). (3-16b)

In summary, the inversion loop that minimizes the difference between the true and the pre-dicted decomposed data by estimating the P and S velocities is as follows:

1. Forward model Φ−mod(z1) and Ψ−mod(z1) in the initial/current velocity models.

2. Compute the gradients.

3. Compute the effect of the gradients (prediction of the residual).

4. Compute the step lengths α.

5. Update the velocities.

6. Repeat steps 1 to 5 until the error reaches to an absolute minimum or the maximumnumber of iterations is reached.

3-1-3 Examples

To test the wavefield tomography method, we apply it to multi-component ocean bottomdata. The modeled data consists of the up-going P- and S-waves at depth level z = 300m. They are modeled in the tau-p domain considering a 1.5D elastic model. In thisdomain, each slowness value (p) corresponds to a certain angle of propagation of a planewave. The synthetic data was computed based on a horizontally layered Earth, wherethe medium parameter vary with respect to depth only. We create a velocity gradientmodel which resembles a true subsurface. The measurements, the modeled data and theresidual before applying wavefield tomography are shown in Figure 3-2. The mismatchbetween the measurements and the modeled data with the initial velocity models is clear.At this stage, the residual is in the same order of the measurements. Note that we only showa portion of the data for illustration purposes. The complete data set is shown in the appendix.

Applying wavefield tomography following equations (3-10) to (3-16) gives the resultsshown in Figure 3-3. Note the great improvement of the predicted data, to the pointthat the residual is almost negligible compared with the measurements’ amplitudes. Theretrieved velocity models, that minimized the objective functions Jp and Js, are smooth

August 13, 2014


representations of the true velocities, see Figure 3-4.

In Figure 3-5, the behaviors of the objective functions are shown. Conversion of theobjective functions to absolute minima takes place at the 8th iteration for P-waves and at the5th iteration for S-waves. This is explained by looking at the initial error, which is higher forthe P-waves due to the larger velocity difference between the true and initial P-wave velocitymodels. The initial S-wave velocity model was chosen to be close to the true velocity modelsince S-wave velocities are usually very small in the near-bottom and their correspondingtwo-way travel times are relatively large. If the initial model of the S-wave velocity is not closeenough to the true model, cycle skipping may occur and wavefield tomography will not be ableto retrieve a reliable velocity model. Hence, careful design of the initial S-wave velocity modelshould be considered. Another option to reduce the risk of getting trapped in a local minimumis to start the tomography process with low frequencies first, so that cycle skipping is avoided.

The argument that the estimated velocity model is not exactly the same as the truevelocity model is valid. This is because the utilized data in the tomography process does notcontain any reflection information from the near-bottom layers. Better velocity estimationscan be found if the primary reflections from within the near-bottom layer were incorporated.This is justified by the higher information content of the reflected waves which propagatemore in the subsurface with different angles. In addition, adding multiples should providemore detailed velocities as shown by [Staal and Verschuur, 2013] for the acoustic case.

3-2 Wavefield composition

Since the retrieved velocities of the ocean floor did not exactly match the actual velocities, wepropose in this section another inversion loop that results in a better estimate of the seafloorvelocites.

The elastic composition equation, (2-48), reads in the rayparameter-frequency domain

(−Txz(z1)

−Tzz(z1)

)= c2

s(z1)

(2 pqp(z1) −(c−2

s (z1)− 2p2)c−2s (z1)− 2p2 2 pqs(z1)

)(Φ+(z1)

Ψ+y (z1)

)+c2

s(z1)

(−2 pqp(z1) −(c−2

s (z1)− 2p2)c−2s (z1)− 2p2 −2 pqs(z1)

)(Φ−(z1)

Ψ−y (z1)

),

(3-17a)

(Vx(z1)

Vz(z1)

)=

1

ρ(z1)

(p −qs(z1)

qp(z1) p

)(Φ+(z1)

Ψ+y (z1)

)+

1

ρ(z1)

(p +qs(z1)

−qp(z1) p

)(Φ−(z1)

Ψ−y (z1)

).

(3-17b)

In order to compose Vx and Vz from their decomposed components using equation (3-17b), itis required to have the 4 potentials as well as the ocean floor parameters. In the tomographyprocess, we only utilized the up-going P- and S-waves to estimate their velocity profiles. For

August 13, 2014

3-2 Wavefield composition 29

0 1 2 3 4 5

x 10−4

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

time

(s)

slowness (s/m)

(a) ϕ−true and ϕ−

mod

−1 0 1 2 3 4 5

x 10−4

1.6

1.65

1.7

1.75

1.8

1.85

1.9

time

(s)

slowness (s/m)

(b) ψ−true and ψ−

mod

−0.2 −0.1 0 0.1 0.2

0.75

0.8

0.85

0.9

0.95

1

time

(s)

amplitude

(c) ∆ϕ− and ϕ−true

−0.1 −0.05 0 0.05 0.1 0.15

1.75

1.76

1.77

1.78

1.79

1.8

1.81

1.82

1.83

time

(s)

amplitude

(d) ∆ψ− and ψ−true

Figure 3-2: Magnified view of the up-going measurements (blue), predicted (red) and residual(black), before applying wavefield tomography. Note that the up-going P- and S-wave residuals are computed using only one trace (p = 0.2622× 10−3 s/m), for thesake of comparison.

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0 1 2 3 4 5

x 10−4

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

time

(s)

slowness (s/m)


mod

0 1 2 3 4 5

x 10−4

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

time

(s)

slowness (s/m)


mod

−0.1 −0.05 0 0.05 0.1 0.15 0.2

0.75

0.8

0.85

0.9

0.95

1

time

(s)

amplitude


−0.1 −0.05 0 0.05

1.75

1.76

1.77

1.78

1.79

1.8

1.81

1.82

1.83

time

(s)

amplitude


Figure 3-3: The measurements (blue), predicted up-going wavefields (red) and residual (black),after applying wavefield tomography. Note how well the predicted up-going wave-fields match the measured up-going wavefields. The trace we used to show theresidual corresponds to (p = 0.2622× 10−3 s/m).

August 13, 2014


1500 1600 1700 1800 1900 2000

50

100

150

200

250

300

velocity (m/s)

dept

h (m

)

(a) cp

200 300 400 500 600 700 800

100

120

140

160

180

200

220

240

260

280

300

velocity (m/s)

dept

h (m

)

(b) cs

Figure 3-4: The true (blue), estimated (red) and initial (green) velocity models for (a) P-waveand (b) S-wave. Note that these are also magnified velocity models to show theupdate levels.

0 5 10 15 200

200

400

600

800

1000

1200

1400

1600

1800

iteration

erro

r

Figure 3-5: Plot of Jp (blue) and Js (red), versus the number of iterations.

August 13, 2014


the sake of simplicity, we assume a known density profile. Using equation (3-17a) with theassumption that the acoustic pressure component (−Tzz(z1)) is known, we compute the down-going P- and S-waves in terms of the up-going P- and S-waves and the acoustic pressure, toget

Ψ+ =

[−TzzA c2

s

− 2Φ− − A

2pqpΨ− +

2pqsA

Ψ−

] [A2 + 4p2qsqp

2A pqp

]−1

(3-18a)

Φ+ =[AΨ+ + 2pqpΦ

− +AΨ−]

[2pqp]−1 , (3-18b)

whereA = c−2

s − 2p2 . (3-18c)

For waves propagating at normal incidence and at 90◦, where p = 0 and q = 0 , the expressionsbecome unstable. Hence, We use a stabilization factor ε to get

Ψ+ =

[−TzzAc2

s

− 2Φ− −(A

2

p

p2 + ε2qp

q2p + ε2

− 2pqsA

)Ψ−

] [A2 + 4p2qsqp

2A pqp

]−1

(3-19a)

Φ+ =[AΨ+ + 2pqpΦ

− +AΨ−] [1

2

p

p2 + ε2qp

q2p + ε2

]. (3-19b)

This means that we can now relate the calculated up-going wavefields Φ−(z1) and Ψ−(z1) tothe measured particle velocities, Vx(z1) and Vz(z1).

3-2-1 Composition/decomposition Inversion

In this inversion loop, we minimize the difference between the observed and modeled verti-cal and horizontal velocity components in a least-squares sense, using the gradient descentmethod. The true data are obtained from the observations in the true model, while the mod-eled data are composed with tomographic estimated velocities of the ocean floor. Hence, theerror is converted to an update to the ocean floor velocities. The objective function reads

J = ||Vxobs(z1)− Vxmod(z1)||2 + ||Vzobs(z1)− Vzmod

(z1)||2. (3-20)

The analytical gradients of J with respect to cp and cs are cumbersome to compute andrequire linearization. Therefore, we estimate the derivatives numerically using the centralfinite difference approximation,

∇Jp =J(cp + ∆cp)− J(cp −∆cp)

2∆cp(3-21a)

∇Js =J(cs + ∆cs)− J(cs −∆cs)

2∆cs. (3-21b)

August 13, 2014


We update the ocean floor velocities with the following equations

cp(z1)new = cp(z1)old + γp∆cp (3-22a)

cs(z1)new = cs(z1)old + γs∆cs . (3-22b)

Again, scale factors γp and γs are obtained by minimizing Jp and Js.

The resulting composition/decomposition inversion loop, which minimizes the differ-ence between the true and the predicted composed data by estimating cp(z1) and cs(z1) is asfollows:

1. Composition of Vz(z1) and Vx(z1) using the current estimated velocities from the wave-field tomography.

2. Compute the gradients.

3. Compute the step lengths γ.

4. Update the velocities.

5. Repeat steps 1 to 4 until the error reaches to an absolute minimum or the maximumnumber of iterations is reached.

3-2-2 Examples

We use the same example we have used for wavefield tomography to implement the compo-sition/decomposition inversion and obtain better estimates of the ocean floor velocities. Wecompose the total wavefields vx and vz with the current estimates from wavefield tomogra-phy, where csest(1) = 200 m/s and cpest(1) = 1750 m/s. The true ocean floor P- and S- wavevelocities are 1700 m/s and 200 m/s, respectively. Following the composition/decomposi-tion inversion scheme, the objective functions reached to an absolute minima after the firstiteration, which is expected since the S-wave velocity of the ocean floor is the same as theestimated one. On the other hand, the P-wave velocity is approximately 50 m/s different fromthe actual ocean floor velocity. Recall from section 2-5, where we tested elastic decompositionwith erroneous velocities, that ±100 m/s error in the decomposition velocity still providesreasonable decomposition results. Similarly, composition with a 50 m/s error should providegood composition results. Figure 3-6 shows the results of composition with the current es-timated velocities, the total measurements, and the residual. It is proven that compositionwith the current velocity estimates provided similar results to the total measurements to highextent. The discrepancy is due to events that do not originally exist in the modeled up-goingwavefields, but are present in the measurements. This is because of capturing the wavefieldsat a deeper depth level below the bottom. An example is the small reflections between 1.9and 2.9 seconds in the measured vx due to the small layers in the near bottom.

August 13, 2014


0 1 2 3 4

x 10−4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

(s)

slowness (s/m)

(a) vztrue and vzmod

−1 0 1 2 3 4 5

x 10−4

0

1

2

3

4

5

6

time

(s)

slowness (s/m)

(b) vxtrue and vxmod

−4 −2 0 2 4 6

x 10−7

0

0.02

0.04

0.06

0.08

0.1

0.12

time

(s)

amplitude

(c) ∆vz and vztrue

−10 −8 −6 −4 −2 0 2 4

x 10−7

1.7

1.8

1.9

2

2.1

2.2

time

(s)

slowness (s/m)

(d) ∆vx and vxtrue

Figure 3-6: The total measurements (blue), predicted data (red) and residual (black), afterapplying composition. Note the good match as well as the consistency of the eventsin the measured and predicted data. The trace chosen to compute the residualcorresponds to (p = 0.266× 10−3 s/m).

August 13, 2014

Chapter 4

Conclusions

In this thesis, we have reviewed the acoustic and elastic wavefield decomposition theoryfollowing [Wapenaar and Berkhout, 1989] and [Schalkwijk, 2001]. By scaling the two-waywavefields, we reached to expressions for the one-way wavefields. Applying the decompositiontheory to synthetic multi-component ocean bottom data provided up- and down-goingacoustic pressure for the acoustic case, and up- and down-going P- and S-waves for theelastic case. The sensitivity of the elastic decomposition theory to the ocean floor velocitiescan be neglected for a range of ±100 m/s velocity errors. Larger velocity errors resulted ininterchanging of the up- and down-going events to the point that the one-way wavefieldsare not reliable anymore. We conclude the importance of valid ocean floor velocity modelssuch that the decomposed wavefields resemble the actual up- and down-going P- and S-waves.

In the third chapter, we have implemented the wavefield tomography, proposed by[Berkhout, 2012] and [Staal and Verschuur, 2013], in the rayparameter-frequency domainbased on a 1.5D model. The methodology showed its ability to retrieve a smooth versionof the near-bottom velocities, which minimized the discrepancy between the measuredand the predicted data. Using the best velocity estimates from wavefield tomography, weapplied composition to form the predicted multi-component ocean bottom data, which wecompare with the measurements to find an optimum velocity estimate of the ocean bottom.From the examples we have shown, the best seafloor estimates are already provided bythe wavefield tomography. This is expected since these estimates are around (±50 m/s)different from the actual seafloor velocities. Based on the tests of elastic decompositionwith erroneous velocities, 100 m/s velocity error is acceptable. This is also proven byhaving the true composition results, when composing the one-way wavefields with slightlyerroneous velocities. However, if there is a large difference between the estimated and thetrue seafloor velocities, the composition/decomposition inversion loop will be able to correctfor that. Then the wavefield tomography and the composition/decomposition inversions canbe run in a flip-flop mode, where the two loops contribute to each others leading to the bestnear-bottom and ocean floor velocity estimates. The results indicate that such combinedinversion scheme will be feasible in practice.

August 13, 2014

36 Conclusions

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Bibliography

[Auld, 1973] Auld, B. A. (1973). Acoustic fields and waves in solids, volume 1. Wiley.

[Berkhout, 2012] Berkhout, A. J. (2012). Combining full wavefield migration and full wave-form inversion, a glance into the future of seismic imaging. GEOPHYSICS, 77(2):S43–S50.

[Cafarelli et al., 2000] Cafarelli, B., Madtson, E., Krail, P., Nolte, B., and Temple, B. (2000).3d gas cloud imaging of the donald field with converted waves. SEG Technical ProgramExpanded Abstracts, pages 1162–1165.

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[Garotta et al., 2002] Garotta, R., Granger, P., and Dariu, H. (2002). Combined interpreta-tion of pp and ps data provides direct access to elastic rock properties. The Leading Edge,21(6):532–535.

[Kennett, 1983] Kennett, B. L. N. (1983). Seismic Wave Propogation in Stratified Media.Cambridge University Press.

[Mancini et al., 2005] Mancini, F., Li, X., Dai, H., and Pointer, T. (2005). Imaging lomondfield using c-wave anisotropic pstm. The Leading Edge, 24(6):614–620.

[Muller, 1985] Muller, G. (1985). The reflectivity method: a tutorial. Journal of Geophysics,58:153–174.

[Schalkwijk et al., 2003] Schalkwijk, K., Wapenaar, C., and Verschuur, D. (2003). Adaptivedecomposition of multicomponent oceanbottom seismic data into downgoing and upgoingp and swaves. GEOPHYSICS, 68(3):1091–1102.

[Schalkwijk, 2001] Schalkwijk, K. M. (2001). Decomposition of multicomponent ocean-bottomdata into P- and S-waves. PhD thesis, TU Delft.

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[Shekar and Tsvankin, 2011] Shekar, B. and Tsvankin, I. (2011). Estimation of shear-waveinterval attenuation from mode-converted data. GEOPHYSICS, 76(6):D11–D19.

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[Staal and Verschuur, 2014] Staal, X. R. and Verschuur, D. J. (2014). Robust velocity esti-mation by joint migration inversion. 76th EAGE Conference and Exhibition.

[Ursin, 1983] Ursin, B. (1983). Review of elastic and electromagnetic wave propagation inhorizontally layered media. GEOPHYSICS, 48(8):1063–1081.

[van Gestel et al., 2008] van Gestel, J., Kommedal, J., Barkved, O., Mundal, I., Bakke, R.,and Best, K. (2008). Continuous seismic surveillance of valhall field. The Leading Edge,27(12):1616–1621.

[Wapenaar and Berkhout, 1989] Wapenaar, C. P. A. and Berkhout, A. J. (1989). ElasticWavefield Extrapolation: Redatuming of Single- and Multi-Component Seismic Data. El-sevier Schience Publ. Co., Inc.

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Bibliography 39

Appendix

For completion, we include in this appendix the complete set of the measured and modeledup-going wavefields before and after applying wavefield tomography, since we only showed amagnified portion for illustration in section 3-1-3. We also show the complete velocity profilesas well.

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40 Bibliography

−1 0 1 2 3 4 5

x 10−4

0

1

2

3

4

5

6

time

(s)

slowness (s/m)


mod

−1 0 1 2 3 4 5

x 10−4

0

1

2

3

4

5

6

time

(s)

slowness (s/m)


mod

−1 0 1 2 3 4 5

x 10−4

0

1

2

3

4

5

6

time

(s)

slowness (s/m)


−1 0 1 2 3 4 5

x 10−4

0

1

2

3

4

5

6

time

(s)

slowness (s/m)


Figure 4-1: Up-going measurements (blue), predicted wavefields (red) and residual (black). (a)and (b) correspond to data before applying wavefield tomography, while (c) and (d)correspond to data after applying wavefield tomography. The difference might notbe noticeable as it is in Figure 3-2 and Figure 3-3.

August 13, 2014

Bibliography 41

1500 2000 2500

100

200

300

400

500

600

700

800

900

1000

velocity (m/s)

dept

h (m

)

(a) P − wavevelocityprofile

0 200 400 600 800 1000

0

200

400

600

800

1000

1200

velocity (m/s)

dept

h (m

)

(b) S-wave velocity profile

Figure 4-2: True (blue), estimated (red) and initial (green) velocity profiles.

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42 Bibliography

August 13, 2014

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