AES/TG/07-27 Simulation of Interferometric Seismo ...€¦ · source types in the interferometric...

AES/TG/07-27 Simulation of Interferometric Seismo-electric Green’s Function Recovery

For the SH-TE propagation mode

August 2007 Sjoerd A.L. de Ridder

Title : Simulation of Interferometric Seismoelectric Green’s

Function Recovery;For the SH-TE propagation mode

Author : Sjoerd A.L. de Ridder, B.Sc.

Date : August 2007Professor : prof.dr.ir. C.P.A. WapenaarSupervisor : dr.ir. E.C. SlobTA Report number : AES/TG/07-27

Postal Address : Section of Earth SciencesDepartment of Applied Earth SciencesThe Netherlands

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Simulation of InterferometricSeismoelectric Green’s Function

Recovery

For the SH-TE propagation mode

Master of Science Thesis

A thesis submitted to the Department of Geotechnology at Delft

University of Technology in partial fulfillment of the requirements for

the degree of Master of Science in Applied Earth Sciences

Sjoerd A.L. de Ridder

August 2007

Delft University of Technology · Department of Geotechnology

Delft University of Technology

Abstract

A recent novel technique known as seismic interferometry makes use of seismic ‘noise’ toreconstruct a Green’s function between two receivers by crosscorrelation. This technique canbe applied for example in permanent subsurface monitoring using passive seismics or thecreation of virtual sources on positions where only recordings were made. The aim of thisthesis is to derive and understand equations for seismoelectric interferometry.

The first part of this thesis focusses on a the calculation of SH-TE seismoelectrical responses ina 2D horizontally stratified earth. We decompose the two-way wave equation for SH-TE wavesinto upgoing and downgoing waves which we relate through a reflectivity formulation. Thereflectivity formulation is based upon reflection matrices only, even though we can simulateboth reflection and transmission experiments. We solve the SH-TE seismoelectric system ina 1D homogeneous world.

In the second part of this thesis we derive interferometric Green’s fucntion representationsfrom reciprocity theorems that relate two different states in one domain. InterferometricGreen’s function representations express the Green’s function between two receivers as a func-tion of crosscorrelations of responses of sources throughout a domain and on it’s boundary.We cast the seismoelectric system in a general diffusion, flow and wave equation and definea Green’s matrix for all different field and source types. Using this formulation we derivea source-receiver reciprocity relation for the Green’s matrix from the convolution type reci-procity theorem. The correlation type reciprocity theorem for the Green’s matrix is modifiedusing source-receiver reciprocity to obtain the interferometric Green’s function representation.

We study the SH-TE seismoelectrical interferometric representation 1D and 2D in homoge-neous media. A seismoelectric interferometric representation was written to recover the causalresponse of the particle velocity at position B due to a electrical current source at positionA, as a function of cross correlations of electric field recordings at A and particle velocityrecordings at B. Provided there exists a dense coverage of sources in the domain and on it’sboundary, the representation was validated in both 1D and 2D.

Approximations to the interferometric representation are investigated by studying the con-tributions of parts of the domain and boundary integrals. It was found that a dominantspurious event resides in the separate contributions of the domain and boundary integrals,

iv Abstract

that destructively interferes when both contributions are combined. The role of differentsource types in the interferometric representation was studied. In a homogeneous mediumthe measured events have propagated either as an electromagnetic wave or as a shear wave.The dominant contribution to the reconstruction of an electromagnetic event is by electro-magnetic sources. Similarly, can the reconstructed shear wave event be attributed mainly dueto seismic sources. In a medium with low electromagnetic and shear wave losses we couldignore the domain integral, this will result in amplitude errors and we will suffer from spuriousevents.

Acknowledgments

This thesis would never be complete without acknowledging Evert Slob. He has been of greatsupport, both technically and mentally, he is a better adviser than any graduate studentcould ever wish for. I would like to thank professor Kees Wapenaar for suggesting the subjectof seismoelectric interferometry, it has been both challenging and interesting. I am greatfullthat both Kees and Evert expressed their confidence in me by suggesting this topic. ProfessorRoel Snieder at the Colorado School of Mines hosted me for a month’s visit to the Center ofWave Phenomena. There was not a day that passed without a discussion with Roel or oneof the students that made my visit worth while. Thanks to all CWP’s students and faculty,especially Kurang, Fan, Steve and Gabi, it was great to see old friends and make new ones. Ithank my committee, professor Hans Bruining, Deyan Dragonov and Christiaan Schoemaker,for their many suggestions for improvement of this manuscript.

During six years of study, I made many friends with whom I share memories that I deeplycherish. Jelmer, Gerson, Ilse, Babet, Floris, Koen, Mara, Menne, Robert, Cheryl, Wouter,and many more. I can only say that it is my dearest wish, that in 10 years time we we’ll besharing even more happy memories. Tonnie, Anela en Niels, my homies from Huize Barabas,made me feel at home in Utrecht. Thanks to them I knew Utrecht was about more than justthe geo-world. I would like to thank my Peruvian friend Raul, together we worked throughthe Delft curriculum that we both found new, chaotic and difficult. Thanks to my geophysicsfriends in Delft; Raul, Joost, Elmer, Nihed, Christiaan, Karel, and more. I had a great time,not in the least during the DOGS drinks and the EAGE conferences.

Thanks to my friends Merijn and Marijn, our path’s parted but friendship remains. Whenwe meet it feels like it was only yesterday that we graduated from high school. My oldestfriends Jelle and Gody, have always been there for me. Lastly and most importantly, thetwo people whom I never forget what they mean to me; thank you Mom and Dad, for yoursupport, encouragement and example.

Sjoerd de RidderDelft, August 2007.

Table of Contents

Abstract iii

Acknowledgments v

List of Symbols xiii

1 Introduction 1

1-1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1-2 Seismoelectric Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1-3 Interferometric Green’s function recovery . . . . . . . . . . . . . . . . . . . . . . 2

1-4 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1-5 Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1-5-1 Fourier transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1-5-2 Transposed and adjoined operators . . . . . . . . . . . . . . . . . . . . . 5

2 Seismoelectric system of equations 7

2-1 Pride’s seismoelectric system of equations . . . . . . . . . . . . . . . . . . . . . 7

2-1-1 The coupling coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2-1-2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2-1-3 Isotropic frequency-independent elastic parameters . . . . . . . . . . . . 10

2-1-4 Electromagnetic equations . . . . . . . . . . . . . . . . . . . . . . . . . 10

2-1-5 Isotropic frequency-independent electromagnetic parameters . . . . . . . 11

2-2 General diffusion, flow and wave equation . . . . . . . . . . . . . . . . . . . . . 12

2-2-1 Symmetry properties of the general diffusion, flow and wave equation . . 14

2-3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2-3-1 Boundary conditions at the pressure-free surface . . . . . . . . . . . . . . 15

viii Table of Contents

I SH-TE Seismoelectrical Modeling in 2D 17

3 The seismoelectric two- and one-way wave equations 19

3-1 Two-way wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3-1-1 Symmetry properties of the two-way system matrix . . . . . . . . . . . . 23

3-2 Decoupling of the SH-TE and P-SV-TM systems . . . . . . . . . . . . . . . . . 23

3-3 One-way wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3-3-1 Homogeneous source-free subdomain . . . . . . . . . . . . . . . . . . . . 26

3-3-2 Symmetry properties of the one-way system matrix . . . . . . . . . . . . 27

3-4 Seismoelectric decomposition for the SH-TE system . . . . . . . . . . . . . . . . 27

3-5 The seismoelectric system in vacuum . . . . . . . . . . . . . . . . . . . . . . . . 29

3-5-1 Diagonalization in a homogeneous source-free subdomain . . . . . . . . . 29

4 Seismoelectric modeling using reflection formalism. 31

4-1 Local and global reflection matrices . . . . . . . . . . . . . . . . . . . . . . . . 31

4-2 Calculation of global reflection matrices . . . . . . . . . . . . . . . . . . . . . . 33

4-3 Homogeneous bounded subdomain with sources . . . . . . . . . . . . . . . . . . 35

4-4 Determination of the wave fields outside the source layer . . . . . . . . . . . . . 36

4-5 Scattering matrix against a pressure-free surface . . . . . . . . . . . . . . . . . . 37

4-6 Seismoelectric modeling scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Examples of seismoelectric simulations in 2D 39

5-1 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5-1-1 Discrete Fourier transformations . . . . . . . . . . . . . . . . . . . . . . 39

5-1-2 Causality trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5-2 Medium parameters for medium type A and B . . . . . . . . . . . . . . . . . . . 42

5-2-1 Reflection matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5-3 Reflection and transmission experiments . . . . . . . . . . . . . . . . . . . . . . 46

II Interferometric Seismoelectric Green’s Function Recovery 53

6 Theory of interferometric Green’s function recovery 55

6-1 Matrix vector representation of the divergence theorem of Gauss . . . . . . . . . 55

6-2 Reciprocity theorem of the convolution type . . . . . . . . . . . . . . . . . . . . 57

6-2-1 Source-receiver reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . 57

6-3 Reciprocity theorem of the correlation type . . . . . . . . . . . . . . . . . . . . . 58

6-4 Interferometric Green’s function representations in 3D . . . . . . . . . . . . . . . 59

6-5 Interferometric Green’s function representations in 2D and 1D . . . . . . . . . . 60

Table of Contents ix

7 SH-TE interferometric Green’s function representations in 1D and 2D 63

7-1 SH-TE seismoelectric coupling in 1D . . . . . . . . . . . . . . . . . . . . . . . . 63

7-2 Convolution-type reciprocity theorem for SH-TE in 1D . . . . . . . . . . . . . . 64

7-3 Correlation-type reciprocity theorem for SH-TE in 1D . . . . . . . . . . . . . . . 65

7-4 Seismoelectric interferometric Green’s function recovery in 1D . . . . . . . . . . 66

7-4-1 Interferometric representation in medium type A and B . . . . . . . . . . 67

7-4-2 Relation between the two retrieved Green’s function matrices . . . . . . . 67

7-5 SH-TE seismoelectric coupling in 2D . . . . . . . . . . . . . . . . . . . . . . . . 68

7-6 Convolution-type reciprocity theorem for SH-TE in 2D . . . . . . . . . . . . . . 69

7-7 Correlation-type reciprocity theorem for SH-TE in 2D . . . . . . . . . . . . . . . 70

7-8 Seismoelectric interferometric Green’s function recovery in 2D . . . . . . . . . . 71

7-8-1 Interferometric representation in medium type A and B . . . . . . . . . . 73

7-8-2 Relation between the two retrieved Green’s function matrices . . . . . . . 73

8 Simulation of interferometric seismoelectric Green’s function recovery 75

8-1 1D Seismoelectric interferometry in homogeneous media. . . . . . . . . . . . . . 75

8-1-1 Results in medium type A . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8-1-2 Results in medium type B . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8-1-3 Middle Riemann sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8-2 Dissecting the 1D interferometric representation . . . . . . . . . . . . . . . . . . 80

8-2-1 Main contributions of the interferometric representation. . . . . . . . . . 81

8-2-2 Correlation gather of the domain integral . . . . . . . . . . . . . . . . . 85

8-2-3 Alternative boundary positions . . . . . . . . . . . . . . . . . . . . . . . 86

8-3 2D Seismoelectric interferometry in a homogeneous medium . . . . . . . . . . . 88

9 Discussion and conclusions 97

9-1 Forward modeling in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

9-2 Seismoelectric interferometric Green’s function representation . . . . . . . . . . . 97

9-3 Numerical evaluation of the Seismoelectric interferometric Green’s function representation 98

9-4 Domain versus boundary integral . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9-4-1 Main contributions to the retrieved result . . . . . . . . . . . . . . . . . 99

9-4-2 Stationary phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9-5 2D versus 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

9-6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

References 103

A Derivation of the seismoelectric two-way wave equation 107

x Table of Contents

B SH-TE Decomposition 113

B-1 Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B-2 Non-zero eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

B-3 Eigenvectors of SH-TE system . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B-3-1 Stability of the scaling terms . . . . . . . . . . . . . . . . . . . . . . . . 118

B-4 The decoupled SH-TE systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

B-4-1 Decoupled SH waves in porous media . . . . . . . . . . . . . . . . . . . 118

B-4-2 Decoupled TE waves in vacuum . . . . . . . . . . . . . . . . . . . . . . 120

C SH-TE general diffusion, flow and wave equation in 2D and 1D 123

C-1 SH-TE general diffusion, flow and wave equation in 2D . . . . . . . . . . . . . . 123

C-2 SH-TE general diffusion, flow and wave equation in 1D . . . . . . . . . . . . . . 125

C-3 Solution to the 1D SH-TE seismoelectric system in a homogeneous domain . . . 125

C-3-1 Green’s matrix for the 1D SH-TE system . . . . . . . . . . . . . . . . . 126

List of Figures

2-1 Interface between two media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4-1 Annotations in a layered medium. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4-2 Calculation scheme of the responses of a source buried in a layered medium. . . . 38

5-1 Wrapping effects in (a) and the results of applying the causality trick in (b). . . . 41

5-2 Exponentials in the causality trick. . . . . . . . . . . . . . . . . . . . . . . . . . 42

5-3 Real and imaginary parts of the seismoelectrical wave velocities. . . . . . . . . . 43

5-4 Loss functions of the SH-TE system. . . . . . . . . . . . . . . . . . . . . . . . . 43

5-5 Reflection matrix, from layer type A against B. . . . . . . . . . . . . . . . . . . 44

5-6 Reflection matrix, from layer type B against A. . . . . . . . . . . . . . . . . . . 45

5-7 Pressure-free surface scattering matrix, from layer type A against vacuum. . . . . 45

5-8 Experiment geometry of a transmission experiment in a homogeneous medium. . 47

5-9 Transmission experiment in a homogeneous medium: f2 → E2. . . . . . . . . . . 47

5-10 Transmission experiment in a homogeneous medium: −Js,e2 → υs

2. . . . . . . . . 48

5-11 Experiment geometry of a reflection experiment above an aquifer. . . . . . . . . 49

5-12 Reflection experiment above an aquifer. . . . . . . . . . . . . . . . . . . . . . . 49

5-13 Experiment geometry of a transmission experiment in an aquifer. . . . . . . . . . 51

5-14 Transmission experiment in an aquifer. . . . . . . . . . . . . . . . . . . . . . . . 51

6-1 Scalar field a(x) in domain D bounded by ∂D. . . . . . . . . . . . . . . . . . . . 56

6-2 Matrix field {a bt}(x) in domain D bounded by ∂D. . . . . . . . . . . . . . . . 57

6-3 The character of two physical states for the convolution type reciprocity theorem. 57

6-4 The character of two physical states for the correlation-type reciprocity theorem. 58

6-5 Geometry for the 3D interferometric Green’s function representation. . . . . . . . 59

6-6 Geometry for the 1D interferometric Green’s function representation. . . . . . . . 60

xii List of Figures

8-1 Symmetric position of receivers and boundary points for 1D interferometry. . . . 76

8-2 Exact and retrieved superposition of Gυ,Je(t) and GE,f (−t) in medium type A. . 76

8-3 Separate contributions of the domain integral and the boundary points. . . . . . 77

8-4 Exact and retrieved superposition of Gυ,Je(t) and GE,f (−t) in medium type B. . 79

8-5 Separate contributions of the domain integral and the boundary points. . . . . . 79

8-6 Recovered signal using alternative sample densities of the Riemann sum. . . . . . 80

8-7 Different segments of the domain integral in the 1D interferometric geometry. . . 81

8-8 The separate contributions of the sources at x3;1 and the sources at x3;2. . . . . 82

8-9 The contributions of the three segments of the domain integral. . . . . . . . . . 82

8-10 Contributions from the different crosscorrelation terms at the boundary point x3;1. 83

8-11 Contributions from the different crosscorrelation terms at the boundary point x3;2. 84

8-12 Contributions from the different crosscorrelations in the domain integral. . . . . . 84

8-13 Correlation gather for seismoelectric interferometry in a 1D homogeneous medium. 85

8-14 Logarithm of the absolute value of the correlation gather in Figure 8-13. . . . . . 86

8-15 Alternative position of receivers and boundary points for 1D interferometry. . . . 87

8-16 Reconstructed signal using two alternative positions of the boundary points. . . . 87

8-17 Geometry for a 2D seismoelectric interferometric experiment. . . . . . . . . . . . 88

8-18 Exact and retrieved superposition of Gυ,Je(t) and GE,f (−t) in 2D. . . . . . . . . 89

8-19 Separate contributions of the domain and boundary integrals. . . . . . . . . . . . 90

8-20 Source contributions with depth to the recovered signal. . . . . . . . . . . . . . 90

8-21 A close look at the time window t = ±0.1 seconds of Figure 8-20. . . . . . . . . 91

8-22 Source contributions with horizontal distance to the recovered signal. . . . . . . 91

8-23 A close look at the time window t = ±0.1 seconds of Figure 8-22. . . . . . . . . 92

8-24 Correlation gather of a horizontal line of sources at x3 = 150.5 meter. . . . . . . 94

8-25 Correlation gather of a horizontal line of sources at x3 = 449.5 meter. . . . . . . 94

8-26 Correlation gather of a vertical line of sources at x1 = −800 meter. . . . . . . . 95



B-1 The nummerical behavior of the ξ scaling terms in the SH-TE eigenvectors. . . . 119

List of Symbols

E : Bulk-averaged electric field strength [V m−1] 2-1-4H : Bulk-averaged magnetic field strength [A m−1] 2-1-4vs : Phase-averaged solid particle velocity [m s−1] 2-1-2vf : Phase-averaged fluid particle velocity [m s−1] 2-1-2w : Bulk-averaged Biot filtration velocity [m s−1] 2-1-2τ b : Bulk-averaged stress [N m−2] 2-1-2pf : Phase-averaged fluid pressure [N m−2] 2-1-2

Js,e : Density of external electric current source [A m−2] 2-1-4Js,m : Density of external magnetic current source [kg s−3A−1] 2-1-4ff : Density of external force applied to fluid phase [N m−3] 2-1-2f b : Density of external force applied to the bulk [N m−3] 2-1-2hb : Density of external deformation rate on the bulk [s−1] 2-1-2qi : Density of volume injection rate in fluid phase [s−1] 2-1-2

ρf : Density of the fluid phase [kg m−3] 2-1-2ρs : Density of the solid phase [kg m−3] 2-1-2ρb : Density of the bulk [kg m−3] 2-1-2Kf : Bulk modulus of fluid phase [N m−2] 2-1-3Ks : Bulk modulus of solid phase [N m−2] 2-1-3Kfr : Bulk modulus of the framework of grains [N m−2] 2-1-3KG : Grassman’s bulk modulus [N m−2] 2-1-3N : Shear modulus of the framework of grains [N m−2] 2-1-3c : Stiffness parameter of the porous solid [N m−2] 2-1-3M : Stiffness parameter of the porous solid [N m−2] 2-1-3d : Stiffness parameter of the porous solid [N m−2] 2-1-3ρE : Effective fluid density [kg m−3 s−1] 2-1-2ρc : Effective bulk density [kg m−3] 2-1-2

L : Dynamic seismoelectric coupling coefficient [m2 s V−1] 2-1-1

xiv List of Symbols

L0 : Static seismoelectric coupling coefficient [m2 s V−1] 2-1-1α∞ : Tortuosity [-] 2-1-1Λ : Volume to surface ration of porous material [m] 2-1-1φ : Porosity as a volume fraction [-] 2-1-1m : Similarity parameter for porous media [-] 2-1-1η : Pore fluid viscosity [N s m−2] 2-1-1δη : Viscous skin depth [m] 2-1-5k : Dynamic permeability [m2] 2-1-1k0 : Static permeability [m2] 2-1-1dl : Debye length [m] 2-1-1ωc : Critical frequency [Radians] 2-1-1ζ : Zeta potential of the double layer [V] 2-1-1C : Electrolyte concentration in the pore fluid [mol liter−1] 2-1-1Nl : Bulk ionic concentration of species l [mol liter−1] 2-1-1zl : Ion valency of ion species l [-] 2-1-1bl : Ion Mobility of ion species l [m s−1N−1] 2-1-5e : Elementary charge

(

1.60217733 · 10−19)

[C] 2-1-1NA : Avogadro’s Number

(

6.022 · 1023)

[-] 2-1-1kb : Boltzmann constant

(

1.380658 · 10−23)

[J K−1] 2-1-1T : Temperature [K] 2-1-1P0 : Parameter in osmotic conductivity [-] 2-1-5

ǫ : Bulk dielectric permittivity [F m−1] 2-1-5µ : Bulk magnetic permeability [H m−1] 2-1-5σe : Bulk electric conductivity [S m−1] 2-1-5

σm : Bulk magnetic conductivity [(S m)−1] 2-1-5ǫ0 : Dielectric permittivity of the vacuum

(

8.854 · 10−12)

[F m−1] 2-1-5

ǫfr : Relative dielectric permittivity of the fluid phase [-] 2-1-5

ǫsr : Relative dielectric permittivity of the solid phase [-] 2-1-5

µ0 : Magnetic permeability of the vacuum(

4π · 10−7)

[H m−1] 2-1-5σf : Conductivity of the fluid phase [S m−1] 2-1-5σem : Double layer electron migration conductivity [S m−1] 2-1-5σos : Osmotic conductivity of the streaming current [S m−1] 2-1-5ε : Effective electrical permittivity [F m−1] 3-1σ : Effective electrical conductivity [S m−1] 3-1εL : Effective electrical permittivity including coupling [F m−1] 3-1σL : Effective electrical conductivity including coupling [S m−1] 2-2

Chapter 1

Introduction

1-1 Motivation

This thesis was sparked by the publication of a generalized theory for Greens function retrievalby crosscorrelation [Wapenaar et al., 2006] for arbitrary diffusion, flow and wave systems.Thus predicting the possibility of retrieving a seismoelectric Green’s function by crosscorre-lating the measurements of the particle velocity at a receiver station A with the measurementsof the electrical field at a receiver station B. This thesis aims to derive and validate the in-terferometric seismoelectric Green’s function representation to understand what seismic andelectromagnetic background noise is needed for the interferometry to deliver the desired seis-moelectric response.

Conventional geophysical techniques aim to obtain a image of the subsurface. Althoughthey succeed in obtaining an image of a wide range of subsurface properties, they providevery little information on hydraulic permeability. Which has been the motivation to performseismoelectric surveys for many authors, who obtained different levels of success. This thesisprovides an alternative way to obtain the seismoelectric response, using alternative positions ofthe source, or using the natural seismic and electromagnetic sources in the subsurface. Usingthis method, we could benefit from a better signal to noise ratio by simply recording for alonger time period. But we likely suffer from a imperfect source distribution. Understandingthe seismoelectric interferometric representation Green’s function representation is a key stepin the development of experiments of seismoelectric interferometry.

The seismoelectric system of Pride [1994] combines Biot’s theory of wave propagation throughsaturated porous media with Maxwell’s equations for the electromagnetic fields. Seismoelec-tric interferometry is an extreme application of Greens function retrieval by crosscorrelation.proper understanding of the seismoelectric interferometric Green’s function representationwill increase our understanding of the boundary conditions for Greens function retrieval bycrosscorrelation.

2 Introduction

1-2 Seismoelectric Phenomena

The first reported observation of seismoelectric phenomena was in 1936 by R.R. Thompsonin the first issue of Geophysics. Thompson [1936, 1939], reported on electrical conductivitychanges upon the passing of a seismic wave. Shortly afterward, Ivanov [1939] reported on themeasurement of a electromagnetic field on the passing of a seismic wave, which later becameknown as the co-seismic electrical field. Frenkel [1944] was the first to attempt to provide atheoretical framework for seismoelectric phenomena. A major corner stone for the derivationof seismoelectric theory for porous saturated media was given by Biot’s his theory for wavepropagation in saturated porous media, [Biot, 1956], [Biot, 1956]. It was until 1994 thatS.R. Pride derived a self-consistent system that describes seismoelectric wave phenomena insaturated porous media. Pride [1994] combined Biot’s theory of wave propagation throughsaturated porous media with Maxwell’s equations for the electromagnetic fields.

1-3 Interferometric Green’s function recovery

The principle for Greens function retrieval by crosscorrelation was first derived by Claerbout[1968], who showed that the reflection response of a horizontally layered medium can besynthesized from the autocorrelation of its transmission response. Only little was reportedon the subject until Weaver and Lobkis [Weaver and Lobkis, 2002] showed how the Green’sfunction emerges in the cross-correlation of diffuse wave-fields. The conjecture of Claerboutthat his relation holds for any arbitrary medium in 3D was confirmed by Wapenaar [2004], whoderived the interferometry relation from reciprocity theorems. The derivation from reciprocitytheorems does not require the wave-field to be diffusive. Another derivation is based uponstationary phase [Snieder, 2004b], showing that certain sources are more important thanother sources. Recent work has shown that the derivation from reciprocity theorems holdsfor for situations where time-reversal invariance does not hold, as for electromagnetic wavesin conducting media [Slob et al., 2007], acoustic waves in attenuating media [Snieder, 2007],or general scalar diffusion phenomena [Snieder, 2006]. Theses developments eventually leadto the derivation of interferometric Green’s function representations for arbitrary diffusion,flow and wave phenomena, [Wapenaar et al., 2006], predicting relations for seismoelectricinterferometry.

1-4 Outline of this thesis

In Chapter 2 we introduce the seismoelectric system of equations and compile them into ageneral matrix-vector equation. In the first chapter of Part I we provide the seismoelectrictwo-way wave equation together with composition and decomposition matrices to decomposethe SH-TE fields into up and down going waves. A reflection formalism is derived in Chapter4 that can be used to compute the responses of sources buried in layered media. Several ex-amples of these simulations are given in Chapter 5. Part II focuses on interferometric Green’sfunction representations and a numerical study of seismoelectric interferometric Green’s func-tion representations. The general theory is introduced in Chapter 6, written in an abstractformulation using the general matrix-vector equation that was introduced in Chapter 2. In

1-5 Notations and definitions 3

Chapter 7 we expand interferometric Green’s function representations for the SH-TE seismo-electric system in 1D and 2D. The anatomy of these representations is studied in Chapter 8,we focused on understanding the complications that arise from ignoring several contributionsin these representations. A discussion and conclusions follow in Chapter 9.

1-5 Notations and definitions

In this thesis, we use a large number of variables and parameters in addition to multiplegeneral vector-matrix equations for the seismoelectric system of equations. Therefore it isimportant to use some general notation conventions.

A position in space is denotes by a triplet of Cartesian coordinates. The position vector isdenoted by x and the vector elements contain the coordinates of the point in space, x =(x1, x2, x3)

t, defined in a right-handed coordinate system and t means matrix transposition(see Section 1-5-2 below), we use xH for the horizontal coordinates x1 and x2. Spatialderivatives are denoted by ∂i, in which the subscript i specifies the coordinate, xi with respectto which the partial differentiation is taken. The time coordinate is denoted by t and ∂t meanstemporal differentiation.

A vector can be written as v = vkik, where the summation convention applies to the repeatedsubscript k and where ik is a unit base vector of the Cartesian reference frame and thesubscript k indicates one of the three base vector directions. Hence, v1 is the i1 componentof the vector v.

Possible additional super- and subscripts are added to denote, for example, a state in reci-procity theorems (A or B), medium phases, when used above a field or source quantity, orsymbols that are added for differentiation between previously defined variables and param-eters. We have the x1 component of the particle velocity in the solid in state A as υs

1,A. Aunderlined variable or matrix denotes a slightly alternative definition from its original version,for example in the system matrices A and A in Section 2-2 and in the source terms f

2and

Je2 defined in Section 7-1 and Appendix B.

We use lower case Latin symbols for variables, except for the electric and magnetic fieldswere we use the capital letters E and H. Matrices and vectors are written in boldface, werethe context separates between matrices and vectors. Some special vectors are denoted byGreek boldface symbols. Operators (matrices) are written in calligraphic script and so arethe matrices that are derived from operators by Fourier transforms, although strictly in theFourier domain they are no longer operators. An exception is the composition operator L,this is to discriminate from the seismoelectric coupling coefficient L.

The Einstein summation convention is used and applies to repeated subscripts, that denotethe elements of a vector or matrix. Lower-case Latin subscripts take on the values 1, 2 and3, while lower-case Greek subscripts take on the values 1 and 2. Hence

∂iwi stands for3∑

i=1

∂iwi, (1-1)

which is the divergence of w.

4 Introduction

Some exceptions have been made in Chapter 4, where we use z as the vertical coordinate.And we do not perform Einstein summation convention over the subscript n that denotes thelayer number and the subscript w that denotes wave type. Some tricky definitions are thoseof the stiffness parameter d, that is denoted in roman-case if used in the isotropic case wheredij = dδij . This is to discriminate from the notion of an infinitesimal small step size, d, in anintegration variable.

1-5-1 Fourier transformations

We use the following definition for the temporal Fourier transformation

F t{f(x, t)} = f(x, ω) =

∫ ∞

−∞f(x, t)e−iωtdt, (1-2)

where ω is angular frequency, t is time, and i =√−1 is the imaginary unit. The frequency

domain function f(x, ω) is the temporal Fourier transform of the time domain function f(x, t).The inverse temporal Fourier transform is,

(

F t)−1 {f(x, ω)} = f(x, t) =

1

2π

∫ ∞

−∞f(x, ω)eiωtdω. (1-3)

Note that ∂t after temporal Fourier transformation is replaced by iω, assuming zero initialconditions. We use the following definition for the spatial Fourier transformation,

Fs{f(x1, x2, x3, t)} = f(k1, x2, x3, t) =

∫ ∞

−∞f(x1, x2, x3, t)e

ik1x1dx1, (1-4)

where k is a wavenumber, x is a spatial coordinate, and f(x1, x2, x3, t) is the time domainfunction that is transformed to the wavenumber domain function f(k1, x2, x3, t). Note thedifferent choice of sign in the exponential of the temporal Fourier transformation. The inversespatial Fourier transformation,

(Fs)−1 {f(k1, x2, x3, t)} = f(x1, x2, x3, t) =1

2π

∫ ∞

−∞f(k1, x3, t)e

−ik1x1dk1. (1-5)

The spatial Fourier transformation can repetitively be applied to each spatial coordinate

(Fs)3 {f(x1, x2, x3, t)} =ˇf(k, t). A spatial derivative with respect to the coordinate xi, ∂i is

after spatial Fourier transformation replaced by −iki. In Chapter 2 we transform our spacetime domain functions to the space frequency domain. In Chapter 3, after dropping the x2

dependence we perform another Fourier transformation and go to the horizontal-wavenumberfrequency domain

(

F t,s)2 {f(x1, x3, t)} = f(k1, x3, ω). The interferometric relations of Part

II of the thesis are all derived in the space frequency domain.

In our calculations we only use positive frequencies and make use of relation 1-6, stating thatfor real functions of t we can rewrite the Fourier transformation over the positive frequenciesonly [Bracewell, 2000],

f(x, t) =1

2π

∫ ∞

−∞f(x, ω)eiωtdω = R

(

1

π

∫ ∞

0f(x, ω)eiωtdω

)

. (1-6)

For a discussion on how we discretise the Fourier transformations and some additional aspectsof how we implement this in our calculations, see Sections 5-1-1 and 5-1-2.

1-5 Notations and definitions 5

1-5-2 Transposed and adjoined operators

Throughout the thesis we make a formal separation between scalar functions and operatorfunctions, or more general between matrices containing algebraic factors and matrices con-taining operators. We use a lower-case superscript t to denote simple matrix transposition,while we use an upper-case superscript T to denote operator matrix transposition. If wedenote a complex conjugation for a scalar function or elements of a matrix we use an astrix∗. When we transpose a matrix and take the complex conjugate of its elements, we use asuperscript H and name it the Hermitian. When we mean complex conjugation and operatormatrix transposition we use a superscript †, and name it the adjoint. In what follows weintroduce the transposed and adjoint of an operator (matrix).

For two vector functions f(xH) and g(xH) of the horizontal coordinates xH we define thebilinear form as

〈f ,g〉b =

∫

S

fT (xH)g(xH)d2xH , (1-7)

and the sesquilinear form as

〈f ,g〉s =

∫

S

f †(xH)g(xH)d2xH . (1-8)

Consider an operator matrix U , we introduce the transposed operator matrix UT via

〈Uf ,g〉b =⟨

f , UTg⟩

b(1-9)

and the adjoint operator matrix U† via

〈Uf ,g〉s =⟨

f , U†g⟩

s. (1-10)

Definitions 1-7 and 1-8 are through an integration over the complete horizontal coordinates,hence transposed and adjoint operators are not defined for an operator acting on the time orvertical-space coordinates. An operator matrix is called symmetric when it obeys

UT = U (1-11)

and it is self-adjoint whenU

† = U . (1-12)

For an operator matrix

U =

(

U11 U12

U21 U22

)

, (1-13)

it follows from relation 1-9 above that

UT =

(

UT11 UT

21

UT12 UT

22

)

. (1-14)

Thus UT is a transposed matrix, containing transposed operators. Through definition 1-10,we have for the adjoint operator matrix U†,

U† =

( (

UT11

)∗ (

UT21

)∗(

UT12

)∗ (

UT22

)∗

)

=(

UT)∗

. (1-15)

Note that the use of T on a matrix that does not contain operators is equal to the use of t,and similarly for † and H .

6 Introduction

Chapter 2

Seismoelectric system of equations

In this Chapter the seismoelectric equations are presented and a general diffusion, flow andwave equation is presented that captures these equations in 22 by 22 matrices. At the endthe boundary conditions used to solve the system of equations are presented for source-freeinterfaces and for a source level in a homogeneous subdomain.

The equations in this Chapter are applicable for 3D seismoelectric wavefieds in inhomogeneousanisotropic prorous media. Nevertheless, some parameters are only specified for isotropicmedia. We use the boundary conditions in Part I to derive a scattering formalism that we usefor seismoelectric modeling in isotropic horizontally-stratified media. And we use the generaldiffusion, flow and wave equation in Part II to derive reciprocity relations and interferometricGreen’s function representations for seismoelectric waves.

2-1 Pride’s seismoelectric system of equations

In this Section we introduce the seismoelectric system of equations as derived by Pride [1994].Pride derived the system starting from Maxwell’s equations for the electromagnetic field andBiot’s equations of elastodynamic waves in porous media. He performed volume averaging ofthe governing equations in a porous medium with coupling through an electric double layersystem around the grains. The final form of the equations take the form of Biot’s equationslinearly coupled with Maxwell’s equations through a coupling coefficient L.

The system of Pride has been derived making numerous assumptions. The most importantones are summarized here. The fluid is assumed to be an ideal electrolyte. Both the solidgrains and all the macroscopic constitutive laws are assumed to be isotropic. In the literatureit is generally assumed that this assumption can be relaxed to include anisotropic macroscopicconstitutive parameters. The frequency dependency of the dynamic permeability and thecoupling coefficient are smooth functions and postulated between continuous current and f∞.Under the assumptions that at the pore and grain scale the dielectric constant of the grainsis much smaller than that of the electrolyte and the thickness of the double layer is muchsmaller than the radii of the curvature of the solid grains, all wave induced diffusion effects are

8 Seismoelectric system of equations

ignored. Scattering at the pore and grain scale is not accounted for, so the highest frequencyof mechanic waves that can be considered is in the order of 106 Hz. No piezoelectric effectsare considered and Lorentz currents are neglected. Only linear disturbances are considered.

2-1-1 The coupling coefficient

The coupling coefficient L is given in the frequency domain by

L = L0

1 + iω

ωc

m

4

(

1 − 2dl

Λ

)2

1 − dl

√

iωρf

η

2

− 12

, (2-1)

where the static coupling coefficient L0 is defined as

L0 = − φ

α∞

ǫ0ǫfζ

η

(

1 − 2dl

Λ

)

. (2-2)

The coupling coefficient is a frequency dependent function of tortuosity α∞, volume-to-surfaceratio of a porous material Λ, fluid relative dielectric permittivity ǫ

fr , dielectric permittivity

in vacuum ǫ0, pore fluid viscosity η, Debye length dl, critical frequency ωc, the density of thefluid phase ρf and the zeta potential of the double layer ζ. Pride and Morgan [1991] havecollected the datasets of previous researchers who determined the streaming potential, or zetapotential, in quartz systems and found

ζ = 8 · 10−3 + 26 · 10−3log10C , (2-3)

were C is the electrolyte concentration in the pore fluid. The assumption that we considerthe fluid to be an ideal electrolyte restricts 10−6 < C < 100 [mol/liter]. The Debye lengthis the characteristic length of the region of influence of the charge of mobile carriers. For anelectrolyte containing L ion species, the Debye length dl is given by

1

(dl)2 =

L∑

i=1

(ezi)2 Ni

ǫ0ǫfkbT, (2-4)

where Ni is the bulk-ionic concentration of species i, zi are the ion valences, e is the elementarycharge, kb is the Boltzmann constant and T is the temperature, we use T = 323K. The bulkionic concentration of ion species i is calculated using

Ni = 103C NA abs(z′i), (2-5)

where z′i is the valency of the conjugate ion. The critical frequency separates low-frequencyviscous flow and high-frequency inertial flow, and is defined as

ωc =φη

α∞k0ρf, (2-6)

were k0 is the static permeability in [m2], and φ is the porosity as a volume fraction [-]. Thisgives us a total of three pore geometry parameters α∞, Λ and k0 that characterize the porous

2-1 Pride’s seismoelectric system of equations 9

material. A parameter, derived from those, is the similarity parameter m [Johnson et al.,1987]

m =φΛ2

α∞k0. (2-7)

Under the assumption that there exists only one scaling function for all porous media, m hasbeen fixed to m = 8 [Johnson, 1989]. We use a Carmen-Kozeny relationship which relates thestatic permeability k0 to the porosity φ cubed, which is representative for clean Fontainebleausandstones [Bourbie et al., 1987],

k0 = 2 · 10−11φ3. (2-8)

The tortuosity α∞ remains the last free pore geometry parameter. In the general anisotropiccase the coupling coefficient is a tensor. Onsager reciprocity is satisfied under the thin doublelayer assumption [Pride, 1994] and we have L = Lt. This means that the coupling coefficientin Biot’s equations is equal to the coupling coefficient in Maxwell’s equations.

2-1-2 Equations of motion

In Pride’s system, the equations of motion are Biot’s equations of motion in porous mediaincluding a coupling term to the electric field. For an arbitrary inhomogeneous anisotropicmedium they are given by

iωρbvs + iωρf w − ∂j τbj = f b, (2-9)

iωρf vs + ηk−1(w − LE) + ∇pf = ff , (2-10)

where w is the Biot filtration velocity w = φ(vf − vs), vs and vf are the averaged solidand fluid particle velocities, E is the averaged electric field strength, τ b the averaged bulkstress and pf is the averaged pressure in the fluid. The source terms ff and f b are thevolume densities of external force applied to the fluid phase and bulk phase, respectively. Theconstitutive parameters ρf , ρs and ρb are anisotropic frequency-dependent density tensors,

and we have ρb = (1 − φ) ρs + φρf . We assume that ρf ={

ρf}t

and ρs = {ρs}t. Which is,

for example, the case when the anisotropy is the result of parallel fine layering at a scale muchsmaller than the wavelength [Schoenberg and Sen, 1983]. The complex frequency-dependent

tensor k is the dynamic permeability tensor of the porous material, with k ={

k}t

. The

stress-strain relations read

− iωτ bj + cjl∂lv

s + dj∇tw = cjlhbl + dj q

i, (2-11)

iωpf + dtl∂lv

s + M∇tw = dtlh

bl + M qi, (2-12)

where hb is the density of external deformation rate on the bulk and qi is the density ofvolume injection rate in the fluid phase, 0 is a 3 × 1 vector of zeros. Our notation has

τ bj =

τ b1j

τ b2j

τ b3j

, hj =

hb1j

hb2j

hb3j

, dj =

d1j

d2j

d3j

and (2-13)


cjl =

c1j1l c1j2l c1j3l

c2j1l c2j2l c2j3l

c3j1l c3j2l c3j3l

. (2-14)

with τ b ={

τ b}t

and hb ={

hb}t

. The stiffness parameters of the porous solid are M ,

d ={

d}t

and cijkl = cjikl = cijlk = cklij .

2-1-3 Isotropic frequency-independent elastic parameters

In this thesis we assume the elastic media parameters to be isotropic. We also assume cer-tain elastic parameters to be frequency independent. For the frequency-dependent dynamicpermeability we assume kij = δij k, with

k = k0

[

(

1 + iω

ωc

4

m

) 12

+ iω

ωc

]−1

, (2-15)

where the static permeability k0 is given in equation 2-8. The frequency-independent stiffnesstensors are dij = δijd and

cijkl = V δijδkl + N (δikδjl + δilδjk) (2-16)

where

V =

(

KG − 2

3N

)

. (2-17)

The frequency-independent shear modulus of the framework of grains, N , is the shear mod-ulus as if the fluid is absent. According to Pride [1994] the frequency-independent elasticparameters KG, d and M are given by,

KG =Kfr + φKf + (1 + φ) Ks∆

1 + ∆, (2-18)

d =Kf + Ks∆

1 + ∆, (2-19)

M =1

φ

Kf

1 + ∆, (2-20)

∆ =Kf

φ (Ks)2

(

(1 − φ) Ks − Kfr)

. (2-21)

where Ks, Kf and Kfr are the frequency-independent compression moduli of the solid, fluidand the framework of the grains, respectively. We also assume isotropic and frequency-independent densities, ρ

fij = δijρ

f , ρsij = δijρ

s and thus ρbij = δijρ

b.

2-1-4 Electromagnetic equations

The other half of Pride’s system are given by Maxwell’s equations with a coupling coefficientto Biot’s equation of motion in porous media. For an arbitrary inhomogeneous anisotropic

2-1 Pride’s seismoelectric system of equations 11

medium Maxwell’s equations are given by

iωǫE + Je −∇× H = −Js,e, (2-22)

iωµH + Jm + ∇× E = −Js,m, (2-23)

with the electric and magnetic current densities given by

Je =(

σe − ηLk−1L

)

E + ηLk−1w, (2-24)

Jm = σmH, (2-25)

where E and H are the averaged electric and magnetic field strengths, Js,e and Js,m arethe external electric and magnetic current source densities. The constitutive parameters σe

and σm are the frequency-dependent electric and magnetic conductivities, ǫ and µ are real-valued frequency-independent dielectric permittivity and magnetic permeability. We haveσe = {σe}t, σm = {σm}t, ǫ = {ǫ}t and µ = {µ}t. Note that all possible relaxationmechanisms are captured in the electric and magnetic conductivity tensors.

2-1-5 Isotropic frequency-independent electromagnetic parameters

We assume the electromagnetic media parameters to be isotropic. The choice for frequency-independent dielectric permittivity and magnetic permeability does not imply any loss ofgenerality, because all relaxation mechanisms are captured in the frequency-dependent elec-trical conductivity and magnetic permeability. We have σe

ij = σeδij , σmij = δij σ

m, ǫij = δijǫ

and µij = δijµ. We neglect magnetic relaxation losses, σm = 0, and approximate the magneticpermeability of the subsurface by the permeability of vacuum µ0. We have

µ = µ0. (2-26)

We calculate the dielectric permittivity according to Pride [1994]

ǫ = ǫ0

(

φ

α∞

(

ǫfr − ǫs

r

)

+ ǫsr

)

, (2-27)

where ǫ0 and µ0 are the electric permittivity and magnetic permeability of the vacuum. And ǫfr

and ǫsr are the fluid and solid relative dielectric constants. The electrical conductivity is com-

posed from the frequency-independent pore-fluid conductivity σf , the frequency-independentdouble layer electron migration conductivity σem and the frequency-dependent osmotic con-ductivity of the streaming current σos. From Pride [1994] we have

σe =φσf

α∞

(

1 +2 (σem + σos)

σfΛ

)

, (2-28)

where we use,

σf =L∑

i=1

(ezi)2 biNi, (2-29)

σem ≈ 2dl

L∑

i=1

(ezi)2 biNi

[

exp

(

− eziζ

2kbT

)

− 1

]

, (2-30)

σos =

(

ǫ0ǫf)2

ζ2

2dlηP0

(

1 − 2

P0

dl

δη

)−1

, (2-31)


where bi are the ion-mobilities and the dimensionless parameter P0 is defined as

P0 =8kbT{dl}2

ǫ0ǫfζ2

L∑

i=1

Ni

[

exp

(

− eziζ

2kbT

)

− 1

]

, (2-32)

and the viscous skin depth δη is defined as

δη =

√

η

iωρf. (2-33)

2-2 General diffusion, flow and wave equation

Following Wapenaar and Fokkema [2004] we capture the seismoelectric system in a generaldiffusion, flow and wave equation. For non-flowing media this equation reads in the frequencydomain as

iωAu + Bu + Dxu = s, (2-34)

where u is a vector containing space and frequency-dependent field quantities, s is a vectorcontaining the source functions, A and B are matrices containing space-dependent materialparameters, and Dx is a matrix containing the spatial differential operators. The iω factorin the first term is the frequency representation of a temporal derivative.

The matrices A, B, Dx the field vector u and the source vector s can be defined such thatthe system represents diffusion processes, acoustic wave propagation, momentum transport,elastodynamic wave propagation, electromagnetic wave propagation or coupled elastodynamicand electromagnetic wave propagation in porous solids. The different systems have differentsizes, the seismoelectric system consists of 22 by 22 matrices and 22 by 1 field and sourcevectors.

Substituting the electromagnetic constitutive relations 2-24 and 2-25 into Maxwell’s equations2-22, 2-23 yields,

iωǫE +(

σe − ηLk−1L

)

E + Lηk−1w −∇× H = −Js,e, (2-35)

iωµH + σmH + ∇× E = −Js,m. (2-36)

We define a conductivity that includes the contribution of the coupling coefficient as σeL =

σe −ηLk−1L. Equations 2-9, 2-10, 2-11, 2-12, 2-35, 2-36 can now be combined to fit into thefollowing form of the general diffusion, flow and wave equation

iω ˆAu + ˆBu + CDxu = C s (2-37)

where

ut =(

Et, Ht, {vs}t,−{τ b1}t,−{τ b

2}t,−{τ b3}t,w, pf

)

, (2-38)

st =(

−{Js,e}t,−{Js,m}t, {f b}t, {hb1}t, {hb

2}t, {hb3}t, {ff}t, qi

)

. (2-39)

2-2 General diffusion, flow and wave equation 13

The matrices A, B, C, Dx are defined as

ˆA =

ǫ O O O O O O 0

O µ O O O O O 0

O O ρb O O O ρf 0

O O O I O O O 0

O O O O I O O 0

O O O O O I O 0

O O {ρf}t O O O O 0

0t 0t 0t 0t 0t 0t 0t 1

, (2-40)

ˆB =

σeL O O O O O ηLk−1 0

O σm O O O O O 0

O O O O O O O 0

O O O O O O O 0

O O O O O O O 0

O O O O O O O 0

−{ηLk−1}t O O O O O ηk−1 0

0t 0t 0t 0t 0t 0t 0t 0

, (2-41)

C =

I O O O O O O 0

O I O O O O O 0

O O I O O O O 0

O O O c11 c12 c13 O d1

O O O c21 c22 c23 O d2

O O O c31 c32 c33 O d3

O O O O O O I 0

0t 0t 0t dt1 dt

2 dt3 0t M

, (2-42)

Dx =

O Dt0 O O O O O 0

D0 O O O O O O 0

O O O D1 D2 D3 O 0

O O D1 O O O O 0

O O D2 O O O O 0

O O D3 O O O O 0

O O O O O O O ∇

0t 0t 0t 0t 0t 0t ∇t 0

. (2-43)

Where

D0 =

0 −∂3 ∂2

∂3 0 −∂1

−∂2 ∂1 0

, Dj =

∂j 0 00 ∂j 00 0 ∂j

and ∇ =

∂1

∂2

∂3

(2-44)

The empty parts of these matrices are filled by appropriately sized null matrices O and null

vectors 0. Finally, to match equation 2-37 to equation 2-34, we define A = C−1 ˆA and

B = C−1 ˆB.


2-2-1 Symmetry properties of the general diffusion, flow and wave equation

The matrices in the general diffusion, flow and wave equation obey symmetry relations. Usingtransposition symmetry of the material parameters in matrices A and B discussed in previousSections, we find

K0AtK0 = A and K0B

tK0 = B, (2-45)

where K0 is defined by

K0 = diag (−1,1,1,−1,−1,−1,1,−1) . (2-46)

The matrix Dx containing spatial derivative operators obeys

K0DxK0 = −Dx = −Dtx. (2-47)

2-3 Boundary conditions

n

DI 2

DI 1

Figure 2-1: Perfectly welded interface with normal vector n between two media D1 and D2 withdifferent medium properties.

At interfaces were the medium parameters change, we need boundary conditions to solvefor the fields at either side of the medium, see Figure 2-3. In general, an interface canbe non-perfect or contain sources. For simplicity we treat two cases, a perfectly weldedsource free interface and a source level in a homogeneous subdomain. For the boundaryconditions across a source-free interface we have the open-pore boundary conditions fromDeresiewicz and Skalak [1963]. In the time domain we have

Normal and shear stresses in the bulk, τ b3, are continuous.

Normal stress in the fluid phase, pf , is continuous.Normal and horizontal velocities in the solid, vs, are continuous.

Normal component of filtration velocity, w3, is continuous.Tangential electric field, E0, is continuous.

Tangential magnetic field, H0, is continuous.

All these quantities are contained in the field vector q, given by

qt =

(

−(

τ b3

)t, pf , (E0)

t , (vs)t , w3, (H0)t

)

, (2-48)

2-3 Boundary conditions 15

where

τ b3 =

τ b13

τ b23

τ b33

,vs =

υs1

υs2

υs3

,E0 =

(

E1

E2

)

,H0 =

(

H2

−H1

)

(2-49)

We see in Chapter 3, how the components of this vector are decomposed in upgoing anddowngoing waves. In Chapter 4 how we make use of these boundary conditions to derive ourscattering formalism. The boundary condition on the field vector q for source free interfacesat x3,n between upper medium n and lower medium n + 1 can be written as,

limx3↓x3,n

q{n+1}(x3) − limx3↑x3,n

q{n}(x3) = 0. (2-50)

Across a level x3,s containing sources, the field vector q does not remain continuous, butinstead experiences a jump. The jump condition is defined by the source vector d, frombelow the source level qb to above the source level qa,

limx3↓x3,s

qb(x3) − limx3↑x3,s

qa(x3) = d(x3,s). (2-51)

The source vector d is defined in chapter 3.

2-3-1 Boundary conditions at the pressure-free surface

We use vacuum to approximate air in simulations containing the earth’s surface. In vacuumthe normal and shear stresses are zero. To solve for the scattering matrix of a porous mediumagainst vacuum, see Section 4-5, we need a special set of boundary conditions. There are noseismic waves and only electromagnetic waves propagate across the pressure-free surface andin vacuum. At the pressure-free surface we have

Normal and shear stresses in the bulk, τ b3, are zero.

Normal stress in the fluid phase, pf , is zero.Tangential electric field, E0, is continuous.

Tangential magnetic field, H0, is continuous.

Part I

SH-TE Seismoelectrical Modeling in2D

Chapter 3

The seismoelectric two- and one-waywave equations

In this chapter the seismoelectric one- and two-way wave equations are introduced. Weconsider horizontally, piecewise homogeneous, layered isotropic media. In those media thetwo-way wave equation can be rearranged into two independent modes of propagation. Inthe SH-TE mode, the horizontally polarized shear waves couple to the transverse polarizedelectrical fields. In the P-SV-TM mode, the pressure waves and vertically polarized shearwaves are coupled to the transverse polarized magnetic fields. The two-way wave equationequates the vertical variations of the continuous field quantities in q, as a function of thehorizontal derivatives, contained in A, of the continuous field quantities and a source vectord. In the frequency domain the two-way wave equation is given by

∂3q = Aq + d. (3-1)

The one-way wave equation contains the field quantities of the two-way wave equation de-composed into upgoing and downgoing waves. The wave vector p contains upgoing anddowngoing waves, the matrix B contains the eigenvalues of A and connects the upgoing anddowngoing waves. The sources are contained in the one-way source vector b. In the horizontalwavenumber-frequency domain the one-way wave equation is given by

∂3p = Bp + b. (3-2)

We derive the decomposition matrix L−1 and the composition matrix L for the SH-TE modeof propagation in the horizontal-wavenumber frequency domain. We further derive one-waywave extrapolators in a homogeneous subdomain by diagonalizing the matrix A. At the endof this chapter we consider the electromagnetic system in vacuum. We connect upgoing anddowngoing waves through a reflection formalism in Chapter 4.

20 The seismoelectric two- and one-way wave equations

3-1 Two-way wave equation

We rearrange the seismoelectric system of equations given in Chapter 2 such that we expressthe vertical derivatives as a function of the horizontal derivatives. To this purpose we eliminatethe field quantities that are not continuous over a source-free interface. According to Section2-3 we capture −τ b

13,−τ b23,−τ b

33, p, E1, E2, υs1, υ

s2, υ

s3, w3, H2,−H1 in the field vector q. The

complete derivation for a isotropic medium starting from the equations of motion 2-9, 2-10,the stress-strain relations 2-11, 2-12 and the electromagnetic equations 2-22 and 2-23, is givenin Appendix A.

We define the effective electric permittivity ε = ǫ + 1iω σe and the effective magnetic

permeability µ = µ + 1iω σm = µ0. We also include the coupling coefficient in the effective

electric permittivity εL = ε − ρEL2. We define the effective fluid density ρE = η

iωkand a

complex density ρc = ρb − (ρf)2

ρE .

We find the seismoelectric two-way wave equation

∂3q = Aq + d. (3-3)

The field vector q and the source vector d are defined as

q =

−τ b3

p

E0

vs

w3

H0

and d =

f b − iωρf kηδαf

fα + 1

iω∂αRαβhbβ

ff3 − 1

iωεLLη

kJ

s,e3

−Js,m0 −

(

∂1

∂2

)

1iωεL

Js,e3

hb3 + e−1

33 e3αhbα

−∂βkη f

fβ + d

M rtβhβ + qi

−Js,e0 − Lγαf

fα −

(

∂2

−∂1

)

1iωµ0

Js,m3

. (3-4)

In d,

(ejl)kl = eijkl = Sδijδkl + N (δikδjl + δilδjk) , (3-5)

in which

S = KG − 2

3N − d2

M(3-6)

and we define

Rαβ = eαβ − eα3e−133 e3β , (3-7)

rα = δα − eα3e−133 δ3. (3-8)

The vectors Js,e0 , J

s,m0 , δi and γi are given by

Js,e0 =

(

Js,e1

Js,e2

)

, Js,m0 =

(

Js,m2

−Js,m1

)

, δi =

δ1i

δ2i

δ3i

, γi =

(

δ1i

δ2i

)

. (3-9)

3-1 Two-way wave equation 21

We have eliminated the fields τ b1, τ b

2, E3, H3, w1 and w2. The source quantities Js,e3 , J

s,m3 ,

ff1 , f

f2 , h1 and h2 are still contained in the source vector d. The eliminated field quantities

can simply be calculated from quantities in q according to,

E3 =1

iωεL

(

∂2H1 − ∂1H2 − Lη

kw3 − J

s,e3

)

, (3-10)

H3 =1

iωµ0

(

∂2E1 − ∂1E2 − Js,m3

)

, (3-11)

w1 =k

η

(

ff1 − ∂1p

f − iωρf υs1

)

+ LE1, (3-12)

w2 =k

η

(

ff2 − ∂2p

f − iωρf υs2

)

+ LE2, (3-13)

−τ b1 =

1

iω

(

e1lhbl + iω

d

Mδ1p

f − e1l∂lvs

)

, (3-14)

−τ b2 =

1

iω

(

e2lhbl + iω

d

Mδ2p

f − e2l∂lvs

)

. (3-15)

The matrix A is composed as

A =

(

A11 A12

A21 A22

)

, (3-16)

where after substitution of equations A-34 - A-53 we find,

A11 =

0 0 −∂1

(

SKc

·)

ρf

ρE ∂1 − ∂1

(

2dNMKc

·)

−iωρf L 0

0 0 −∂2

(

SKc

·)

ρf

ρE ∂2 − ∂2

(

2dNMKc

·)

0 −iωρf L−∂1 −∂2 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

, (3-17)


A12 =

−iωρc + 1iω (∂1 (ν1∂1·) + ∂2 (N∂2·))

1iω (∂2 (ν2∂1·) + ∂1 (N∂2·))

0000

(3-18)

1iω (∂2 (N∂1·) + ∂1 (ν2∂2·)) 0

−iωρc + 1iω (∂1 (N∂1·) + ∂2 (ν1∂2·)) 0

0 −iωρb

0 −iωρf

0 00 0

(3-19)

0 0 00 0 0

−iωρf 0 0

−iωρE(

1 + ρE

εLL2)

ρE

εLL∂1

ρE

εL∂2

−∂1

(

ρE

εLL·)

−iωµ0 + 1iω∂1

(

1εL

∂1·)

1iω∂1

(

1εL

∂2·)

−∂2

(

ρE

εLL·)

1iω∂2

(

1εL

∂1·)

−iωµ0 + 1iω∂2

(

1εL

∂2·)

, (3-20)

A21 =

− iωN

0 0 0

0 − iωN 0 0

0 0 − iωKc

iωdMKc

0 0 iωdMKc

−iω(

d2

M2Kc+ 1

M

)

+ ∂β

(

1iωρE ∂β ·

)

0 0 0 L∂1

0 0 0 L∂2

(3-21)

0 00 00 0

−∂1

(

L·)

−∂2

(

L·)

−iωε + 1iω∂2

(

1µ0

∂2·)

− 1iω∂2

(

1µ0

∂1·)

− 1iω∂1

(

1µ0

∂2·)

−iωε + 1iω∂1

(

1µ0

∂1·)

, (3-22)

3-2 Decoupling of the SH-TE and P-SV-TM systems 23

A22 =

0 0 0 −∂1 0 00 0 0 −∂2 0 0

−(

SKc

)

∂1 −(

SKc

)

∂2 0 0 0 0

∂1

(

ρf

ρE ·)

− 2dNMKc

∂1 ∂2

(

ρf

ρE ·)

− 2dNMKc

∂2 0 0 0 0

iωρf L 0 0 0 0 0

0 iωρf L 0 0 0 0

, (3-23)

where

ν1 = 4N

(

S + N

Kc

)

, (3-24)

ν2 = 2N

(

S

Kc

)

and (3-25)

Kc = S + 2N. (3-26)

3-1-1 Symmetry properties of the two-way system matrix

The matrix A obeys similar symmetry properties as the matrices A, B and Dx in the generaldiffusion, flow and wave equation of Section 2-2. The matrix A obeys [Wapenaar, 1996]

ATN0 = −N0A and A

†M0 = −M0A

∗. (3-27)

The time domain equivalent of the medium state A∗

is a medium with time-reversed mediumparameters, i.e., a lossy medium becomes effectual. The matrices N0 and M0 are given by

N0 =

(

O In

−In O

)

and M0 =

(

O In

In O

)

, (3-28)

where In is the identity matrix of size n. For the seismoelectric matrix given in Section 3-1we have n = 6. The SH-TE system of Section 3-4 has n = 2 and the separate TE and SHsystems each have n = 1.

3-2 Decoupling of the SH-TE and P-SV-TM systems

In a stratified earth there are two independent modes of propagation. The system can bedecoupled by using a cylindrical coordinate transformation [White and Zhou, 2006], or todefin line sources. Interferometric Green’s function recovery requires source coverage overlarge volumes and surfaces. Therefore we define line sources in the x2 direction. This sets allderivatives with respect to x2 to zero, ∂2 = 0. With that, we rearrange the matrix A suchthat it decouples into a block diagonal matrix of two sub matrices,

∂3q = Aq + d, (3-29)

where

A =

(

Ashte O

O Apsvtm

)

, q =

(

qshte

qpsvtm

)

and d =

(

dshte

dpsvtm

)

. (3-30)


We choose the order in the field vectors qshte and qpsvtm such that they represent the orig-

inal order in the field vector q. In that way, the sub matrices Ashte and Apsvtm exhibit

the same symmetry properties of the original matrix A. Alternative arrangements canbe chosen [Haartsen, 1995], [Haartsen and Pride, 1997], [van der Burg, 2002], [Shaw, 2004],[White and Zhou, 2006], such that the sub matrices Ashte and Apsvtm become anti-diagonal.This simplifies the linear algebra methods that are employed to decompose into upgoing anddowngoing waves [Ursin, 1983], but we show in Appendix B that this is not strictly necessary.

We find

qshte =(

−τ b23, E2, υ

s2,−H1

)t, (3-31)

qpsvtm =(

−τ b13,−τ b

33, p, E1, υs1, υ

s3, w3, H2

)

, (3-32)

Ashte =

0 −iωρf L −iωρc + 1iω (∂1 (N∂1·)) 0

0 0 0 −iωµ0

−iω 1N 0 0 0

0 −iωε + 1iω

(

∂1

(

1µ0

∂1·))

iωρf L 0

, (3-33)

Apsvtm =

0 −∂1

(

SKc

·)

ρf

ρE ∂1 − ∂1

(

2dNMKc

·)

−iωρf L−∂1 0 0 00 0 0 00 0 0 0

−iω 1N 0 0 0

0 −iω 1Kc

iω dMKc

0

0 iω dMKc

−iω(

d2

M2Kc+ 1

M

)

+ ∂1

(

1iωρE ∂1·

)

−∂1

(

L·)

0 0 0 0

(3-34)

−iωρc + 1iω (∂1 (ν1∂1·)) 0 0 0

0 −iωρb −iωρf 0

0 −iωρf −iωρE(

1 + ρE

εLL2)

ρE

εLL∂1

0 0 −∂1

(

ρE

εLL·)

−iωµ0 + 1iω∂1

(

1εL

∂1·)

0 −∂1 0 0

− SKc

∂1 0 0 0

∂1

(

ρf

ρE ·)

− 2dNMKc

∂1 0 0 0

0 0 0 0

.

The decoupled source vectors are given by

dshte =

f b2 − ρf

ρE ff2 − 1

iω∂1N(

h12 + h21

)

Js,m1

h23 + h32

−Js,e2 − Lf

f2 + ∂1

1iωµ0

Js,m3

(3-35)

3-3 One-way wave equation 25

and

dpsvtm =

f b1 − ρf

ρE ff1 − 1

iω∂1

(

2N h11 +(

S − S2

Kc

)(

h11 + h22

))

f b3 − ρf

ρE ff3

ff3 − ρE

εLLJ

s,e3

−Js,m2 − ∂1

1iωεL

Js,e3

h31 + h13

h33 +(

SKc

)(

h11 + h22

)

qi − ∂11

iωρE ff1

−Js,e1 − Lf

f1

. (3-36)

In the remaining of Part I we only consider the SH-TE mode, therefore by default we referto the SH-TE system when we omit the subscript shte in the matrix A, field vector q andsource vector d.

3-3 One-way wave equation

We give a short summary of one-way wave theory, for a more thorough treatment, seeWapenaar and Berkhout [1989]. Starting from the two-way wave equation 3-1 in thehorizontal-wavenumber frequency domain (see Section 1-5-1), we diagonalize the system ma-trix A 3-37. Matrix L is a matrix whose columns are the eigenvectors of the system matrixA, the diagonal matrix H contains the eigenvalues of the system,

H = L−1AL. (3-37)

The field vector q transforms into the vector p, upon multiplication by L−1. Similarly thesource vector d transforms into the vector b. We have

d = Lb and q = Lp. (3-38)

Inserting equations 3-38 and 3-37 into the two-way wave equation 3-1 we obtain

∂3p = Bp + b, (3-39)

with the one-way first oder differential operator

B = H − L−1∂3L. (3-40)

The general solution of equation 3-39 reads [Wapenaar and Berkhout, 1989]

p(x3) = W(x3, x3,0)p(x3,0) +

∫ x3

x3,0

W(x3, x′3)b(x′

3)dx′3, (3-41)

were W(x3, x3,0) is defined by

W(x3, x3,0) =∞∑

m=0

(x3 − x3,0)m

m!Bm(x3,0) (3-42)


and Cm is recursively defined as

Bm+1(x3,0) = ∂3Bm(x3)∣

∣

∣

x3,0

+ Bm(x3,0)B1(x3,0), (3-43)

where

B0(x3,0) = I. (3-44)

3-3-1 Homogeneous source-free subdomain

For the special case of a homogeneous source-free subdomain, the solution to equation 3-39greatly simplifies. We can see that the vertical derivatives in equations 3-39, 3-43 and thesource terms in equations 3-39, 3-41 disappear. We find

∂3p = Hp, (3-45)

with solution

p(x3) = W(x3, x3,0)p(x3,0). (3-46)

The matrix W extrapolates the one-way wave fields in p from depth x3,0 to x3. This is onlyvalid in homogeneous source-free subdomains.

For Cm(x3,0) in equation 3-42 we find that equation 3-43 simplifies to Cm+1 = CmC1, orCm = C

m1 = H

m. With this simplification of Cm, we may represent W(x3, x3,0) in equation

3-46 symbolically as

W(x3, x3,0) = exp{

H(x3 − x3,0)}

. (3-47)

We show below that due to our choice of the structure of matrix L, the eigenvalue matrix H

has a structure

H =

(

H+

0

0 H−

)

, (3-48)

where H−

= −H+. Hence the wavefield extrapolator matrix W has a similar form

W =

(

W+

0

0 W−

)

. (3-49)

We can see that according to 3-47 and 3-48 we have

W(z2, z1) =(

W(z1, z2))−1

. (3-50)

We now see how the matrix L−1 decomposes the two-way field quantities into upgoing anddowngoing waves, therefore we refer to the matrix L and L−1 as the composition and decom-position matrices respectively. The one-way wave and source vectors are written as

p =

(

p+

p−

)

and b =

(

b+

b−

)

. (3-51)

3-4 Seismoelectric decomposition for the SH-TE system 27

3-3-2 Symmetry properties of the one-way system matrix

We do not perform flux normalization on our decomposition matrices, but choose a velocitynormalization or electric field normalization instead. However, in homogeneous subdomains,the one-way system matrix B still obeys a very similar symmetry relation as the two-waysystem matrix,

BTN0 = −N0B and B

†J0 = −J0B

∗. (3-52)

Where the matrices N0 and J0 are given by

N0 =

(

O In

−In O

)

and J0 =

(

In O

O −In

)

, (3-53)

where In is the identity matrix of size n. For the seismoelectric matrix given in Section 3-1 wehave n = 6. The SH-TE system of Section 3-4 has n = 2 and the separate TE and SH systemseach have n = 1. The symmetry properties given in equations 3-52 are symmetry properties of

the operator H =√

ω2

c2− k2

1. These are discussed by Wapenaar [1996] and Grimbergen et al.

[1998]. The time-reversed operator H∗ is given by a time-reversed propagation velocity c∗,defined in the time domain in a medium with adjoint medium parameters.

3-4 Seismoelectric decomposition for the SH-TE system

In Section 3-2 we derived the system matrix A for the SH-TE coupling mode, see matrix3-33. In the horizontal-wavenumber frequency domain Ashte is given by

A =

0 −iωρf L −iωρc +ik2

1

ω N 00 0 0 −iωµ0

−iω 1N 0 0 0

0 −iωε +ik2

1

ω1µ0

iωρf L 0

. (3-54)

The solutions to the eigenvalue problem are contained on the diagonal of H. In AppendixB we derive the wave velocities, the eigenvalues and eigenvectors of matrix A. Horizontallypropagating waves have zero vertical slowness and follow from the trivial solution of theeigenvalue problem. There are 4 roots to the zero-eigenvalue problem,

2

c2=

ρc

N+ εµ0 ±

√

(

ρc

N− εµ0

)2

− 4µ0

N

(

ρf L)2

. (3-55)

The plus sign is associated with the velocity of the SH-wave, csh, and the minus sign withthe velocity of the TE wave, cte, as can be seen when the coupling coefficient is set to zero,L = 0.

In Appendix B-2 we derive the non-zero eigenvalues of the seismoelectric SH-TE system.There are 4 non-zero solutions to the eigenvalue problem that represent plane wave solutionseach propagating with a non-zero vertical slowness as eigenvalue. All four non-zero eigenvalues

have the form of iH±w = ∓iHw, with Hw =

√

ω2

c2w− k2

1, where cw is the velocity of either a


shear horizontal (w = sh) or an transverse electric wave (w = te). We have to choosethe branch of the square root such that propagating waves decay. The real part of theexponentials in equation 3-47 need to be negative for propagating waves. Thus we choose

Im

{√

ω2

c2w− k2

1

}

≤ 0. We find for the eigenvalue matrix,

H =

iH+sh 0 0 0

0 iH+te 0 0

0 0 iH−sh 0

0 0 0 iH−te

. (3-56)

The arrangement follows from our eigenvector arrangement in the matrix L and is chosen suchthat we decompose our two-way wave fields into shear waves and electric waves, respectively,downgoing and upgoing.

For a general eigenvalue iH±w we find a general eigenvector a±

w

a±w =

± Hw

ω N

ξw

1

± Hw

ω1µ0

ξw

, (3-57)

with

ξsh =µ0ρ

f L(εµ0 − 1

c2sh

)and ξte =

( Nc2te

− ρc)

ρf L. (3-58)

We arrange the eigenvectors into L as L =(

a+sh, a+

te, a−sh, a−

te

)

. With that, the compositionmatrix is given by

L =

Hsh

ω N Hte

ω N − Hsh

ω N − Hte

ω N

ξsh ξte ξsh ξte

1 1 1 1Hsh

ω1µ0

ξshHte

ω1µ0

ξte − Hsh

ω1µ0

ξsh − Hte

ω1µ0

ξte

. (3-59)

The decomposition matrix is found inverting equation 3-59,

L−1 =1

2(

ξsh − ξte

)

− ωHsh

ξte

N 1 −ξteω

Hshµ0

ωHte

ξsh

N −1 ξsh − ωHte

µ0

ωHsh

ξte

N 1 −ξte − ωHsh

µ0

− ωHte

ξsh

N −1 ξshω

Hteµ0

. (3-60)

The decomposition matrix is normalized to the particle velocity υs2. Alternative normal-

izations, for example power-flux normalization, are discussed by Haartsen [1995]. One-wayreciprocity theorems would require power-flux normalization [Wapenaar, 1996] because theone-way formalism has to be energy conservative for reciprocity to hold. We use reciprocityrelations based upon the two-way field quantities in u and q so the normalization of L isarbitrary.

3-5 The seismoelectric system in vacuum 29

The one-way wave vector and one-way source vector are given as

p =

p+sh

p+te

p−shp−te

and b =

b+sh

b+te

b−shb−te

. (3-61)

For the wavefield extrapolator matrix W in a homogeneous source-free subdomain we find

W(x3, x3,0) =

eiH+sh

(x3−x3,0) 0 0 0

0 eiH+te(x3−x3,0) 0 0

0 0 eiH−

sh(x3−x3,0) 0

0 0 0 eiH−

te(x3−x3,0)

. (3-62)

3-5 The seismoelectric system in vacuum

The air cannot sustain shear stresses and we neglect pressures. There is no linear seismo-electric coupling in the air and we approximate air by vacuum. The dielectric permittivityis ε = ǫ0. There are only two equations left that represent the TE waves in vacuum. Thematrix A reduces to a 2×2 matrix, see Appendix B-4-2. We have the two-way wave equationof the TE system in vacuum as

q = Aq + d, (3-63)

with

A =

(

0 −iωµ0

−iωǫ0 +ik2

1

ω1µ0

0

)

, (3-64)

q =

(

E2

−H1

)

and d =

(

Js,m1

−Js,e2

)

. (3-65)

3-5-1 Diagonalization in a homogeneous source-free subdomain

We diagonalize the system matrix 3-64 in a homogeneous source-free subdomain

H = L−1AL. (3-66)

The zero-eigenvalue problem confirms the velocity of electromagnetic waves

c20 =

1

ǫ0µ0. (3-67)

For the non-zero eigenvalues we find

H =

(

iH+ 0

0 iH−

)

. (3-68)


The general eigenvalue iH± = ∓iH with H =√

ω2

c20− k2

1 corresponds to a general eigenvector

a± given as

a± =

(

1

± Hω

1µ0

)

, (3-69)

which has been normalized to the electrical field in the x2 direction. The different normaliza-tion of the eigenvectors in vacuum and in the porous medium is of no consequences as longas we work with composed field quantities. All normalizations are captured in the reflectionand transmission relationships as derived in Chapter 4.

For the composition and decomposition matrices L and L−1 of the TE system in vacuum weagain choose L = (a+, a−) and find

L =

(

1 1Hω

1µ0

− Hω

1µ0

)

and L−1 =1

2

(

1 ωH

1µ0

1 − ωH

1µ0

)

. (3-70)

Application of L−1 on the field vector q and source vector d, leads to

p =

(

p+

p−

)

and b =

(

b+

b−

)

(3-71)

In a homogeneous source-free subdomain we can extrapolate upgoing or downgoing wave-fieldswith the wavefield extrapolator

W(x3, x3,0) =

(

eiH+(x3−x3,0) 0

0 eiH−(x3−x3,0)

)

. (3-72)

Chapter 4

Seismoelectric modeling usingreflection formalism.

In this chapter we introduce a reflection formalism for simulation of reflection and transmissionexperiments in horizontally stratified media. We start by defining local and global reflectionmatrices and show how the global reflection matrices can be calculated from the local mediumparameters. Then we provide a general description of the relationship between upgoing anddowngoing waves in a homogeneous, but bounded, subdomain with a source. We also showhow the field can be propagated to a neighboring bounded source-free subdomain. In thesedescriptions we only use medium parameters, local and global reflection matrices, therebyavoiding the use of transmission matrices. The first to give the procedure for the scalarglobal reflection coefficient of a three layer medium was Airy [1833], this was later writtenin matrix form. To our knowledge we are the first to extend this procedure to calculate theresponse of a source buried inside a stack of layers. Examples of seismoelectric responsescalculated with the equations derived in this chapter are provided in Chapter 5.

We use a right handed Cartesian coordinate system with z axis positive downward. Subdo-mains are labeled Dn, with n denoting the layer number. A subdomain Dn is bounded belowby interface labeled n, with z coordinate zn. The upper half-space is vacuum, this subdomainhas label D0, the associated pressure-free surface has z coordinate z0. The lower half-space isnumbered DN . See Figure 4-1. All derivations are performed in the horizontal wavenumberfrequency domain.

4-1 Local and global reflection matrices

Our reflection formalism is based upon the use of reflection matrices. The local downgoingreflection matrix of interface n at depth zn, below layer n and above layer n is defined as,

p−n (zn) = r+

n (zn)p+n (zn). (4-1)

32 Seismoelectric modeling using reflection formalism.

DD 0

DD 1

DD n

DD n+1

DD N−1

DD N

z0

z1

zn

zn+1

zN−2

zN−1

Air

Porous Medium

Figure 4-1: Layer and interface annotations: Subdomains are labeled Dn, with n denoting thelayer number. A subdomain Dn is bounded below by interface labeled n, with z coordinate zn.The upper half-space is vacuum, this subdomain has label D0, the associated pressure-free surfacehas z coordinate z0. The lower half-space is numbered DN .

The local upgoing reflection matrix of interface n − 1 at depth zn−1, below layer n − 1 andabove layer n + 1 is defined as,

p+n (zn−1) = r−n (zn−1)p

−n (zn−1). (4-2)

The z dependence of the reflection matrices is explicitly mentioned, because these reflectionmatrices can be extrapolated in homogeneous source-free subdomains to different depth levelswhere they still connect the same upgoing and downgoing waves. The local reflection matricesstrictly only account for one reflection at one interface.

We define a global reflection matrix that accounts for all multiple scattering behind the locallyreflecting interface. For the global downgoing reflection matrix we have,

p−n (zn) = R+

n (zn)p+n (zn), (4-3)

and for the global upgoing reflection matrix

p+n (zn−1) = R−

n (zn−1)p−n (zn−1). (4-4)

It is important to note that these global reflection matrices include all multiple reflectionsof the incoming wave. Not included are waves coming from sources below or above the levelat which the global downgoing or upgoing reflection matrix is defined. Also excluded aresecondary incoming waves. For example, incoming waves that have been reflected by theglobal downgoing or upgoing reflection matrix defined at a certain reference depth level andthat are reflected above or below that certain depth level to return at the reference depth level.

4-2 Calculation of global reflection matrices 33

However, all phenomena can be included in the calculations if we follow a certain scheme asoutlined in Section 4-6.

In homogeneous source-free subdomains we can extrapolate local or global reflection matricesaway from the interface using the extrapolators as defined in equation 3-49. For the incidentand reflected wave-fields in the global downgoing reflection matrix 4-3 we have,

p−n (z) = W

−(z, zn)p−

n (zn), (4-5)

p+n (z) = W

+(z, zn)p+

n (zn). (4-6)

Substituting equations 4-5 and 4-6 into equation 4-3 we have

W−(z, zn)p−

n (z) = R+n (zn)W

+(z, zn)p+

n (z). (4-7)

We rewrite equation 4-7 making use of the symmetry property of the wavefield extrapolatorin equation 3-50, as

p−n (z) = W

+(zn, z)R+

n (zn)W+(zn, z)p+

n (z). (4-8)

We recognize that we have for Rn at a depth z in homogeneous subdomain Dn,

R+n (z) = W

+(zn, z)R+

n (zn)W+(zn, z). (4-9)

The result in equation 4-9 shows how a downgoing reflection matrix can be extrapolated in ahomogeneous subdomain. For the upgoing reflection matrix in the same layer we can followa similar derivation and find,

R−n (z) = W

−(zn−1, z)R−

n (zn−1)W−(zn−1, z). (4-10)

4-2 Calculation of global reflection matrices

We need to calculate the global reflection matrices from the medium parameters in regionsoutside the layer with the source. To this aim we write the boundary conditions from Section2-3 of an interface n,

limz↓zn

qn+1(z) − limz↑zn

qn(z) = 0. (4-11)

The layer n + 1 does not contain a source, thus evaluating the limits we can write,

qn(zn) = qn+1(zn), (4-12)

we substitute qn = Lpn and find,

Lnpn(zn) = Ln+1pn+1(zn). (4-13)

We split the composition matrix L into the columns that multiply into p+ and that multiplyinto p− according to,

Lnpn(zn) = L+n p+

n (zn) + L−n p−

n (zn). (4-14)


With the definitions of the columns in L for the SH-TE system given by the general eigenvectora±

n in Section 3-4 we have,

L+ =(

a+sh, a+

te

)

, (4-15)

L− =(

a−sh, a−

te

)

. (4-16)

The next modification is to rearrange the components of the general eigenvector a±n such that

the first two do not change sign for upgoing or downgoing waves and the last two do changetheir sign,

a± =

(

a′

a′′±

)

. (4-17)

For the SH-TE system we have,

a± =

a3

a1

a±2a±4

. (4-18)

These modifications define a composition operator written as,

L =

(

L′+ L′−

L′′+ L′′−

)

=

(

L′+ L′+

L′′+ −L′′+

)

. (4-19)

We can omit the superscript + in the right-hand side of equation 4-19. Implementing thisrearrangement into equation 4-13 gives,

L′np

+n (zn) + L′

np−n (zn) = L′

n+1p+n+1(zn) + L′

n+1p−n+1(zn), (4-20)

L′′np

+n (zn) − L′′

np−n (zn) = L′′

n+1p+n+1(zn) − L′′

n+1p−n+1(zn). (4-21)

Using the definition of the global reflection matrix 4-3 we have,

p−n (zn) = R+

n (zn)p+n (zn), (4-22)

p−n+1(zn) = R+

n+1(zn)p+n+1(zn), (4-23)

substituting 4-22 and 4-23 into equations 4-20 and 4-21 gives,

L′n

(

I + R+n (zn)

)

p+n (zn) = L′

n+1

(

I + R+n+1(zn)

)

, (4-24)

L′′n

(

I − R+n (zn)

)

p+n (zn) = L′′

n+1

(

I − R+n+1(zn)

)

. (4-25)

Now we can solve for R+n (zn) as a function of R+

n+1(zn). We find,

R+n (zn) =

[(

L′n − L

′′n

)

+(

L′n + L

′′n

)

R+n+1(zn)

] [(

L′n + L

′′n

)

+(

L′n − L

′′n

)

R+n+1(zn)

]−1,

(4-26)where we defined,

L′n =

[

L′n

]−1L′

n+1, (4-27)

L′′n =

[

L′′n

]−1L′′

n+1. (4-28)

4-3 Homogeneous bounded subdomain with sources 35

Note that we have,

R+n+1(zn) = W

+(zn+1, zn)R+

n+1(zn+1)W+(zn+1, zn). (4-29)

Equation 4-26 is a matrix form of the equation given by Fokkema and Ziolkowski [1987]. Withequation 4-26 we can calculate the downgoing global reflection matrix in any layer startingfrom the bottom interface N − 1, where R+

N (zN ) = 0. In that case equation 4-26 reduces toan expression for a local reflection matrix of interface n,

r+n =

[

L′n − L

′′n

] [

L′n + L

′′n

]−1. (4-30)

Equation 4-26 can be cast such that we recognise the local reflection matrix, like in the scalarversion given by Fokkema and Ziolkowski [1987], we can write,

R+n (zn) =

[

r+n (zn) +

(

L′n + L′′

n

)

R+n+1

(

L′n + L′′

n

)−1] [

I +(

L′n − L′′

n

)

R+n+1

(

L′n + L′′

n

)−1]−1

.

(4-31)

To calculate the upgoing global reflection matrix, we simply switch the geometry verticallyand start our recursion formula with the local upgoing reflection matrix of the pressure-freesurface.

4-3 Homogeneous bounded subdomain with sources

In a homogeneous layer n we have a source at source level zs, which does not coincide withthe interfaces bounding the layer. We need to distinguish between the wave fields above thesource level with superscript a and below the source level with superscript b. The upperbounding interface is at z = zn−1 and the lower bounding interface at z = zn.

From the boundary condition equation 2-51 in Section 2-3 we have,

limz↓zs

qbn(z) − lim

z↑zs

qan(z) = d(zs). (4-32)

If we now apply the decomposition operator Ln to the two-way fields in equation 4-32, wefind the jump condition in one-way wave quantities. Substitute qn = Lnpn and d = Lnb into4-32 and evaluate at the limits. We arrive at,

Lnpbn(zs) − Lnp

an(zs) = Lnb(zs). (4-33)

If we divide Ln out of the equation, we arrive at two relationships between pa,+n , p

a,−n , p

b,+n

and pb,−n as a function of the decomposed source terms b+ and b− at zs,

pb,+n (zs) = pa,+

n (zs) + b+(zs), (4-34)

pb,−n (zs) = pa,−

n (zs) + b−(zs), (4-35)

together with the definitions for the global upgoing and downgoing reflection matrices,

pb,−n (zs) = R+

n (zs)pb,+n (zs), (4-36)

pa,+n (zs) = R−

n (zs)pa,−n (zs). (4-37)


We defined R+n and R−

n under and above the source level, respectively, they cannot be ex-trapolated across the source level using equations 4-9 and 4-10 . We can solve this system forp

b,−n , p

b,+n , p

a,−n and p

a,−n , this gives

pb,−(zs) =[

I − R+n (zs)R

−n (zs)

]−1 [

R+n (zs)b

+ − R+n (zs)R

−n (zs)b

−]

, (4-38)

pb,+(zs) =[

I − R−n (zs)R

+n (zs)

]−1 [

b+ − R−n (zs)b

−]

, (4-39)

pa,−(zs) =[

I − R+n (zs)R

−n (zs)

]−1 [

R+n (zs)b

+ − b−]

, (4-40)

pa,+(zs) =[

I − R−n (zs)R

+n (zs)

]−1 [

R−n (zs)R

+n (zs)b

+ − R−n (zs)b

−]

. (4-41)

Note that equations 4-38 and 4-39 satisfy equation 4-36 and equations 4-40 and 4-41 satisfyequation 4-37. This set provides the basis of our modeling schemes that incorporate multiplescattering.

4-4 Determination of the wave fields outside the source layer

We need an equation that relates the upgoing and downgoing wave field amplitudes in twoadjacent layers n and n + 1 that can be used to propagate known wave field amplitudes inlayer n across interface n to layer n + 1. Following the same procedure as in previous Section4-2, we start from the boundary condition at a source-free interface below the source level,

limz↓zn

qn+1(z) − limz↑zn

qn(z) = 0. (4-42)

Again we substitute qn = Lpn into the boundary condition, and we evaluate the limits,

Ln+1pn+1(zn) = Lnpn(zn). (4-43)

We reverse the left-hand and right-hand sides of this equation, and use the re-arranged com-position matrix L given in equation 4-19 to split the equation. We need only one of theresulting two equations,

L′np

+n (zn) + L′

np−n (zn) = L′

n+1p+n+1(zn) + L′

n+1p−n+1(zn). (4-44)

We solve this equation for p+n+1(zn) making use of the definition of L

′n in equation 4-27. We

find

p+n+1(zn) =

[

I + R+n+1,N (zn)

]−1 [

L′n

]−1 [

I + R+n,N (zn)

]

p+n (zn). (4-45)

Note again that we have

R+n+1(zn) = W

+(zn+1, zn)R+

n+1(zn+1)W+(zn+1, zn). (4-46)

Equation 4-45 can be used to propagate a known downgoing wave field amplitude acrossan interface. To obtain the upgoing wave field in the next layer we simply use the globalreflection matrix in that layer according to,

p−n+1(zn) = R+

n+1(zn)p+n+1(zn), (4-47)

with R+n+1,N (zn) defined in equation 4-46.

4-5 Scattering matrix against a pressure-free surface 37

4-5 Scattering matrix against a pressure-free surface

We approximate the air in the upper half-space of our earths model with vacuum. Onlyelectromagnetic waves propagate in vacuum and there are no seismic waves. Therefore weneed a special scattering matrix for the pressure-free surface at z = z0. There are two wavetypes in the vacuum p+

0,te and p−0,te, in the subsurface there are four wave types p+1,sh, p+

1,te,

p−1,sh and p−1,te. The scattering matrix is 3 × 3,

p−0,te(z0)

p+1,sh(z0)

p+1,te(z0)

=

r+0,te−te(z0) t−te−sh(z0) t−te−te(z0)

t+sh−te(z0) r−1,sh−sh(z0) r−1,sh−te(z0)

t+te−te(z0) r−1,te−sh(z0) r−1,te−te(z0)

p+0,te(z0)

p−1,sh(z0)

p−1,te(z0)

. (4-48)

We ignore the definitions of transmission coefficients and follow van der Burg [2002] and Shaw[2004] to solve for the pressure-free surface local reflection matrix. From Section 2-3-1 we haveat the interface of a pressure-free surface,

0

E2

−H1

air

=

τ b23

E2

−H1

subsurface

. (4-49)

If we want to compose these from the one-way wave types that exist in the vacuum and inthe subsurface we write,

0 0a+

0,te,1 a−0,te,1

a+0,te,2 a−0,te,2

(

p+0,te(z0)

p−0,te(z0)

)

=

a+1,sh,1 a+

1,te,1 a−1,sh,1 a−1,te,1

a+1,sh,2 a+

1,te,2 a−1,sh,2 a−1,te,2

a+1,sh,4 a+

1,te,4 a−1,sh,4 a−1,te,4

p+1,sh(z0)

p+1,te(z0)

p−1,sh(z0)

p−1,te(z0)

,

(4-50)where a±0,te,i is given by a±

te in equation 3-69 and a±1,w,i is given by a±w in equation 3-57.

Equation 4-50 is rearranged to mimic equation 4-48,

0 a+1,sh,1 a+

1,te,1

−a−0,te,1 a+1,sh,2 a+

1,te,2

−a−0,te,2 a+1,sh,4 a+

1,te,4

p−0,te(z0)

p+1,sh(z0)

p+1,te(z0)

=

0 −a−1,sh,1 −a−1,te,1

a+0,te,1 −a−1,sh,2 −a−1,te,2

a+0,te,2 −a−1,sh,4 −a−1,te,4

p+0,te(z0)

p−1,sh(z0)

p−1,te(z0)

.

(4-51)Using the results of van der Burg [2002], we perform a division with the matrix in the left-hand side of equation 4-51, we find for the upgoing reflection matrix

r−0,sh−sh =1

Λ

(

H1,sh

(

H1,te + H0,te

)

ξte − H1,te

(

H1,sh − H0,te

)

ξsh

)

, (4-52)

r−0,sh−te =1

Λ

(

H1,te

(

H1,te + H0,te

)

ξte − H1,te

(

H1,te − H0,te

)

ξte

)

, (4-53)

r−0,te−sh =1

Λ

(

H1,sh

(

H1,sh − H0,te

)

ξsh − H1,sh

(

H1,sh + H0,te

)

ξsh

)

, (4-54)

r−0,te−te =1

Λ

(

H1,sh

(

H1,te − H0,te

)

ξte − H1,te

(

H1,sh + H0,te

)

ξsh

)

, (4-55)


whereΛ = H1,shξte

(

H1,sh + H0,te

)

− H1,teξsh

(

H1,sh + H0,te

)

. (4-56)

The first subscript of the operator H denotes the layer under consideration. For example,H1,w is the square root operator for wavetype w in the first subsurface layer, while H0,w isthe square root operator for wavetype w in the vacuum. The upgoing local reflection matrixat the pressure-free surface that initializes the recursive scheme for upgoing global reflectionmatrices in the stack of layers is given in equations 4-55, 4-54, 4-53 and 4-52.

4-6 Seismoelectric modeling scheme

Our seismoelectric simulation scheme is based upon the building blocks derived in this chapter.The first step is to calculate the global reflection matrices at the boundaries of the sourcelayer. This is done iteratively, using equation 4-26 or 4-31, starting from the local reflectionmatrices at the outer interfaces. For the scattering matrix of the pressure-free surface weuse the equations of Section 4-5. In step two we use equation 4-39 or 4-40 in Section 4-3to calculate the wavefield in the source layer. We need the wavefield in the direction of thereceiver, at the same side of the source level as the receiver level. Step three is to propagatethe wavefield across the interfaces until we reach the receiver layer, this is done as a functionof the global reflection matrices that we have already calculated in step one. The wavefield inthe receiver layer is completely determined by reflecting the wavefield once, such that we haveboth upgoing and downgoing components. A schematic overview of the path of calculation isgiven in Figure 4-2.

zs

zr

z0

zN−1

Air

Porous Medium

Figure 4-2: Calculation scheme of the seismoelectric responses of a source buried in a layeredmedium. Step 1; iteratively calculate the global reflection matrices, from the outer interfacesup to the source layer. Step 2; solve the wavefield inside the source layer. Step 3; iterativelypropagate the known field from the source layer to the receiver layer. Step 4; solve the field insidethe receiver layer.

Chapter 5

Examples of seismoelectric simulationsin 2D

In this chapter we give several examples of seismoelectric simulations using the one-way waveequations of Chapter 3 and the reflection formalism of Chapter 4. We have an exact solutionin the frequency-wavenumber domain, that we compute on a discretised grid and transformnumerically to the time-space domain. When we discretise the wavenumber-frequency grid, wediscretise the source, and introduce a periodicity in the time-space solution. After describinghow we use Matlab’s [X] = fft(x) and [x] = ifft(X) functions to perform discrete Fouriertransformations, we describe the causality trick that damps the introduced periodicity intime. In the next section we introduce the media types used throughout this thesis. In thelast section we show the results of a transmission experiment in a homogeneous medium. Theresults of a reflection and transmission experiment in a more complex medium containing twointerfaces, the upper half-space being air approximated by vacuum and the lower half-spacebeing a porous medium.

5-1 Numerical implementation

5-1-1 Discrete Fourier transformations

We start from exact relations in the horizontal wavenumber frequency domain as definedby equations 1-2 and 1-4. To transform back to the space-time domain we need to employequations 1-3 and 1-5. However, we compute the solution along a discrete time and spaceaxes defined by,

t = [−NT : NT − 1] ∆t, (5-1)

x = [−NX : NX − 1] ∆x. (5-2)

40 Examples of seismoelectric simulations in 2D

Correspondingly, the angular frequency and wavenumber axes are also discretised,

ω = [−NT : NT − 1] ∆ω, (5-3)

k = [−NX : NX − 1] ∆x. (5-4)

If we substitute equations 5-1, 5-2, 5-3 and 5-4 into our definitions of the Fourier transfor-mations 1-2 and 1-3 and replace the integrals by summations over frequencies and times,tn = n∆t, ωm = m∆ω, f(x, f) → fm(x) and f(x, t) → fn(x) we arrive at

fm(x) =NT−1∑

n=−NT

fn(x)e−i(∆ω∆t n m)∆t, (5-5)

fn(x) =1

2π

NT−1∑

m=−NT

fm(x)ei(∆ω∆t n m)∆ω. (5-6)

Note that we only discretise the time and frequency axes at this point. It is easy to seethat the periodicity of the Fourier kernel is ∆ω∆t = 2π

2NT . We also have that the angularfrequency is linked to the frequency through ω = 2πf (with f the frequency in Hz), so wehave ∆ω = 2π∆f . In Matlab, we cannot use an index to an array that is negative or zero. Wefirst rearrange the summation to run from 0 to 2π, in stead of from −π to π used in 5-5 and5-6, by replacing the negative counters with m = 2NT + [−NT : −1 : −1] and then we addone to the index counters. Changing the range from −π to π into 0 to 2π, and the reverse, isperformed by the standard Matlab functions [X] = fftshift(x) and [X] = ifftshift(x). Wearrive at

fm

(x) = ∆t

2NT∑

n=1

fn(x)e−i 2π2NT

(n−1)(m−1), (5-7)

fn(x) = ∆f

2NT∑

m=1

fm(x)ei 2π2NT

(n−1)(m−1). (5-8)

We employ Matlab’s [X] = fft(x) and [x] = ifft(X) functions, they perform a transformationof vectors with length N of the form

X(k) =N∑

j=1

x(j)e(−2πi/N)(j−1)(k−1), (5-9)

x(j) =1

N

N∑

k=1

X(k)e(2πi/N)(j−1)(k−1). (5-10)

If we apply the [x] = ifft(X) transformation to our exact solution in the frequency domain,we have Nt = 2NT = 1

∆t∆f and we need to apply an amplitude correction to the functionoutput. We match the temporal Fourier transformation 5-7 to the transformation performed

by [x] = ifft(X) by defining f′= f

∆t . This means we have to scale the function return by1

∆t . Since we use only positive frequencies, we have to padd our array with zeros in thenegative frequencies and multiply the return of ifft(X) by 2 according to equation 1-6. Asimilar argument can be made for the spatial Fourier transformation. However, because ofthe different sign convention we employ [x] = fft(X) to transform back to the space domain.Hence we need to perform an amplitude correction ∆k

2π .

5-1 Numerical implementation 41

5-1-2 Causality trick

If we discretise the solution in the wavenumber-frequency domain, we introduce a periodicityin the source position in the space-time domain window. This means that events leavingat the maximum positive recording times return in the solution after traveling through thenegative times. Events leaving the window at maximum offset continue in the negative offsets.This wrapping effect can be seen in Figure 5-1(a), where we see the response of a f b

2 sourcerecorded by a υs

2 receiver-line after it is propagated over a distance of 150 m. There existsan exact trick that damps the anti-causal events in our space-time window. We add a small

fb2 → vs

2

offset [m]

time

[s]

−500 −400 −300 −200 −100 0 100 200 300 400 500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(a) Events wrapping in space and time.

fb2 → vs

2

offset [m]

time

[s]

−500 −400 −300 −200 −100 0 100 200 300 400 500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(b) Events wrapping in space, after applying thecausality trick.

Figure 5-1: Wrapping effects in (a) and the results of applying the causality trick in (b).

negative imaginary number to our real frequency array. Using complex frequencies meansthat we have a Fourier transformed equivalent of a time function multiplied with a dampingexponential in time (black curve in Figure 5-2). The damping factor is equal to the imaginaryfrequency value. The anti-causal events are damped twice, before they reoccur in the positivetime window. We multiply our time domain return of the discrete Fourier transformation bya growing exponential (blue curve in Figure 5-2) that undoes the effect of the damping thatwe introduced in the frequency domain on the causal events (red curve in Figure 5-2). Notethat the curves in Figure 5-2 only show the first order wrap around, the higher order wraparounds are damped even stronger. We can get rid of the wrapping in space by choosing aspace interval large enough to encompass the first arrivals in the time interval.


−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

Angle in Fourier kernel [π]

Am

plitu

de

Figure 5-2: Exponentials in the causality trick: Black curve: damped exact solution in thefrequency domain. Blue curve: growing exponential to correct for the damping. Red Curve:result after correcting for the damping.

5-2 Medium parameters for medium type A and B

Throughout this thesis we make use of two sets of medium parameters, sets A and B. Theseare the same two media types as used by Shaw [2004]. In both media the seismoelectriccoupling is present, but in medium B this coupling is considerably higher than in medium A.In Table 5-1 we list the properties of the two medium types.

Table 5-1: Medium characteristics of two porous media.

Parameters Symbol Medium A Medium B

Porosity φ 40 % 20 %Fluid density ρf 1000 kg m−3 1000 kg m−3

Solid density ρs 2700 kg m−3 2700 kg m−3

Bulk Modulus of the solid Ks 4 · 1010 N m−2 4 · 1010 N m−2

Bulk Modulus of the fluid Kf 2.2 · 109 N m−2 2.2 · 109 N m−2

Bulk Modulus of the framework Kfr 4 · 109 N m−2 4 · 109 N m−2

Shear Modulus of the framework N 9 · 109 N m−2 9 · 109 N m−2

Viscosity η 10−3 N s m−2 10−3 N s m−2

Relative permittivity of the solid ǫs 4 F m−1 4 F m−1

Relative permittivity of the fluid ǫf 80 F m−1 80 F m−1

Tortuosity α∞ 3 3Similarity parameter m 8 8Electrolyte concentration C 10−6 Mol l−1 10−2 Mol l−1

Electrolyte NaCl NaCl

Na+ Cl−

Ion mobility 1 bl 5.2·10−8 m s−1 N−1 7.9·10−8 m s−1 N−1

1from Nabighian and Corbett [1987]

5-2 Medium parameters for medium type A and B 43

100 200 300 4000

1

2

3x 10

7c

em

Frequency [Hz]

velo

city

[m/s

]

100 200 300 4000

2

4

x 105

cem

Frequency [Hz]

velo

city

[m/s

]

100 200 300 4000

1000

2000

csh

Frequency [Hz]

velo

city

[m/s

]

100 200 300 4000

1000

2000

csh

Frequency [Hz]

velo

city

[m/s

]

Medium type A Medium type B

Figure 5-3: Real (blue curves) and imaginary (red curves) parts of the wave velocities. The realand imaginary parts of the electromagnetic wave speed in medium B overlap.

100 200 300 4003.74

3.75

3.76x 10

−6

Frequency [Hz]

−ω

Imag

(ε")

100 200 300 400

8.4347

8.4347

8.4347x 10

−3

Frequency [Hz]

−ω

Imag

(ε")

100 200 300 4001.295

1.296x 10

−8

Frequency [Hz]

Rea

l(L)

100 200 300 4002.07452.07462.0747

x 10−9

Frequency [Hz]

Rea

l(L)

100 200 300 400

024

Frequency [Hz]

−Im

ag(ρ

c )

100 200 300 400

0

0.5

1

Frequency [Hz]

−Im

ag(ρ

c )

Medium type A Medium type B

Figure 5-4: Loss functions appearing in Sections 7-4 and 7-8; imaginary parts of ε and ρc, real

part of L.

Medium type A corresponds to a clean sandstone, with 40% porosity, containing clean waterwith a very low salinity. Medium type B corresponds to the same type of sandstone, but witha porosity of 20% containing a NaCl brine of 10−2 [Mol/l] (a concentration of 584 [mg/l]).We use equation 3-55 to compute the wave velocities of electromagnetic waves and seismicwaves. In Figure 5-3 we plot the real (in blue) and imaginary parts (in red). Note how theshear waves have a small imaginary part, but the electromagnetic waves have comparablereal and imaginary parts, implying the strong diffusive character of the electromagnetic field.


The dispersion of shear waves is a little higher in medium type A then in medium type B.In Figure 5-4 we plot the loss terms in the SH-TE system that appear in the interferometricGreen’s function representations of Sections 7-4 and 7-8. We only plot the parts that act aslosses in the volume integrals 7-15 and 7-43, the imaginary parts of ε and ρc and the real partof L.

5-2-1 Reflection matrices

To complete the characteristics of the medium parameters we show the reflection matricesthat occur for the geometries of this thesis, using equation 4-30 to compute the local reflectionmatrix. In figures 5-5 and 5-6 we give the reflection matrices of the interface between mediumtype A and medium type B approached from both sides.

The amplitude of the reflection coefficient is plotted versus the ray parameter times theabsolute wave velocity of the incident wave. There exists a very simple relationship betweenray parameter and horizontal wavenumber, given by k1 = ωp1. The ray parameter is themeasure for the direction of propagation of a plane wave, according to

p1 =sin (θ)

c, (5-11)

where θ is the propagation angle and c is the wave propagation velocity. Note in Figures 5-5and 5-6 how the conversion from an incident shear wave to an outgoing electromagnetic waveonly takes place for very hight angles of incidence. While the conversion from an incidentelectromagnetic wave to an outgoing shear wave takes place for a wide angle of incidents.This effect is even stronger for the reflection against the pressure-free surface interface frombelow, see figure 5-7. We use the equations of van der Burg [2002] given in Section 4-5 tocompute the local upgoing reflection matrix of the pressure-free surface using porous mediumtype A.

0 1 2 30

0.5

1

SH −> SH

P |Csh|

|Rsh

,sh|

0 1 2 30

0.5

1

1.5

2

2.5TE −> SH

P |Csh|

|Rsh

,em

|

0 100 200 3000

1

2

3x 10

−11 SH −> TE

P |Cte|

|Rem

,sh|

0 100 200 3000

0.5

1TE −> TE

P |Cte|

|Rem

,em

|

Figure 5-5: Reflection matrix, from layer type A against B. Horizontal axis is the ray parametertimes the absolute wave velocity of the incident wave, in medium type A.


5-3 Reflection and transmission experiments

In this section we show examples of a seismoelectrical experiment to familiarize the readerwith the interpretation of electroseismograms. The first example is the simplest case possible;a transmission experiment in a homogeneous medium. We have one source and at a distanceof 150 meter we place a line of receivers, see Figure 5-8. Blue rays are shear waves and red raysare electromagnetic waves. We always record two events, event number one, labeled 1, travelswith the shear velocity, while event number two, labeled 2, travels with the electromagneticwave velocity. This means that both electromagnetic and seismic sources emit electromagneticand shear waves. We use a source that emits a Ricker wavelet Srw, see equation 5-12, withcenter angular frequency ω0 of 800 Radians, corresponding to approximately 127 Hz. InFigure 5-9 we see the x2 component of the electric field, E2, recording as a response of a forcesource in the x2 direction, f

2. In Figure 5-10 we see a recording of the x2 component of the

particle velocity in the solid, υs2, as a response to an electric current source acting in the x2

direction, Je2. Both events are hyperbolic, however the hyperbolicity of the electromagnetic

event is very small due to the velocity of the electromagnetic wave. Thus we see a nearly flatelectromagnetic event and a hyperbolic shear wave event.

Srw =2√π

ω2

ω30

exp

(−ω2

ω20

)

. (5-12)

5-3 Reflection and transmission experiments 47

*

∇∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇

Porous medium A

1 2

150

[m]

Figure 5-8: Experiment geometry of a transmission experiment in a homogeneous medium, 2events, see text.

fb2 → E

2

offset [m]

time

[s]

−500 −400 −300 −200 −100 0 100 200 300 400 500

0

0.05

0.1

0.15

0.2

0.25

2

1

Figure 5-9: Transmission experiment in a homogeneous medium f2 → E2; 1: shear wave orco-seismic event, 2: electromagnetic wave event.


−Je2 → vs

2

offset [m]

time

[s]

−500 −400 −300 −200 −100 0 100 200 300 400 500

0

0.05

0.1

0.15

0.2

0.25

2

1

Figure 5-10: Transmission experiment in a homogeneous medium −Js,e2 → υs

2; 1: shear waveevent, 2: electromagnetic wave event.

Note the similarity between Figure 5-9 and 5-10, they are equal except for a signswitch. This is a manifestation of source-receiver reciprocity, see Section 7-6,Gυs,Je

(xA,xB) = −GE,f (xB,xA).

The second example is a more realistic geometry corresponding to an aquifer with a brine 100[m] beneath the pressure-free surface, see Figure 5-11. There are two interfaces in the geom-etry, thus we have multiple arrivals. The receivers are placed infinitively close to, but underthe pressure-free surface. In the first simulation the source is at an infinitesimal distancebelow the receivers. The source and receiver ghosts of the pressure-free surface are indistin-guishable from their primaries. Figure 5-11 shows the geometry and the primary arrivals,blue rays are shear waves and red rays are electromagnetic waves, at each interface there is aconversion and the number of possible rays paths double. Figure 5-12 shows the simulationof a reflection experiment using a current source, −J

s,e2 , in the x2 direction and we measure

the x2 component of the velocity in the solid, υs2.


Air

Porous medium A

Porous medium B

*∇ ∇ ∇ ∇ ∇∇ ∇ ∇ ∇−Je

2υ

2

100

[m]

1 2

4 35 6

Figure 5-11: Experiment geometry of a reflection experiment above an aquifer.

−Je2 → vs

2

offset [m]

time

[s]

−500 −400 −300 −200 −100 0 100 200 300 400 500

0

0.05

0.1

0.15

0.2

2/41

5/6

3

5/6m

Figure 5-12: Reflection experiment above an aquifer, 5 events see text.


In figure 5-12 we see 5 events, the labels correspond to the ray paths in figure 5-11. Flatevent labeled ’2/4’ is the combination of all rays that only traveled as an electromagneticwave. Event ’1’ is the direct shear wave emitted by the electrical source that travels along thesurface. Event ’5/6’ is the combination of the converted waves at the interface. The strongestcomponent leaves the source as an electromagnetic wave and converts at the subsurfaceinterface everywhere simultaneously, the result is a vertically upgoing almost plane wave. Itarrives at the one-way travel time of the subsurface interface. If we write travel-time, wemean shear wave travel time really. Because the travel time associated with the propagationof an electromagnetic wave is negligible on this time scale. Event ’3’ is a shear wave emittedfrom the electrical source and reflecting at the subsurface interface. It arrives at two-waytravel time of the subsurface interface. Event ’5/6m’ is the multiple of event ’5/6’, it made afull shear wave multiple between the two interfaces, and therefore arrived at two-way traveltime after event ’5/6’.

For the second simulation we place the source 50 meter beneath the subsurface interface. Infigure 5-13 we show the geometry together with the primaries. In figure 5-14 we see 7 events,the labels correspond to the ray paths in figure 5-13. Event ’2’ is the combination of all therays that travel purely as an electromagnetic wave. Event ’3’ is the arrival of the transmittedconversion on the subsurface interface. Event ’1’ is the co-seismic electrical field of the directshear wave. Event ’1m1’ is the manifestation of the electromagnetic wave that is created whenthe direct shear wave hits the pressure-free surface. It is really a multiple of event ’1’, and atzero offset it coincides with its primary. Event ’3m1’ is the multiple of event ’3’, it arrives attwo-way shear wave travel time of the subsurface interface. Event ’1m2’ is another multipleof event ’1’, it made a full shear wave multiple between the two interfaces. Event ’3m2’ is thesecond order multiple of event ’3’.


Air

Porous medium A

Porous medium B

*

∇ ∇ ∇ ∇ ∇ ∇∇ ∇ ∇ ∇

f2

E2

100

[m]

50 [m

]

1 23 4

Figure 5-13: Experiment geometry of a transmission experiment in an aquifer.

fb2 → E

2

offset [m]

time

[s]

−500 −400 −300 −200 −100 0 100 200 300 400 500

0

0.05

0.1

0.15

0.2

0.25

2

3

1m11

3m1

1m2

3m2

Figure 5-14: Transmission experiment in an aquifer, 7 events see text.

Part II

Interferometric Seismoelectric Green’sFunction Recovery

Chapter 6

Theory of interferometric Green’sfunction recovery

In this chapter we introduce the interferometric Green’s function representation for the generalflow, diffusion and wave equation as derived by Wapenaar et al. [2006] from the reciprocitytheorems of Wapenaar and Fokkema [2004]. By writing the interferometric Green’s functionrepresentations for the general flow, diffusion and wave equation, it went beyond the scope ofelastodynamic wave fields by Wapenaar and Fokkema [2006]. It validated the representationsfor the electromagnetic wave field [Slob et al., 2007] and representations for a diffusive system[Snieder et al., 2006]. Moreover, it provided interferometric Green’s function representationsfor the coupled elastodynamic and electromagnetic wave-fields, that are the subject of Chapter7. For simplicity, we only consider the theory for non-flowing media, but by Wapenaar et al.[2006] it has also been derived for flowing media.

A reciprocity theorem interrelates two independent states in one and the same domain[de Hoop, 1966], [Fokkema and van den Berg, 1993], the basis for reciprocity theorems isthe theorem of Gauss. In Section 6-1 the scalar theorem of Gauss is modified to applyto vectorial systems of arbitrary lengths. The results are used to derive a convolution-typereciprocity theorem in Section 6-2 and a correlation-type reciprocity theorem in Section 6-3.The convolution-type reciprocity theorem leads to the well-known source-receiver reciprocityrelation, for the special case in which both states have the same medium parameters. Thecorrelation-type reciprocity theorem is modified using the source-receiver reciprocity to findan interferometric Green’s function matrix representation.

6-1 Matrix vector representation of the divergence theorem ofGauss

Reciprocity theorems are derived using the theorem of Gauss. For a scalar field a = a(x) indomain D bounded by a boundary ∂D with outward pointing normal vector n, see Figure 6-1,

56 Theory of interferometric Green’s function recovery

the divergence theorem of Gauss reads∫

D

∂iad3x =

∮

∂D

anid2x. (6-1)

In the derivations of reciprocity theorems we need a divergence theorem for the matrix Dx

n

DD

∂ DD

a(x)

Figure 6-1: Scalar field a = a(x) in domain D bounded by ∂D with outward pointing normalvector n.

appearing in the general diffusion, flow and wave equation 2-34 instead of the operator ∂i

in equation 6-1. Let DIJ denote the I th element from the Jth column of the matrix Dx,from symmetry relation 2-47 we have DIJ = DJI . We define a matrix Nx that contains thecomponents of a normal vector ni arranged in the same way as the partial spatial derivatives∂i in the matrix Dx. We replace the scalar field a, by a matrix field formed by the dyadicproduct of two vectors aI(x) and bJ(x). This generalizes equation 6-1 to

∫

D

DIJ (aIbJ) d3x =

∮

∂D

aIbJNIJd2x. (6-2)

The left hand-side contains the inner product of Dx with the matrix formed by the dyadicproduct of the vectors a and b, written as Dx ·

(

a bt)

. We evaluate the left-hand side ofequation 6-2 using the product rule of DIJ (aIbJ) and the symmetry property of DIJ . Weobtain

DIJ (aIbJ) = aIDIJbJ + (DJIaI) bJ

Dx ·(

a bt)

= atDxb + (Dxa)tb. (6-3)

When we rewrite the subscript notation on the left-hand side of equation 6-2 to a matrix-vector product we obtain the divergence theorem of Gauss in matrix-vector form, see alsoFigure 6-2,

∫

D

Dx ·(

a bt)

d3x =

∫

D

atDxb + (Dxa)tbd3x =

∮

∂D

atNxbd2x. (6-4)

This equation is used in the derivation of the reciprocity theorem of the correlation typein Section 6-3. We consider a variant of equation 6-4 which we use in the derivation of thereciprocity theorem of the convolution type in Section 6-2. We replace a by K0a, where K0 isthe real-valued diagonal matrix introduced in equation 2-46. Using equation 2-47 we obtain

∫

D

Dx ·(

K0a bt)

d3x =

∫

D

atK0Dxb + (Dxa)tK0bd3x =

∮

∂D

atK0Nxb d2x. (6-5)

6-2 Reciprocity theorem of the convolution type 57

n

DD

∂ DD

{a b t} (x)

Figure 6-2: Matrix field {a bt}(x) in domain D bounded by ∂D with normal vector n.

6-2 Reciprocity theorem of the convolution type

n

DD

∂ DDu

s

A

B

A

A

A

A

u

s

A

B

B

B

B

B

Figure 6-3: The general character of two physical states A and B for the reciprocity theorem ofthe convolution type in domain D.

We consider two physical states in domain D, they are distinguished with subscripts A andB. The material parameters and the source functions may be different in both states but forsimplicity we consider non-flowing media, see Figure 6-3. We consider the interaction quantityDx ·

(

K0uAutB

)

, using equation 6-5 and the general flow, diffusion and wave equation 2-34together with the symmetry properties for the system matrices 2-45 and 2-47 to expand theproducts. We find

∮

∂D

utAK0NxuBd2x =

∫

D

[

utAK0sB − st

AK0uB

]

d3x +

∫

D

utAK0

[

iω(

AA − AB

)

+(

BA − BB

)]

uBd3x. (6-6)

This is the unified reciprocity theorem of the convolution type.

6-2-1 Source-receiver reciprocity

We consider a special case of the reciprocity theorem of the convolution type, when we haveequal medium parameters in both states AA = AB = A and BA = BB = B. We choose point


xA and xB both within D. And we substitute the source vectors in both states by a frequency-independent point-source matrix sA,B → Iδ (x − xA,B). We consequentially have to replace

the field vectors in both states by Green’s function matrices u (x,xA,B, ω) → G (x,xA,B, ω).The diagonal of this Green’s function matrix corresponds to the Green’s function of theequivalent source field, while the off-diagonal elements are conversions of a certain emittedfield type to another field type. See Section 7-1 equation 7-27 for the Green’s function matrixof the SH-TE system. With these replacements, equation 6-6 becomes a reciprocity relationfor the Green’s function matrix of the convolution type. If we integrate over the whole space,rD → ∞, the boundary integral on the left-hand side of equation 6-6 disappears on accountof the radiation condition [Bleistein, 1984], more generally, since this boundary integral is apropagation invariant, as long as D includes xA and xB, it vanishes for any ∂D. The result is

K0Gt(xA,xB, ω)K0 = G(xB,xA, ω). (6-7)

From equation 6-7, we see that G(xA,xB, ω) commutes to G(xB,xA, ω) if we switch sourceand receiver positions and types. There is a possible sign switch modulated by K0. SeeSections 7-2 and 7-6 for the specific source-receiver reciprocity relations of the SH-TE seis-moelectric system in 1D and 2D.

6-3 Reciprocity theorem of the correlation type

n

DD

∂ DDu

s

A

B

A

A

A

A

u

s

A

B

B

B

B

B

*

*

*

*

Figure 6-4: The general character of two physical states A and B for the reciprocity theorem ofthe correlation type in domain D.

For the reciprocity theorem of the correlation type we consider two independent states A andB in the domain D, but take the time-reversed state in A, see Figure 6-4. We consider theinteraction quantity Dx ·

(

u∗A ut

B

)

and use the general flow, diffusion and wave equation 2-34together with the symmetry properties for the system matrices 2-45 and 2-47 to expand theproducts. We find

∫

∂D

u†ANxuBd2x =

∫

D

[

u†AsB + s

†AuB

]

d3x −∫

D

u†A

[

iω(

AB − A†A

)

+(

BB + B†A

)]

uB d3x. (6-8)

This is the unified reciprocity theorem of the correlation type. The right-hand side of equa-tion 6-8 contains a volume integral that does not disappear when we choose equal medium

6-4 Interferometric Green’s function representations in 3D 59

parameters. This is an important different property of the correlation type reciprocity the-orem with the convolution type reciprocity theorem. When we consider the situation inwhich the wave-felds, medium parameters and source functions in both states are identicalwe find the power balance for the general flow, diffusion and wave equation. Therefore thecorrelation-type reciprocity theorem is also referred to as the power reciprocity theorem.

6-4 Interferometric Green’s function representations in 3D

n∂ DD

DD∇∇

xA

∇∇x

B

Figure 6-5: Geometry for the 3D interferometric Green’s function representation.

We now modify the reciprocity theorem of the correlation type such that it can be used forinterferometric recovery of the Green’s function matrix. We start by introducing the samesource matrix and Green’s function matrix of Section 6-2-1 into equation 6-8. We choosepoints xA and xB, where x = (x1, x2, x3), both within D and define both states A and B withthe same medium parameters, thus AA = AB = A and BA = BB = B. We find

G(xA,xB, ω) + G†(xB,xA, ω) =∮

∂D

G†(x,xA, ω)NxG(x,xB, ω)d2x

+

∫

D

G†(x,xA, ω)[

iω(

AB − A†A

)

+(

BB + B†A

)]

G(x,xB, ω) d3x. (6-9)

The first integral on the right-hand side of equation 6-9 is over crosscorrelations of Green’sfunctions from sources at xA and xB measured at the boundary ∂D and the second integral onthe right-hand side of equation 6-9 is over crosscorrelations of Green’s functions from sourcesat xA and xB measured throughout the domain D. We transpose both sides of equation 6-9and use the source-receiver reciprocity relation 6-7 together with the symmetry properties forthe system matrices A and B in equation 2-45. The result is

G(xB,xA, ω) + G†(xA,xB, ω) =

−∮

∂D

G(xB,x, ω)NxG†(xA,x, ω)d2x

+

∫

D

G(xB,x, ω)[

iω(

AB − A†A

)

+(

BB + B†A

)]

G†(xA,x, ω) d3x. (6-10)


The integrations on the right-hand side of equation 6-10 are of Green’s functions of sources onthe boundary ∂D and throughout the volume D measured at xA and xB. This step is the keyto the modification of the reciprocity theorems to a relation that can be applied for interfero-metric Green’s function recovery. Equation 6-10 tells us that we can retrieve a component ofthe Green’s function matrix and the time-reversed of its reciprocal counterpart, if we cross-correlate observations of certain sources throughout the domain D and on its boundary ∂D,with appropriate weighting functions. The specific relationship between the two retrievedGreen’s functions is given by the source-receiver reciprocity relation 6-7. The importanceof the different integrals are studied in Chapter 8. We call equation 6-10 an interferometricGreen’s function representation, because it represents the Green’s function between xA andxB as an integral of cross-correlations of observed green’s functions at point xA and xB.

6-5 Interferometric Green’s function representations in 2D and 1D

To evaluate the interferometric Green’s function representation for the SH-TE mode of propa-gation, we need to rewrite the representation in 2D. The volume domain D reduces to a surfacedomain S. Again we choose point xA and xB, where x = (x1, x3), both in S. Equation 6-10reduces to

G(xB,xA, ω) + G†(xA,xB, ω) =

−∮

∂S

G(xB,x, ω)NxG†(xA,x, ω)dx

+

∫

S

G(xB,x, ω)[

iω(

AB − A†A

)

+(

BB + B†A

)]

G†(xA,x, ω) d2x, (6-11)

where Nx contains the components of the normal vector n = (n1, n3) on the line ∂S. Accord-ing to equation 6-11, we can retrieve a component of the Green’s function matrix and thetime-reversed of its reciprocal counterpart, if we crosscorrelate observations of certain sourcesall over the domain S and on its boundary ∂S, with appropriate weighting functions.

We can simplify the representation even further by going to a 1D system. We choose

LI

x3

n n∇∇x

3,A

∇∇x

3,Bx

3;1x

3;2

Figure 6-6: Geometry for the 1D interferometric Green’s function representation.

x3;1 < x3,A/B < x3;2, see Figure 6-6. The surface domain S reduces to a line domain L andthe boundary integral reduces to a sum over two points on the edges of L. We take the

6-5 Interferometric Green’s function representations in 2D and 1D 61

x3-coordinate as the remaining coordinate. Equation 6-11 reduces to

G(x3,B, x3,A, ω) + G†(x3,A, x3,B, ω) =

−2∑

k=1

[

G(x3,B, x3;k, ω)N3,kG†(x3,A, x3;k, ω)

]

+

∫ x3;2

x3;1

G(x3,B, x3, ω)[

iω(

AB − A†A

)

+(

BB + B†A

)]

G†(x3,A,x, ω) dx3, (6-12)

where N3,k is a matrix containing zeros and ones, we have N3,1 = −N3,1. According toequation 6-12, we can retrieve a component of the Green’s function matrix and the time-reversed of its reciprocal counterpart, if we crosscorrelate observations of certain sources allalong the domain L and on the domain boundary edge points, with appropriate weightingfunctions.

Chapter 7

SH-TE interferometric Green’sfunction representations in 1D and 2D

In this section we derive interferometric Green’s function representations for the SH-TE sys-tem in 1D and 2D. There is a high degree of similarity in the equations and both derivationsfollow the same line and can be read independently. We start writing the SH-TE systemequations in a matrix form, such that the equations of the Chapter 7 can be applied to theseparate systems. In what follows we generalize the system equations by assuming that thereare elastic and magnetic losses, i.e. N and µ are frequency dependent. We briefly discuss thecontributions of the separate integrals in the interferometric representations, a more extensivediscussion follows in the next chapter with numerical examples.

7-1 SH-TE seismoelectric coupling in 1D

Starting from the 22 equations in the matrices in chapter 2 we drop the x1 and x2 dependence,see derivations in Appendix C-2. The SH-TE system in 1D is governed by the fields E2, H1,υs

2 and τ b23. The field w2 does not obey an independent differential equation, therefore we

eliminated w2 from the system. The SH-TE system in 1D can be captured in the generalflow, diffusion and wave equation as

iω

ε 0 −ρf L 00 µ 0 0

ρf L 0 ρc 0

0 0 0 N−1

E2

H1

υs2

−τ b23

+

0 −∂3 0 0−∂3 0 0 00 0 0 ∂3

0 0 ∂3 0

E2

H1

υs2

−τ b23

=

−Js,e2

−Js,m1

f2

hb

3

,

(7-1)

where we combined the matrices A and B, by defining ε = ǫ + 1iω σe and µ = µ + 1

iω σm. Thesystem matrices of the SH-TE system obey the same symmetry properties as those of the

64 SH-TE interferometric Green’s function representations in 1D and 2D

complete seismoelectric system matrices, see Section 2-2-1. We have

K0AtK0 = A, (7-2)

with

K0 = diag (−1, 1, 1,−1) . (7-3)

The field and source vectors are

u =

E2

H1

υs2

−τ b23

and s =

−Js,e2

−Js,m1

f2

hb

3

. (7-4)

We defined new electrical −Js,e2 , force f

2and deformation rate h3 sources, according to

Js,e2 = J

s,e2 + Lf

f2 , f

2= f b

2 − ρf

ρE ff2 and h

b

3 = hb32 + hb

23. The force on the fluid phase ff isnot an independent source type in the SH-TE system, see Section C-1, because there are nowaves traveling only through the fluid phase.

We replace the source vector s by a 4×4 point-source matrix Iδ(x3−x3,s) and correspondingly,the field vector u observed at x3,r is replaced by a 4 × 4 Green’s function matrix,

G(x3,r, x3,s, ω) =

GE,Je

GE,JmGE,f GE,h

GH,Je

GH,JmGH,f GH,h

Gυ,Je

Gυ,JmGυ,f Gυ,h

Gτ,Je

Gτ,JmGτ,f Gτ,h

(x3,r, x3,s, ω). (7-5)

We omitted all unnecessary superscripts and directional subscripts of the fields and sources.In the notation of subsequent sections we will also omit the ω dependence. Solutions forthe SH-TE seismoelectric Green’s functions in 1D in a homogeneous domain are given inAppendix C-3.

7-2 Convolution-type reciprocity theorem for SH-TE in 1D

In this section, we use the equations of Section 6-2 to find a convolution-type reciprocitytheorem for SH-TE waves in 1D in the frequency domain. If we substitute the field vector fromequation 7-4 for the two different states A and B into the interaction quantity Dx ·

(

K0uAutB

)

we find

∂3

[

E2,AH1,B − H1,AE2,B + τ b23,Aυs

2,B − υs2,Aτ b

23,B

]

. (7-6)

We apply the product rule to equation 7-6 and substitute the differential equations 7-1, afterintegration over the domain L from x3;1 to x3;2, we apply the theorem of Gauss to the sum

7-3 Correlation-type reciprocity theorem for SH-TE in 1D 65

containing the differential operator ∂3 and we find

2∑

k=1

[


2,B − υs2,Aτ b

23,B

]

n3;k =

∫ x3;2

x3;1

[

E2,AJs,e2,B − J

s,e2,AE2,B + J

s,m1,A H1,B − H1,AJ

s,m1,B

+ υs2,Af

2,B− f

2,Aυs

2,B + τ b23,Ah3,B − h3,Aτ b

23,B

]

dx3

+iω

∫ x3;2

x3;1

[

−E2,A (εA − εB) E2,B + E2,A

(

ρfALA − ρ

fBLB

)

υs2,B + H1,A (µA − µB) H1,B

+ υ2,A

(

ρfALA − ρ

fBLB

)

E2,B + υs2,A (ρc

A − ρcB) υs

2,B − τ b23,A

(

N−1A − N−1

B

)

τ b23,B

]

dx3.

(7-7)

This is the convolution-type reciprocity theorem for 1D SH-TE seismoelectric waves. Wesubstitute the SH-TE Green’s function matrix 7-5 in 1D and the symmetry matrix 7-3 intothe source-receiver relation for the Green’s matrix equation 6-7, to we find the source-receiverreciprocity relations as

GE,Je

(x3,A, x3,B) = GE,Je

(x3,B, x3,A) , GH,Je

(x3,A, x3,B) = −GE,Jm

(x3,B, x3,A), (7-8)

Gυ,Je

(x3,A, x3,B) = −GE,f (x3,B, x3,A) , Gτ,Je

(x3,A, x3,B) = GE,h(x3,B, x3,A), (7-9)

GH,Jm

(x3,A, x3,B) = GH,Jm

(x3,B, x3,A) , Gυ,f (x3,A, x3,B) = Gυ,f (x3,B, x3,A), (7-10)

Gτ,h(x3,A, x3,B) = Gτ,h(x3,B, x3,A) , Gυ,Jm

(x3,A, x3,B) = GH,f (x3,B, x3,A), (7-11)

Gτ,Jm

(x3,A, x3,B) = −GH,h(x3,B, x3,A) , Gτ,f (x3,A, x3,B) = −Gυ,h(x3,B, x3,A). (7-12)

7-3 Correlation-type reciprocity theorem for SH-TE in 1D

In this section, we use the equations of Section 6-3 to find a correlation-type reciprocitytheorem for 1D SH-TE waves in the frequency domain. If we substitute the field vector fromequation 7-4 for the two different states A and B into the interaction quantity Dx ·

(

uAutB

)

we find

−∂3

[

E∗2,AH1,B + H∗

1,AE2,B + τ b∗23,Aυs

2,B + υs∗2,Aτ b

23,B

]

. (7-13)

We apply the product rule to equation 7-13 and substitute the differential equations 7-1, afterintegration along the length of L from x3;1 to x3;2, we apply the theorem of Gauss to the sum


containing the differential operator ∂3 and we find

−2∑

k=1

[

E∗2,AH1,B + H∗


2,B + υs∗2,Aτ b

23,B

]

n3;k =

∫ x3;2

x3;1

[

−Js,e∗2,A E2,B − E∗

2,AJs,e2,B − J

s,m∗1,A H1,B − H∗

1,AJs,m1,B

+ υs∗2,Af

2,B+ f

∗2,A

υs2,B − h

∗3,Aτ b

23,B − τ b∗23,Ah3,B

]

dx3

−iω

∫ x3;2

x3;1

[

E∗2,A (εB − ε∗A) E2,B − E∗

2,A

(

ρfBLB + ρ

fAL∗

A

)

υs2,B + H∗

1,A (µB − µ∗A) H1,B

+ υ∗2,A

(

ρfBLB + ρ

fAL∗

A

)

E2,B + υs∗2,A (ρc

B − ρc∗A ) υs

2,B + τ b∗23,A

(

N−1B − N−1∗

A

)

τ b23,B

]

dx3.

(7-14)

This is the correlation-type reciprocity theorem for SH-TE seismoelectric waves in 1D. Notethat the domain integral on the right-hand side of equation 7-14 does not disappear if wechoose equal medium parameters in both states. This domain integral is over the loss functionsof the system and over the coupling coefficient, that governs the conversion of electromagneticenergy to elastodynamic energy and vice versa.

7-4 Seismoelectric interferometric Green’s function recovery in 1D

From equation 6-12 we can derive an interferometric representation of the SH-TE Green’sfunction matrix in 1D. We substitute the Green’s function matrix 7-5 and medium parametermatrix of equation 7-1 into equation 6-12, to find

GIJ(x3,B, x3,A) + G∗JI(x3,A, x3,B) =

2∑

k=1

n3;k

[

GJm

I2 (x3,B, x3;k)GJe∗J1 (x3,A, x3;k) + G

Je

I1 (x3,B, x3;k)GJm∗J2 (x3,A, x3;k)

−GhI4(x3,B, x3;k)G

f∗J3(x3,A, x3;k) − G

f

I3(x3,B, x3;k)Gh∗J4(x3,A, x3;k)

]

+iω2

∫ x3;2

x3;1

[

GJe

I1 (x3,B, x3)iIm{ε}GJe∗J1 (x3,A, x3) − G

Je

I1 (x3,B, x3)Re{ρf L}Gf∗J3(x3,A, x3)

+Gf

I3(x3,B, x3)Re{ρf L}GJe∗J1 (x3,A, x3) + G

f

I3(x3,B, x3)iIm{ρc}Gf∗J3(x3,A, x3)

+GJm

I2 (x3,B, x3)iIm{µ}GJm∗J2 (x3,A, x3) + G

hI4(x3,B, x3)iIm{N−1}Gh∗

J4(x3,A, x3)]

dx3,

(7-15)

in which GIJ(x3,A, x3,B) denotes the I th element of the J th column of the Green’s functionmatrix 7-5 at x3,B yielding a response due to a point-source at x3,A. The domain integral onthe right-hand sides of equation 7-15 is a sum of crosscorrelations of observed responses ofsources weighted by the loss and seismoelectric coupling functions. This integral not onlycompensates for the losses in L, but also contributes to the seismoelectric coupling in L.The summation of crosscorrelations of responses of sources on the end points of L accountsfor the contributions to the Green’s functions from losses and seismoelectric coupling outside

7-4 Seismoelectric interferometric Green’s function recovery in 1D 67

the domain L.

We can use 7-15 for an interferometric expression for every element of the Green’s functionmatrix G(x3,B, x3,A). For example, for the particle velocity υs

2 in the solid at observationlocation x3,B due to an electrical current source −J

s,e2 at observation location x3,A, we have

Gυs,Je

(x3,B, x3,A) + GE,f∗(x3,A, x3,B) =2∑

k=1

n3;k

[

Gυ,Jm

(x3,B, x3;k)GE,Je∗(x3,A, x3;k) + Gυ,Je

(x3,B, x3;k)GE,Jm∗(x3,A, x3;k)

−Gυ,h(x3,B, x3;k)GE,f∗(x3,A, x3;k) − Gυ,f (x3,B, x3;k)G

E,h∗(x3,A, x3;k)]

+iω2

∫ x3;2

x3;1

[

Gυ,Je

(x3,B, x3)iIm{ε}GE,Je∗(x3,A, x3) − Gυ,Je

(x3,B, x3)Re{ρf L}GE,f∗(x3,A, x3)

+Gυ,f (x3,B, x3)Re{ρf L}GE,Je∗(x3,A, x3) + Gυ,f (x3,B, x3)iIm{ρc}GE,f∗(x3,A, x3)

+Gυ,Jm

(x3,B, x3)iIm{µ}GE,Jm∗(x3,A, x3) + Gυ,h(x3,B, x3)iIm{N−1}GE,h∗(x3,A, x3)]

dx3.

(7-16)

7-4-1 Interferometric representation in medium type A and B

We simplify equation 7-15 for the porous medium types A and B of Chapter 5 that haveµ = µ0 and obey perfect elasticity N = N . The volume integral over the magnetic and elasticloss functions disappears. We consider all the sources to emit a Ricker wavelet srw(ω) witha power spectrum Srw(ω), thus we replaced the source vector s of equation 7-4 by a 4 × 4point-source matrix Iδ(x3 − x3,s)s

rw. We would find for the particle velocity υs2 at x3,B due

to a current source −Js,e2 at x3,A,{

Gυs,Je

(x3,B, x3,A) + GE,f∗(x3,A, x3,B)}

Srw =

2∑

k=1

n3;k

[

Gυ,Jm

(x3,B, x3;k)GE,Je∗(x3,A, x3;k) + Gυ,Je

(x3,B, x3;k)GE,Jm∗(x3,A, x3;k)

−Gυ,h(x3,B, x3;k)GE,f∗(x3,A, x3;k) − Gυ,f (x3,B, x3;k)G

E,h∗(x3,A, x3;k)]

Srw

+iω2

∫ x3;2

x3;1

[

Gυ,Je

(x3,B, x3)iIm{ε}GE,Je∗(x3,A, x3) − Gυ,Je

(x3,B, x3)Re{ρf L}GE,f∗(x3,A, x3)

+Gυ,f (x3,B, x3)Re{ρf L}GE,Je∗(x3,A, x3, ω) + Gυ,f (x3,B, x3)iIm{ρc}GE,f∗(x3,A, x3)]

Srwdx3.

(7-17)

In Chapter 8 we use this equation to create an example of the SH-TE seismoelectric interfer-ometric representation in 1D.

7-4-2 Relation between the two retrieved Green’s function matrices

The left-hand side of equation 7-15 is a sum of the causal Green’s function matrix and thetime reversed of it’s reciprocal Green’s function matrix. Using the source-receiver reciprocity


relations 7-8 to 7-12, we can explore the left-hand side of equation 7-15 in the frequencydomain. If there is a sign switch between GIJ(x3,B, x3,A) and G∗

JI(x3,A, x3,B) we retrievean imaginary signal, this means that the retrieved signal is asymmetric in the time domain.While if there is no sign switch between GIJ(x3,B, x3,A) and G∗

JI(x3,A, x3,B), we retrievea purely real signal, which means that in the time domain we have a symmetric function.Applying the source-receiver relations 7-8 to 7-12 to the left-hand of equation 7-15, yields

G(x3,B, x3,A) + G†(x3,A, x3,B) =

2

Re{GE,Je} iIm{GE,Jm} iIm{GE,f} Re{GE,h}iIm{GH,Je} Re{GH,Jm} Re{GH,f} iIm{GH,h}iIm{Gυ,Je} Re{Gυ,Jm} Re{Gυ,f} iIm{Gυ,h}Re{Gτ,Je} iIm{Gτ,Jm} iIm{Gτ,f} Re{Gτ,h}

(x3,B, x3,A). (7-18)

7-5 SH-TE seismoelectric coupling in 2D

In Chapter 3 we have seen that in 2D the seismoelectric system decouples into the SH-TEand P-SV-TM modes of propagation. The SH-TE mode is governed by the fields E2, H1, H3,υs

2, −τ b21, and −τ b

23. The coupling coefficient in the 3D seismoelectric system is incorporatedin the equations for w, the differential velocity of the solid and fluid phases, w. In AppendixC-1 we see that in 2D the field w2 is does not obey an independent differential equation. Weeliminated w2 from the system of equations and defined new source functions that incorporatethe force on the fluid phase. The SH-TE system in 2D can be captured in the general flow,diffusion and wave equation as

iωAu + Bu + Dxu = s. (7-19)

Where the field u and source vectors s are defined by

ut =(

E2, H1, H3, υs2,−τ b

21,−τ b23

)

(7-20)

st =(

−Je2,−Jm

1 ,−Jm1 , f

2, h

b

1, hb

3

)

(7-21)

With the new source types Js,e2 = J

s,e2 + Lf

f2 , f

2= f b

2 − ρf

ρE ff2 , h

b

3 = hb32 + hb

23 and

hb

1 = hb12 + hb

21. The system matrices A, B and Dx are given by

A =

ǫ 0 0 −ρf L 0 00 µ 0 0 0 00 0 µ 0 0 0

ρf L 0 0 ρt 0 00 0 0 0 1

N 00 0 0 0 0 1

N

, B =

σe 0 0 0 0 00 σm 0 0 0 00 0 σm 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

(7-22)

and Dx =

0 −∂3 ∂1 0 0 0−∂3 0 0 0 0 0∂1 0 0 0 0 00 0 0 0 ∂1 ∂3

0 0 0 ∂1 0 00 0 0 ∂3 0 0

. (7-23)

7-6 Convolution-type reciprocity theorem for SH-TE in 2D 69

The system matrices of the SH-TE system obey the same symmetry properties as those ofthe entire seismoelectric system, see Section 2-2-1. We have

K0AK0 = A and K0BK0 = B. (7-24)

The matrix Dx, containing spatial derivatives, obeys

K0DxK0 = −Dx = −Dtx, (7-25)

where K0 is defined by

K0 = diag (−1, 1, 1, 1,−1,−1) . (7-26)

We replace the source vector s by a 6× 6 point-source matrix Iδ(x−xs) and correspondinglyreplace the field vector u observed at xr by a 6 × 6 Green’s function matrix,

G(xr,xs, ω) =

GE,Je

GE,Jm1 GE,Jm

3 GE,f GE,h1 GE,h3

GH1,Je

GH1,Jm1 GH1,Jm

3 GH1,f GH1,h1 GH1,h3

GH3,Je

GH3,Jm1 GH3,Jm

3 GH3,f GH3,h1 GH3,h3

Gυ,Je

Gυ,Jm1 Gυ,Jm

3 Gυ,f Gυ,h1 Gυ,h3

Gτ21,Je

Gτ21,Jm1 Gτ21,Jm

3 Gτ21,f Gτ21,h1 Gτ21,h3

Gτ23,Je

Gτ23,Jm1 Gτ23,Jm

3 Gτ23,f Gτ23,h1 Gτ23,h3

(xr,xs, ω). (7-27)

Where we omitted all unnecessary directional subscripts of the fields in the superscripts. Inthe notation of subsequent sections we will also omit the ω dependence. In part I of thisthesis, we further reduced this set by eliminating the fields H3 and τ b

21 from the field vectoru. This way we obtained the two-way wave equation that we used to construct seismoelectricGreen’s functions for the SH-TE system in 2D in horizontally-stratified media.

7-6 Convolution-type reciprocity theorem for SH-TE in 2D

In this section we use the equations of Section 6-2 to find a convolution-type reciprocitytheorem for SH-TE waves in 2D in the frequency domain. If we substitute the field vector fromequation 7-21 for the two different states A and B into the interaction quantity Dx·

(

K0uAutB

)

,we find

∂3

[

E2,AH1,B − H1,AE2,B + τ b23,Aυs2,B − υs

2,Aτ b23,B

]

+∂1

[

H3,AE2,B − E2,AH3,B + τ b21,Aυs

2,B − υs2,Aτ b21,B

]

. (7-28)

We apply the product rule to equation 7-28 and substitute the differential equations 7-22and 7-23, after integration over the domain S, we apply the theorem of Gauss to the integral


containing the differential operators ∂3 and ∂1 and we find∮

∂S

[{


2,B − υs2,Aτ b

23,B

}

n3

+{

H3,AE2,B − E2,AH3,B + τ b21,Aυs

2,B − υs2,Aτ b

21,B

}

n1

]

dx =∫

S

[

E2,AJs,e2,B − J

s,e2,AE2,B + J


s,m1,B + J


s,m3,B

+υs2,Af

2,B− f

2,Aυs

2,B + τ b23,Ah3,B − h3,Aτ b

23,B + τ b21,Ah1,B − h1,Aτ b

21,B

]

d2x

+iω

∫

S

[

−E2,A (εA − εB) E2,B + E2,A

(

ρfALA − ρ

fBLB

)

υs2,B

+H1,A (µA − µB) H1,B + H3,A (µA − µB) H3,B

+υs2,A

(

ρfALA − ρ

fBLB

)

E2,B + υs2,A (ρc

A − ρcB) υs

2,B

−τ b23,A

(

N−1A − N−1

B

)

τ b23,B − τ b

21,A

(

N−1A − N−1

B

)

τ b21,B

]

d2x.

(7-29)

This is the convolution-type reciprocity theorem for 2D SH-TE seismoelectric waves. Wesubstitute the SH-TE Green’s function matrix 7-27 in 2D and the symmetry matrix 7-26 intothe source-receiver relation for the Green’s matrix equation 6-7, to we find the source-receiverreciprocity relations as

GE,Je

(xA,xB) = GE,Je

(xB,xA) , GH1,Jm1 (xA,xB) = GH1,Jm

1 (xB,xA), (7-30)

GH3,Jm3 (xA,xB) = GH3,Jm

3 (xB,xA) , GH1,Jm3 (xA,xB) = GH3,Jm

1 (xB,xA), (7-31)

GH1,Je

(xA,xB) = −GE,Jm1 (xB,xA) , GH3,Je

(xA,xB) = −GE,Jm3 (xB,xA), (7-32)

Gυs,Jm1 (xA,xB) = GH1,f (xB,xA) , Gυs,Jm

3 (xA,xB) = GH3,f (xB,xA), (7-33)

Gτ23,Je

(xA,xB) = GE,h3(xB,xA) , Gτ,Je21(xA,xB) = GE,h1(xB,xA), (7-34)

Gτ23,Jm

(xA,xB) = −GH,h3(xB,xA) , Gτ21,Jm

(xA,xB) = −GH,h1(xB,xA), (7-35)

Gτ23,Jm3 (xA,xB) = −G

H,h3

3 (xB,xA) , Gτ21,Jm3 (xA,xB) = −G

H,h1

3 (xB,xA), (7-36)

Gυ,f (xA,xB) = Gυ,f (xB,xA) , Gτ23,h3(xA,xB) = Gτ23,h3(xB,xA), (7-37)

Gτ21,h1(xA,xB) = Gτ21,h1(xB,xA) , Gτ23,h1(xA,xB) = Gτ21,h3(xB,xA), (7-38)

Gτ23,f (xA,xB) = −Gυ,h3(xB,xA) , Gτ21,f (xA,xB) = −Gυ,h1(xB,xA), (7-39)

Gυ,Je

(xA,xB) = −GE,f (xB,xA). (7-40)

7-7 Correlation-type reciprocity theorem for SH-TE in 2D

In this section we use the equations of Section 6-3 to find a correlation-type reciprocitytheorem for SH-TE waves in 2D in the frequency domain. We substitute the field vector fromequation 7-21 for two different states A and B into the interaction quantity Dx ·

(

uAutB

)

, tofind

− ∂3

[

E∗2,AH1,B + H∗


2,B + υs∗2,Aτ b

23,B

]

+∂1

[

H∗3,AE2,B + E∗

2,AH3,B − τ b∗21,Aυs

2,B − υs∗2,Aτ b

21,B

]

. (7-41)


We apply the product rule and substitute the differential equations equations 7-22 and 7-23,after integration over the domain S, we apply the theorem of Gauss to the integral containingthe differential operators ∂3 and ∂1 and we find

∮

∂S

[

−{

E∗2,AH1,B + H∗


2,B + υs∗2,Aτ b

23,B

}

n3

+{

H∗3,AE2,B + E∗

2,AH3,B − τ b∗21,Aυs∗

2,B − υs∗2,Aτ b

21,B

}

n1

]

dx =∫

S

[

−Js,e∗2,A E2,B − E∗

2,AJs,e2,B − J

s,m∗1,A H1,B − H∗

1,AJs,m1,B − J

s,m∗3,A H3,B − H∗

3,AJs,m3,B

+υs2,Af

2,B+ f

∗2,A

υs2,B − τ b∗

23,Ahb

3,B − hb∗3,Aτ b

23,B − τ b∗21,Ah

b

1,B − hb∗1,Aτ b

21,B

]

d2x

−iω

∫

S

[

E∗2,A (εB − ε∗A) E2,B − E∗

2,A

(

ρfBLB + ρ

f∗A L∗

A

)

υs2,B

+H∗1,A (µB − µ∗

A) H1,B + H∗3,A (µB − µ∗

A) H3,B

+υs∗2,A

(

ρfBLB + ρ

f∗A L∗

A

)

E2,B + υs∗2,A (ρc

B − ρc∗A ) υs

2,B

+τ b∗23,A

(

N−1B − N−1∗

A

)

τ b23,B + τ b∗

21,A

(

N−1B − N−1∗

A

)

τ b21,B

]

d2x.

(7-42)

This is the correlation-type reciprocity theorem for SH-TE seismoelectric waves in 2D. Notethat the right-hand side of equation 7-42 does not disappear if we choose equal mediumparameters in both states. The right-hand side of equation 7-42 contains a volume integralover the loss functions of the system and over the coupling coefficient, that governs theconversion of electromagnetic energy to elastodynamic energy and vice versa.

7-8 Seismoelectric interferometric Green’s function recovery in 2D

From equation 6-11 we can derive an interferometric representation of the SH-TE Green’sfunction matrix in 2D. We substitute the Green’s function matrix 7-27 and the medium


parameter matrices 7-23 into equation 6-11, to find

GIJ(xB,xA) + G∗JI(xA,xB) =

−∮

∂S

[

−GJm1

I2 (xB,x)n3GJe∗J1 (xA,x) − G

Je

I1 (xB,x)n3GJm1 ∗

J2 (xA,x)

+Gh3

I6(xB,x)n3Gf∗J4(xA,x) + G

f

I4(xB,x)n3Gh3∗J6 (xA,x)

+GJe

I1 (xB,x)n1GJm3 ∗

J3 (xA,x) + GJm3

I3 (xB,x)n1GJe∗J1 (xA,x)

+Gf

I4(xB,x)n1Gh∗1

J5(xA,x) + Gh1

I5(xB,x)n1Gf∗J4(xA,x)

]

dx

+iω2

∫

S

[

GJe

I1 (xB,x)iIm{ε}GJe∗J1 (xA,x) − G

Je

I1 (xB,x)Re{ρf L}Gf∗J4(xA,x)

+Gf

I4(xB,x)Re{ρf L}GJe∗J1 (xA,x) + G

f

I4(xB,x)iIm{ρc}Gf∗J4(xA,x)

+GJm

1

I2 (xB,x)iIm{µ}GJm1 ∗

J2 (xA,x) + GJm

3

I3 (xB,x)iIm{µ}GJm3 ∗

J3 (xA,x)

+Gh1

I5(xB,x)iIm{N−1}Gh1∗J5 (xA,x) + G

h3

I6(xB,x)iIm{N−1}Gh3∗J6 (xA,x)

]

d2x.

(7-43)

In which GIJ(xA, xB) denotes the I th element of the J th column of the Green’s functionmatrix 7-27 at xB yielding a response due to a point-source at xA. The domain integral overS on the right-hand side of equation 7-43 is a sum of crosscorrelations of observed responsesof sources weighted by the loss and seismoelectric coupling functions. This integral notonly compensates for the losses inside S, but also contributes to the seismoelectric couplinginside S. The crosscorrelations of responses of sources on the boundary ∂S accounts for thecontributions to the Green’s functions from losses and seismoelectric coupling outside S.

We can use equation 7-43 for an interferometric expression for every element of the Green’sfunction matrix G(xB, xA). For example, for the particle velocity υs

2 in the solid at observationlocation xB due to an electrical current source −J

s,e2 at observation location xA, we have

Gυ,Je

(xB,xA) + GE,f∗(xA,xB) =∮

∂S

[

Gυ,Jm1 (xB,x)n3G

E,Je∗(xA,x) + Gυ,Je

(xB,x)n3GE,Jm

1 ∗(xA,x)

−Gυ,h3(xB,x)n3GE,f∗(xA,x) − Gυ,f (xB,x)n3G

E,h3∗(xA,x)

−Gυ,Je

(xB,x)n1GE,Jm

3 ∗(xA,x) − Gυ,Jm3 (xB,x)n1G

E,Je∗(xA,x)

−Gυ,f (xB,x)n1GE,h1∗(xA,x) − Gυ,h1(xB,x)n1G

E,f∗(xA,x)]

dx

+iω2

∫

S

[

Gυ,Je

(xB,x)iIm{ε}GE,Je∗(xA,x) − Gυ,Je

(xB,x)Re{ρf L}GE,f∗(xA,x)

+Gυ,f (xB,x)Re{ρf L}GE,Je∗(xA,x) + Gυ,f (xB,x)iIm{ρc}GE,f∗(xA,x)

+Gυ,Jm1 (xB,x)iIm{µ}GE,Jm

1 ∗(xA,x) + Gυ,Jm3 (xB,x)iIm{µ}GE,Jm

3 ∗(xA,x)

+Gυ,h3(xB,x)iIm{N−1}GE,h3∗(xA,x) + Gυ,h1(xB,x)iIm{N−1}GE,h1∗(xA,x)]

d2x.

(7-44)


7-8-1 Interferometric representation in medium type A and B

We simplify equation 7-15 this for porous medium types A and B of Chapter 5 that haveµ = µ0 and obey perfect elasticity N = N . The volume integral over the magnetic and elasticloss functions disappears. We separate the boundary integral into a horizontal part S3 andvertical part S1. The horizontal boundaries have normal vector n3 = 1 for the lower boundaryand n3 = −1 for the upper boundary. The vertical boundaries have normal vector n1 = −1for the left side and n1 = 1 for the right side boundary. We consider all the sources to emita Ricker wavelet srw(ω) with a power spectrum Srw(ω), thus we replaced the source vector s

of equation 7-21 by a 6× 6 point-source matrix Iδ(x−xs)srw. We would find for the particle

velocity υs2 at x3,B due to a current source −J

s,e2 at x3,A,

{

Gυ,Je

(xB,xA) + GE,f∗(xA,xB)}

Srw =∮

∂S3

[

Gυ,Jm1 (xB,x)n3G

E,Je∗(xA,x) + Gυ,Je

(xB,x)n3GE,Jm

1 ∗(xA,x)

−Gυ,h3(xB,x)n3GE,f∗(xA,x) − Gυ,f (xB,x)n3G

E,h3∗(xA,x)]

Srwdx

−∮

∂S1

[

Gυ,Je

(xB,x)n1GE,Jm

3 ∗(xA,x) + Gυs,Jm3 (xB,x)n1G

E,Je∗(xA,x)

+Gυ,f (xB,x)n1GE,h1∗(xA,x) + Gυ,h1(xB,x)n1G

E,f∗(xA,x)]

Srwdx

+iω2

∫

S

[

Gυ,Je

(xB,x)iIm{ε}GE,Je∗(xA,x) − Gυ,Je

(xB,x)Re{ρf L}GE,f∗(xA,x)

+Gυ,f (xB,x)Re{ρf L}GE,Je∗(xA,x) + Gυ,f (xB,x)iIm{ρc}GE,f∗(xA,x)]

Srwd2x.

(7-45)

In Chapter 8 we use this equation to recover the SH-TE seismoelectric Green’s function in2D in homogeneous media.

7-8-2 Relation between the two retrieved Green’s function matrices

The left-hand side of equation 7-43 is a sum of the causal Green’s function matrix and thetime reversed of it’s reciprocal Green’s function matrix. Using the source-receiver reciprocityrelations 7-30 to 7-40, we can explore the left-hand side of equation 7-43 in the frequencydomain. If there is a sign switch between GIJ(x3,B, x3,A) and G∗

JI(x3,A, x3,B) we retrievean imaginary signal, this means that the retrieved signal is asymmetric in the time domain.While if there is no sign switch between GIJ(x3,B, x3,A) and G∗

JI(x3,A, x3,B), we retrievea purely real signal, which means that in the time domain we have a symmetric function.


Applying the source-receiver relations 7-30 to 7-40 to the left-hand of equation 7-43, yields

G(xB,xA) + G†(xA,xB) =

2

Re{GE,Je} iIm{GE,Jm1 } iIm{GE,Jm

3 } iIm{GE,f} Re{GE,h1} Re{GE,h3}iIm{GH1,Je} Re{GH1,Jm

1 } Re{GH1,Jm3 } Re{GH1,f} iIm{GH1,h1} iIm{GH1,h3}

iIm{GH3,Je} Re{GH3,Jm1 } Re{GH3,Jm

3 } Re{GH3,f} iIm{GH3,h1} iIm{GH3,h3}iIm{Gυ,Je} Re{Gυ,Jm

1 } Re{Gυs,Jm3 } Re{Gυ,f} iIm{Gυ,h1} iIm{Gυ,h3}

Re{Gτ21,Je} iIm{Gτ21,Jm1 } iIm{Gτ21,Jm

3 } iIm{Gτ21,f} Re{Gτ21,h1} Re{Gτ21,h3}Re{Gτ23,Je} iIm{Gτ23,Jm

1 } iIm{Gτ23,Jm3 } iIm{Gτ23,f} Re{Gτ23,h1} Re{Gτ23,h3}

(xB,xA).

(7-46)

Chapter 8

Simulation of interferometricseismoelectric Green’s function

recovery

This chapter contains several examples of the interferometric representations of Chapter 7. Weconsider a homogeneous world and perform the experiments separately in the two media typesintroduced in Chapter 5. The 1D examples have been created using the Green’s functions ofAppendix C-3. The 2D example is created using the one-way wave equations of Chapter 3.We describe the observations of our experiments, paying particular attention to the role of thedifferent integrals of interferometric representations. All events in the 1D and 2D correlationshave been consistently labeled. There are two physical events, see Section 5-3, the first arriveswith the electromagnetic wave velocity and is labeled a. The second event arrives with theshear wave velocity and is labeled b. There is a spurious event in the separate contributionsof the domain and boundary integrals, this event is labeled c. The events are not alwayscontinuous in the figures, in that case they have been assigned an extra subscript that isexplained in the text.

8-1 1D Seismoelectric interferometry in homogeneous media.

We consider the situation of 2 receivers, labeled A and B, at a distance of 150 meter, seeFigure 8-1. The domain L is chosen to extend 325 meter away from the receivers. We placetwo source types at 8000 source positions evenly spaced by 10 cm, between the end pointsof L, thus performing a middle Riemann sum to evaluate the domain integral. Each sourceis weighted as prescribed in the 1D interferometric representation equation 7-17. At eachof the end points we place 8 sources. At receiver A we measure the electrical field and atreceiver B we measure the particle velocity, evaluating the interferometric representation forGυ,Je

(xB,xA, t) + GE,f (xA,xB,−t) in 1D given by equation 7-17. Thus we should recoverthe sum of two Green’s functions, the causal response of the particle velocity at B due to a

76 Simulation of interferometric seismoelectric Green’s function recovery

∇∇x

3,A

∇∇x

3,Bx

3;1x

3;2

325 m 150 m 325 m

Figure 8-1: Symmetric position of receivers and boundary points for 1D interferometry; receiverdistance is 150 m, the boundary points are 325 meter away from the receivers. Examples areshown in Figures 8-2, 8-3, 8-4 and 8-5.

electrical current source at B plus the time-reversed response of the electrical field at A as aresult of a force source at B. All sources emit a Ricker wavelet, equation 5-12, with a centralradial frequency of ω0 = 800 radians. Thus the recovered Green’s functions are convolvedwith the autocorrelation of the source function, see equation 7-17. The Green’s functionswere calculated using a time sampling of ∆t = 0.001 seconds.

8-1-1 Results in medium type A

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−5

0

5

x 10−12

time [s]

GvJ

e (t)+

GE

f (−t)

exactretrieved

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−1

−0.5

0

0.5

1x 10

−15

time [s]

GvJ

e (t)+

GE

f (−t)

exactretrieved

a

cr

cl

b−

b+

Medium type A

Figure 8-2: Exact and retrieved superposition of Gυ,Je

(xB ,xA, t) and GE,f (xA,xB ,−t) inmedium type A, using two source types at 8000 source positions evenly spaced over 800 meter.Receiver offset is 150 meter as indicated in Figure 8-1.

We perform the computations using the medium parameters as described for medium type Ain Section 5-2. We recover the Green’s functions reasonably well and find two physical eventsin each Green’s function, see Figure 8-2. The second event, labeled b, arrives at t = ±0.07

8-1 1D Seismoelectric interferometry in homogeneous media. 77

seconds and is three orders of magnitude smaller than the first event, labeled a, that arrivesaround t = 0 seconds. Event a is actually the superposition of the electromagnetic arrivals inGυ,Je

(xB,xA, t) and GE,f (xA,xB,−t). There are two visible spurious events, labeled cr andcl in positive times at t = 0.15 seconds and t = 0.22 seconds, they are about half as strongas event b, see Section 8-1-3 for their origin. In Figure 8-3 we see the separate contributionsof the boundary points and the domain integral. We can see in the top panel of Figure8-3 that the amplitudes of both integrals are strong for event a but they are reversed inpolarity. The contribution from the domain integral is stronger than the desired event, hencethe contribution of the boundary is subtracted, this results in the perfect reconstruction ofevent a. Event b+ in Gυ,Je

(xB,xA, t) is reconstructed in a similar way. The contribution ofthe domain integral is stronger than that of the boundary points and has the correct phase asevent b+ of the exact Green’s function. The contribution of the boundary integral is weakerand reversed in polarity with respect to the contribution of the domain integral, they subtractand perfectly reconstruct the shear wave event in positive times. Event b− in GE,f (xA,xB,−t)

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−5

0

5

x 10−12

time [s]

GvJ

e (t)+

GE

f (−t)

boundarydomain

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−1

0

1

2x 10

−15

time [s]

GvJ

e (t)+

GE

f (−t)

boundarydomain

ac

lc

r

b−

b+

Medium type A

Figure 8-3: Separate contributions of the domain integral (red dashed curve) and the boundarypoints (blue continuous curve) to the reconstruction of the Green’s functions in Figure 8-2.

is reconstructed differently. The contribution of the boundary points seem to be equal (oneyesight) to the contribution of the boundary to event b+ in Gυ,Je

(xB,xA, t). However, thetotal contribution of the domain integral is a lot smaller to b− than to b+ in Figure 8-3. Butthe contributions of the domain and boundary integrals add constructively to reconstruct theevent b− in GE,f (xA,xB,−t), which is anti-symmetric to the event b+ in Gυ,Je

(xB,xA, t).


8-1-2 Results in medium type B

We perform the same computations as above, in the medium with characteristics described asmedium type B in Section 5-2. We recover the Green’s functions even better then in mediumtype A using the same geometry, see Figure 8-1. Events b+ and b− have slightly larger arrivaltimes then the second events in medium type A, they are two orders of magnitude smallerthan event a. Spurious events cr and cl are three orders of magnitude smaller than eventsb− and b+. In Figure 8-5 we see the separate contributions of the boundary points and thedomain integral. We see that the contributions to the reconstructed Green’s functions behavesimilarly as in medium type A. But we do notice that the relative contribution of the boundarypoints to event, a, is clearly smaller than the relative contribution of the domain integral.It can also be observed that event a is three orders of magnitude smaller in medium type Bthan in medium type A, while the seismic event is only one order of magnitude smaller. Butwe see that the relative contribution of the domain integral to event b− in GE,f (xA,xB,−t)is smaller than it was in medium type A.

8-1 1D Seismoelectric interferometry in homogeneous media. 79

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−5

0

5

x 10−15

time [s]

GvJ

e (t)+

GE

f (−t)

exactretrieved

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−1

−0.5

0

0.5

1x 10

−16

time [s]

GvJ

e (t)+

GE

f (−t)

exactretrieved

a

b−

b+

Medium type B


(xB ,xA, t) and GE,f (xA,xB ,−t) inmedium type B, using two source types at 8000 source positions evenly spaced over 800 meter.Receiver offset 150 meter as indicated in Figure 8-1.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−5

0

5

x 10−15

time [s]

GvJ

e (t)+

GE

f (−t)

boundaryvolume

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−1

0

1

2x 10

−16

time [s]

GvJ

e (t)+

GE

f (−t)

boundaryvolume

ac

rc

l

b−

b+

Medium type B

Figure 8-5: Separate contributions of the domain integral (red dashed curve) and the boundarypoints (blue continuous curve) to the reconstruction of the Green’s functions in Figure 8-4.


8-1-3 Middle Riemann sum

The spurious events in the contributions of the domain integral and the boundary pointsshould cancel out exactly when combined. In these simulations they do so only approximately,because we evaluated the domain integral by performing a middle Riemann sum. The middleRiemann sum divides the integrand in equally spaced regions, the value of the integrand overthat region is approximated by the exact value in the middle of that region weighted by thesize of the region. The smaller the regions are, the more exact the Riemann sum is. In Figure8-6 we choose three different source spacings; 1.0 meter, 0.5 meter and 0.1 meter and evaluatethe 1D interferometric integral equation 7-17 for both media types. We show the exact resultwith a blue continuous line, there are no physical events at times later than t = 0.1 seconds.However, there are two visible remainders of the imperfect cancellation of spurious events cr

and cl arriving at times later than t = 0.1 seconds. This remainder is larger in medium typeA than in medium type B in absolute sense and relative to event, b+, near 0.07 seconds.

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−2

−1

0

1

2x 10

−14

GvJ

e (t)+

GE

f (−t)

exact∆ x

3,s = 1.0 m

∆ x3,s

= 0.5 m

∆ x3,s

= 0.1 m

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−2

−1

0

1

2x 10

−16

time [s]

GvJ

e (t)+

GE

f (−t)

exact∆ x

3,s = 1.0 m

∆ x3,s

= 0.5 m

∆ x3,s

= 0.1 m

Medium type A

Medium type B

a cl

crb

+

ac

lc

rb

+

Figure 8-6: Recovered signal using three different sample densities of the Riemann sum; ∆s = 1meter, ∆s = .5 meter, ∆s = 0.1 meter.

8-2 Dissecting the 1D interferometric representation

In this section, we take a closer look to the different contributions from the boundarypoints and different parts of the domain integral, to the reconstruction of Gυ,Je

(xB,xA, t) +GE,f (xA,xB,−t) in medium type A. We distinguish between the contributions of the differ-ent sides of the boundary contribution, from x3;1 and from x3;2. The domain integral can bedissected in three contributions, from the segment left of receiver A, the segment between thetwo receivers and the segment part on the right side of receiver B, see Figure 8-7.

8-2 Dissecting the 1D interferometric representation 81

In Figure 8-8 we see the contribution of the sources at x3;1 ( blue continuous curve) and atx3;2 (red dashed curve). We see how they each contribute equally to event, a. Each of theboundary points creates its own spurious event, cr and cl, both in positive times. Event cl isconstructed by the sources at the left boundary point, x3;1. Event cr is constructed by sourcesat the right boundary point, x3;2. The contribution from the sources at x3;1 (continuousblue curve) ends up in the Green’s function of waves traveling in the same direction fromGυ,Je

(xB,xA, t) and vice versa for the contribution of x3;2 (dashed red curve) to the Green’sfunction GE,f (xA,xB,−t). Lastly, we notice how the significant contributions from x3;1 seemto be all causal with a zero-phase wavelet, but the contributions from x3;2 are both causaland anti-causal.

The contributions of the three segments of the domain integral are shown in Figure 8-9. Fromeach segment we can distinguish three events. Segment labeled 2 has both causal and anti-causal contributions, but it only contributes to physical events. Its contribution to event b− ,in GE,f (xA,xB,−t) is two orders of magnitude smaller than the contribution made by segment3, which is still an order of magnitude smaller than the contribution of the boundary points,see Figures 8-8 and 8-5. In GE,f (xA,xB,−t) the domain and boundary contributions haveequal polarity. The contribution of segment 2 to event b+ in Gυ,Je

(xB,xA, t) is almost equalto the contribution by segment 1, but reversed in polarity. Therefore they nearly cancel, theresult is an event three orders of magnitude smaller, that still contributes strongly to the totalreconstructed event b+, see Figures 8-2 and 8-3. All three segments contribute to event a, butthe contribution of segment 1 is at least an order of magnitude smaller than the contributionsof segments 2 and 3, which are approximately equal, all three segments contribute with thesame polarity to event a. The spurious events cr and cl, come from the two outer segments.The arrivals from segment 1, within the first 7 to 8 orders, are all causal with a zero-phasewavelet, while the contributions from segment 3 are both causal and anti-causal.

8-2-1 Main contributions in crosscorrelations of the interferometric representa-tion.

In this subsection we analyse the contributions of the different terms in the boundary anddomain integrals. The interferometric representation, equation 7-17, contains 8 differentcrosscorrelations of Green’s functions. The arrivals of shear or electromagnetic waves in theGreen’s functions are stronger or weaker for different source types. The crosscorrelations

∇∇x

3,A

∇∇x

3,Bx

3;1x

3;2

1; 325 m 2; 150 m 3; 325 m

Figure 8-7: Three different segments of the domain integral. The first segment (labeled 1) islocated left of receiver A and is 325 meter long, the second segment (labeled 2) lies in betweenthe two receivers and is 150 meter long, the third segment (labeled 3) is located right of receiverB and is 325 meter long.


−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−2

0

2

x 10−12

time [s]

GvJ

e (t)+

GE

f (−t)

left boundaryright boundary

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

−5

0

5

x 10−16

time [s]

GvJ

e (t)+

GE

f (−t)

a cl

cr

b−

b+

Figure 8-8: The separate contributions of the sources at x3;1 (blue continuous curve) and thesources at x3;2 (red dashed curve) to the total boundary contribution in Figure 8-3.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−4

−2

0

2

4x 10

−12

time [s]

GvJ

e (t)+

GE

f (−t)

line segment 1line segment 2line segment 3

−0.1 −0.08 −0.06 −0.04

−2

0

2

x 10−17

time [s]

GvJ

e (t)+

GE

f (−t)

−0.1 −0.08 −0.06 −0.04−8

−6

−4

−2

0

2x 10

−19

time [s]

a cl

cr

b−

b+

b+

Figure 8-9: The contributions of the three segments to the total domain contribution in Figure8-3.


in the domain integral are also weighted by the loss functions and the coupling coefficient.Therefore certain terms in the representation contribute stronger to the reconstruction ofcertain events than others terms do. The contributions of each of the four crosscorrelationterms, equation 8-1 to 8-4, from the left boundary point, x3;1, are shown in Figure 8-10. Thecontributions of each of the four crosscorrelation terms, equation 8-1 to 8-4, from the leftboundary point, x3;2, are shown in Figure 8-11.

−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−2

0

2

x 10−12

time [s]

GvJ

e (t)+

GE

f (−t)

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−1

−0.5

0

0.5

1x 10

−15

time [s]

GvJ

e (t)+

GE

f (−t)

boundary x3;1

term 1term 2term 3term 4

a cl

b+

Medium type A

Figure 8-10: Contributions from the different crosscorrelation terms at the boundary point x3;1,to the total contribution coming from the boundary point x3;1. The crosscorrelation terms aredefined in equations 8-1 to 8-4.

The terms of Figures 8-10 and 8-11 are given by;

term1 : Gυ,Jm

(x3,B, x3;k)GE,Je∗(x3,A, x3;k), (8-1)

term2 : Gυ,Je

(x3,B, x3;k)GE,Jm∗(x3,A, x3;k), (8-2)

term3 : −Gυ,h(x3,B, x3;k)GE,f∗(x3,A, x3;k), (8-3)

term4 : −Gυ,f (x3,B, x3;k)GE,h∗(x3,A, x3;k). (8-4)

We see in Figures 8-10 and 8-11 that terms 2 and 3 contribute equally to the spuriousevents cr and cl, terms 1 and 2 contribute equally to event a, terms 3 and 4 contributeequally to event b+ and b−. In Figure 8-12 we display the contributions of each of the fourcrosscorrelation terms, equation 8-5 to 8-8, to the total contribution coming from the domainintegral.


−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−2

0

2

x 10−12

time [s]

GvJ

e (t)+

GE

f (−t)

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−1

−0.5

0

0.5

1x 10

−15

time [s]

GvJ

e (t)+

GE

f (−t)

boundary x3;2

term 1term 2term 3term 4

a cr

b−

Medium type A

Figure 8-11: Contributions from the different crosscorrelation terms at the boundary point x3;2,to the total contribution coming from the boundary point x3;2. The crosscorrelation terms aredefined in equations 8-1 to 8-4.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−2

0

2

x 10−12

time [s]

GvJ

e (t)+

GE

f (−t)

domainterm 1term 2term 3term 4

−0.1 −0.05 0 0.05 0.1−5

0

5x 10

−17

time [s]

GvJ

e (t)+

GE

f (−t)

0 0.05 0.1 0.15−1

−0.5

0

0.5

1

x 10−15

time [s]

GvJ

e (t)+

GE

f (−t)

a cr

cl

b−

b+

b+

Medium type A

Figure 8-12: Contributions from the different crosscorrelations in the domain integral, to thetotal contribution of the domain integral. The crosscorrelation terms are defined in equations 8-5to 8-8.


The terms of Figure 8-12 are given by

1 : Gυ,Je

(x3,B, x3)iIm{ε}GE,Je∗(x3,A, x3), (8-5)

2 : −Gυ,Je

(x3,B, x3)Re{ρf L}GE,f∗(x3,A, x3), (8-6)

3 : Gυ,f (x3,B, x3)Re{ρf L}GE,Je∗(x3,A, x3), (8-7)

4 : Gυ,f (x3,B, x3)iIm{ρc}GE,f∗(x3,A, x3). (8-8)

The 4th term almost completely reconstructs event b−, while the 3rd term almost completelyaccounts for event b+. Event a is also dominantly constructed from the contribution of term3.

8-2-2 Correlation gather of the domain integral

Offset [m]

time

[s]

−500 −400 −300 −200 −100 0 100 200 300 400 500

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x 3,A

x 3,B

1) 2)

a

c

Figure 8-13: Correlation gather for seismoelectric interferometry compiled for 10000 sourcepositions in a domain stretching from −500 meter to 500 meter. The receiver locations at −75meter and 75 meter are indicated by dashed lines, and labeled x3,A and x3,B . Two events can bedistinguished, events a and c, see text. Two alternative boundary points can be take such thatevent b hits both boundaries simultaneously. One possible choice is levels labeled 1 and 2, seeFigure 8-16.

It is very useful to study a correlation gather, because it gives visual insight on stationaryphases and contributions from different regions of the source coverage. Usually the correlationgather is created from the sources on the boundary only, but since we deal with a seismoelectricdiffusive system in 1D we can collect a gather of the sources in the domain integral. Thereceivers are 150 meter apart from each other in medium type B, the correlation gathercovers a total length of 1000 meter and is compiled for 10000 source positions. In Figure 8-13


the gather is shown on a linear gray scale with a clip value of 0.0001 of the maximum value.We can only distinguish two events. Event a, is nearly stationary for all offsets, see Section9-4-2, and arrives around t = 0 seconds. Event b, is not visible in Figure 8-13. Event c, isnon-stationary for all offsets except for sources near receiver B it gives a strong contributionat t = 0 seconds. This event resides in positive times but is not symmetric around the mid-point of the receivers, it intersects with the contribution from the boundaries at differenttimes for symmetrically positioned boundaries. The further away we choose the boundarypositions from receiver B, the later the arrival times of the strong spurious events in theseparate contributions from the domain and boundary integrals.

Offset [m]

time

[s]

−500 −400 −300 −200 −100 0 100 200 300 400 500

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5 −34

−32

−30

−28

−26

−24

−22

−20

−18

−16x 3,

A

x 3,B

a

c

b

a

c

b

Figure 8-14: Logarithm of the absolute value of the Correlation gather in Figure 8-13. We cannow distinguish three events, labeled a, b and c, see text.

In Figure 8-14 we see the logarithmic value of the absolute amplitude of the correlation gatherin Figure 8-13. We can see a third event labeled, b, that is several orders of magnitude smallerthan events a and c. This event is stationary for the parts of the domain that lay outsideof the receivers span. It is non-stationary in between the two receivers and arrives at causaltimes if we are at x3 < x3,A and at anti-causal times for x3 > x3,B.

8-2-3 Alternative boundary positions

There are two distinct boundary positions that give a special alignment of physical andspurious events. The first has been proposed in the correlation gather Figure 8-13. If wechoose our boundaries at equal distances from receiver B, for example as in Figure 8-15,we would find that the two strong spurious events in positive times overlap, see top panel ofFigure 8-16. The other choice is to place the boundary points just outside the receivers, it can


easily be seen in Figures 8-13 and 8-14 that the spurious events would arrive simultaneouslywith the two physical events, see bottom panel of Figure 8-16. Remember that we have toinclude both receivers inside L for the interferometric representation to hold, see Section 6-5.

∇∇x

3,A

∇∇x

3,Bx

3;1x

3;2

325 m 150 m 325 m

Figure 8-15: Alternative position of receivers and boundary points for 1D interferometry; bound-ary points are taken symmetrically around receiver B at a distance of 325 meter. Example is shownin Figure 8-16.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−0.5

0

0.5

x 10−14

time [s]

GvJ

e (t)+

GE

f (−t)

boundaryvolume

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−0.5

0

0.5

x 10−14

time [s]

GvJ

e (t)+

GE

f (−t)

boundaryvolume

a cr+c

lb

−b

+

a+cr

b−

b++c

l

Figure 8-16: Reconstructed signal using two alternative positions of the boundaries. Top panel;boundary points are both at a distance of 325 meter from receiver B. Bottom panel; boundarypoints are just besides the receivers.


8-3 2D Seismoelectric interferometry in a homogeneous medium

We consider the situation of two receivers in a homogeneous world. The x1 and x3 axes beingthe horizontal distance and depth. We position receiver A at a depth of 225 meter below theupper domain boundary and receiver B positioned 150 meter directly below receiver A. Thelower domain boundary is located 600 meter below the upper boundary. See Figure 8-17 forthe experimental setup. We use the modeling method as described in Part I of this thesis,but deal with a rather simplified case because there are no heterogeneities. We evaluate theinterferometric representation for Gυ,Je

(xB,xA, t)+GE,f (xA,xB,−t) in 2D given by equation7-45. We place sources spaced as 1 per squared meter on an area of 1794×600 squared meters.All sources emit a Ricker wavelet, equation 5-12, with a central angular frequency of ω = 800radians, the recovered Green’s functions are convolved with the autocorrelation of the sourcewavelet. The Green’s functions were calculated using a time sampling of ∆t = 0.001 seconds.We omitted the side boundary ∂S1 for the calculations in this section. We used the mediumparameters of medium type B, see Section 5-2.

∇

∇

1

23 4 5

1792 [m]

600

[m]

150

[m]A

B

Figure 8-17: Geometry for 2D seismoelectric interferometric experiment. The dotted lines labeled1, 2, 3, 4 and 5, denote the locations of the correlation gathers in Figures 8-24, 8-25 8-26, 8-27and 8-28, respectively.

8-3 2D Seismoelectric interferometry in a homogeneous medium 89

−0.1 −0.05 0 0.05 0.1 0.15−1.5

−1

−0.5

0

0.5

1

1.5x 10

−17

time [s]

GvJ

e (t)+

GE

f (−t)

exactretrieved

a

b−

b+b

sb

s


(t) and GE,f (−t) in medium type B in2D.

In Figure 8-18 we see the exact and retrieved super position of Gυ,Je

(xB,xA, t) +GE,f (xA,xB,−t). We recover the second physical events, labeled b− and b+, arriving neart = ±0.07 seconds, very well. But we do not recover the first event, labeled a, so well. In ad-dition to that, we see two small spurious events, labeled bs, arriving at early times t = ±0.02seconds. In Figure 8-19 we plotted the separate contributions from the boundary integral andthe domain integral. We see strong spurious events in the separate contributions from thedomain and boundary integrals that cancel each other after combination. The cancellation isimperfect, as in the 1D computations, and some spurious events remain in the end result. InFigures 8-20 and 8-21 we can see the contribution from each source level with depth. Figure8-20 is a 2D version of Figure 8-13, but it is a correlation gather in the sense that it is acompilation of summed horizontal correlation gathers. In 2D we encounter some additionalspurious events with respect to the 1D results due to the limited source aperture in horizontaldirection. Event a in Figures 8-20 and 8-21 form the first arrival, event b forms the secondphysical arrival. Event c is canceled by the boundary contributions from the upper and lowerboundaries. Events bs and cs are spurious events, whose origin we can see later in Figures8-24 and 8-25.


−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−6

−4

−2

0

2

4

6

x 10−18

time [s]

GvJ

e (t)+

GE

f (−t)

exactvolumeboundary

a

b−

b+

bs

bs

cb

ct

Figure 8-19: Boundary integral, domain integral contributions to the exact retrieved signal.

depth [m]

time

[s]

0 100 200 300 400 500 600

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

a

b

c

cs

Figure 8-20: Source contributions with depth to the recovered signal in Figure 8-18; the domainintegral crosscorrelations have been summed in the horizontal direction for each depth level,creating one trace for each depth level. Annotations see text.


depth [m]

time

[s]

50 100 150 200 250 300 350 400 450 500 550

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

a

b

bs

b

bs

c

Figure 8-21: A close look at the time window t = −0.1 to t = 0.1 seconds of Figure 8-20,brightening has been adjusted to enhance arrivals b and bs. Annotations see text.

position [m]

time

[s]

−800 −600 −400 −200 0 200 400 600 800

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

a

b−

b+

c cb

ct

Figure 8-22: Source contributions with horizontal distance to the recovered signal in Figure8-18; the domain integral crosscorrelations have been summed in the vertical direction for eachhorizontal position, creating one trace for each horizontal position. Annotations see text.


position [m]

time

[s]

−800 −600 −400 −200 0 200 400 600 800

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

a

b−

b+

c

Figure 8-23: A close look at the time window t = −0.1 to t = 0.1 seconds of Figure 8-22,brightening has been adjusted to enhance arrivals b1 and b2. Annotations see text.


In Figures 8-22 and 8-23 we show the contributions per horizontal distance, we compiledsummed vertical correlation gathers. Events b− and b+, are clearly constructed by sourceswith a lateral position close to the line crossing both receivers. This is not the case for eventa, that has contributions from sources outside the considered domain. Events c, ct and cb areappearances of the spurious event in the domain integral that is canceled by the boundaryintegral. Event c is the stationary phase in the vertical correlation gathers, of the spuriousevent in the separate contribution from the domain integral. Events ct and cb are the spuriousevents in the separate domain contribution that arise that the top and bottom of the domain.We can see that the spurious event caused by neglecting the boundary integral over ∂S1

arrives even later than the window shown in Figure 8-19. There is an additional spuriousevent clearly visible in Figure 8-23, this ‘cross’ is caused by cross-correlations remnants of thespatial and temporal wrap arounds in the calculations of the 2D Green’s functions.

Finally, we show five cross sections from the cube of crosscorrelation traces through thedomain. Their locations are shown in Figure 8-17. The horizontal correlation gathers arechosen above and below the receivers, respectively. Note how event b is located in positivetimes for the sources above the receivers and in negative times for sources below the receivers.We can also see that event a is stationary with respect to horizontal position. Event b becomesstationary with respect to horizontal position, for sources near the line crossing both receivers.Spurious event c, arrives at increasingly later times for the sources away from the receivers.Figures 8-26, 8-27 8-28 are three vertical correlation gathers, they have been chosen at oneside of the receivers because the behavior of the correlations is symmetric in x1 as can be seenin Figure 8-22. Events a, b and c can be seen, but only event a is stationary with respectto depth. We can see how event b gradually moves from positive times for sources above thereceivers to negative times for sources below the receivers.


horizontal position [m]

time

[s]

−800 −600 −400 −200 0 200 400 600 800

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

a

b

c

Figure 8-24: Correlation gather of horizontal surface of sources at x3 = 150 meter. Annotationssee text.

horizontal position [m]

time

[s]

−800 −600 −400 −200 0 200 400 600 800

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

a

b

c

Figure 8-25: Correlation gather of horizontal surface of sources at x3 = 450 meter. Annotationssee text.


depth [m]

time

[s]

0 100 200 300 400 500 600

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

a

ba

b

c

Figure 8-26: Correlation gather of a vertical surface of sources at x1 = −800 meter. Annotationssee text.

depth [m]

time

[s]

0 100 200 300 400 500 600

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

a

ba

b

c



depth [m]

time

[s]

0 100 200 300 400 500 600

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

a

ba

b

c


Chapter 9

Discussion and conclusions

We discuss the results of Part I only briefly, because the results are relatively straightforward.The 1D results of Part II are discussed with particular attention to how the events were createdby the interferometric representation. We discuss the relative contributions of different partsof the boundary and domain integrals. The relative contribution of the different terms underthe integrations of both the domain and the boundary are discussed. Finally we explain the2D results using the concepts we have learned from the 1D simulations.

9-1 Forward modeling in 2D

The derivation of the two-way wave equation was already performed by previous researchers[White and Zhou, 2006], [Wapenaar, 2007, in preparation], in this work we included a smallbut valuable extension by including deformation rate sources. This completes the two-waywave equation and enables us to evaluate the 2D interferometric representations in Part II.The causality trick has been applied to successfully minimize temporal wrapping caused bythe improper discretisation of the solution. Spatial wrap around can be avoided choosing anappropriate space-time window as already discussed by Haartsen and Pride [1997]. Howeverthey did not note that the ’flat’ arrivals, i.e. with a very small hyperbolicity due to therelatively fast electromagnetic speed, also wrap around. Events that are even in space, seethe flat arrivals in Figures 5-12, 5-14, would become enhanced, while flat events that areuneven in space, for example in Gυ,Jm

3 , would become faded, especially on the flanks.

9-2 Seismoelectric interferometric Green’s function representation

In Chapter 7 we derived specific representations of Green’s functions between two receivers, asa function of sources in and on a domain and it’s boundary that encloses both receivers. Theboundary integral in equation 6-10 is a summation of 4 different source types, simplif param-eters and source functions, we drop the subscripts A and B and the reciprocity theorem of the

98 Discussion and conclusions

correlation-type reduces to Poynting’s theorem. The domain integral over the term containingthe coupling coefficient reduces a domain integral over 2Im{υs∗

2,rE2,r}Re{ρf L}. In words thisterm constitutes the energy that is transferred to heat in the coupling of electromagnetic andseismic fields. It is interesting to see that this term is weighted by the conservative strengthof the coupling coefficient, so this term is only as strong as the coupling between the seismicand electromagnetic fields.

9-3 Numerical evaluation of the Seismoelectric interferometricGreen’s function representation

The interferometric representation of Gυ,Je

(xB,xA, t)+GE,f (xA,xB,−t) has been shown validfor the bandwidth of the source function, in both 1D and 2D, see Figures 8-2, 8-4 and 8-18.But we can safely say that it is valid for the entire bandwidth for which the seismoelectricsystem of equations is valid. Our numerical simulations of the interferometric representationscarried additional spurious events besides the physical events that are predicted by the exactdirect modeled Green’s functions. All those spurious events can easily be attributed to ei-ther a numerical artifact or an artifact arising from the approximations made in the discreteinterferometric representations. With numerical artifacts we mean events that were causedpurely by the way we calculate the responses at the receivers. For example the weak remnantof temporal wrapping caused when using a discrete Fourier transformation. Artifacts causedby approximations of the interferometric representations include those of ignoring the sides∂S1 and of the finite sampling of the Riemann sum that we use to approximate the domainand boundary integrations. The most obvious observation is that both the domain integraland the boundary integral contain a very strong spurious event that interfere destructively.In Figure 8-6 we see that for this destructive interference to lead to a spurious event smallerthan the second physical event, the Riemann sum has to be sampled quit dense. Definitelywith ∆xs ≤ 1 meter for the media types as defined in section 5-2. But this will depends onthe receiver offset and the distance to the boundaries. There are two such strong events, oneassociated with each boundary. They both arrive in positive times for the interferometricrepresentation for Gυ,Je

(xB,xA, t) + GE,f (xA,xB,−t). Since the left-hand side of the inter-ferometric representations in matrix form, see equation 6-10, is Hermitian the right-hand sidealso is. If we evaluate GE,f (xB,xA, t)+Gυ,Je

(xA,xB,−t), both strong spurious events wouldreside in the negative time window. This is good news, we can use this knowledge to interpretspurious events arising when we ignore the domain or boundary integral.

9-4 Domain versus boundary integral

For common seismic interferometry, the domain integral can be ignored because the lossesare low. The effect of ignoring the losses would only affect the amplitudes of the arrivals, andnot create any additional spurious events [Snieder, 2007]. The same has been concluded formildly diffusive electromagnetic systems [Slob et al., 2006], with the provision that ignoringthe surface integral did not lead to more spurious events than those that had been created bysimplifying the surface integral under a far field approximation. We did not perform a far field

9-4 Domain versus boundary integral 99

approximation of the boundary integral, so this connot yet be concluded for seismoelectricinterferometry.

In a homogeneous medium the surface integral disappears if our domain covers the entireinfinite medium, provided the medium has one or more non-zero loss terms. Then, evenwhen the domain only covers part of the infinite medium, the contribution of the boundaryintegral decreases with distance due to the losses in the system. If the boundary is far awaywith respect to the receiver offset and the losses in the domain are high enough, the spuriousevents that remain after neglecting the boundary integral are negligible.

In Figures 8-3 and 8-5 we can see that the contribution of the domain integral to the sec-ond physical event in positive times has the correct phase but the amplitude is too strong,while the contribution to the second physical event in negative times is too weak. The seg-ments of the domain integral outside the receiver span complement the contribution of thereceiver span. If the domain integral does not extend over all space, the boundary integralcomplements the domain integral. We recover an asymmetric signal, see Section 7-4-2, thecontributions of the domain segments outside the receiver span and additional contributionsof the boundary integral to have equal polarity for both second physical events, see Figure 8-7.This cannot easily be extended for inhomogeneous media, because separation of the domainin three segments is no longer valid due to reflections and transmissions. The sensitivity of theamplitude errors introduced when we omit the contributions of the domain integral dependon the wave-type of the arrival. We can see in Figures 8-3 and 8-5 how the contributions ofthe boundary integral, relatively to the domain integral, is of larger contribution in mediumtype B than in medium type A. But the domain integral contribution is of higher importanceto the first physical events around t = 0 second in medium type B than in medium typeA. In Section 5-3 we discussed how the second event travels as a shear wave and the firstevent travels as an electromagnetic wave. The shear wave losses are concluded to be higherin medium type A than in medium type B, see Section 5-2, while the electromagnetic lossesare higher in medium type B than in medium type A. The conclusion is that the ampli-tude errors arising from ignoring the domain and or boundary integral contribution will ingeneral be different for all events. Depending on the wave-type and of the receivers to theboundaries. However, this does not mean that ignoring the domain integral will cause the am-plitude of the second physical events in both positive and negative times to decrease. In factfor Gυ,Je

(xB,xA, t) + GE,f (xA,xB,−t) the amplitude of the second event in Gυ,Je

(xB,xA, t)will increase, while the amplitude of the second event in GE,f (xA,xB,−t) will decrease.

9-4-1 Main contributions to the retrieved result

Even though we need the entire representation to retrieve the exact Green’s functions, thereare certain parts of the representation that contribute significantly less than other parts.In Figures 8-10 and 8-11, we see how terms 3 and 4 create the second physical events inGυ,Je

(xB,xA, t)+GE,f (xA,xB,−t) and how term 1 and 2 create the first physical events thatarrive around t = 0 second. Terms 1 and 2 are boundary integrals over electromagnetic sources(see equations 8-1 and 8-2), they create the event that arrived with the electromagnetic wavespeed. While the seismic sources (see equations 8-3 and 8-4) create the event arriving withshear wave velocity. Ignoring one type of boundary sources would inevitably lead to missingeither electromagnetic arrivals or seismic arrivals in the contribution of the boundary integral.


The contribution of the domain integral is almost completely given by the 3rd term,Gυ,f (x3,B,x3)Re{ρf L}GE,Je∗(x3,A,x3). This term contains recordings of two different fieldsat two different receiver positions of two different source types at the same source positionacting simultaneously. These Green’s functions are each very strong, because they measurethe field type associated with the source type, i.e. E2 response of Je

2 and υs2 response of f

2.

The weakest contribution comes from the 2nd term, that contains recordings of two differentfields at two different receiver positions of two different source types at the same source posi-tion acting simultaneously. But each recorded field is the response of the other source type,i.e. E2 response of f

2and υs

2 response of Je2. Some events that arise in these crosscorrela-

tions are to weak to be visible in the figures of Chapter 7. The third term has no significantcontribution to the second event in negative times. The total but weak contribution of thedomain integral to reconstructing the second event in negative times comes completely fromthe 4th term. The 4th term most dominantly compensates for the shear wave losses insidethe domain and is the most important contribution to events b+ and b−. The 1st term mostdominantly compensates for electromagnetic losses inside the domain, it is the most impor-tant contribution to event a. There will be a loss of amplitudes and imperfect destructiveinterference of spurious events, but judging from the small contributions of the 1st and 4th

terms in the domain integral, they can probably be ignored if the boundary integral is closeenough. The 2nd and 3rd term cannot be ignored. Although the 2nd term is very weak, it isclosely associated with the 3rd term. This can be seen when we complex conjugate the entireexpression, the 2nd term transforms in the 3rd term and vice versa. This is the key to theHermitian property of equations 7-15 and 7-43. The condition set by Wapenaar [2004] thatthe loss matrix should be diagonal for the interferometric representation to become a directcrosscorrelation of two Green’s functions seems to be hard to realise. Simply ignoring the offdiagonal terms would seriously affect the amplitude and phase of the recovered events, thespurious event c would also remain in the recovered result.

9-4-2 Stationary phases

A stationary phase in the context of seismic interferometry has been used to refer to positionsof sources on the boundary integral for which the phase of certain events in the correlationgather becomes stationary [Snieder, 2004a], [Snieder, 2006], crosscorrelations of recordingsdue to these sources have stationary arrival times. In 1D the boundary integral consists ofjust two points, that logically are at the stationary phase for all events. But since we deal witha domain integral over the loss functions, we see that both physical events are stationary forline segments 1 and 3. In Figure 8-14, we can see that event labeled b, corresponding to thesecond physical event in positive and negative times, has stationary contributions from theline segments outside the receiver span. The same can be said for event labeled a in Figures8-13 or 8-14, corresponding to the first physical event, but since it’s arrival time is almostzero seconds, the non-stationary behavior for line segment 2 is not visible. It is interestingto note that the spurious event, labeled c is non-stationary in the entire domain. Choosing alarger domain would not only decrease the amplitude of the spurious events in the separateboundary and domain integrals, but also increase the arrival time.

9-5 2D versus 1D 101

9-5 2D versus 1D

An important step toward real situations is to understand the interferometric representationin 2D. Fortunately, the results are very similar to 1D, but because we omitted the boundaryintegral over ∂S1 we expect some spurious events. In Figure 8-18 we see that we reconstructthe second physical events very well, but reconstruct the first event with a too strong ampli-tude. In Figures 8-22 and 8-23 it is simple to see the consequences of modeling in 2D. Firstly,we see how the vertical line with x1 = 0 meter contains all stationary phases, see also Figures8-24 and 8-25, outside this line all phases are non-stationary. This is a direct consequenceof the positioning of both receivers on a vertical line. However, sources positioned in thelimits x1 → ±∞ are at stationary phases for all physical events. however, source contribu-tions decreases with distance because of the losses in the system, if we extend our domain inhorizontal distances far enough we can exclude the sides of the boundary. We see how eventa in Figures 8-22 and 8-23 is clearly cut-off due to the limited horizontal aperture. Eventc has a very strong contribution to the reconstruction of the first arrival. The amplitudeof the contribution of event c is decreased by event a, since event a was not reconstructedperfectly, the combination with c will lead to an event that is too strong. We can also seetwo small spurious events at t = ±0.02 seconds in Figure 8-18, these are spurious event dueto the limited horizontal aperture, see Figures 8-23 and 8-24 or 8-25. Figure 8-23 also givesus a good insight on the reconstruction of event b+. We can see the strong event, labeled b+,that in 1D is created by the sources in the segment of the receiver span. We see a very strongevent that will be decreased by the contributions of sources outside the receiver span and onthe boundary integral. The spurious event c is also non-stationary in the x1 direction, theseshould be canceled by destructive interference with the contributions from the ∂S1 bound-ary. The stationary contributions to the physical events remain stationary lines, they lie ina vertical line with x1 = 0 meter. We can conclude from Figure 8-23, that the line segmentsoutside the receiver span are the stationary lines for events a and b. The boundary integralcontains stationary phases for the physical events, where the stationary line of the domainintersects the boundary.

9-6 Conclusions

We validated the interferometric Greens function representation for Gυ,Je

(xB,xA, t) +GE,f (xA,xB,−t). We have shown that we need a dense distribution of sources through-out a domain and on its boundary to obtain exact reconstruction. The electromagneticsources of the representation contribute dominantly to the reconstruction of the diffusive EMfield. The seismic sources contribute dominantly to the reconstruction of the shear wave field.Sources near receivers cannot be neglected for the reconstruction of Gυ,Je

. A strong non-stationary event is identified in the domain integral and boundary integral contribution toGυ,Je

(xB,xA, t), it is absent in the reciprocal Green’s function GE,f (xA,xB,−t). A numericalstudy should be conducted to generalize these conclusions to inhomogeneous media.

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106 References

Appendix A

Derivation of the seismoelectrictwo-way wave equation

In this appendix we will derive the seismoelectric two-way wave equation of chapter 3 following[Wapenaar, 2007, in preparation], prepublished by Shaw [2004]. The two-way wave equation3-1, A-1 equates the vertical variation of the contiuous field quantities as a function of thehorizontal derivatives of the contiuous field quantities and a source vector,

∂zq = Aq + d. (A-1)

The field quantities that are contiuous over a horizontal source free interface are capturedin the vector q, see section 2-3. The two-way wave equation equates the vertical variationsin q as a function of the horizontal derivatives of q in A and a two way source-vector d.We consider an isotropic medium, in addition to the parameters given in section 2-1-3 and

2-1-5 we define ρE = η

iωk, ε = ǫ + 1

iω σe and εL = ε − ρEL2, and in isotropic media we have

ρEij = ρEδij , εij = εδij and εL,ij = εLδij .

We rewrite the equations of motion 2-9 and 2-10 as

iωρbvs + iωρfδjwj − ∂j τbj = f b, (A-2)

iωρfδtiv

s +η

k

(

wi − L(

γtiE0 + δ3iE3

))

+ ∂ip = ffi , (A-3)

where the vectors vs, τ bj , E0, f b, δi and γi are defined as

vs =

υs1

υs2

υs3

, τ bj =

τ b1j

τ b2j

τ b3j

, E0 =

(

E1

E2

)

, f b =

f b1

f b2

f b3

, δi =

δ1i

δ2i

δ3i

, γi =

(

δ1i

δ2i

)

.

(A-4)

108 Derivation of the seismoelectric two-way wave equation

Before we rewrite the stress-strain relations 2-11 and 2-12 in a similar form, we eliminate theterm ∂kwk from equation 2-11, using equation 2-12. This yields

−iωτ bij + eijkl∂lυ

sk − iω

d

Mδij p = eijklh

bkl, (A-5)

with eijkl defined as

eijkl = cijkl −d2

Mδijδkl. (A-6)

The stress-strain relations A-5 and 2-12 are now written as

− iωτ bj + ejl∂lv

s − iωd

Mδj p = ejlh

bl , (A-7)

iωp + dδtk∂kv

s + M∂kwk = dTl hb

l + M qi, (A-8)

with matrix ejl and vector 0 defined as

eij =

e1i1j e1i2j e1i3j

e2i1j e2i2j e2i3j

e3i1j e3i2j e3i3j

. (A-9)

Note thatet

ij = eji. (A-10)

The electromagnetic field equations 2-35 and 2-36 are rewritten as

iωεLE0 +η

kLγαwα + ∂3H0 −

(

∂2

−∂1

)

H3 = −Js,e0 , (A-11)

iωεL +η

kLw3 − (∂1 ∂2) H0 = −J

s,e3 , (A-12)

iωµ0H0 + ∂3E0 −(

∂1

∂2

)

E3 = −Js,m0 , (A-13)

iωµ0H3 − ∂3E0 − (∂1 ∂2) E0 = −Js,m3 , (A-14)

were the vectors H0, Js,e0 , J

s,m0 are defined as

H0 =

(

H2

−H1

)

, Js,e0 =

(

Js,e1

Js,e2

)

, Js,m0 =

(

Js,m2

−Js,m1

)

. (A-15)

Next we separate the vertical derivatives in from the lateral derivatives, according to

− ∂3τb3 = −iωρbvs − iωρf (δαwα + δ3w3) + ∂ατ b

α + f b, (A-16)

∂3p = −ωρfδt3v

s − η

k

(

w3 − LE3

)

+ ff3 , (A-17)

∂3vs = e−1

33

(

iωτ b3 + iω

d

Mδ3p − e3β∂βv

s

)

+ e−133 e3lh

bl , (A-18)

∂3w3 = − iω

Mp − d

M

(

δt

β∂βvs + δt

3∂3vs)

− ∂βwβ + dTl hb

l + M qi, (A-19)

∂3E0 = −ωµ0H0 +

(

∂1

∂2

)

E3 − Js,m0 , (A-20)

∂H0 = −iωεLE0 −η

kLγαwα +

(

∂2

−∂1

)

H3 − Js,e0 . (A-21)

109

Using equations A-3 and A-7, we can eliminate the terms δαwα and ∂ατα from equationA-16, yielding

− ∂3τb3 = −iωρbvs + iωρfδα

(

iωρf η

kδt

αvs +k

η∂αp − Lγt

αE0

)

− iωρfδ3w3

+1

iω∂α

(

eαβ∂βvs + eα3∂3v

s − iωd

Mδαp

)

+ f b − iωρf k

ηδαff

α

+1

iω∂αeαlh

bl , (A-22)

or upon substitution of equation A-18

− ∂3τb3 = ∂α

(

eα3e−133 τ b

3

)

+ iωρf k

ηδα∂αp − 1

iω∂α

(

iωd

Mrαp − Rαβ∂βv

s

)

−iω

(

ρbI3 − iω(

ρf)2 k

ηδαδt

α

)

vs − iωρfδ3w3 − iωρf LδαγtαE0

+f b − iωρf k

ηδαff

α +1

iω∂αRαβh

bβ , (A-23)

where I3 is a 3 × 3 indentity matrix and

Rαβ = eαβ − eα3e−133 e3β , (A-24)

rα = δα − eα3e−133 δ3. (A-25)

Note that Rtαβ = Rβα. On account of equation A-10, Using equation A-12, we eliminate E3

from equation A-17, yielding

∂3p = −iωρfδt3v

s − η

k

(

1 +1

iωεL

η

kL2

)

w3 +1

iωεL

η

kL (∂1 ∂2) H0 −

1

iωεL

η

kLJ

s,e3 + f

f3 . (A-26)

Using equations A-18 and A-3, we eliminate the terms ∂3vs and ∂βwβ from equation A-19,

according to

∂3w3 = − d

Mδt

3e−133

(

iωτ b3 + iω

d

Mδ3p

)

− iω

Mp + ∂β

(

k

η∂β p + iωρf k

ηδt

βvs − Lγt

βE0

)

d

Mrtβ∂βv

s − ∂βk

ηf

fβ +

d

Mrtβhβ + qi. (A-27)

Using equation A-12, we can eliminate E3 from equation A-21, yielding

∂3E0 = −iωµ0H0 +

(

∂1

∂2

)

1

iωεL(∂1 ∂2) H0

−(

∂1

∂2

)

1

iωεL

η

kLw3 − J

s,m0 −

(

∂1

∂2

)

1

iωεLJ

s,e3 . (A-28)

Using equations A-14 and A-3, we eliminate H3 and wα from equation A-21, yielding

∂3H0 = −iωεLE0 +η

kLγα

(

iωρf k

ηδt

αvs +k

η

(

∂αp − ffα

)

− LγtαE0

)

− Js,e0

+

(

∂2

−∂1

)

1

iωµ0(∂2 − ∂1) E0 −

(

∂2

−∂1

)

1

iωµ0J

s,m0 . (A-29)


Equations A-23, A-26, A-28, A-18, A-27 and A-29 are now combined into the seismoelectrictwo-way wave equation, according to

∂zq = Aq + d, (A-30)

where the wave vector q and the source vector d are defined as

q =

−τ b3

p

E0

vs

w3

H0

, d =

f b − iωρf kηδαf

fα + 1

iω∂αRαβhbβ

ff3 − 1

iωεLLη

kJ

s,e3

−Js,m0 −

(

∂1

∂2

)

1iωεL

Js,e3

hb3 + e−1

33 e3αhbα

qi − ∂βkη f

fβ + d

M rtβhβ

−Js,e0 − Lγαf

fα −

(

∂2

−∂1

)

1iωµ0

Js,m3

, (A-31)

and the operator matrix A as

A =

(

A11 A12

A21 A22

)

, (A-32)

with

A11 =

A1111 A

1211 A

1311

A2111 A

2211 A

2311

A3111 A

3211 A

3311

, A12 =

A1112 A

1212 A

1312

A2112 A

2212 A

2312

A3112 A

3212 A

3312

, (A-33)

A21 =

A1121 A

1221 A

1321

A2121 A

2221 A

2321

A3121 A

3221 A

3321

, A22 =

A1122 A

1222 A

1322

A2122 A

2222 A

2322

A3122 A

3222 A

3322

.

Where

111

A1111 = −∂α

(

eα3e−133 ·)

, (A-34)

A1211 = iωρf k

ηδα∂α − ∂α

(

d

Mrα·)

, (A-35)

A1311 = −iωρf Lδαγt

α, (A-36)

A1112 =

1

iω∂α (Rαβ∂β ·) − iω

(

ρbI3 − iω(

ρf)2 k

ηδαδt

α

)

, (A-37)

A1212 = −iωρfδ3, (A-38)

A2112 = −iωρfδt

3, (A-39)

A2212 = −η

k

(

1 +1

iωεL

η

kL2

)

, (A-40)

A2312 =

1

iωεL

η

kL (∂1∂2) , (A-41)

A3212 = −

(

∂1

∂2

)

1

iωε

η

kL, (A-42)

A3312 = −iωµ0I2 +

(

∂1

∂2

)

1

iωεL(∂1∂2) , (A-43)

A1121 = −iωe−1

33 , (A-44)

A1221 = iω

d

Me−1

33 δ3, (A-45)

A2121 = iω

d

Mδt

3e−133 , (A-46)

A2221 = −iω

d2

M2δt

3e−133 δt

3 −iω

M+ ∂β

(

k

η∂β·)

, (A-47)

A2321 = −∂βLγt

β, (A-48)

A3221 = Lγα∂α, (A-49)

A3321 = −iωεLI2 +

(

∂2

−∂1

)

1

iωµ0(∂2 − ∂1) −

η

kL2γαγt

α, (A-50)

A1122 = −e−1

33 e3β∂β , (A-51)

A2122 = ∂β

(

iωρf k

ηδt

β

)

− d

Mrtβ∂β, (A-52)

A3122 = iωρf Lγαδt

α, (A-53)

where I2 is a 2× 2 identity matrix. The submatrices that have not been listed are zero. Thestiffness matrix ejl reads, according to equations A-6, A-9 and 2-16

(ejl)kl = eijkl = Sδijδkl + N (δikδjl + δilδjk) , (A-54)

with

S = KG − 2

3N − d2

M. (A-55)


We have eliminated the fields τ b1, τ b

2, E3, H3, w1 and w2. The source quantities Js,e3 , J

s,m3 , f

f1 ,

ff2 , hb

1 and hb2 are still contained in the source vector d. The eliminated field quantities can

simply be calculated, using equations 2-22, 2-23, 2-10 and A-7, from quantities in q accordingto

E3 =1

iωεL

(

∂2H1 − ∂1H2 − Lη

kw3 − J

s,e3

)

, (A-56)

H3 =1

iωµ0

(

∂2E1 − ∂1E2 − Js,m3

)

, (A-57)

w1 =k

η

(

ff1 − ∂1p

f − iωρf υs1

)

+ LE1, (A-58)

w2 =k

η

(

ff2 − ∂2p

f − iωρf υs2

)

+ LE2, (A-59)

−τ b1 =

1

iω

(

e1lhbl + iω

d

Mδ1p

f − e1l∂lvs

)

, (A-60)

−τ b2 =

1

iω

(

e2lhbl + iω

d

Mδ2p

f − e2l∂lvs

)

. (A-61)

Appendix B

SH-TE Decomposition

B-1 Velocities

We start from the matrix Ashte of the SH-TE mode 3-33 in the horizontal-wavenumberfrequency domain it is given by

Ashte =


1

ω N 00 0 0 −iωµ0

−iω 1N 0 0 0

0 −iωε +ik2

1

ω1µ0

iωρf L 0

. (B-1)

When we omit the subscript shte, by default we mean the SH-TE system in this appendix.

To solve for upgoing and downgoing waves, we determine the eigenvalues of the system matrix∣

∣

∣A − Iλ

∣

∣

∣= 0. The trivial solution of a zero-eigenvalue gives the velocity for a wave traveling

purely in the horizontal direction. Hence the velocity in a layer is found solving∣

∣

∣A∣

∣

∣ = 0,

∣

∣

∣

∣

∣

∣

∣

∣

∣


1

ω N 00 0 0 −iωµ0

−iω 1N 0 0 0

0 −iωε +ik2

1

ω1µ0

iωρf L 0

∣

∣

∣

∣

∣

∣

∣

∣

∣

= 0. (B-2)

We perform Laplace development over the first and last columns, this leads to

(ω2 µ0

N)

∣

∣

∣

∣

∣

−iωρf L −iωρc +ik2

1

ω N

−iωε +ik2

1

ω1µ0

iωρf L

∣

∣

∣

∣

∣

= 0, (B-3)

since for ω 6= 0 the factor ω2 µ0

N 6= 0, we have∣

∣

∣

∣

∣


1

ω N

−iωε +ik2

1

ω1µ0

iωρf L

∣

∣

∣

∣

∣

= 0, (B-4)

114 SH-TE Decomposition

thus

ω2(

ρf)2

L2 −[(

−iωρc +ik2

1

ωN

)(

−iωε +ik2

1

ω

1

µ0

)]

= 0 (B-5)

or as quadratic ink21

ω2 ,

(

N

µ0

)

k41

ω4+

[

−(

1

µ0ρc + εN

)]

k21

ω2+(

ρf2L2 + (ρc)2 ε)

= 0. (B-6)

The general solution to a quadratic equation ax2 + bx + c = 0 is given as x = −b±√

b2−4ac2a .

Using this to solve fork21

ω2 in equation B-6 gives

2k2

1

ω2=

ρc

N+ εµ0 ±

√

(

ρc

N− εµ0

)2

− 4µ0

N

(

ρf L)2

. (B-7)

For horizontally propagating plane waves we havek21

ω2 = 1c2

, hence for the velocities of SH-TEseismoelectric waves we have

2

c2=

ρc

N+ εµ0 ±

√

(

ρc

N− εµ0

)2

− 4µ0

N

(

ρf L)2

. (B-8)

The plus sign is associated with the velocity of the SH-wave csh and the minus sign with thevelocity of the cte wave, as can be seen when the coupling coefficient is set to zero L = 0.These velocities are a function of the medium parameters of the layer under consideration.

B-2 Non-zero eigenvalues

We look for non-zero solutions of the equation∣

∣

∣A − Iλ∣

∣

∣ = 0. Thus we solve for

∣

∣

∣

∣

∣

∣

∣

∣

∣

−λ −iωρf L −iωρc +ik2

1

ω N 00 −λ 0 −iωµ0

−iω 1N 0 −λ 0

0 −iωε +ik2

1

ω1µ0

iωρf L −λ

∣

∣

∣

∣

∣

∣

∣

∣

∣

= 0, (B-9)

performing Laplace development over the 4th column leads to

− iωµ0

∣

∣

∣

∣

∣

∣

∣


1

ω N

−iω 1N 0 −λ

0 −iωε +ik2

1

ω1µ0

iωρf L

∣

∣

∣

∣

∣

∣

∣

(B-10)

−λ

∣

∣

∣

∣

∣

∣

∣


1

ω N

0 −λ 0−iω 1

N 0 −λ

∣

∣

∣

∣

∣

∣

∣

= 0.

B-2 Non-zero eigenvalues 115

In the first determinant of equation B-11 we perform Laplace development over the middlecolumn, and in the second determinant we perform Laplace development over the first column

ω2µ0ρf L∣

∣

∣

∣

−iω 1N −λ

0 iωρf L

∣

∣

∣

∣

− (−iωµ0)

(

−iωε +ik2

1

ω

1

µ0

)

∣

∣

∣

∣

∣

−λ −iωρc +ik2

1

ω N

−iω 1N −λ

∣

∣

∣

∣

∣

+λ2

∣

∣

∣

∣

−λ 00 −λ

∣

∣

∣

∣

− λ

(

−iω1

N

)

∣

∣

∣

∣

∣


1

ω N

−λ 0

∣

∣

∣

∣

∣

= 0

(B-11)

or

− ω4 µ0

N

(

ρf L)2

− (−iωµ0)

(

−iωε +ik2

1

ω

1

µ0

)

λ2

+ (−iωµ0)

(

−iωε +ik2

1

ω

1

µ0

)(

−iωρc +ik2

1

ωN

)(

−iω1

N

)

+λ4 − λ2

(

−iω1

N

)(

−iωρc +ik2

1

ωN

)

= 0. (B-12)

From equation B-12 which we extract a quadratic function in λ2

λ4 +

[

(

ω2µ0ǫ − k21

)

+

(

ω2 ρc

N− k2

1

)]

λ2

+

[

ω4 µ0

N

(

ρf L)2

+(

ω2µ0ǫ − k21

)

+

(

ω2 ρc

N− k2

1

)]

= 0, (B-13)

which we solve again as a quadratic function;

2λ2 = −ω2

(

ρc

N+ εµ0

)

+ 2k21 ± ω2

√

(

ρc

N− εµ0

)2

− 4µ0

N

(

ρf L)2

. (B-14)

Equation B-14 can be simplified using the expression B-8 for the seismoelectric velocities,thisleads to

k21 − λ2

w =ω2

c2w

. (B-15)

We define an operator H =√

ω2

c2w− k2

1, such that our eigenvalues are

iHsh, iHte, −iHsh and − iHte. (B-16)

In what follows we will use a generalized eigenvalue iH±w = ∓i

√

ω2

c2w− k2

1 = ∓iHw. In which

the superscript ± denotes upgoing or downgoing waves (+ denotes down) according to thechoice of the sign of the square root operator made in section 3-4 and the subscript w denotesthe wavetype. We have diagonalized the matrix A and find H, where

H =

iH+sh 0 0 0

0 iH+te 0 0

0 0 iH−sh 0

0 0 0 iH−te

. (B-17)


B-3 Eigenvectors of SH-TE system

We need the eigenvectors to diagonalize the system matrix A and decompose the SHTE fields−τ b

23, E2, υs2 and −H1 in upgoing and downgoing waves. We start from a general eigenvalue

iH±w and derive a general eigenvector a±

n . From the eigenvector problem(

A − H

)

a±w = 0

we have four equations for each eigenvalue,

− iH±wa±1,w − iωρf La±2,w +

(

−iωρc +ik2

1

ωN

)

a±3,w = 0, (B-18)

−iH±wa±2,w − iωµ0a

±4,w = 0, (B-19)

−iω1

Na±1,w − iH±

wa±3,w = 0, (B-20)(

−iωε +ik2

1

ω

1

µ0

)

a±2,w + iωρf La±3,w − iH±wa±4,w = 0. (B-21)

We solve this set of equations, normalizing to a±3,n = 1, from equation B-20 we find

a±1,w = −(

H±w

ω

)

N, (B-22)

using equation B-19 we find

a±4,w = −H±w

ω

1

µ0a±2,w. (B-23)

Finally, substituting equations B-22 and B-23 into equation B-18 we find

a±2,w =

(

1c2w

− ρc

N

)

1N ρf L

. (B-24)

Summarizing, we have

a±w =

± Hw

ω N

ξ1w

1

± Hw

ω1µ0

ξ1w

, (B-25)

where

ξ1w =

(

1c2w

− ρc

N

)

1N ρf L

. (B-26)

There is another expression for the scaling term ξ. Which we find when we substitute equationB-23 in equation B-21

(

−iωε +ik2

1

ω

1

µ0

)

a±3,w + iωρf L − iH±w

(

− H±w

µ0ωa±2,w

)

= 0. (B-27)

B-3 Eigenvectors of SH-TE system 117

Again normalize to a±3,w = 1, we find

a±2,w =µ0ρ

f Lεµ0 − 1

c2w

. (B-28)

This choice leads to

a±w =

± Hw

ω N

ξ2w

1

± Hw

ω1µ0

ξ2w

, (B-29)

where

ξ2w =

µ0ρf L

(

εµ0 − 1c2w

) . (B-30)

If we equate B-26 and B-30 we obtain B-8.

There is an expression for a±n from Haartsen [1995], Pride and Haartsen [1996],

Haartsen and Pride [1997]. This one was previously also used by Shaw [2004], van der Burg[2002] and White and Zhou [2006], rearranging and rewriting for the fields in our SH-TEtwo-way field vector q gives

a±w =

± Hw

ω N

ξ3w

1

± Hw

ω1µ0

ξ3w

, (B-31)

where

ξ3w =

µ0LρE(

N − ρbc2w

)

εLµ0ρf c2w − ρf

. (B-32)

We now equate ξ1w and ξ3

w, this is a little tedious,

Nc2w

− ρc

ρf L=

µ0LρE(

ρbc2w − N

)

εLµ0ρf c2w − ρf

, (B-33)

with

ρc =

(

ρb −(

ρf)2

ρE

)

and εL = ε − ρEL2. (B-34)

Cross-multiplying the denominators gives(

N

c2w

− ρc

)

((

ε − ρEL2)

µ0ρf c2

w − ρf)

=(

µ0LρEρbc2w − N

)(

ρf L)

. (B-35)

We divide by ρf and develop all multiplication terms

Nεµ0 − ρcεµ0c2w − NρEL2µ0 + ρcρEL2µ0c

2w − N

c2w

+ ρc =

µ0L2ρEρbc2w − Nµ0L2ρE , (B-36)


summing out the Nµ0L2ρE term, dividing by c2w and rearranging gives

N1

c4w

− (ρc + εµ0N)1

c2w

−(

εµ0ρc + ρbρEL2µ0 − ρcρEL2µ0

)

= 0. (B-37)

Now we replace the ρc in the last term with ρb − (ρf)2

ρE and we find

N1

c4w

− (ρc + εµ0N)1

c2w

− εµ0ρc + µ0

(

ρf L)2

= 0. (B-38)

Which is again the quadratic function in 1c2w

found in section B-1 deriving the seismoelectric

wave velocities cw, thus ξ1w = ξ3

w.

B-3-1 Stability of the scaling terms

We have shown that all three scaling terms B-26, B-30 and B-32 are equal. However,this does not mean that they can all be applied for seismoelectric modeling. The thirdexpression for the eigenvectors using ξ3

w, see equation B-32 has been used successfully byprevious authors [Haartsen, 1995], [Pride and Haartsen, 1996], [Haartsen and Pride, 1997],[van der Burg, 2002], [Shaw, 2004], [White and Zhou, 2006]. The first two expressions forthe eigenvectors using ξ1

w or ξ2w, see equations B-26 and B-30, are both esthetically preferred

above the one one used by previous authors. However the terms ξ1w and ξ2

w exhibit numericalinstabilities. Through trial and error we found that if we combine ξ1

te and ξ2sh we can construct

stable composition and decomposition matrices. In figure B-1 we plot ξ1sh, ξ1

te, ξ2sh, ξ2

te, ξ3sh

and ξ3te in the first subsurface layer and see the instability issues.

B-4 The decoupled SH-TE systems

The seismoelectric SH-TE system consist of a SH polarized shear wave traveling throughin the solid part of the porous medium and a TE polarized electromagnetic wave. Theyare coupled the terms ±iωρf L in the SH-TE system matrix Ashte. If we set the couplingcoefficient zero, we see how the matrix Ashte decouples into a SH system and a TE systemwith matrices Ash and Ate. matrices Ash and Ate.

B-4-1 Decoupled SH waves in porous media

The two-way wave equation of the SH system in porous media is given by

∂3q = Aq + d, (B-39)

with

A =

(

0 −iωρc +ik2

1

ω N

−iω 1N 0

)

, (B-40)

B-4 The decoupled SH-TE systems 119

0 5 10−1.5589

−1.5589

−1.5589

−1.5589

−1.5589

−1.5589

−1.5589x 10

8ξ

1,sh

frequency [Hz]

ampl

itude

0 5 10−1.5592

−1.559

−1.5588

−1.5586

−1.5584x 10

8ξ

2,sh

frequency [Hz]

ampl

itude

0 5 10−1.5589

−1.5589

−1.5589

−1.5589

−1.5589

−1.5589

−1.5589x 10

8ξ

3,sh

frequency [Hz]

ampl

itude

0 5 10

−7.26

−7.255

−7.25

−7.245

−7.24x 10

−5ξ

1,te

frequency [Hz]am

plitu

de

0 5 10−7.255

−7.255

−7.255

−7.255x 10

−5ξ

2,te

frequency [Hz]

ampl

itude

0 5 10−7.255

−7.255

−7.255

−7.255x 10

−5ξ

3,te

frequency [Hz]

ampl

itude

Figure B-1: Overview of the nummerical behavior of the three different ξ terms, calculated from0 Hz to 10 Hz using the media parameters of the layer type A.


q =

(

−τ b23

υs2

)

and d =

(

f b2 − ρf

ρE ff2 + k1

ω N(

h12 + h12

)

h23 + h32

)

. (B-41)

The velocities of SH waves csh is found evaluating∣

∣

∣A

∣

∣

∣= 0 for horizontally propagating waves

( 1c2

=k21

ω2 ), we find

1

c2sh

=k2

1

ω2=

ρc

N. (B-42)

We solve te eigenvalue problem∣

∣

∣A − Iλ

∣

∣

∣= 0 and find

λ2 + ω2 ρc

N− k2

1 = 0, (B-43)

or using B-42 we write

−λ2 =ω2

csh− k2

1 (B-44)

Thus we have two eigenvalues iH and −iH, which we collect as iH± = ∓i√

ω2

c2sh

− k21.

Substituting the eigenvalue iH± in the eigenvector problem(

A − H

)

a± = 0 we find two

equations

− iH±a±1 +

(

−iωρc +ik2

1

ωN

)

a±2 = 0, (B-45)

−iω1

Na±1 − iH± = 0. (B-46)

we these two equations normalizing to a±2 = 1, and find

a± =

(

± Hω N

1

)

. (B-47)

To complete these equations we also give the composition and decomposition matrices, al-though we do not use these in our modeling scheme. We arrange the eigenvectors into thecolumns of L as L = (a+, a−). We have L and L−1 given by

L =

(

Hω N − H

ω N

1 1

)

and L−1 =1

2

(

ωH

1N 1

− ωH

1N 1

)

. (B-48)

B-4-2 Decoupled TE waves in vacuum

The other seismoelectric wave type TE also propagates independently when the couplingcoefficient is set to zero. For simplicity we will evaluate this wave in vacuum. This in

B-4 The decoupled SH-TE systems 121

addition to L = 0 we also have ε = ǫ0, µ = µ0 and σm = 0. The two-way wave equation forTE electromagnetic waves becomes

q = Aq + d, (B-49)

with

A =

(

0 −iωµ0

−iωǫ0 +ik2

1

ω1µ0

0

)

, (B-50)

q =

(

E2

−H1

)

and d =

(

Js,m1

−Js,e2

)

. (B-51)

We evaluate∣

∣

∣A∣

∣

∣ = 0 for horizontally propagating plane waves 1c20

=k21

ω2 and find the electro-

magnetic wave velocity

(iωµ0)

(

−iωǫ0 +ik2

1

ω

1

µ0

)

= 0, (B-52)

1

c20

=k2

1

ω2= ǫ0µ0. (B-53)

Following the same procedure as in above sections to find expressions for the eigenvalues , weagain find

λ2 + (iωµ0)

(

−iωǫ0 +ik2

1

ω

1

µ0

)

= 0, (B-54)

or1

c20

=k2

1

ω2− λ2

ω2. (B-55)

We see again that we have two eigenvalues iH and −iH, which we collect as iH± =

∓i√

ω2

c20− k2

1.

Substitutying the general eigenvalue H± into the eigenvector problem we find two equations

− iH± − iωµ0 = 0 (B-56)

−iωǫ0 +ik2

1

ω

1

µ0− iH± = 0 (B-57)

We solve these equations normalizing for = 1 and find

a± =

(

1

± Hω

1µ0

)

(B-58)

If we arrange the eigenvectors into the columns of L as L = (a+, a−). We find for thecomposition and decomposition matrices

L =

(

1 1Hω

1µ0

− Hω

1µ0

)

and L−1 =1

2

(

1 ωHµ0

1 − ωHµ0

)

(B-59)

Appendix C

SH-TE general diffusion, flow andwave equation in 2D and 1D

We start from the seismoelectric general diffusion, flow and wave equation matrices of section2-2. In 2D and 1D the SH-TE system decoupled from the P-SV-TM system. We need a generaldiffusion, flow and wave equation for the SH-TE system to employ the general derivations ofWapenaar et al. [2006] to find interferometric Green’s function representations.

C-1 SH-TE general diffusion, flow and wave equation in 2D

Expanding the equations of Pride with ∂2 = 0 for the fields that govern the SH-TE systemwe find

iωǫE2 +(

σe − ηL2k−1)

E2 + ηLk−1w2 − ∂3H1 + ∂1H3 = −Js,e2 , (C-1)

iωµH1 + σmH1 − ∂3E2 = −Js,m1 , (C-2)

iωµH3 + σmH3 + ∂1E2 = −Js,m3 , (C-3)

iωρbυs2 + iωρf w2 − ∂1τ21 − ∂3τ23 = f b

2 , (C-4)

−iωτ12 + N∂1υs2 = N hb

21 + N hb12, (C-5)


12 + N hb21, (C-6)


32 + N hb23, (C-7)


23 + N hb32, (C-8)

iωρf υs2 + ηk−1

(

w2 − LE2

)

= ff2 . (C-9)

Equations C-5 and C-7 are equal to equations C-6 and C-8, therefore we omit equations C-5and C-8. We rewrite equation C-9 as

w2 = LE2 − iωk

ηρf vs

2 +k

ηf

f2 . (C-10)

124 SH-TE general diffusion, flow and wave equation in 2D and 1D

We use equation C-10 to eliminate w2 in equations C-1 and C-4. We normalize equations C-6and C-7 by N . We find

iωǫE2 + σeE2 − iωρf Lυs2 − ∂3H1 + ∂1H3 = −

(

Js,e2 + Lf

f2

)

, (C-11)

iωρcυs2 + iωρf LE2 − ∂1τ

b21 − ∂3τ

b23 = f b

2 − ρf

ρEf

f2 , (C-12)

−iω1

Nτ21 + ∂1v

s2 = hb

12 + hb21, (C-13)

−iω1

Nτ23 + ∂3v

s2 = hb

32 + hb23, (C-14)

where ρE = 1iω

ηk and ρc = ρb − (ρf)

2

ρE .

We define 4 alternative source types, the force on the fluid phase is not an independent sourcetype in the SH-TE system, therefore we include it in the electrical current density and the

force on the bulk as: Js,e2 = J

s,e2 + Lf

f2 and f

2= f b

2 − ρf

ρE ff2 . The deformation rate sources

hb

1 and hb

3 have been defined for notational convenience as ˆhb1 = hb

12 + hb21 and ˆhb

3 = hb32 + hb

23.Equation C-11, C-12, C-3 C-3, C-13 and C-14 can now be captured into the general diffusionflow and wave equation as

iωAu + Bu + Dxu = s, (C-15)

where the field vector u and the source vector s are defined by

uT =(

E2, H1, H3, vs2,−τ b

21,−τ b23

)

(C-16)

and

sT =(

−Js,e2 ,−Jm

1 ,−Jm3 , f

2, h

b

1, hb

3

)

. (C-17)

The matrices A, B and Dx are defined by

A =

ǫ 0 0 −ρf L 0 00 µ 0 0 0 00 0 µ 0 0 0

ρf L 0 0 ρc 0 00 0 0 0 1

N 00 0 0 0 0 1

N

, B =

σe 0 0 0 0 00 σm 0 0 0 00 0 σm 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

and Dx =

0 −∂3 ∂1 0 0 0−∂3 0 0 0 0 0∂1 0 0 0 0 00 0 0 0 ∂1 ∂3

0 0 0 ∂1 0 00 0 0 ∂3 0 0

. (C-18)

C-2 SH-TE general diffusion, flow and wave equation in 1D 125

C-2 SH-TE general diffusion, flow and wave equation in 1D

We start from equations C-11, C-12, C-13 and C-14, setting ∂1 = 0 results to four equations

iωǫE2 + σeE2 − iωρf Lυs2 − ∂3H1 = −J

s,e2 , (C-19)

iωµH1 + σmH1 − ∂3E2 = −Js,m1 , (C-20)

iωρcυs2 + iωρf LE2 − ∂3τ

b23 = f

2, (C-21)

−iω1

Nτ23 + ∂3v

s2 = h

b

3. (C-22)

We capture equations C-19, C-20, C-21 and C-22 into the general flow, diffusion and waveequation C-15 with field and source vectors given by

u =

E2

H1

vs2

−τ b23

and s =

−Js,e2

−Js,m1

f2

hb

3

. (C-23)

And the matrices A, B and Dx by

A =

ǫ 0 −ρf L 00 µ 0 0

ρf L 0 ρc 00 0 0 1

N

, B =

σe 0 0 00 σm 0 00 0 0 00 0 0 0

(C-24)

and Dx =

0 −∂3 0 0−∂3 0 0 00 0 0 ∂3

0 0 ∂3 0

. (C-25)

C-3 Solution to the 1D SH-TE seismoelectric system in a homo-

geneous domain

We can solve equations C-19 to C-22 for the fields υs2, E2, H1 and τ b

23. We find that all fieldquantities obey a fourth order ODE.

∂43E2 − {iω}2

(

ρc

N+ εµ

)

∂23E2 + {iω}4

(

ρcµε

N+

ρf Lµ

N

)

E2 =

[

iω3 ρcµ

N− iωµ∂2

3

]

(

−Js,e2

)

+

[

iω2 ρc

N∂3 − ∂3

3

]

(

−Js,m1

)

+

[

iω3 µρf LN

]

(

f2

)

−[

iω2ρf Lµ∂3

] (

hb

3

)

, (C-26)


∂43H1 − {iω}2

(

ρc

N+ εµ

)

∂23H1 + {iω}4

(

ρcµε

N+

ρf Lµ

N

)

H1 =

[

iω2 ρc

N∂3 − ∂3

3

]

(

−Js,e2

)

−

iωε∂2

3 − iω3

ρcε

N+

(

ρf L)2

N

(

−Js,m1

)

+

[

iω2 ρf LN

∂3

]

(

f2

)

−[

iωρf L∂23

] (

hb

3

)

, (C-27)

∂43 υs

2 − {iω}2

(

ρc

N+ εµ

)

∂23 υs

2 + {iω}4

(

ρcµε

N+

ρf Lµ

N

)

υs2 =

−[

iω3 ρf Lµ

N

]

(

−Js,e2

)

−[

iω2 ρf LN

∂3

]

(

−Js,m1

)

+

[

iω3 εµ

N− iω

1

N∂2

3

]

(

f2

)

+[

∂33 − iω2εµ∂3

]

(

hb

3

)

, (C-28)

∂43 τ b

23 − {iω}2

(

ρc

N+ εµ

)

∂23 τ b

23 + {iω}4

(

ρcµε

N+

ρf Lµ

N

)

τ b23 =

+[

iω2ρf Lµ∂3

] (

−Js,e2

)

+[

iωρf L∂23

] (

−Js,m1

)

+[

∂33 − iω2εµ∂3

]

(

f2

)

+

[

−iωρc∂23 + iω3

(

εµρc +(

ρf L)2

µ

)]

(

hb

3

)

, (C-29)

where ε = ǫ + 1iω σe and µ = µ + 1

iω σm. The left-hand sides of equations C-26 to C-29 can besimplified to

[

∂3 −iω

csh

] [

∂3 +iω

csh

] [

∂3 −iω

cte

] [

∂3 +iω

cte

]

u. (C-30)

Which is a 4th order wave equation for all the fields in the field vector u. And the wavevelocities csh and cte are given by

2

c2sh

=ρc

N+ εµ +

√

(

ρc

N− εµ

)2

− 4µ0

N

(

ρf L)2

, (C-31)

2

c2te

=ρc

N+ εµ −

√

(

ρc

N− εµ

)2

− 4µ0

N

(

ρf L)2

. (C-32)

C-3-1 Green’s matrix for the 1D SH-TE system in a homogeneous domain

We defined the Green’s matrix in section 7-1 by replacing the 4 × 1 source vector by a 4 × 4source matrix. From equations C-26 to C-29 we see that the solutions to the Green’s functionsare defined by a scalar part of the Green’s function Gs multiplied with the correct wighting

C-3 Solution to the 1D SH-TE seismoelectric system in a homogeneous domain127

factor for each source type. The weighting factors are defined on the right-hand sides ofequations C-26 to C-29. The Green’s matrix elements are given by

GE,Je

2,2 =

[

iω3µρc

N− iωµ∂2

3

]

Gs, (C-33)

GE,Jm

2,1 =

[

iω2 ρc

N∂3 − ∂3

3

]

Gs, (C-34)

GE,f

2,2 =

[

iω3 ρf LN

µ

]

Gs, (C-35)

GE,hb

2,23 = −[

iω2ρf Lµ∂3

]

Gs, (C-36)

GH,Je

1,2 =

[

iω2 ρc

N∂3 − ∂3

3

]

Gs, (C-37)

GH,Jm

1,1 = −

iωε∂2

3 − iω3

ρc

Nε +

(

ρf L)2

N

Gs, (C-38)

GH,f

1,2 =

[

iω2 ρf LN

∂3

]

Gs, (C-39)

GH,hb

1,23 = −[

iωρf L∂23

]

Gs, (C-40)

Gυs,Je

2,2 = −[

iω3 ρf µLN

µ

]

Gs, (C-41)

Gυs,Jm

2,1 = −[

iω2 ρf LN

∂3

]

Gs, (C-42)

Gυs,f

2,2 =

[

iω3 εµ

N− iω

1

N∂2

3

]

Gs, (C-43)

Gυs,hb

2,23 =[

∂33 − iω2εµ∂3

]

Gs, (C-44)

Gτb,Je

23,2 =[

iω2ρf Lµ∂3

]

Gs, (C-45)

Gτb,Jm

23,1 =[

iωρf L∂23

]

Gs, (C-46)

Gτb,f

23,2 =[

∂33 − iω2εµ∂3

]

Gs, (C-47)

Gτb,hb

23,23 = −[

iωρc∂23 − iω3

(

εµρc +(

ρf L)2

µ

)]

Gs. (C-48)

The scalar part of the Green’s matrix is given by the inverse of the 4th order wave equationC-30. The scalar Green’s function obeys

[

∂23 −

(

iω

csh

)2][

∂23 −

(

iω

cte

)2]

Gs(x3, ω) = δ(x3 − x3,s). (C-49)

Equation C-49 can be algebraically manipulated in the vertical wavenumber-frequency ma-nipulated to

Gs(x3, ω) =[

{ik3}2 − {iωssh}2]−1 [

{ik3}2 − {iωste}2]−1

δ(x3 − x3,s). (C-50)


Where we defined the slowness ssh;te as ssh;te = c−1sh;te. Equation C-50 can also be written as

Gs(x3, ω) = −[

{iωssh}2 − {iωste}2]−1

[

(

{ik3}2 − {iωste}2)−1

−(

{ik3}2 − {iωssh}2)−1

]

δ(x3 − x3,s).

This implies that the scalar Green function is the sum of two Green functions, each satisfyinga modified Helmholtz equation

[

∂3∂3 − {iωssh}2]

Gssh;te(x3, ω) = −δ(x3 − x3,s). (C-51)

The solution to this equation is well known,

Gssh;te(x3, ω) =

exp (−iωssh;te |x3 − x3,s|)2iωssh;te

, (C-52)

and the complete scalar Green function is given by,

Gs(x3, ω) =Gs

te(x3, ω) − Gssh(x3, ω)

{iωssh}2 − {iωste}2 . (C-53)

The first two derivatives of the partial Green functions are given by

∂3Gssh;te(x3, ω) = −1

2sign(x3 − x3,s)exp (−iωssh;te |x3 − x3,s|) , (C-54)

∂3∂3Gssh;te(x3, ω) = −δ(x3 − x3,s) + {iωssh;te}2 Gs

sh;te(x3, ω), (C-55)

and hence the first three derivatives acting on the full scalar Green function are given by

∂3Gs(x3, ω) = −sign(x3 − x3,s)

{iωste} Gte − {iωssh} Gsh

{iωssh}2 − {iωste}2 , (C-56)

∂3∂3Gs(x3, ω) =

{iωste}2 Gte − {iωssh}2 Gsh

{iωssh}2 − {iωste}2 , (C-57)

∂3∂3∂3Gs(x3, ω) = −sign(x3 − x3,s)

{iωste}3 Gte − {iωssh}3 Gsh

{iωssh}2 − {iωste}2 . (C-58)

This completes the solution.

AES/TG/07-27 Simulation of Interferometric Seismo ...€¦ · source types in the interferometric...

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