Master of Science in Applied Geophysics Research …...Master of Science in Applied Geophysics...

Master of Science in Applied Geophysics

Research Thesis

Subsampling and Interpolating the

Injection and Recording Surfaces in the

Exact Boundary Condition Method

Daniel Ortiz Rubio

August 8, 2014

Subsampling and Interpolating the

Injection and Recording Surfaces in the

Exact Boundary Condition Method

Master of Science Thesis

for the degree of Master of Science in Applied Geophysics at

Delft University of Technology

ETH Zurich

RWTH Aachen University

by

Daniel Ortiz Rubio

August 8, 2014

Department of Geoscience & Engineering · Delft University of Technology

Department of Earth Sciences · ETH Zurich

Faculty of Georesources and Material Engineering · RWTH Aachen University

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

ETH Zurich

All rights reserved.

No part of the material protected by this copyright notice may be reproduced or utilized in any

form or by any means, electronic or mechanical, including photocopying or by any information

storage and retrieval system, without permission from this publisher.

Printed in Switzerland

IDEA LEAGUE

JOINT MASTER’S IN APPLIED GEOPHYSICS

Delft University of Technology, The Netherlands

ETH Zurich, Switzerland

RWTH Aachen, Germany

Dated: August 8, 2014

Supervisor(s):

Filippo Broggini

Johan Olof Anders Robertsson

Committee Members:

Filippo Broggini

iv

Johan Olof Anders Robertsson

Dr. Auke Barnhoorn

August 8, 2014

Abstract

In many geophysical applications, it is often desirable to restrict the modeling of the wave

propagation inside a perturbed domain which is embedded in a larger background domain. The

exact boundary condition (EBC) method yields the exact solution to the wave equation within

such a spatially limited subdomain while accounting for all the interactions with the background

domain. The EBC method relies on a Kirchhoff-type integral extrapolation which updates the

boundary condition along the perturbed domain at each time step of the simulation, and on a

set of pre-computed Green’s functions which are used during the extrapolation process. The

implementation of the EBC algorithm in a finite-difference (FD) time-domain (TD) scheme

introduces significant computational cost. In this thesis, a systematic reduction of the number

of receivers and sources along the recording and the injection surfaces (required by the EBC

method) has been implemented and assessed in order to reduce the computational cost of the

EBC implementation in the FDTD scheme. This study is based on simple two dimensional

velocity and density models with square-shaped and rectangular geometries. Results show a

significant reduction in computational time for both the Green’s functions computations and

the extrapolation process. However, the accuracy of the reconstructed wavefields is affected

by the subsampling process, and in particular when subsampling along the recording surface.

More significantly, the presence of spatially aliased wavefields leads to an erroneous evaluation

of the extrapolation integral since the implicit up/down separation in the Kirchhoff integral

breaks down. Finally, interpolation of the subsampled wavefields on both the recording and

the injection surfaces improves the accuracy of the reconstructed wavefields. Provided that the

spatial Nyquist sampling criterion is not violated, we find that computational savings of up to

49% on the Green’s functions computations and 76% on the EBC computations are possible

compared to using all gridpoints everywhere in the FD scheme. We expect these savings to

become even greater in 3D.

August 8, 2014

vi Abstract

August 8, 2014

Acknowledgements

First of all, I would like to dedicate some words for all people who participated in this project.

My first mention is to Filippo whose tremendous assistance made this thesis possible, and in

particular helping me to improve my writing skills. Second, to Johan for his enthusiasm into

the project encouraging and supporting me from before my arrival in Zurich until the very

end. To Dirk-Jan van Manen for his friendly and devoted support which has been extremely

useful, specially with his theoretical insights. Marlies Vasmel for her availability for questions

and discussions all the way through the thesis, from the very first day until the very last one.

I would like to not forget Heinrich Horstmeyer, for his committed assistance with any kind of

issue, and in particular with our workstations.

I would like to thank as well all my fellow students for these fantastic two years full of joy

and new adventures. I am confident that this new ”family” will remain for a long time. To

my colleagues in FO67, for the good times we have shared but specially for those stressful and

not so pleasant moments. Personally, I am very grateful to Wynze Meijer for helping me out

whenever I encountered problems, always with a big smile upon his face regardless his own

stress.

To E.ON and RWTH Aachen University’s Education Fund for sponsoring my studies during

this master program.

Last but not least, I would like to thank my family who always supported and motivated me

to work hard and learn as much as possible. Finally, to Marina Fernandez (also family) for the

instructive example offered of never giving up throughout her PhD thesis.

Swiss Federal Institute of Technology Daniel Ortiz Rubio

August 8, 2014

viii Acknowledgements

August 8, 2014

Contents

Abstract v

Acknowledgements vii

1 Introduction 1

1-1 Seismic wave propagation and the exact boundary condition method . . . . . . . . . 1

1-2 Thesis objectives and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory 5

2-1 Acoustic wave propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2-2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2-3 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2-4 Exact boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2-4-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2-4-2 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2-4-3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2-4-4 Discretization of the extrapolation equation . . . . . . . . . . . . . . . . . . 13

3 Subsampling of the EBC extrapolation surface integral 15

3-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3-2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3-2-1 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3-2-2 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3-3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

August 8, 2014

x Contents

3-3-1 Square Homogeneous 2D example . . . . . . . . . . . . . . . . . . . . . . . 18

3-3-2 Square Heterogeneous 2D example . . . . . . . . . . . . . . . . . . . . . . . 23

3-3-3 Rectangular Homogeneous 2D example . . . . . . . . . . . . . . . . . . . . 27

3-3-4 Rectangular Heterogeneous 2D example . . . . . . . . . . . . . . . . . . . . 31

3-3-5 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Interpolation of the EBC extrapolation surface integral 43

4-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4-2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Conclusions 49

5-1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5-2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Bibliography 53

August 8, 2014

List of Figures

2-1 Interface ∂D either representing a free surface or a rigid boundary between two mediaD and D′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2-2 Configuration illustrating the theory of exact boundary conditions. The rays labeled(1) and (2) denote waves that cross the boundary ∂Dinj leaving and entering thedomain D, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3-1 Velocity and density model for the homogeneous case with the square-shaped geom-etry. The blue star denotes the source location. . . . . . . . . . . . . . . . . . . . . 19

3-2 Snapshots of modeled pressure for a homogeneous case with the square-shaped ge-ometry. Left panel: modeled pressure in the reference full model. Centre panel:modeled pressure in the truncated domain. Right panel: difference between the firsttwo panels multiplied by a factor of 1012. . . . . . . . . . . . . . . . . . . . . . . . 20

3-3 RMS error due to subsampling of the injection surface for a homogeneous case withthe square-shaped geometry. The dashed line connects the RMS error of the testswith sources on all corners. The vertical red dashed line indicates the Nyquist spatialsampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3-4 Evolution of the RMS error as function of time for a subsampling factor of 4 of theinjection (left), recording (center), and both (right) surfaces for the square-shapedhomogeneous model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3-5 Comparison between the observed and the theoretical savings in computational timefor the square-shaped homogeneous model. The theoretical savings curve has beencalculated with equation 3-3 (left panel) and equation 3-4 (right panel). Left panel:the GF computation. Right panel: the EBC computation. . . . . . . . . . . . . . . . 24

3-6 Velocity and density model in a simple heterogeneous case with the square-shapedgeometry. Two anomalies are considered for both models, one inside cint = 2500m/s and ρint = 5000 kg/m3, and one outside cext = 2500 m/s and ρext = 5000kg/m3, respectively. The blue star denotes the source location. . . . . . . . . . . . . 27

August 8, 2014

xii List of Figures

3-7 Velocity and density model for the homogeneous medium with a rectangular geom-etry. For display purposes, the vertical dimension is exaggerated by a factor of ten.The blue star denotes the source location. . . . . . . . . . . . . . . . . . . . . . . . 30

3-8 Evolution of the RMS error as a function of time for the rectangular configuration ina homogeneous model. Upper row, from left to right, subsampling of the recordingsurface: h=2, v=1; and h=2, v=2. Lower row, from left to right, subsampling onthe recording surface: Rec.F h=4, v=1 and subsampling of both surfaces: Sour.Fh=2, v=2 Rec.F h=2, v=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3-9 Velocity and density model for the heterogeneous medium with a rectangular ge-ometry. For display purposes, the vertical dimensions is exaggerated by a factor often. Two anomalies are considered for both models, one inside cint = 2500 m/sand ρint = 5000 kg/m3, and one outside cext = 2500 m/s and ρext = 5000 kg/m3,respectively. The blue star denotes the source location. . . . . . . . . . . . . . . . . 32

3-10 RMS evolution as a function of time for the rectangular configuration in a hetero-geneous model. From left to right: subsampled approach with Sour.F h=1, v=16,Rec.F h=1, v=4; incomplete surface with two vertical sides missing; and incompletesurface with one vertical side missing. . . . . . . . . . . . . . . . . . . . . . . . . . 34

3-11 Diagram illustrating the extrapolation process. Recording surface fully sampled. Toppanel: a purely outgoing wavefield propagating from D to D′ (labelled (1)) is ex-trapolated from ∂Drec to ∂Dinj after it reaches ∂Drec (labelled (2)). Bottom panel:a purely incoming wavefield propagating from D′ to D (labelled (3)) is extrapolatedfrom ∂Drec to ∂Dextp after it reaches ∂Drec (labelled (4)). The star denotes a sourcelocated along ∂Dinj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3-12 Extrapolation integral results for a purely incoming wavefield. Top row: the extrap-olation to ∂Dextp. Bottom row: the extrapolation to ∂Dinj . From left to right: thereference wavefield, the extrapolated wavefield, and the difference between them. . 38

3-13 Extrapolation integral results for a purely incoming wavefield when the recordingsurface is not subsampled (above) and when the recording surface is subsampled bya factor of 2 (below). From left to right: the reference wavefield, the extrapolatedwavefield, and the difference between them. . . . . . . . . . . . . . . . . . . . . . 39

3-14 Diagram illustrating the extrapolation process. Recording surface subsampled with afactor Rec.F = 2. A purely incoming wavefield (labelled (1)) propagating from D′ toD is extrapolated from ∂Drec to ∂Dinj (labelled (2)). Label (3): injected wavefieldfrom the source placed along ∂Dinj is extrapolated from ∂Drec to ∂Dinj (labelled(4)). The star denotes a source located along ∂Dinj . . . . . . . . . . . . . . . . . . 39

3-15 Evolution of the RMS error for the homogeneous case with the square-shaped geom-etry. Left panel: subsampling the recording surface by a factor of Rec.F = 2. Rightpanel: subsampling the recording surface by a factor of Rec.F = 4. Threshold forthe RMS error of 55%. Time window of 0.6 s for each distance such that (for all thecurves) the last value of the RMS error is the same (i.e., 55%). . . . . . . . . . . . . 40

3-16 Incoming plane waves at the recording surface for the extrapolation test. Top row:take off angle of 40◦. Bottom row: take off angle of 80◦. From left to right: Rickerwavelet with a central frequency of 25 Hz and Ricker wavelet with a central frequencyof 40 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3-17 Extrapolation integral results for a purely incoming plane wave when the recordingsurface is subsampled by a factor of 2. Top row: take off angle of 40◦. Bottom row:take off angle of 80◦. From left to right: Ricker wavelet with a central frequency of25 Hz and Ricker wavelet with a central frequency of 40 Hz. . . . . . . . . . . . . . 42

August 8, 2014

List of Figures xiii

4-1 RMS error due to subsampling of the injection surface for a homogeneous case withthe square-shaped geometry. Solid line: connects the RMS errors of the tests beforeinterpolation has been applied to the injection surface. Dashed line: connects theRMS errors of the tests after interpolation has been applied to the injection surface.The vertical red dashed line indicates the Nyquist spatial sampling. . . . . . . . . . . 45

4-2 Comparison between the observed and the theoretical savings in computational timefor the EBC computations for the square-shaped homogeneous model. Dashed line:theoretical savings after subsampling the injection surface (calculated with equation3-4). Black solid line: observed savings after subsampling the injection boundarybut before interpolation. Red solid line: observed savings after subsampling andinterpolation have been implemented along the injection surface. . . . . . . . . . . . 46

August 8, 2014

xiv List of Figures

August 8, 2014

List of Tables

2-1 States A and B in the representation of the acoustic pressure for the truncated domain. 11

2-2 States A and B in the representation of the acoustic pressure for the linked domain. 12

2-3 Extrapolation process with N=3 time steps. . . . . . . . . . . . . . . . . . . . . . . 14

3-1 Terminology used in all the tables showing results in Chapters 3 and 4. . . . . . . . 17

3-2 FD scheme parameters used in all the simulations carried out in Chapters 3 and 4. . 19

3-3 Geometrical parameters for the square-shaped model used in Chapters 3 and 4. . . . 19

3-4 Homogeneous case with the square-shaped geometry. Subsampling the injectionsurface. First column: subsampling factor applied to the injection surface. Secondand third columns: number of sources and receivers used on the test, respectively.Fourth and fifth columns: savings in computational time for the Green’s functionsand EBC calculations, respectively. Sixth column: maximum RMS error due tosubsampling of the injection surface. . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3-5 Homogeneous case with the square-shaped geometry. Subsampling the recordingsurface. First column: subsampling factor applied to the recording surface. Secondand third columns: number of sources and receivers used on the test, respectively.Fourth and fifth columns: savings in computational time for the Green’s functionsand EBC calculations, respectively. Sixth column: maximum RMS error due tosubsampling of the recording surface. . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3-6 Homogeneous case with the square-shaped geometry. Subsampling both the injectionand the recording surfaces. First and second columns: subsampling factors applied tothe injection and recording surfaces, respectively. Third and fourth columns: num-ber of sources and receivers used on the test, respectively. Fifth and sixth columns:savings in computational time for the Green’s functions and EBC calculations, re-spectively. Seventh column: maximum RMS error due to subsampling of both theinjection and recording surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

August 8, 2014

xvi List of Tables

3-7 Heterogeneous case with the square-shaped geometry. Subsampling the injectionsurface. First column: subsampling factor applied to the injection surface. Secondand third columns: number of sources and receivers used on the test, respectively.Fourth and fifth columns: savings in computational time for the Green’s functionsand EBC calculations, respectively. Sixth column: maximum RMS error due tosubsampling of the injection surface. . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3-8 Heterogeneous case with the square-shaped geometry. Subsampling the recordingsurface. First column: subsampling factor applied to the recording surface. Secondand third columns: number of sources and receivers used on the test, respectively.Fourth and fifth columns: savings in computational time for the Green’s functionsand EBC calculations, respectively. Sixth column: maximum RMS error due tosubsampling of the recording surface. . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3-9 Heterogeneous case with the square-shaped geometry. Subsampling both the in-jection and the recording surfaces. First and second columns: subsampling fac-tors applied to the injection and recording surfaces, respectively. Third and fourthcolumns: number of sources and receivers used on the test, respectively. Fifth andsixth columns: savings in computational time for the Green’s functions and EBC cal-culations, respectively. Seventh column: maximum RMS error due to subsamplingof both the injection and recording surfaces. . . . . . . . . . . . . . . . . . . . . . . 26

3-10 Geometrical parameters for the rectangular model used in Chapters 3. . . . . . . . . 28

3-11 Homogeneous case with the rectangular geometry. Subsampling the injection surface.First and second columns: subsampling factors applied to the injection surface alongthe horizontal and vertical sides, respectively. Third and fourth columns: number ofsources used on the test along the horizontal and vertical sides, respectively. Fifthand sixth columns: number of receivers used on the test along the horizontal andvertical sides, respectively. Seventh and eighth columns: savings in computationaltime for the Green’s functions and EBC calculations, respectively. Ninth column:maximum RMS error due to subsampling of the injection surface. . . . . . . . . . . 28

3-12 Homogeneous case with the rectangular geometry. Subsampling the recording sur-face. First and second columns: subsampling factors applied to the recording surfacealong the horizontal and vertical sides, respectively. Third and fourth columns: num-ber of sources used on the test along the horizontal and vertical sides, respectively.Fifth and sixth columns: number of receivers used on the test along the horizontal andvertical sides, respectively. Seventh and eighth columns: savings in computationaltime for the Green’s functions and EBC calculations, respectively. Ninth column:maximum RMS error due to subsampling of the recording surface. . . . . . . . . . . 29

3-13 Homogeneous case with the rectangular geometry. Subsampling both the injectionand the recording surfaces. First and second columns: subsampling factors appliedto the injection surface along the horizontal and vertical sides, respectively. Thirdand fourth columns: subsampling factors applied to the recording surface along thehorizontal and vertical sides, respectively. Fifth and sixth columns: number of sourcesused on the test along the horizontal and vertical sides, respectively. Seventh andeighth columns: number of receivers used on the test along the horizontal and verticalsides, respectively. Ninth and tenth columns: savings in computational time for theGreen’s functions and EBC calculations, respectively. Eleventh column: maximumRMS error due to subsampling of both the injection and recording surface. . . . . . 29

August 8, 2014

List of Tables xvii

3-14 Heterogeneous case with the rectangular geometry. Subsampling the injection sur-face. First and second columns: subsampling factors applied to the injection surfacealong the horizontal and vertical sides, respectively. Third and fourth columns: num-ber of sources used on the test along the horizontal and vertical sides, respectively.Fifth and sixth columns: number of receivers used on the test along the horizontal andvertical sides, respectively. Seventh and eighth columns: savings in computationaltime for the Green’s functions and EBC calculations, respectively. Ninth column:maximum RMS error due to subsampling of the injection surface. . . . . . . . . . . 33

3-15 Heterogeneous case with the rectangular geometry. Subsampling the recording sur-face. First and second columns: subsampling factors applied to the recording surfacealong the horizontal and vertical sides, respectively. Third and fourth columns: num-ber of sources used on the test along the horizontal and vertical sides, respectively.Fifth and sixth columns: number of receivers used on the test along the horizontal andvertical sides, respectively. Seventh and eighth columns: savings in computationaltime for the Green’s functions and EBC calculations, respectively. Ninth column:maximum RMS error due to subsampling of the recording surface. . . . . . . . . . . 35

3-16 Heterogeneous case with the rectangular geometry. Subsampling both the injectionand the recording surfaces. First and second columns: subsampling factors appliedto the injection surface along the horizontal and vertical sides, respectively. Thirdand fourth columns: subsampling factors applied to the recording surface along thehorizontal and vertical sides, respectively. Fifth and sixth columns: number of sourcesused on the test along the horizontal and vertical sides, respectively. Seventh andeighth columns: number of receivers used on the test along the horizontal and verticalsides, respectively. Ninth and tenth columns: savings in computational time for theGreen’s functions and EBC calculations, respectively. Eleventh column: maximumRMS error due to subsampling of both the injection and recording surface. . . . . . 36

3-17 FD parameters used in the extrapolation integral test. . . . . . . . . . . . . . . . . . 36

3-18 First column: distances between ∂Drec and ∂Dinj . Second and third columns: sim-ulation times at which the RMS error starts to significantly increase because ofsubsampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4-1 Homogenous case with the square-shaped geometry. Interpolating the injection sur-face to obtain a source at each gridpoint position. The recording surface is not sub-sampled. The relative difference, Rel.Diff, is a comparison between the Max.Errorsobtained before (b) and after (a) interpolation has been implemented. Positive re-sults mean that the Max.Error increases, whereas negatives results mean that theMax.Error decreases. The Saved Time is compared to the reference computationaltime and positive values mean larger savings in time. . . . . . . . . . . . . . . . . . 47

4-2 Homogenous case with the square-shaped geometry. Interpolating the recordingsurface to obtain a receiver at each gridpoint position. The injection surface is notsubsampled. The relative difference, Rel.Diff, is a comparison between the Max.Errorsobtained before (b) and after (a) interpolation has been implemented. Positiveresults mean that the Max.Error increases, whereas negatives results mean that theMax.Error decreases. Note that for this test, the savings in time (column 4 and 5)are insignificant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4-3 Homogenous case with a square-shaped geometry. Interpolating both the recordingand the injection surfaces to obtain receivers and sources at each gridpoint position,respectively. The relative difference, Rel.Diff, is a comparison between the Max.Errorsobtained before (b) and after (a) interpolation has been implemented. Positiveresults mean that the Max.Error increases, whereas negatives results mean that theMax.Error decreases. The Saved Time is compared to the reference computationaltime and positive values mean larger savings in time. . . . . . . . . . . . . . . . . . 48

August 8, 2014

xviii List of Tables

August 8, 2014

Chapter 1

Introduction

1-1 Seismic wave propagation and the exact boundary condition

method

Numerical modeling of seismic wave propagation is a fundamental tool in geophysics (Nihei

et al., 2012). Modeling is key to understand the physics of wave propagation in real earth media,

to image sub-surface structures, to invert for or characterize sub-surface properties, to locate

earthquakes, to generate synthetic data for research, etc. The value of numerical modeling for

applications on many scales, ranging from shallow applications in engineering or environmental

geophysics, to global scale seismology, can therefore hardly be overstated. Various applications

require the recalculation of the seismic response after model updates within a spatially limited

domain embedded in an unperturbed larger domain. For example, in 4D or time-lapse seismics,

the region where model parameters change is restricted to a certain limited volume. In full

waveform inversion, some model parameters of the model space need to be updated at each

iteration, whereas the rest of the model parameters may remain unchanged (Virieux and Operto,

2009). For this kind of scenarios, it would be beneficial to restrict the size of the simulation

domain to the subvolume of interest avoiding the necessity to run the simulation on the complete

model. By doing so, significant savings in the number of calculations and memory for storage of

variables can be achieved, in particular when the wavefield needs to be recomputed several times

(e.g., many iterations in an inversion scheme) or when the volumes where the model alterations

occur are much smaller than the background domain. To accurately reconstruct the wavefield

August 8, 2014

2 Introduction

inside the subdomain of interest, all of its interactions with the background domain need to be

taken correctly into account.

Various approaches have been developed to model the propagation of an acoustic wavefield in a

limited subvolume, while accounting for its interaction with a larger background domain (Schus-

ter, 1985; Chapman and Coates, 1994; Robertsson and Chapman, 2000). Yet, none of these

approaches perfectly reconstruct the so-called high-order long-range interactions, which corre-

spond to multiple interactions between the subdomain (where the model parameters change)

and the surrounding model. The exact boundary condition (EBC) method presented by van Ma-

nen et al. (2007) and Vasmel et al. (2013) allows exact recomputation of the acoustic wavefield

inside a spatially limited domain while accounting for all its interactions with the background

domain. Additionally, the method does not put any restrictions on size, magnitude, or shape

of the anomalies. The EBC method relies on a Kirchhoff-type integral extrapolation to update

the boundary conditions along the domain of interest at each time step of the simulation. The

Green’s functions required for the extrapolation process are only computed once in advance, and

they do not need to be recomputed if model alterations are restricted only to a set of spatially

limited subvolumes.

However, the implementation of the EBC method in a time-domain (TD) finite-difference (FD)

scheme can be computationally demanding. In particular, the Green’s functions computations

carried out on the full background domain and the extrapolation process computed at every

time step of the simulation introduce large costs.

1-2 Thesis objectives and outline

The primary objective of this work is to investigate ways to significantly reduce the cost asso-

ciated with the EBC implementation in a FD scheme by increasing the spatial sampling of the

extrapolation integral. We carefully examine the effects of subsampling on the accuracy of the

reconstructed wavefields. In addition, we investigate employing interpolation of wavefields to

improve the accuracy of results obtained after subsampling.

The background theory of the exact boundary conditions is presented in Chapter 3. We then

present the results based on a variety of 2D acoustic models. First, in Section 3-3-1 we focus on

a square-shaped homogeneous model where we apply subsampling to the extrapolation integral.

After, in Section 3-3-2, subsampling is tested on a simple heterogeneous model which includes

one anomaly of strong contrast in material properties inside and outside the homogeneous

model used in Section 3-3-1. With this heterogeneous model, we assess the ability of the EBC

algorithm to adequately incorporate the long-term high-order interactions.

August 8, 2014

1-2 Thesis objectives and outline 3

In Section 3-3-3, we extend the work to a rectangular configuration and apply subsampling

to the vertical edges of the rectangular model in order to approximate an open surface which

closely resembles a marine seismic data acquisition setup. We analyze the introduced errors and

compare them with a simulation without sources and receivers on the vertical edges. Again,

for the rectangular configuration we test the simple heterogeneous model used in Section 3-3-2.

In Section 3-3-5, we proceed to analyze the errors introduced by subsampling in detail, with

special emphasis on the extrapolate integral. In Chapter 4, interpolation of the wavefields is

implemented for the simple square-shaped homogeneous model previously studied. A cubic

spline interpolation technique is utilised aiming to increase the accuracy of the reconstructed

wavefields after subsampling. Furthermore, interpolation after subsampling is of special interest

because in various applications we are constrained by the number of sources and receivers

available (e.g., laboratory environment). Finally, in Chapter 5 we present the conclusions from

the study we undertook and an outlook for future research focussing on further possibilities to

improve the accuracy of the reconstructed wavefields after subsampling.

August 8, 2014

4 Introduction

August 8, 2014

Chapter 2

Theory

In this chapter, we briefly review the theory of wave propagation in acoustic media and introduce

its boundary conditions and Green’s functions. Finally, we discuss the theory behind the exact

boundary conditions.

2-1 Acoustic wave propagation

Any mechanical disturbance in a solid, liquid, or gaseous medium is followed by a restoring

force driving the system to its equilibrium. The propagation of this mechanical disturbance

through the medium is what is called wave propagation. Here, we focus on acoustic waves,

since we only consider compressional forces propagating through an inhomogeneous medium

(Rayleigh, 1878). The underlying assumption is that the medium can be represented as

an instantaneously reacting fluid which does not support shear stress. Moreover, this work

is based on 2D wave propagation. The spatial coordinates are defined as x = (x1,x3),

where x1 is the horizontal coordinate and x3 the vertical one. The positive direction of the

x3-axis points downwards. The starting points to describe the linear wave motion are the

equation of continuity and the equation of motion (Wapenaar and Berkhout, 1989), respectively:

1

K

∂p

∂t+∇ · ~v = q (2-1)

August 8, 2014

6 Theory

ρ∂~v

∂t+∇p = ~f, (2-2)

where p is the pressure field, t is time, ~v is the particle velocity, ρ is the particle density, K = ρc2

is the bulk modulus, q is the volume density of injection rate, and ~f is the volume density of

external force. The volume density of injection rate is representative for the action of acoustic

sources of the “monopole” type, whereas the volume density of external force will be employed

to represent the action of acoustic sources of the “dipole” type. Combining these two equations,

we obtain the linear 2D two-way wave equation for the acoustic pressure p:

ρ∇ ·(

1

ρ∇p)− ρ

K

∂2p

∂t2= −s, (2-3)

with

s = ρ∂q

∂t− ρ∇ ·

(1

ρ~f

), (2-4)

where s represents a source distribution in terms of the volume density of volume injection rate

q and the volume density force ~f .

2-2 Boundary conditions

In practice, acoustic media with different material properties are often in contact along inter-

faces. This is shown in Figure 2-2, where ∂D is an interface that separates two different media

represented by D and D′, respectively. Across this interface, the constitutive parameters present

a jump discontinuity. From the equations of motion and continuity, it follows that some com-

ponents of the particle velocity and the pressure also show a jump discontinuity across such

interface. For this reason, the pressure and/or the particle velocity are not continuously differ-

entiable, hence equations 2-1 and 2-2 do not hold anymore. Therefore, to interrelate the acoustic

wavefield at either side of the interface ∂D, we need to enforce specific boundary conditions. In

this work, two types of boundary conditions will be particularly important:

1. ∂D is a free boundary. This situation describes, for example, what happens at the interface

between water and air. Hence, this particular boundary condition will be of interest when

we model or study marine seismic data acquisition. In this case, the pressure along ∂Dvanishes:

pD = pD′ = 0 on ∂D . (2-5)

August 8, 2014

2-3 Green’s functions 7

Exact boundary conditions and their applications:A tutorial — 2/7

2.2 ConfigurationWe consider the configuration shown in Figure 1. Forthe sake of simplicity, we focus on a two-dimensionalconfiguration; however, the theory of EBCs is also validin one and three dimensions. The domain D is boundedby the surface ∂Dinj and the normal n points away fromD. The domain D′ is the complement of D ∪ ∂Dinj inR2 (R3 in three dimensions). The surface ∂Drec doesnot correspond with a physical surface. First, we definea physical domain which, in practice, can be associatedwith a wave propagation laboratory bounded by rigidboundaries, such as the the walls of a water tank. Thephysical domain corresponds to the domain D, where∂Dinj is a rigid boundary. Then, we introduce an ex-tended domain defined in D ∪ D′. This new domain isequal to the physical domain inside D and characterizedby its own material properties in D′. The surface ∂Dinj

does not correspond with a physical surface in the ex-tended domain.

NOTE The normal to the surface must point out-ward, away from D, because the global form ofRayleigh’s theorem uses Gauss’s theorem. I am notsure if it is correct to have an inward normal, as inVasmel et al. (2013).

A conventional wave propagation laboratory consistsof a water tank (e.g., 2 m x 2 m x 2 m), where sourcesand receivers can be placed on the walls of the tankand/or in the water. Samples of various dimension andcomposition, such as rocks, are immersed in in the waterand probed by a wave field generated by user definedsources. One of the main limitations of a laboratory ofthis kind is its finite size. If we want to probe the samplewith a source characterized by a wavelet with a wavelength in the order of 50 m, then the wave field scatteredby the sample is masked by the high-amplitude reflectiondue to the walls of the tanks. The research question thatwe want to address is: Can we transform the physicaldomain into the extended domain, so that we can usesources characterized by a wave length comparable tothe dimension of the physical domain? EBCs providethe answer to this question as we show in the remainderof the paper.

2.3 RepresentationsTo derive the theory behind EBCs, our starting point isthe global form of Rayleigh’s recpirocity theorem of theconvolution type (de Hoop, 1995) applied to the domainD: !

∂Dinj

{pAvi,B − vi,ApB} nid2x = (1)

jω

"

D{(κA − κB) pApB − (ρA − ρB) vi,Avi,B} d3x

+

"

D{pAqB − vi,Afi,B − qApB + fi,Avi,B} d3x,

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

Figure 1. Configuration used to present the theory ofexact boundary conditions. The rays labeled (1) and(2) denote waves that cross the boundary ∂Dinj leavingand entering the domain D, respectively.

where the subscripts A and B denote the wave fields andmedium parameters for two different states.

Now, we derive general representations of the convo-lution type for the acoustic pressure inside the domainD. First, we choose for state A the actual acoustic wavefields propagating in the physical domain due to a sourceof volume injection rate and for state B we considerthe Green’s functions in the physical domain due to apoint source of volume injection rate, as described inTable 1. The surface ∂Dinj corresponds to the physicalboundaries of the wave propagation laboratory; hence,on ∂Dinj , both the Green’s functions and the actualwave fields satisfy rigid boundary conditions. Here, weconsider rigid boundary conditions on ∂Dinj , but we em-phasize that the applications of EBCs are not limited tothis configuration. EBCs can be applied also when theboundary ∂Dinj is not characterized by specific bound-ary conditions and the physical domain is also definedin D′. In fact, EBCs can be used to link two domainscharacterized by very different properties.







∂Dinj


jω

"


+

"


x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′










∂Dinj


jω

"


+

"


x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′










∂Dinj


jω

"


+

"


x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′




Figure 2-1: Interface ∂D either representing a free surface or a rigid boundary between twomedia D and D′.

2. ∂D is a rigid boundary and it describes the behaviour of a fluid bounded by a surface that

cannot be deformed. This is the situation of water contained in a water tank with rigid

walls. In this case, the normal component of the particle velocity along ∂D vanishes:

~vD · n = ~vD′ · n = 0 on ∂D . (2-6)

2-3 Green’s functions

An acoustic Green’s function represents the impulse response of a fluid medium and satisfyies

the two-way wave equation in which the source distribution has been replaced by an impulsive

source at x′:

∇ ·(

1

ρ(~x)∇Gp,q(x,x′, t)

)− 1

K(~x)

∂2Gp,q(x,x′, t)

∂t2= −δ(x− x′)

∂δ(t)

∂t. (2-7)

Comparing equation 2-7 to equation 2-3, Gp,q(x,x′, t) is the Green’s function replacing the

pressure field p, and δ(x−x′)∂δ(t)∂t is the impulsive source replacing the volume density of volume

injection rate. Following the notation of de Hoop (1995), the Green’s function Gp,q(x,x′, t)

denotes the acoustic pressure observed at x (superscript p) due to an impulsive point source of

volume injection rate at x′ (superscript q). The Green’s functions corresponding to the particle

velocity are derived from the Green’s functions corresponding to the pressure response:

∂

∂tGv,qi (x,x′, t) = −1

ρ∇Gp,q(x,x′, t), (2-8)

where the subscript i denotes the component of the velocity along xi. Similarly, the Green’s

function Gp,fm (x,x′, t) satisfies equation 2-7, but with a different source term: ∂∂x′m

(δ(x−x′)

ρ

).

August 8, 2014

8 Theory

The Green’s function Gp,fm (x,x′, t) corresponds to the acoustic pressure measured at x (super-

script p) due to an impulsive point source of force at x′ (superscript f) oriented in the xm

direction. The particle velocity can be derived from the pressure response:

∂

∂tGv,fi,m(x,x′, t) = −1

ρ

(∇Gp,fm (x,x′, t) + δi,mδ(x− x′)

). (2-9)

Considering equation 2-3 and equation 2-7, the relationship between the acoustic wavefield p

and its associated Green’s functions Gp,q(x,x′, t) and Gp,fm (x,x′, t), for a volume D enclosing

the source distribution s(x′, t′) (de Hoop, 1995), is

p(x, t) =

∫

ts∈Rdts

∫

xs∈D

{Gp,q(x,xs, t− ts)q(xs, ts) +Gp,fm (x,xs, t− ts)~f(xs, ts)

}d3xs. (2-10)

A similar relationship can be derived for the particle velocity and its associated Green’s

functions Gv,q(x,xs, t) and Gv,f (x,xs, t).

2-4 Exact boundary conditions

2-4-1 Introduction

Now we consider the situation where a domain D with parameters KD and ρD is embedded in

a larger domain D′, characterised by different material parameters KD′ and ρD′ . Moreover, we

emphasise that both domains can be heterogeneous, hence waves can scatter in and out from one

domain to the other. This is illustrated in Figure 2-2, where the rays labelled (1) and (2) denote

waves crossing the boundary outward and inward, respectively. No restrictions are introduced

on the magnitude, shape, or size of the heterogeneities. For the purpose of modeling wave

propagation through the entire domain D∪D′, we follow the wave propagation theory described

in Section 2-1. We use, for instance, a time-domain (TD) finite-difference (FD) algorithm to

solve equation 2-3 everywhere on D∪D′ (Robertsson and Blanch, 2011; van Manen et al., 2005).

However, sometimes our interest is restricted to the smaller subvolume D only, where we assume

that the model parameters change (van Manen et al., 2007). Solving equation 2-3 inside D only

is not sufficient because any wave propagating from the smaller domain to the larger one and

scattering back to the smaller domain, the so-called high-order long-range interactions, labelled

(1) and (2) in Figure 2-2, are not accounted for by solving equation 2-3 inside D only. The exact

August 8, 2014

2-4 Exact boundary conditions 9

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

Figure 2-2: Configuration illustrating the theory of exact boundary conditions. The rayslabeled (1) and (2) denote waves that cross the boundary ∂Dinj leaving andentering the domain D, respectively.

boundary condition (EBC) approach introduced in the next subsection and presented in van

Manen et al. (2007) and Vasmel et al. (2013) enables restricting the numerical simulation to

limited subvolumes while perfectly accounting for all interactions of the propagating wavefield

with its background volume.

The main advantage enabled by restricting the modeling to a smaller subvolume is a considerable

reduction of computational cost in terms of the number of calculations, while still being able to

accurately reproduce the wavefield inside the domain of interest including the interaction with

the surrounding domain. Additionally, we reduce the amount of storage and memory needed

for the material parameters and wavefield variables required by the FD algorithm. Therefore,

significant savings in the number of calculations and memory for storage can be achieved, for

instance, when the goal is to recompute the wavefield inside the limited subvolume (e.g., 4D or

time-lapse seismics).

August 8, 2014

10 Theory

2-4-2 Configuration

We now again consider the configuration shown in Figure 2-2 and introduce the definitions of

the terms used to describe the EBC method. The region where model parameters change is

referred to as the altered or truncated domain D and is embedded in a larger domain D′ referred

to as background domain. For simplicity, we describe the situation where all the changes in

medium parameters are limited to a single subvolume. However, the EBC theory also holds for

situations where multiple altered subdomains exist.

In order to account for all the interactions between the altered domain D and its background

domain D′, two surfaces are required: the inner recording surface ∂Drec, which does not

correspond with a physical surface, and the outer injecting or injection ∂Dinj surface. The

terminology “recording a wavefield” refers to the action of measuring the value of the wavefield

with an array of receivers placed along the recording surface, ∂Drec. The terminology “injecting

a wavefield” refers to introducing a wavefield, previously computed and stored, along the closed

injection surface, ∂Dinj , surrounding the injection subvolume D.

2-4-3 Formulation

To derive the exact boundary conditions, two states A and B are defined. First, we choose

for state A the actual acoustic wavefield propagating through D due to a source of volume

injection rate. For state B, we consider the Green’s functions in D due to an impulsive point

source of volume injection rate, as summarized in Table 2-1 . For both states, we impose a

rigid boundary along the injection surface. According to equation 2-6, this means that the

particle velocity for both the actual wavefield and the Green’s function vanishes along ∂Dinj .Note that the application of EBC’s is not limited to this configuration and, in particular, a free

boundary could have also been considered. Following Vasmel et al. (2013), the exact boundary

conditions are derived from Rayleigh’s reciprocity theorem (de Hoop, 1995) applied to the

configuration shown in Figure 2-2:

August 8, 2014


Table 2-1: States A and B in the representation of the acoustic pressure for the truncateddomain.

State A State B

wavefields {p, vi}(x, t) {Gp,q, Gv,qi }(x,x′, t)Medium parameters {ρ,K}(x) {ρ,K}(x)Source functions {q, 0}(xs, ts) {δ(x− x′), δ(t− 0)}BCs on ∂Dinj vi|∂Dinj

ni = 0 Gv,qi |∂Dinjni = 0

Domain D

∮

∂D{Ct(pA, ~vk,B;x, t)− Ct(pB, ~vk,A;x, t)} nkd2x = (2-11)

∫

D

{(ρB − ρA)∂tCt(~vi,A, ~vm,B;x, t)−

(1

K B− 1

K A

)∂tCt(pA, pB;x, t)

}d3x

+

∫

D{Ct(fm,A, ~vm,B,x, t) + Ct(pA, qB,x, t)− Ct(fi,B, ~vi,A,x, t)− Ct(pBqA,x, t)} d3x.

Note that we have used the so-called time convolution operator Ct (de Hoop, 1995) to simplify

the expression of equation 2-11. However, from now on, when we refer to the convolution

operation, we will write it in its integral form as shown in equation 2-12. After inserting the

quantities presented in Table 2-1 into equation 2-11 and enforcing the boundary condition on

∂Dinj , for any point x′ inside D, we obtain

p(x′, t) =

∫ t

0

∫

DGp,q(x′,xs, t− ts)q(xs, ts)d3xdts, (2-12)

where source-receiver reciprocity, G(xs,x′, t, ts) = G(x′,xs, t, ts), has been used.

Now, since our goal is to model the wave propagation inside the truncated domain D including

the long-range high-order interactions with the background domain D′, we need to define a

second state A where the two domains D and D′ are merged together, acting as one larger

domain. We emphasize that the actual computations are restricted to the truncated domain

D, but we want to model the wavefield as if it was propagating in the larger domain D ∪ D′.Therefore, for this second state, the actual wavefield is continuous through ∂Dinj because we

do not enforce any specific boundary condition. The Green’s functions state remain unchanged

and we repeat it in Table 2-2. Again, no restriction has been introduced and a free surface or

August 8, 2014

12 Theory

Table 2-2: States A and B in the representation of the acoustic pressure for the linked domain.

State A State B

wavefields {p, vi}(x, t) {Gp,q, Gv,qi }(x,x′, t)Medium parameters {ρ,K}(x) {ρ,K}(x)Source functions {q, 0}(xs, ts) {δ(x− x′), δ(t− 0)}BCs on ∂Dinj None Gv,qi |∂Dinj

ni = 0

Domain D

a combination of a free surface and a rigid boundary could also have been implemented. After

inserting the quantities presented in Table 2-2 into equation 2-11 and enforcing the boundary

condition on ∂Dinj , for any point x′ inside D, we obtain the expression for the acoustic pressure

of the linked domain,

p(x′, t) = (2-13)∫ t

0

∫

DGp,q(x′,xs, t− ts)q(xs, ts)d3xdts

−∫ t

0

∮

∂Dinj

{Gp,q(x′,x, t− t′)vi(x, t′)

}nid

2xdt′.

Comparing the expression for the obtained acoustic pressure in the linked domain (equation

2-13) to the one obtained for the truncated domain (equation 2-12) the difference resides in the

surface integral term

−∫ t

0

∮

∂Dinj

{Gp,q(x′,x, t− t′)vi(x, t′)

}nid

2xdt′, (2-14)

which corresponds to a distribution of monopole sources along ∂Dinj with source strength

vi(x, t′) · ni. Hence, in order to link the truncated domain D, surrounded by a rigid boundary,

with its background domain D′, an array of monopole sources along ∂Dinj is required. By

imposing the rigid boundary along D, we spatially truncate the area where we carry out our

wavefield modelling introducing undesired reflections caused by the implementation of such a

boundary. These reflections are artefacts that must be removed. In order to eliminate these

undesired reflections, we need to inject a term weighted by vi(x, t′) · ni at the exact same time

as the energy response for this velocity arrives at ∂Dinj (see equation 2-15). To predict the

exact arriving energy to ∂Dinj ahead of time, p(xrec, t) and vm(xrec, t) need to be recorded first

along the recording surface ∂Drec and then extrapolated to ∂Dinj . Using an equation similar

August 8, 2014


to equation (4) of van Manen et al. (2007), we preform an extrapolation forward in time:

vi(xinj , t′) = (2-15)

∫ t′

0

∮

∂Drec

{Gv,qi (xinj ,xrec, t′ − t)vm(xrec, t)

+ Gv,fi,m(xinj ,xrec, t′ − t)p(xrec, t)}nmd2xrecdt,

where Gv,qi and Gv,fi,m are the particle velocity responses due to a monopole and a dipole impulse

function at ∂Drec, respectively. These Green’s functions are computed in advance through

numerical modeling on a model corresponding to the linked domain D ∪ D′ outside ∂Drec. In

fact, for the extrapolation process, the media for which the Green’s functions are calculated

must be identical only between the ∂Drec and ∂Dinj surfaces in the truncated and background

domains. Moreover, the structure of the model inside ∂Drec is irrelevant since the total effect of

the contributions in equation 2-15 is zero (Fokkema and van den Berg, 1993; Thomson, 2012).

Note that by using these Green’s functions, Gv,qi and Gv,fi,m, we account for all the interactions

with the background domain since they contain all the information from the outside of the

truncated domain D. For the particular case where we need to recompute the Green’s functions

(e.g., 4D or time-lapse seismics), because all the changes in the model parameters occur inside

D, these Green’s functions need to be calculated only once resulting into significant savings in

computational cost and memory.

2-4-4 Discretization of the extrapolation equation

Throughout the application of the EBC method implemented in a finite-difference FD code, a

configuration of sources and receivers exist on the injection and recording surface respectively.

Along the inner recording surface ∂Drec, receivers record the wavefield at each grid point of the

FD scheme, separated by the spatial sampling ∆xrec, resulting in a total number of receivers

N◦Rec. At each time step k of the simulation, the Green’s functions are used to extrapolate the

recorded wavefield from ∂Drec to ∂Dinj for all future time steps l > k. At each time step, the

extrapolated wavefield is stored at each grid point of the FD scheme along ∂Dinj , separated by

the spatial sampling ∆xinj , and injected back into the altered domain (see Table 2-3). In total

N◦Sour sources are used to inject the wavefield back into the truncated domain from each grid

point of the FD scheme along the outer injection surface ∂Dinj . The spatial sampling is the

same everywhere in the FD scheme, therefore ∆xrec = ∆xinj = ∆x. The recursive discretized

August 8, 2014

14 Theory

Table 2-3: Extrapolation process with N=3 time steps.

k l ˆvi(x, l, k)

0 0 ˆvi(x, 0, 0) injected1 ˆvi(x, 1, 0) stored2 ˆvi(x, 2, 0) stored

1 1 ˆvi(x, 1, 1) = ˆvi(x, 1, 0) +∮∂Drec

... injected

2 ˆvi(x, 2, 1) = ˆvi(x, 2, 0) +∮∂Drec

... stored

2 2 ˆvi(x, 2, 2) = ˆvi(x, 2, 1) +∮∂Drec

... injected

expression of equation 2-15 is

ˆvi(xinjn , l, k) = ˆvi(x

injn , l, k − 1) + (2-16)

N◦Rec∑

n=1

{ˆGv,qm (xinjn ,xrecn , l, k)ˆvm(xrecn , k)

+ ˆGv,fi,m(xinjn ,xrecn , l, k)ˆp(xrecn , k)}nm∆xrec∆t,

where ˆ is used to denote sampled quantities, the discrete-time indices l and k correspond to

t′ and t, respectively, ∆t is the time sampling, and nm is the normal to the recording surface

∂Drec. In the first term of the right-hand side of equation 2-16, we observe that the injected

wavefield at a time k depends on the values of the stored wavefield at a time k − 1 and,

implicitly, on earlier time steps as described in Table 2-3. The second term on the right-hand

side of equation 2-16 describes to the process of recording and extrapolating the wavefield from

all points xrecn along ∂Drec to a single point xinjn along ∂Dinj .

August 8, 2014

Chapter 3

Subsampling of the EBC extrapolation

surface integral

The EBC method changes the nature of the computations that need to be carried out. The

simulation on the smaller subdomain D relies on the computation of the Green’s functions in ad-

vance and the extrapolation process at every time step of the simulation. Both the calculations

of the Green’s functions and the extrapolation process can be computationally demanding. In

this chapter, an approach to reduce the computational time (number of calculations) and mem-

ory required for storage is introduced addressing the cost of computation of Green’s functions

as well as the extrapolation step.

3-1 Introduction

In the previous chapter, we explained that the process of computing Green’s functions and

extrapolating the wavefield to link two domains can be computationally very demanding. The

cost is particularly high since FD calculations typically are oversampled such that the spatial

sampling ∆x is much smaller than what is needed for the extrapolation of the wavefield in or-

der to minimize numerical dispersion. In this chapter, we study whether subsampling both the

recording and injection surfaces (guided by the Nyquist sampling criterion (Robinson and Clark,

1991)) can reduce the computational cost in terms of time and memory without substantially

August 8, 2014

16 Subsampling of the EBC extrapolation surface integral

compromising on the quality of the recomputed wavefields inside D. First, we consider a sim-

ple square-shaped homogeneous case with an equal number of grid points along the horizontal

and vertical dimensions. Because of its symmetry, we focus on subsampling first the recording

surface, then the injection surface, and finally both surfaces simultaneously. Afterwards, we

consider a rectangular surface where the horizontal dimension is ten times larger than the verti-

cal one. Here, we also subsample each dimension separately trying to approximately reproduce

a situation where on two sides (the vertical ones) there are no sources nor receivers. Finally,

using the same geometries, we test the subsampling in a simple heterogeneous medium.

3-2 Numerical implementation

In Section 2-4-4, the discretized version of the extrapolation equation was introduced (equation

2-16). A key parameter governing the computational time and memory savings is the spatial

sampling ∆x. We therefore consider reducing the number of sources and receivers on both

boundaries ∂Drec and ∂Dinj by increasing their spatial sampling. We denote the new sampling

intervals as ∆xrec and ∆xinj , respectively. For the outer surface ∂Dinj , we use the so-called

subsampling source factor, Sour.F, whereas for the inner one we use the subsampling receiver

factor, Rec.F (see Table 3-1). Although the physics behind the EBC methodology (see Section

2-4-3) does not change once the subsampling has been implemented, some considerations need

to be kept in mind. When we subsample the recording surface, we use fewer receivers to

record the energy arriving at ∂Drec. In order to correct for this missing energy, we increase the

strength of each receiver by multiplying each recorded term ˆGv,qi (xinjn ,xrecn , l, k) · ˆvm(xrecn , k) by

the subsampling factor Rec.F (in the 2D case we study here):

vi(xinjn , l, k) ≈

N−1∑

l=k

N◦RecS∑

n=1

{ˆGv,qi (xinjn ,xrecn , l, k)ˆvm(xrecn , k) (3-1)

+ ˆGp,fi,m(xinjn ,xrecn , l, k)ˆp(xrecn , k)}nm∆xrec,

where now ∆xrec is the new receiver sampling interval after subsampling the recording surface,

∆xrec = Rec.F · ∆x, and N◦Rec S= N◦Rec/Rec.F is the number of receivers along ∂Drec after

subsampling. The same sampling argument applies to subsampling the injection surface ∂Dinj ,

August 8, 2014

3-2 Numerical implementation 17

where the subsampled monopole source array injects energy from fewer sources onto ∂Dinj :

−N◦SourS∑

n=1

{Gp,q(x′n,x

injn , l − k)ˆvi(x

injn , k)

}ni∆xinj , (3-2)

where now ∆xinj is the new injection sampling interval after subsampling the injection surface,

∆xinj = Sour.F · ∆x, and N◦SourS =N◦Sour/Sour.F is the number of sources along ∂Dinj after

subsampling.

Table 3-1: Terminology used in all the tables showing results in Chapters 3 and 4.

N◦Rec Number of receivers before subsamplingN◦Sour Number of sources before subsamplingSour.F Source factorRec.F Receiver factorMax.Error Maximum errorRel.Diff Relative differenceSaved.Time Saved timeGF Green’s functionsEBC Exact boundary condition simulationh Horizontalv Verticala After interpolationb Before interpolation

3-2-1 Accuracy

For an EBC computation where the injection and recording surfaces utilize the same sampling

interval as in the FD computation, we have already seen that the result is close to machine

precision accuracy. To quantify the error introduced because of subsampling these surfaces, we

use the maximum value of the Root Mean Square (RMS) error of the difference between the

reference solution for the full domain and the EBC solution, normalized by the maximum value

of the wavefield anywhere in the full domain1. At each time step of the simulation, the RMS

error is computed for all grid points in the truncated domain D.

1This value is constant for all the examples considered in this work and has a magnitude of 9.35× 105 Pa.

August 8, 2014


3-2-2 Computational cost

The number of Green’s functions needed is 2×N◦Rec×N◦Sour, where the factor 2 accounts

for the monopole and dipole responses. Thus, significant savings can clearly be achieved by

reducing the number of receivers and sources N◦Rec and N◦Sour.

Subsampling has a strong impact on the number of computations in two different stages of

the modeling process. First, during the computation of the Green’s functions required by

the extrapolation process (equation 2-16), where the cost scales linearly with the subsampling

factor Rec.F (or reciprocally, with Sour.F). For each of the receiver locations along ∂Drec, we

compute the Green’s functions for all time steps of the simulation and store them at all the

source locations along ∂Dinj . Although recording and storing the Green’s functions does not

contribute significantly to the computational cost, storage and memory requirements can be

significant.

The second stage where subsampling saves computational time is during the execution of the

EBC simulation. For each source along ∂Dinj , we record the wavefields p and vm at each grid

point of the subsampled surface ∂Drec, then we convolve these two wavefields with the pre-

computed Green’s functions between the source location on ∂Dinj and each receiver location on

∂Drec, and finally we sum along the receivers placed all over ∂Drec. Then, we have to repeat

this process for all sources along ∂Dinj . Thus, the savings at this stage depend equally on both

the subsampling factors Rec.F and Sour.F.

3-3 Results

3-3-1 Square Homogeneous 2D example

Subsampling of injection and recording surfaces is investigated through a simple example using

a staggered finite-difference (FD) approximation of the two-dimensional acoustic wave equation.

The accuracy of the approximation is second-order in both time and space, meaning that the

first-order derivatives of the wavefield constituents p and ~v are computed using a two-points

stencil (Robertsson and Blanch, 2011). The most important FD parameters are shown in Table

3-2.

The model consists of a 2D square-shaped homogeneous medium embedded in a larger homo-

geneous background domain bounded everywhere by rigid boundaries, as shown in Figure 3-1.

The dimensions of the truncated domain D and the background domain D′ are listed in Table

3-3. This example and the ones in the following subsections comprise spatial dimensions that

August 8, 2014

3-3 Results 19

Table 3-2: FD scheme parameters used in all the simulations carried out in Chapters 3 and 4.

Variable Magnitude Units Description

cno 0.5 - Courant numberdt 5×10−4 s Time samplingnt 120 - Number of time stepsfc 50 Hz Central frequency of the Ricker wavelet∆x 2 m Horizontal spatial sampling∆y 2 m Vertical spatial samplingλmin/ max (∆x,∆y) 10 - Number of grid points per minimum wavelength

Table 3-3: Geometrical parameters for the square-shaped model used in Chapters 3 and 4.


LD′x 200 m Horizontal dimension of domain D′LD′y 200 m Vertical dimension of domain D′LDx 100 m Horizontal dimension of domain DLDy 100 m Vertical dimension of domain D

are typical of a near surface survey (on the order of hundreds of meters). However, the EBC

method can of course also be applied to models with dimensions on the order of several thousand

of meters, more typical of hydrocarbon exploration seismic acquisition surveys (Vasmel et al.,

2014). The values of the velocity and the density in the homogeneous medium are c0 = 2000

m/s and ρ0 = 1500 kg/m3, respectively. A volume injection-rate point source is located near the

top left corner of the truncated domain D and this completely illuminates the model. A Ricker

wavelet source function with a central frequency of fc = 50 Hz has been used and, in order to

X (m)

Y (

m)

Density model

0 50 100 150 200

0

50

100

150

200

kg/m

3

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

Velocity model

0 50 100 150 200

0

50

100

150

200

m/s

2000

2100

2200

2300

2400

2500

Figure 3-1: Velocity and density model for the homogeneous case with the square-shapedgeometry. The blue star denotes the source location.

August 8, 2014


ensure stability during the simulation, we have chosen a Courant number of cno = 0.5 with a

simulation time step of dt = 5×10−4 s. All computations are carried out at ten gridpoints per

minimum wavelength, ∆x = λmin10 where λmin = cmin

fmax. In this example, cmin = 2000 m/s and

fmax ≈ 100 Hz; hence, λmin ≈ 20 m and ∆x ≈ 2 m.

The resulting wavefield after t = 0.6 s (which corresponds to nt =120 time steps) is shown in

Figure 3-2. In the right panel of Figure 3-2, we show the difference between the EBC solution

and a reference solution for the full domain where the computational error inside the altered

domain D is insignificant (close to machine precision accuracy).

Full grid: Time 0.6 s

Y (

m)

X (m)a)

50 100 150

50

100

150

Y (

m)

Truncated grid: Time 0.6 s

X (m)b)

50 100 150

50

100

150

X (m)c)

Y (

m)

Difference x 1012

Time 0.6 s

50 100 150

50

100

150

Pa

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−7

Figure 3-2: Snapshots of modeled pressure for a homogeneous case with the square-shapedgeometry. Left panel: modeled pressure in the reference full model. Centre panel:modeled pressure in the truncated domain. Right panel: difference between thefirst two panels multiplied by a factor of 1012.

In a series of tests, the injection surface ∂Dinj was subsampled up to a factor of Sour.F = 6,

resulting in a maximum distance between sources along ∂Dinj of ∆xinj = 12 m and a maximum

RMS error of 5% inside D (see Table 3-4). Note that Sour.F = 6 corresponds to a situation

where ∂Dinj is sampled slightly sparser than the Nyquist sampling criterion of two sources per

minimum wavelength (i.e., Sour.F = 5). Further values of Sour.F, up to source distances twice

as large as the Nyquist spatial sampling, were also tested (rows 6 and 7 on Table 3-4).

The RMS error as a function of Sour.F is shown in Figure 3-3. A noticeable feature of the curve

shown in Figure 3-3 is the different behaviour of the RMS error when the subsampling factor is

lower or higher than Sour.F = 5 (the Nyquist sampling criterion). From Sour.F = 2 to Sour.F

= 5, the RMS error curve appears to grow slowly with a linear increase, whereas the error

increases faster when the Nyquist criterion is violated. The second characteristic feature of the

curve in Figure 3-3 is the consistency of the subsampling implementation when the 4 corners

August 8, 2014

3-3 Results 21

Table 3-4: Homogeneous case with the square-shaped geometry. Subsampling the injectionsurface. First column: subsampling factor applied to the injection surface. Secondand third columns: number of sources and receivers used on the test, respectively.Fourth and fifth columns: savings in computational time for the Green’s functionsand EBC calculations, respectively. Sixth column: maximum RMS error due tosubsampling of the injection surface.

Sour.F N◦Sour N◦Rec Saved.Time (%) Max.Error (%)

GF EBC

1 51 41 0 0 3.4×10−14

2 26 41 1.4 48 0.294 13 41 1.6 73 1.05 11 41 1.7 76 2.66 9 41 2.0 81 5.08 7 41 2.1 85 1110 6 41 2.1 88 16

of the square-shaped surfaces are all sampled with a source. This is shown by the dashed line

in Figure 3-3 which connects the values of the RMS error for Sour.F = 2, 5, and 10. For the

subsampling factors Sour.F = 4, 6, 8, 9, the top right and bottom left and right corners are

not sampled with a source, and for the subsampling factor Sour.F = 7 only the bottom right

corner is not sampled with a source. Thus, when the implementation of the subsampling is not

consistent and not all corners are sampled with a source, the RMS errors do not lay along the

dashed line (values of the RMS error for Sour.F = 3, 4, 6, 7, 8, 9). Additionally, note that we

have discretized the integral over the injection surface using the mid-point rule approximation,

hence the sampling of the corners is different for each subsampling factor. In the chosen EBC

implementation, we use rigid boundaries on the edge of the truncated domain. In principle,

we can choose to use other boundary conditions as well and in particular absorbing boundary

conditions.

Table 3-5 summarizes the error introduced when subsampling the recording surface ∂Drec. When

the recording surface ∂Drec is subsampled by a factor of Rec.F = 2, which corresponds to a

sampling interval two and a half times smaller than ∆xNyq, the maximum RMS error is still

very small. However, for Rec.F ≥ 4, the introduced error is unacceptably in excess of 10%

(see column 6 of Table 3-5). Figure 3-4 shows the RMS error of snapshots as a function of

simulation time for a subsampling factor of 4 of the injection and recording surfaces. For the

injection surface, the introduced error is small and stable throughout the entire simulation (left

panel in Figure 3-4). However, for the recording surface, the RMS error grows considerably after

0.4 s (middle panel in Figure 3-4). Thus, we note that subsampling the recording surface ∂Drec

August 8, 2014


2 4 6 8 100

5

10

15

20

Sour.F

Max R

MS

err

or

(%)

Nyquistsampling

Figure 3-3: RMS error due to subsampling of the injection surface for a homogeneous casewith the square-shaped geometry. The dashed line connects the RMS error ofthe tests with sources on all corners. The vertical red dashed line indicates theNyquist spatial sampling.

is less robust compared to subsampling the injection surface ∂Dinj . This is analyzed further

and discussed in greater detail in Section 3-3-5.

Table 3-6 summarizes the results obtained when subsampling both the injection and the record-

ing surfaces with different sampling intervals. All the results with Rec.F = 2 are accurate with

values of the RMS error below 5%, whereas for the results with Rec.F = 4 the RMS error starts

to grow following a behaviour similar to Figure 3-4. In particular, for Sour.F = 2 and Rec.F

= 4, the simulation is accurate until t = 0.53 s, which corresponds to 88% of the total simu-

lation time, and for the simulation with Sour.F = 4 and Rec.F = 4 the simulation is accurate

up to t = 0.45 s, which corresponds to 75% of the total simulation time. We note that the

combination of Sour.F = 6 and Rec.F = 2 is a particularly attractive choice for subsampling

since the introduced error is below 5% and resulting in GF and EBC time savings of 74% and

90%, respectively. The small difference between the RMS error obtained for the simulation with

Sour.F = 6 and Rec.F = 1 (row 5 of Table 3-4) and the simulation with Sour.F = 6 and Rec.F

= 2 (row 4 of Table 3-6) is due to the fluctuations of the RMS error at the end of the simulation.

August 8, 2014

3-3 Results 23

0 0.2 0.4 0.60

20

40

60

80

100

RM

S (

%)

Injection surface subsampled: Sour.F=4

time (s)0 0.2 0.4 0.6

0

20

40

60

80

100

RM

S (

%)

Recording surface subsampled: Rec.F=4

time (s)0 0.2 0.4 0.6

0

20

40

60

80

100

RM

S (

%)

Both surfaces subsampled: Sour.F=4 Rec.F=4

time (s)

Figure 3-4: Evolution of the RMS error as function of time for a subsampling factor of 4 of theinjection (left), recording (center), and both (right) surfaces for the square-shapedhomogeneous model.

As previously discussed in Section 3-2-2, the observed savings in computational time for the

Green’s functions calculation are much greater when subsampling ∂Drec (see column 4 of Tables

3-4 and 3-5). In fact, the computational time savings related to the calculation of the Green’s

functions remain approximately constant for the subsampling of ∂Dinj2. The savings increase

when we subsample ∂Drec and follows this expression:

tGF,saving =

(1− 1

Rec.F

). (3-3)

For the computation time of the EBC calculation (column 5 of Tables 3-4 and 3-5) both subsam-

plings of ∂Drec and ∂Dinj are approximately significant resulting in the following time savings

(see Figure 3-5):

tEBC,saving = 1−(

1

Rec.F · Sour.F

). (3-4)

3-3-2 Square Heterogeneous 2D example

Now, we present the results for a simple square-shaped heterogeneous model whose dimensions

are the same as the ones from the homogeneous example. The objective of this simulation is to

study the effect of subsampling when the model presents strong contrasts in material properties

inside the truncated domain as well as outside. In this example, we include two heterogeneities,

2Note that this statement holds only if we do not apply reciprocity. If the Green’s functions are calculatedreciprocally, the same argument applies to subsampling ∂Dinj .

August 8, 2014


2 4 6 8 10

50

60

70

80

90

Rec.F

Tim

e s

avin

gs (

%)

GF computation

2 4 6 8 10

50

60

70

80

90

Subsamplig factor

Tim

e s

avin

gs (

%)

EBC computation

Recording surface

Injection surface

Theoretical savings

Theoretical savings

Observed savings

Figure 3-5: Comparison between the observed and the theoretical savings in computationaltime for the square-shaped homogeneous model. The theoretical savings curvehas been calculated with equation 3-3 (left panel) and equation 3-4 (right panel).Left panel: the GF computation. Right panel: the EBC computation.

Table 3-5: Homogeneous case with the square-shaped geometry. Subsampling the recordingsurface. First column: subsampling factor applied to the recording surface. Secondand third columns: number of sources and receivers used on the test, respectively.Fourth and fifth columns: savings in computational time for the Green’s functionsand EBC calculations, respectively. Sixth column: maximum RMS error due tosubsampling of the recording surface.

Rec.F N◦Sour N◦Rec Saved.Time (%) Max.Error (%)

GF EBC

1 51 41 0 0 3.4×10−14

2 51 21 49 49 0.224 51 11 71 74 665 51 9 76 78 776 51 7 84 78 2.3×102

one outside and one inside D, as shown in Figure 3-6. The values of the velocity and density

of the anomaly in the background domain D′ are cext = 2500 m/s and ρext = 5000 kg/m3,

respectively. The values for the anomaly inside the truncated domain are cint = 2500 m/s and

ρint = 5000 kg/m3, respectively. The results obtained with this simulation are summarized in

Tables 3-7, 3-8, and 3-9. We observe a consistent behaviour of the RMS error and computational

time as for those obtained for the homogeneous square-shaped model. We observe that the RMS

errors are slightly larger compared to the ones obtained for the homogeneous model when the

sampling used does not violate the Nyquist criterion. On the other hand, when subsampling

with Sour.F ≥ 4 or Rec.F ≥ 4 (or a combination of both), the RMS error is much larger than in

the simulation with the homogeneous square-shaped model. This can be observed comparing

the last rows of Tables 3-7, 3-8 and 3-9 with the same row of Tables 3-4, 3-5, and 3-6. This

August 8, 2014

3-3 Results 25

Table 3-6: Homogeneous case with the square-shaped geometry. Subsampling both the injec-tion and the recording surfaces. First and second columns: subsampling factorsapplied to the injection and recording surfaces, respectively. Third and fourthcolumns: number of sources and receivers used on the test, respectively. Fifth andsixth columns: savings in computational time for the Green’s functions and EBCcalculations, respectively. Seventh column: maximum RMS error due to subsam-pling of both the injection and recording surfaces.

Sourc.F Rec.F N◦Sour N◦Rec Saved.Time (%) Max.Error (%)

GF EBC

1 1 51 41 0 0 3.4×10−14

2 2 26 21 49 74 0.334 2 13 21 52 86 1.16 2 9 21 74 90 5.02 4 26 11 69 86 664 4 13 11 49 93 3.7×102

example demonstrates a major advantage of the EBC method confirming that it handles very

well the so-called long-term high-order interactions with anomalies located outside the domain

D.

Table 3-7: Heterogeneous case with the square-shaped geometry. Subsampling the injectionsurface. First column: subsampling factor applied to the injection surface. Secondand third columns: number of sources and receivers used on the test, respectively.Fourth and fifth columns: savings in computational time for the Green’s functionsand EBC calculations, respectively. Sixth column: maximum RMS error due tosubsampling of the injection surface.

Sour.F N◦Sour N◦Rec Saved.Time (%) Max.Error(%)

GF EBC

1 51 41 0 0 3.7×10−14

2 26 41 4.1 48 0.354 13 41 1.8 73 1.15 11 41 1.6 75 3.56 9 41 2.1 81 5.98 7 41 2.2 83 1210 6 41 2.2 88 22

August 8, 2014


Table 3-8: Heterogeneous case with the square-shaped geometry. Subsampling the recordingsurface. First column: subsampling factor applied to the recording surface. Secondand third columns: number of sources and receivers used on the test, respectively.Fourth and fifth columns: savings in computational time for the Green’s functionsand EBC calculations, respectively. Sixth column: maximum RMS error due tosubsampling of the recording surface.


GF EBC

1 51 41 0 0 3.7×10−14

2 51 21 40 50 0.244 51 11 71 74 795 51 9 77 79 1.2×102

6 51 7 81 84 7.4×102

Table 3-9: Heterogeneous case with the square-shaped geometry. Subsampling both the in-jection and the recording surfaces. First and second columns: subsampling factorsapplied to the injection and recording surfaces, respectively. Third and fourthcolumns: number of sources and receivers used on the test, respectively. Fifth andsixth columns: savings in computational time for the Green’s functions and EBCcalculations, respectively. Seventh column: maximum RMS error due to subsam-pling of both the injection and recording surfaces.

Sour.F Rec.F N◦Sour N◦Rec Saved.Time (%) Max.Error (%)

GF EBC

1 1 51 41 0 0 3.7×10−14

2 2 26 21 40 74 0.514 2 13 21 48 86 1.26 2 9 21 47 90 142 4 26 11 69 86 904 4 13 11 68 93 2.0×105

August 8, 2014

3-3 Results 27

X (m)

Y (

m)

Density model

0 50 100 150 200

0

50

100

150

200

kg

/m3

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

Velocity model

0 50 100 150 200

0

50

100

150

200

m/s

2000

2100

2200

2300

2400

2500

Figure 3-6: Velocity and density model in a simple heterogeneous case with the square-shapedgeometry. Two anomalies are considered for both models, one inside cint = 2500m/s and ρint = 5000 kg/m3, and one outside cext = 2500 m/s and ρext = 5000kg/m3, respectively. The blue star denotes the source location.

3-3-3 Rectangular Homogeneous 2D example

In this example, we study a rectangular model whose horizontal dimension is ten times larger

than the vertical one. The analysis of this example is of interest for two main reasons. First,

it usually is very difficult to have a perfectly squared recording surface surrounding the area

of interest as in the examples studied in the previous subsections. A squared distribution of

receiver can be easily implemented in a laboratory environment, but not in a conventional

seismic survey. Second, in marine seismic data acquisition, the receiver arrays are placed along

horizontal lines only, resulting in an open recording surface.

In this section, we distinguish between the horizontal and the vertical subsampling factors for

both the recording and the injection surfaces, where h refers to horizontal and v to vertical

dimension. We focus on subsampling specially the vertical dimension in order to approximate

the scenario discussed above where no receivers are located along the vertical edges of ∂Drec.This scenario corresponds either to a marine seismic survey where survey vessels tow steamers

suspended below the surface or to seabed seismic surveys. In both situations, p and ~v data

is usually acquired along a line resulting in one horizontal layer of receivers. In this example,

the two receivers arrays are only along the horizontal edges of the recording surface and no

recording array exists along the vertical dimension. Even though the EBC theory introduced in

Section 2-4-3 requires that the truncated domain is surrounded by a closed array of receivers,

it is of fundamental importance to determine how much error is introduced when we violate an

important assumption using an incomplete rectangular surface.

The FD scheme parameters used in this simulation are the same as the ones used in the

August 8, 2014


Table 3-10: Geometrical parameters for the rectangular model used in Chapters 3.

Variable Magnitude Units

LD′x 500 m

LD′y 50 mLDx 300 mLDy 30 m

Table 3-11: Homogeneous case with the rectangular geometry. Subsampling the injectionsurface. First and second columns: subsampling factors applied to the injectionsurface along the horizontal and vertical sides, respectively. Third and fourthcolumns: number of sources used on the test along the horizontal and verticalsides, respectively. Fifth and sixth columns: number of receivers used on the testalong the horizontal and vertical sides, respectively. Seventh and eighth columns:savings in computational time for the Green’s functions and EBC calculations,respectively. Ninth column: maximum RMS error due to subsampling of theinjection surface.


h v h v h v GF EBC

1 1 151 16 101 4 0 0 1.2×10−13

1 2 151 8 101 4 0.13 4.0 0.191 4 151 4 101 4 1.2 6.4 1.92 1 76 16 101 4 3.7 44 0.202 2 76 8 101 4 4.5 49 0.254 1 38 16 101 4 6.1 67 0.67

August 8, 2014

3-3 Results 29

Table 3-12: Homogeneous case with the rectangular geometry. Subsampling the recordingsurface. First and second columns: subsampling factors applied to the recordingsurface along the horizontal and vertical sides, respectively. Third and fourthcolumns: number of sources used on the test along the horizontal and verticalsides, respectively. Fifth and sixth columns: number of receivers used on the testalong the horizontal and vertical sides, respectively. Seventh and eighth columns:savings in computational time for the Green’s functions and EBC calculations,respectively. Ninth column: maximum RMS error due to subsampling of therecording surface.

Rec.F N◦Rec N◦Sour Saved.Time (%) Max.Error (%)

h v h v h v GF EBC

1 1 151 16 101 4 0 0 1.2×10−13

1 2 151 16 101 2 2.6 0.32 1.91 4 151 16 101 1 3.6 1.0 2.62 1 151 16 51 4 48 57 2.0×102

2 2 151 16 51 2 50 59 2.0×102

4 1 151 16 26 4 72 78 9.1×104

Table 3-13: Homogeneous case with the rectangular geometry. Subsampling both the injectionand the recording surfaces. First and second columns: subsampling factors ap-plied to the injection surface along the horizontal and vertical sides, respectively.Third and fourth columns: subsampling factors applied to the recording surfacealong the horizontal and vertical sides, respectively. Fifth and sixth columns:number of sources used on the test along the horizontal and vertical sides, re-spectively. Seventh and eighth columns: number of receivers used on the testalong the horizontal and vertical sides, respectively. Ninth and tenth columns:savings in computational time for the Green’s functions and EBC calculations,respectively. Eleventh column: maximum RMS error due to subsampling of boththe injection and recording surface.

Sour.F Rec.F N◦Sour N◦Rec Saved.Time (%) Max.Error(%)

h v h v h v h v GF EBC

1 1 1 1 151 16 101 4 0 0 1.2×10−13

1 2 1 2 151 8 101 2 2.5 5.5 1.91 4 1 4 151 4 101 1 4.1 8.5 3.41 16 1 4 151 1 101 1 5.2 10 152 1 1 2 76 16 101 2 5.3 45 2.02 1 2 1 76 16 51 4 50 76 4.3×102

2 2 2 2 76 8 51 2 51 79 4.3×102

August 8, 2014


X (m)

Y (

m)

Density model

0 100 200 300 400 500

0

10

20

30

40

50

kg/m

3

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

Velocity model

0 100 200 300 400 500

0

10

20

30

40

50

m/s

2000

2100

2200

2300

2400

2500

Figure 3-7: Velocity and density model for the homogeneous medium with a rectangular ge-ometry. For display purposes, the vertical dimension is exaggerated by a factorof ten. The blue star denotes the source location.

square-shaped example, whereas the dimensions of the model are different and shown in Table

3-10. The results obtained with this example are summarized in Tables 3-11, 3-12, and 3-13.

It is not surprising to observe that the evolution of the RMS error, as well as the time savings

(both for GF and EBC) show the same behaviour throughout the subsampling process as

observed in the square-shaped configuration, (Tables 3-4 and 3-5). Again, all the introduced

errors when subsampling ∂Dinj are relatively low (all below 2%) and the computational time

is reduced significantly up to 70% (Table 3-11). These results clearly show that, in a more

complex and larger geometry where data volumes could be significant, subsampling ∂Dinjwould be very beneficial in terms of computational savings. The results obtained for the

subsampling of ∂Drec with a horizontal factor Rec.F=2 (Table 3-12) are significantly different

in terms of the RMS errors obtained at the end of the simulation compared to the ones

obtained with the square-shaped example. However, throughout 80% of the total simulation,

the maximum RMS error is below 10% (see Figure 3-8). It is interesting to observe that the

evolution of the RMS error throughout the simulation (Figure 3-8) is very similar to the one

observed in the squared-shaped example (Figure 3-4). Hence, the mechanism by which the

error is introduced is expected to be similar. Looking at Table 3-13, we observe that the best

combination of subsampling factors in terms of a compromise between the saved time and the

maximum introduced error is when subsampling with a factor 2 both the horizontal injection

surface ∂Dinj and the vertical recording surface ∂Drec. In such a situation, the introduced

error is sufficiently small (2%) and the savings in computational time are up to 45%.

August 8, 2014

3-3 Results 31

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

RM

S (

%)

Recording surface subsampled: Rec.F h=2 v=1

time (s)0 0.1 0.2 0.3 0.4 0.5 0.6

0

20

40

60

80

100

RM

S (

%)


time (s)

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

RM

S (

%)


time (s)0 0.1 0.2 0.3 0.4 0.5 0.6

0

20

40

60

80

100

RM

S (

%)

Both surfaces subsampled: Sour.F h=2 v=2 Rec.F h=2 v=2

time (s)

Figure 3-8: Evolution of the RMS error as a function of time for the rectangular configurationin a homogeneous model. Upper row, from left to right, subsampling of therecording surface: h=2, v=1; and h=2, v=2. Lower row, from left to right,subsampling on the recording surface: Rec.F h=4, v=1 and subsampling of bothsurfaces: Sour.F h=2, v=2 Rec.F h=2, v=2.

3-3-4 Rectangular Heterogeneous 2D example

In the last example presented in this chapter, we include heterogeneities in the rectangular model

introduced in the previous section (see Figure 3-9). The FD parameters and the magnitude of

the included heterogeneities are the same as the ones from previous examples. Tables 3-14, 3-15,

and 3-16 show a similar behaviour, in terms of the evolution of the RMS error as well as the time

savings (both for the GF and EBC computations), to the ones obtained for the homogeneous

rectangular model. The RMS errors are slightly different compared to the ones obtained for the

homogeneous model for the subsampling factor Rec.F = 1, whereas the differences are much

greater for Rec.F ≥ 2. This can be observed comparing the last three rows of Tables 3-12 and

3-15 and the last two rows of Tables 3-13 and 3-16.

As discussed in the previous subsection, it is of fundamental importance to analyze the accuracy

of the subsampling approach as an approximation of an incomplete rectangular surface. Thus,

we compare the result for Sour.F h=1, v=16 and Rec.F h=1, v=4 (row 4 of Tables 3-13 and

3-16) to the maximum RMS error obtained when not considering any vertical side along D,

Max.Error = 19% (computed with a specific simulation). The differences between the maximum

RMS errors are relatively small supporting the choice of the subsampling approach. Finally, in

August 8, 2014


X (m)

Y (

m)

Density model

0 100 200 300 400 500

0

10

20

30

40

50

kg/m

3

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

Velocity model

0 100 200 300 400 500

0

10

20

30

40

50

m/s

2000

2100

2200

2300

2400

2500

Figure 3-9: Velocity and density model for the heterogeneous medium with a rectangulargeometry. For display purposes, the vertical dimensions is exaggerated by a factorof ten. Two anomalies are considered for both models, one inside cint = 2500 m/sand ρint = 5000 kg/m3, and one outside cext = 2500 m/s and ρext = 5000 kg/m3,respectively. The blue star denotes the source location.

Figure 3-10, we show the RMS evolution as a function of time for the subsampled approach, the

incomplete model on both vertical sides, and the incomplete model on the right vertical side for

this heterogeneous medium. We observe that the three different configurations show a similar

RMS error evolution as a function of time. A noticeable feature of the curves is the jump in

RMS error that takes place when the energy scatters back into D for the first time. This jump

indicates that all the energy is introduced at once, and after that the error grows slowly. For

both the subsampling approach and the incomplete rectangular surface on both vertical sides,

the RMS error jump occurs around t = 0.1 s; for the incomplete rectangular surface on the right

side, it occurs around t = 0.16 s. The delay on the third configuration is due to the fact that

the energy has to travel through the entire domain D before scattering back along the edges of

the full domain.

3-3-5 General results

Summarizing the results obtained and presented in this chapter, we observe that the evolution

of the RMS error exhibits a different behaviour when subsampling the recording surface ∂Drecor the injection surface ∂Dinj . We conclude that subsampling the recording surface is a more

critical process than subsampling the injection surface. This is a somewhat expected result

because the recording surface plays a fundamental role in the extrapolation process (equation

2-15). The recording surface implicitly separates each wavefield that reaches the boundary

into outgoing and incoming wavefields. An outgoing wavefield corresponds to the energy that

leaves the truncated domain D and propagates towards D′ . On the other hand, an incoming

August 8, 2014

3-3 Results 33

Table 3-14: Heterogeneous case with the rectangular geometry. Subsampling the injectionsurface. First and second columns: subsampling factors applied to the injectionsurface along the horizontal and vertical sides, respectively. Third and fourthcolumns: number of sources used on the test along the horizontal and verticalsides, respectively. Fifth and sixth columns: number of receivers used on the testalong the horizontal and vertical sides, respectively. Seventh and eighth columns:savings in computational time for the Green’s functions and EBC calculations,respectively. Ninth column: maximum RMS error due to subsampling of theinjection surface.


h v h v h v GF EBC

1 1 151 16 101 4 0 0 4.9×10−14

1 2 151 8 101 4 3.3 4.6 0.191 4 151 4 101 4 2.9 7.3 1.82 1 76 16 101 4 2.7 42 0.322 2 76 8 101 4 4.7 49 0.334 1 38 16 101 4 2.8 67 0.89

wavefield consists of the energy entering the truncated domain D from D′. To carefully explain

why subsampling affects the extrapolation process, we now focus only on the top edge of the

truncated domain (Figure 3-11). First, we assume that a purely outgoing wavefield propagates

towards ∂Drec (labelled (1) in Figure 3-11). When the outgoing wavefield reaches ∂Drec, our

algorithm implements equation 2-16 to extrapolate the wavefield to ∂Dinj (labelled (2) in Figure

3-11). Without subsampling, the extrapolation integral correctly yields a result different from

zero because the initial wavefield was outgoing.

Now, we consider a purely incoming wavefield propagating from D′ and entering the truncated

domain D (labelled (3) in Figure 3-11). When this incoming wavefield reaches ∂Drec, it is ex-

trapolated above ∂Drec (labelled (4) in Figure 3-11). Since the initial wavefield is only incoming,

the extrapolation process should yield zero because there is no outgoing energy that needs to be

extrapolated towards ∂Dinj . To validate our reasoning, we have prepared a simple test based on

a homogeneous medium with c = 1500 m/s. We consider two surfaces: one corresponds to the

recording surface ∂Drec, and the other one is an observation boundary, ∂Dextp, located below

∂Drec. Additionally, a point source is located along the injection surface ∂Dinj . The length

of ∂Drec is Lx = 750 m. The distances between the source located on ∂Dinj and ∂Drec, and

between ∂Drec and ∂Dextp are the same and equal to Ly = 250 m. In this scenario, the point

source represents one of the sources that would be placed along the injection boundary. The

source signature is a Ricker wavelet with a central frequency of fc = 25 Hz and the Green’s

functions used for the extrapolation process are far-field approximations to the free-space 2D

August 8, 2014


0 0.2 0.4 0.60

20

40

60

80

100

RM

S (

%)

Subsampling: Sour.F h=1 v=16 Rec.F h=1 v=4

time (s)0 0.2 0.4 0.6

0

20

40

60

80

100R

MS

(%

)Incomplete surface with both vertical sides missing

time (s)0 0.2 0.4 0.6

0

20

40

60

80

100

RM

S (

%)

Incomplete surface with one vertical sides missing

time (s)

Figure 3-10: RMS evolution as a function of time for the rectangular configuration in a het-erogeneous model. From left to right: subsampled approach with Sour.F h=1,v=16, Rec.F h=1, v=4; incomplete surface with two vertical sides missing; andincomplete surface with one vertical side missing.

Green’s functions. The most important FD parameters are listed in Table 3-17. Note that

the EBC algorithm implemented in our modeling code only extrapolates wavefields from ∂Drecto ∂Dinj while the simple code used for this test allows to extrapolate a wavefield to different

surfaces (e.g., to ∂Dextp).

In Figure 3-12, we show the results of the extrapolation process for a purely incoming wavefield.

In the top-left panel we show the reference wavefield measured (not extrapolated) along ∂Dextp.The top-middle panel shows the result for the process of extrapolating the incoming wavefield

from ∂Drec to ∂Dextp. The top right-panel shows the difference between the reference wavefield

and the extrapolated wavefield along ∂Dextp. The extrapolation produces an excellent result.

The artefacts present on both sides of the middle and right panels of the top row are due to the

finite length of ∂Drec and the fact that we are truncating the extrapolation integral. On the

bottom row of Figure 3-12, we show the results of the extrapolation of the incoming wavefield

from ∂Drec to ∂Dinj (labelled (4) in Figure 3-11) are shown. The panel on the left of Figure 3-12

shows the reference wavefield measured along ∂Dinj . Since ∂Drec is not a real boundary, the

incoming wavefield does not reflect along ∂Drec and therefore the reference outgoing wavefield

along ∂Dinj should be zero. The bottom-middle panel shows the result for the extrapolation

to ∂Dinj . As discussed above, this result is correctly zero since the initial source wavefield is

purely incoming. The accuracy of the extrapolation result along ∂Dinj is illustrated in the

August 8, 2014

3-3 Results 35

Table 3-15: Heterogeneous case with the rectangular geometry. Subsampling the recordingsurface. First and second columns: subsampling factors applied to the recordingsurface along the horizontal and vertical sides, respectively. Third and fourthcolumns: number of sources used on the test along the horizontal and verticalsides, respectively. Fifth and sixth columns: number of receivers used on the testalong the horizontal and vertical sides, respectively. Seventh and eighth columns:savings in computational time for the Green’s functions and EBC calculations,respectively. Ninth column: maximum RMS error due to subsampling of therecording surface.


h v h v h v GF EBC

1 1 151 16 101 4 0 0 4.9×10−14

1 2 151 16 101 2 3.4 1.6 2.01 4 151 16 101 1 3.2 1.9 2.82 1 151 16 51 4 46 58 2.82 2 151 16 51 2 50 60 3.44 1 151 16 26 4 72 78 1.6×103

last panel of the bottom row of Figure 3-12 where the difference between the reference and

the extrapolated wavefield is shown. Again, the weak artefacts present in the extrapolated

wavefield are due to the truncation of the extrapolation integral. Now, in order to verify that

subsampling of the recording surface indeed affects the accuracy of the extrapolation process,

we carry out a similar test as above but we subsample the recording surface with a factor

Rec.F = 2 and show the results in Figure 3-13. To facilitate the comparison, the top row

shows again the results obtained without subsampling (i.e., the bottom row of Figure 3-12).

The left panel on the bottom row shows the reference outgoing wavefield along ∂Dinj which,

as discussed above, is zero because the initial wavefield is incoming. However, in the bottom-

middle panel of Figure 3-13, we observe that the extrapolation process yields an inaccurate result

when subsampling ∂Drec. The mechanism by which ∂Drec implicitly separates each wavefield

into its incoming and outgoing components stops working adequately when subsampling the

recording surface. Because of aliasing of the incoming wavefield which leads to an apparent

outgoing wavefield. Therefore, when part of the incoming energy (labelled (1) in Figure 3-14) is

erroneously interpreted as outgoing, the recording surface will extrapolate it upwards to ∂Dinj(labelled (2) in Figure 3-14). Because there is no real outgoing energy arriving to ∂Dinj from

∂Drec, the extrapolated energy from ∂Drec to ∂Dinj (corresponding to the bottom-middle panel

of Figure 3-13) will by injected back towards D by the EBC algorithm (labelled (3) in Figure

3-14). Additionally, this injected energy travels towards the recording surface and hence it is

erroneously used in the extrapolation process (labelled (4) in Figure 3-14). Thus, this process

August 8, 2014


Table 3-16: Heterogeneous case with the rectangular geometry. Subsampling both the injec-tion and the recording surfaces. First and second columns: subsampling factorsapplied to the injection surface along the horizontal and vertical sides, respec-tively. Third and fourth columns: subsampling factors applied to the record-ing surface along the horizontal and vertical sides, respectively. Fifth and sixthcolumns: number of sources used on the test along the horizontal and verticalsides, respectively. Seventh and eighth columns: number of receivers used on thetest along the horizontal and vertical sides, respectively. Ninth and tenth columns:savings in computational time for the Green’s functions and EBC calculations,respectively. Eleventh column: maximum RMS error due to subsampling of boththe injection and recording surface.

Sour.F Rec.F N◦Sour N◦Rec Saved.Time (%) Max.Error (%)

h v h v h v h v GF EBC

1 1 1 1 151 16 101 4 0 0 4.9×10−14

1 2 1 2 151 8 101 2 3.4 5.4 2.01 4 1 4 151 4 101 1 5.9 8.1 3.41 16 1 4 151 1 101 1 6.5 11 132 1 1 2 76 16 101 2 3.7 45 2.02 1 2 1 76 16 51 4 49 77 1.1×103

2 2 2 2 76 8 51 2 52 79 1.1×103

will be repeated each time an incoming wavefield reaches ∂Drec leading to a continuous increase

of the error.

Another parameter that affects the evolution of the RMS error is the distance between the

recording and the injection surfaces. We applied subsampling factors Rec.F = 2 and 4 with

distances between the two surfaces of 10 m, 12 m, 14 m, 20 m, and 40 m and the results are

showed in Table 3-18. In Figure 3-15, we show the evolution of the RMS error for the different

distances between the recording and injection surfaces. In order to create the graphs in Figure

3-15, we have picked 55% as threshold for the RMS error and then we have selected a time

Table 3-17: FD parameters used in the extrapolation integral test.


dt 2×10−3 s Time samplingnt 2×1014 - Number of time stepsfc 25 Hz Central frequency of the Ricker waveletfNyq 250 Hz Nyquist frequency∆x 10 m Horizontal spatial sampling along ∂Drec∆xNyq 15 m Nyquist spatial sampling along ∂Drec

August 8, 2014

3-3 Results 37

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

(1)!

(2)!

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

extp!

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

inj!

rec!

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

inj!

rec!

extp!

(3)!

(4)!

Figure 3-11: Diagram illustrating the extrapolation process. Recording surface fully sampled.Top panel: a purely outgoing wavefield propagating from D to D′ (labelled (1))is extrapolated from ∂Drec to ∂Dinj after it reaches ∂Drec (labelled (2)). Bottompanel: a purely incoming wavefield propagating from D′ to D (labelled (3)) isextrapolated from ∂Drec to ∂Dextp after it reaches ∂Drec (labelled (4)). The stardenotes a source located along ∂Dinj .

window of 0.6 s for each distance in such a way that (for all the curves) the last value of the

RMS error is the same (i.e., 55%). This particular way to display the different curves allows

us to compare the rate of increase introduced by the subsampling. The results show that the

larger the distance between the surfaces the gentler the growth is. Since the aliased energy

due to subsampling of ∂Drec leads to apparent outgoing and incoming wavefields, the frequency

content of the source wavefield is also a key parameter governing the error introduced in the

extrapolation process. For a specific subsampling factor, the higher the frequency content the

more aliased energy is present. Thus, higher frequencies introduce larger errors during the

extrapolation process. Additionally, the angle of incidence of the incoming wavefield arriving at

∂Drec is a factor that must be taken into account to understand the error introduced because

of subsampling. To illustrate this conjecture, we run the extrapolation tests again, but now

we use a plane wave as incoming wavefield instead of the point source used before. We focus

our analysis on the frequency content and on the angle of incidence at ∂Drec of the plane wave.

August 8, 2014


offset (m)

time

(s)

Reference wavefield below

−500 0 500

0

0.2

0.4

0.6

0.8

1

offset (m)

time

(s)

Extrap. wavefield below

−500 0 500

0

0.2

0.4

0.6

0.8

1

offset (m)

time

(s)

Difference

−500 0 500

0

0.2

0.4

0.6

0.8

1

Pa

−0.2

−0.1

0

0.1

0.2

offset (m)

time

(s)

Reference wavefield above

−500 0 500

0

0.2

0.4

0.6

0.8

1

offset (m)

time

(s)

Extrap. wavefield above

−500 0 500

0

0.2

0.4

0.6

0.8

1

offset (m)

time

(s)

Difference

−500 0 500

0

0.2

0.4

0.6

0.8

1

Pa

−0.2

−0.1

0

0.1

0.2

Figure 3-12: Extrapolation integral results for a purely incoming wavefield. Top row: theextrapolation to ∂Dextp. Bottom row: the extrapolation to ∂Dinj . From leftto right: the reference wavefield, the extrapolated wavefield, and the differencebetween them.

In Figure 3-16, we show the different incoming wavefields used for these tests. The first three

panels on the top of Figures 3-17 show the results of the extrapolation integral when the angle

of incidence of the incoming plane wave is 40◦ and fc = 35 Hz. Since subsampling ∂Drec with

Rec.F = 2 already yields a sampling distance between receivers ∆x ≥ ∆xNyq, the aliased energy

introduces the small observed errors. If the frequency content of the plane wave increases, more

aliased energy is introduced and therefore the error also increases (see top right panel of Figure

3-17). On the bottom row of Figure 3-17, we observe the effect or the error introduced by

the extrapolation integral due to an increase of the angle of incidence. The higher the angles

are, the more aliased energy is introduced. Also in this case, a higher frequency content of the

incoming wavefield reduces the accuracy of the extrapolation process (bottom row of Figure

3-17).

August 8, 2014

3-3 Results 39

offset (m)

time

(s)


−500 0 500

0

0.2

0.4

0.6

0.8

1

offset (m)

time

(s)

Extrap. wavefield above

−500 0 500

0

0.2

0.4

0.6

0.8

1

offset (m)

time

(s)

Difference

−500 0 500

0

0.2

0.4

0.6

0.8

1

Pa

−0.2

−0.1

0

0.1

0.2

offset (m)

time

(s)


−500 0 500

0

0.2

0.4

0.6

0.8

1

offset (m)

time

(s)

Extrap. wavefield above: Rec.F = 2

−500 0 500

0

0.2

0.4

0.6

0.8

1

offset (m)

time

(s)

Difference

−500 0 500

0

0.2

0.4

0.6

0.8

1

Pa

−0.2

−0.1

0

0.1

0.2

Figure 3-13: Extrapolation integral results for a purely incoming wavefield when the recordingsurface is not subsampled (above) and when the recording surface is subsampledby a factor of 2 (below). From left to right: the reference wavefield, the extrap-olated wavefield, and the difference between them.

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

inj!

rec!(4)!

(3)! (2)! (1)!

x3

x1 n

D

∂Dinj

∂Drec

⋆

(2)(1)

scatterer

scatterer

D′

Figure 3-14: Diagram illustrating the extrapolation process. Recording surface subsampledwith a factor Rec.F = 2. A purely incoming wavefield (labelled (1)) propagatingfrom D′ to D is extrapolated from ∂Drec to ∂Dinj (labelled (2)). Label (3):injected wavefield from the source placed along ∂Dinj is extrapolated from ∂Drec

to ∂Dinj (labelled (4)). The star denotes a source located along ∂Dinj .

August 8, 2014


0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100RMS error evolution for different surfaces distances. Rec.F=2

RM

S (

%)

Time (s)

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100RMS error evolution for different surfaces distances. Rec.F=4

RM

S (

%)

Time (s)

Distance 10m

Distance 12m

Distance 14m

Distance 20m

Distance 40m

Distance 10m

Distance 12m

Distance 14m

Distance 20m

Distance 40m

Figure 3-15: Evolution of the RMS error for the homogeneous case with the square-shapedgeometry. Left panel: subsampling the recording surface by a factor of Rec.F= 2. Right panel: subsampling the recording surface by a factor of Rec.F = 4.Threshold for the RMS error of 55%. Time window of 0.6 s for each distancesuch that (for all the curves) the last value of the RMS error is the same (i.e.,55%).

Table 3-18: First column: distances between ∂Drec and ∂Dinj . Second and third columns:simulation times at which the RMS error starts to significantly increase becauseof subsampling.

Distance (m) Time (s)

Rec.F = 2 Rec.F = 4

10 0.90 0.4012 1.00 0.5314 1.05 0.5520 1.25 0.5840 2.40 1.00

August 8, 2014

3-3 Results 41

offset (m)

tim

e (

s)

Incident pressure: ideg = 40, freq = 25 Hz

−500 0 500

0

0.5

1

1.5

2

Pa

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

offset (m)

tim

e (

s)


−500 0 500

0

0.5

1

1.5

2

Pa

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

offset (m)

tim

e (

s)


−500 0 500

0

0.5

1

1.5

2

Pa

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

offset (m)

tim

e (

s)


−500 0 500

0

0.5

1

1.5

2

Pa

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Figure 3-16: Incoming plane waves at the recording surface for the extrapolation test. Toprow: take off angle of 40◦. Bottom row: take off angle of 80◦. From left toright: Ricker wavelet with a central frequency of 25 Hz and Ricker wavelet witha central frequency of 40 Hz.

August 8, 2014


offs

et (m

)

time (s)

Fully

sam

ple

d

−500

0500

0

0.51

1.52

offs

et (m

)

time (s)

Subsam

ple

d: R

ec.F

= 2

−500

0500

0

0.51

1.52

offs

et (m

)

time (s)

Diffe

rence

−500

0500

0

0.51

1.52

Pa

−0.0

4

−0.0

2

0 0.0

2

0.0

4

offs

et (m

)

time (s)

Fully

sam

ple

d

−500

0500

0

0.51

1.52

offs

et (m

)

time (s)

Subsam

ple

d: R

ec.F

= 2

−500

0500

0

0.51

1.52

offs

et (m

)

time (s)

Diffe

rence

−500

0500

0

0.51

1.52

Pa

−0.0

4

−0.0

2

0 0.0

2

0.0

4

offs

et (m

)

time (s)

Fully

sam

ple

d

−500

0500

0

0.51

1.52

offs

et (m

)

time (s)

Subsam

ple

d: R

ec.F

= 2

−500

0500

0

0.51

1.52

offs

et (m

)

time (s)

Diffe

rence

−500

0500

0

0.51

1.52

Pa

−0.0

4

−0.0

2

0 0.0

2

0.0

4

offs

et (m

)

time (s)

Fully

sam

ple

d

−500

0500

0

0.51

1.52

offs

et (m

)

time (s)

Subsam

ple

d: R

ec.F

= 2

−500

0500

0

0.51

1.52

offs

et (m

)

time (s)

Diffe

rence

−500

0500

0

0.51

1.52

Pa

−0.0

4

−0.0

2

0 0.0

2

0.0

4

Figure

3-17:

Extra

polatio

nin

tegral

results

for

ap

urely

inco

min

gp

lan

ew

avew

hen

the

recordin

gsu

rfaceis

sub

samp

ledby

afacto

rof

2.

Top

row:

takeoff

an

gle

of

40 ◦.

Botto

mrow

:ta

keoff

an

gleof

80 ◦.F

romleft

torigh

t:R

ickerw

avelet

with

acen

tral

frequ

ency

of25

Hz

an

dR

ickerw

aveletw

itha

central

frequ

ency

of40

Hz.

August 8, 2014

Chapter 4

Interpolation of the EBC extrapolation

surface integral

In the previous chapter, we showed that significant savings in time and memory can be achieved

by subsampling the recording and injection surfaces. However, we also showed that the accuracy

of the computed wavefields inside the truncated domain D decreases considerably, in particu-

lar when subsampling the recording surface. In this chapter, we employ interpolation of the

wavefields on both surfaces aiming to improve the accuracy of the computed wavefields inside

D.

4-1 Introduction

Interpolation is a mathematical method that allows one to construct new data points within

the range of a discrete set of existing data points. In our case, the discrete set of existing data

points is composed of the values of the wavefields p and ~v measured along the recording surface

(after subsampling) and of the values of the extrapolated wavefields on the injection surface

(after subsampling). The new constructed data points behave as additional receivers or sources

depending whether the interpolation has been applied to the recording or injection surface.

By creating additional receivers and sources, we aim to better reconstruct the wavefields p

and ~v inside the truncated domain searching for the optimal compromise between savings

in computational time and the introduced error. Note that, after interpolation has been

August 8, 2014

44 Interpolation of the EBC extrapolation surface integral

implemented, both the recording and the injection surfaces are sampled with a receiver and

a source at each grid point of the FD scheme, respectively. In principle, in accordance with

the Nyquist sampling theorem, we should be able to accurately reconstruct any wavefield

that does not contain aliased energy. However, for the wavefields that contain aliased energy

due to subsampling, interpolation does not recover the lost wavenumbers and therefore the

wavefields can not be adequately reconstructed. We consider the square-shaped homogeneous

configuration to test the interpolation approach.

4-2 Results

To quantify the accuracy of the interpolation approach, we compute the difference of the RMS

error before and after interpolation and we normalize it by the RMS error obtained before

interpolation. We refer to this quantity as the relative difference of the RMS error, Rel.Diff

(see column 8 of Tables 4-1 and 4-2 and column 9 of Table 4-3). The examples presented in

this chapter utilise cubic spline interpolation because it provides a good compromise in terms

of computational time and accuracy.

First, we apply interpolation to the extrapolated values of the wavefields on the injection surface.

The results are shown in Table 4-1. Analysing these results, we conclude that interpolation

reduced the introduced errors for all tested configurations (> 10%). This can also be observed

in Figure 4-1, where the results after interpolation (dashed line) are all below the results before

interpolation (solid line). Note that, before interpolation, the injection process followed equation

3-2, whereas now it again follows the discretized version of equation 2-14 since all the sources

along ∂Dinj are active. A second characteristic feature of the dashed curve in Figure 4-1 is that

the accuracy of the interpolated results rapidly decreases for simulations where the subsampling

initially violates the Nyquist sampling theorem (Sour.F ≥ 5). A particularly successful result

is obtained for the configuration initially subsampled with a factor Sour.F = 2. In this case,

the error has decreased by 93% and the reconstructed wavefield almost perfectly matches the

reference one, resulting in a maximum RMS error of 0.02%.

The observed savings in computational time for the Green’s functions calculation (column 4 of

Table 4-1) are insignificant (≤ 2%) as expected from expression 3-3 with Rec.F = 1. For the

computation time of the EBC calculation (column 5 of Table 4-1), the savings are significant

as they reflect the amount of sources used along ∂Dinj during the extrapolation process (see

expression 3-4). Figure 4-2 shows the savings in time for the EBC computation after interpola-

tion (red solid line) compared to the ones obtained before interpolation (black solid line). It is

August 8, 2014

4-2 Results 45

2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

Sour.F

Ma

x R

MS

err

or

(%)

Before interpolation

After interpolation

Nyquistsampling

Figure 4-1: RMS error due to subsampling of the injection surface for a homogeneous casewith the square-shaped geometry. Solid line: connects the RMS errors of thetests before interpolation has been applied to the injection surface. Dashed line:connects the RMS errors of the tests after interpolation has been applied to theinjection surface. The vertical red dashed line indicates the Nyquist spatial sam-pling.

worthwhile to observe that the difference between these two curves is not systematic as they in-

terchange each other such that the total cost of the interpolation procedure is insignificant. The

curve for the theoretical savings is calculated following equation 3-4 for different subsampling

factors.

Second, we apply interpolation to the recorded values of the wavefields p and ~v on the recording

surface such that we can extrapolate them from all the gridpoints along ∂Drec. The results

are shown in Table 4-2. Note that, before interpolation, the extrapolation process followed

equation 3-1 whereas now it again follows equation 2-16. The most interesting feature observed

in Table 4-2 is that the reconstruction of the wavefields after interpolation is very promising for

the simulations that do not violate the Nyquist criterion (Rec.F ≤ 5). Of particular interest is

the result obtained when subsampling the recording surface with Rec.F = 4 since interpolation

has reconstructed the wavefield particularly well (see columns 6 and 7 in Table 4-2). On the

other hand, for Rec.F ≥ 5, interpolation is unable to correctly reconstruct the wavefields since

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2 3 4 5 6 7 8 9 1045

50

55

60

65

70

75

80

85

90

Subsamplig factor

Tim

e s

avin

gs (

%)

EBC computation

After interpolation

Before interpolation

Theoretical savings

Figure 4-2: Comparison between the observed and the theoretical savings in computationaltime for the EBC computations for the square-shaped homogeneous model.Dashed line: theoretical savings after subsampling the injection surface (calcu-lated with equation 3-4). Black solid line: observed savings after subsamplingthe injection boundary but before interpolation. Red solid line: observed savingsafter subsampling and interpolation have been implemented along the injectionsurface.

aliased energy is present and this yields a rapid increase of the RMS error.

Interpolating the recording surface after subsampling does not yield any savings in time for

the Green’s functions computation (see column 4 in Table 4-2). This is because interpolation

occurs after recording the wavefields p and ~v along ∂Drec, and therefore all the pre-computed

Green’s functions are used again for the extrapolation process (in accordance with equation

3-1). Furthermore, since no subsampling has been applied to the injection surface, no savings

in time for the EBC computations were achieved (see expression 3-4). Yet, interpolating the

recording surface after subsampling is of great interest for particular applications where it is

not possible to densely sample the recording surface with receivers (Vasmel et al., 2013) and

(Robertsson et al., 2008).

Finally, the results for the interpolation of both the injection and recording surfaces are shown

in Table 4-3. The simulations where the recording surface was subsampled with a factor of

Rec.F = 4 now becomes particularly interesting (see rows 3 and 4 of Table 4-3). The RMS error

has been reduced to 2.1% (compared to 367% before interpolation ) with savings in time for

the EBC computation of up to 76%.

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4-2 Results 47

Table 4-1: Homogenous case with the square-shaped geometry. Interpolating the injectionsurface to obtain a source at each gridpoint position. The recording surface isnot subsampled. The relative difference, Rel.Diff, is a comparison between theMax.Errors obtained before (b) and after (a) interpolation has been implemented.Positive results mean that the Max.Error increases, whereas negatives results meanthat the Max.Error decreases. The Saved Time is compared to the reference com-putational time and positive values mean larger savings in time.

Sour.F Rec.F Saved.Time (%) Max.Error (%) Rel.Diff (%)

b a GF EBC b a

2 1 1 1.5 47 0.29 0.02 -934 1 1 1.6 72 1.0 0.88 -135 1 1 0.21 76 2.6 1.2 -406 1 1 1.4 78 5.0 3.5 -318 1 1 0.67 84 11 7.2 -3410 1 1 0.56 86 16 9.8 -39

Table 4-2: Homogenous case with the square-shaped geometry. Interpolating the recordingsurface to obtain a receiver at each gridpoint position. The injection surface isnot subsampled. The relative difference, Rel.Diff, is a comparison between theMax.Errors obtained before (b) and after (a) interpolation has been implemented.Positive results mean that the Max.Error increases, whereas negatives results meanthat the Max.Error decreases. Note that for this test, the savings in time (column4 and 5) are insignificant.


b a GF EBC b a

1 2 1 3.0 1.3 0.22 0.05 -771 4 1 1.6 1.1 66 0.78 -991 5 1 0.02 0.05 77 96 241 6 1 5.9 0.75 2.3×102 2.5×106 1.1×104

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Table 4-3: Homogenous case with a square-shaped geometry. Interpolating both the record-ing and the injection surfaces to obtain receivers and sources at each gridpointposition, respectively. The relative difference, Rel.Diff, is a comparison betweenthe Max.Errors obtained before (b) and after (a) interpolation has been imple-mented. Positive results mean that the Max.Error increases, whereas negativesresults mean that the Max.Error decreases. The Saved Time is compared to thereference computational time and positive values mean larger savings in time.


b a b a GF EBC b a

2 1 2 1 0.11 48 0.33 0.05 -854 1 2 1 3.5 72 1.1 0.87 -172 1 4 1 0.99 48 66 0.78 -994 1 4 1 0.54 76 3.7×102 2.1 -99

August 8, 2014

Chapter 5

Conclusions

In this thesis we presented the results of an investigation we carried out on the effect of sub-

sampling both the recording and injection surfaces of the exact boundary condition (EBC)

algorithm. The subsampling study was based on synthetic models using a time-domain (TD)

finite-difference (FD) scheme. Results from subsampling both surfaces showed that significant

savings in time for both the Green’s functions and EBC computations can be achieved while

still being able to accurately reproduce the wavefields inside the truncated domain.

5-1 Conclusions

Modeling of seismic wave propagation inside subvolumes of a large model is of interest in various

fields, such as inversion, full waveform inversion, 4D, or time-lapse seismics. However, to obtain

an accurate reconstruction of the wavefield inside the subvolume of interest, all interactions

with its background domain must be taken into account. The exact boundary condition (EBC)

method provides an exact solution to reconstruct the wavefield inside the subvolume of interest

by accounting for all the interactions of the propagating wavefield with its background domain.

The method relies on a set of Green’s functions that are computed in advance and used during

the simulation inside the subvolume, and on an extrapolation process. Both the calculations of

the Green’s functions and the extrapolation process can be computationally demanding.

In this thesis, we showed that subsampling the recording and injection surfaces results in sig-

nificant savings in time for the two computationally demanding processes. The cost for the

August 8, 2014

50 Conclusions

Green’s functions calculations scales linearly with the number of receivers along the recording

surface. Subsampling ∂Drec provides savings in time up to 50% while still obtaining accurate

results. The cost for the extrapolation process depends linearly on both the number of receivers

along ∂Drec and the number of sources along ∂Dinj . Thus, for instance, savings of up to 90%

can be achieved when subsampling ∂Drec with Rec.F = 2 and ∂Dinj with Sour.F = 6.

We also showed that subsampling both the recording and injection surfaces decreases the ac-

curacy of the reconstructed wavefields inside the truncated domain D. However, subsampling

does not equally affect the accuracy of the reconstructed wavefields when it is implemented

along ∂Drec or along ∂Dinj . Specially significative, is the effect of subsampling the recording

surface on the accuracy of the results. During the extrapolation process, aliasing of the incident

wavefield because of subsampling leads to apparent outgoing waves that are erroneously extrap-

olated to the injection surface. Therefore, subsampling ∂Drec with a sampling interval coarser

than the Nyquist criterion yields simulations that eventually become extremely inaccurate. We

demonstrated that several parameters, such as the frequency content of the source, the angle

of incidence of the wavefield reaching ∂Drec, and the distance between the recoding and the

injection surface, are important to understand the evolution of the introduced RMS error. The

higher frequency components of the source and the wider angles of incidence of the wavefield

at ∂Drec turned out to have the strongest effect on the error evolution. Finally, for large dis-

tances between the recording and injection surfaces, longer simulation times were required to

observe a significant increase of the error. In contrast to the results obtained after subsampling

the recording surface, the errors introduced when subsampling the injection surface are much

smaller and stable. Even subsampling the injection surface beyond the Nyquist criterion re-

sulted in acceptable errors: for instance, a RMS error of 15% for a sampling interval twice as

large as the Nyquist criterion.

To confirm that the EBC method provides the correct solution for the reconstructed wavefield

inside D, we added strong contrasts in material properties to a simple square-shaped homoge-

nous medium. The strong scattering caused by the anomaly placed outside D was correctly

accounted for by the algorithm, confirming that the EBC method adequately incorporates the

long-term high-order interactions. Furthermore, the RMS error evolution as a function of time

for subsampling of the recording and injection surfaces was in very good agreement with the

results obtained in the homogeneous model.

For some applications of the EBC method, it is of interest to study the accuracy of its imple-

mentation when we have open recording and injection surfaces. To test this, we approximate the

open surface by subsampling the vertical edges, such that after subsampling these have almost

no sources nor receivers on both the injection and recording surfaces, respectively. Our results

August 8, 2014

5-2 Outlook 51

showed a highly similar evolution of the RMS error as function of time for the subsampled

test compared to an open surface test with no vertical sides along D. For this configuration,

the same heterogeneities as for the square-shaped model were tested showing consistent results

compared to the ones obtained in the homogeneous model.

In the final chapter, we assessed the impact of interpolation of the subsampled wavefields on

the injection and the recording surfaces aiming for the best compromise between the savings in

computational cost and accuracy of the reconstructed wavefield inside the truncated domain. In

general, all errors decreased, especially those introduced after subsampling the recording surface.

However, for those configurations where the wavefield on the recording surface was aliased (after

subsampling), interpolation could not improve the results. In terms of computational time,

results were different depending on which boundary the wavefields were interpolated. For the

injection surface, the savings in the EBC computation were barely affected since interpolation

is applied after the extrapolation process. For the recording surface, there were no savings for

the Green’s functions computation as all the Green’s must be computed upfront in order to be

used during the extrapolation process.

5-2 Outlook

We have demonstrated that subsampling can significantly reduce the computational cost of

the exact boundary condition method. Yet, there are several possibilities for further research

that could lead to an even better understanding of the errors introduced after subsampling.

First of all, it would be of great interest to investigate different integration techniques for the

discretization of the integrals along the injection and recording surfaces. The work presented

in this thesis is based on a mid-point rule integration technique, but Gauss quadrature, for

example, could help to improve the accuracy of the reconstructed wavefields.

The analysis of the obtained errors after subsampling the recording surface was based on the

Nyquist sampling theorem. In this work, we assumed the maximum frequency to be fmax = 2·fcto determine the minimum subsampling factor by which we considered aliasing to be present.

However, since the factor of 2 is merely a rule of thumb, it would be instructive to carry out

a proper analysis of the frequency spectrum of the source. Additionally, since the maximum

frequencies were the ones introducing the larger errors, it could be beneficial to apply a low-pass

filter to the extrapolated wavefields to observe its effect on the accuracy of the reconstructed

wavefields.

The interpolation technique used in the last chapter of this thesis provided a general improve-

ment to the accuracy of the reconstructed wavefield inside the truncated domain. However, only

August 8, 2014

52 Conclusions

a spline interpolation method has been tested. It would be interesting to implement different

interpolation techniques to analyze whether the errors can be further reduced.

To conclude, the 2D EBC algorithm implemented in this thesis should be extended to 3D in

order to facilitate the application of the EBC method to actual field data (e.g., in a full waveform

inversion scheme). Furthermore, in 3D, the savings in time and memory for storage of both the

Green’s functions and the EBC computation would be of the order of n4 compared to n3 for a

2D implementation when the spatial sampling is increased by a factor of n. Therefore, in 3D,

the benefits introduced by the subsampling approach would be even more significant than those

presented in this 2D study.

August 8, 2014

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