SimulationofSeismicWave Propagation onIrregularGridskaeserm/index.php_files/SOURCES/diplom.pdf · SimulationofSeismicWave Propagation onIrregularGrids Diplomarbeit von MartinAndreasK˜aser

Simulation of Seismic Wave

Propagationon Irregular Grids

Diplomarbeit

von

Martin Andreas Kaser

November 1999

Institut fur

Allgemeine und Angewandte Geophysik

der

Ludwig-Maximilians-Universitat

Munchen

ii

Acknowledgments

An erster Stelle mochte ich meinem Betreuer Prof. Heiner Igel danken, dermir die Moglichkeit eroffnet hat, in ein vollig neues Gebiet der Geophysikeinzusteigen. Obwohl die Arbeitsgruppe Numerische Geophysik am Anfangmeiner Diplomarbeit nur aus uns beiden bestand und erst im Aufbau be-griffen war, hatte er trotz aller organisatorischen Aufgaben stets die Zeitund Geduld, mir bei meinen Fragen weiter zu helfen. Seine große Erfahrungsowie seine herausfordernden Ideen haben mich sehr beeindruckt und mo-tiviert, auftretende Probleme analytisch anzugehen und zu losen.Desweiteren mochte ich mich bei Prof. Helmut Gebrande bedanken, derdurch interessante Vorschlage und Hinweise zur Verbesserung der Arbeitbeigetragen hat.Besonderer Dank gilt meinem Kommilitonen Gunnar Jahnke fur die unzahligenTips, die er mir bei Soft- und Hardware-Problemen geben konnte, und furdas harmonische und erfolgreiche Zusammenarbeiten mit ihm. Dies gilt auchfur alle anderen Kommilitonen und Mitarbeiter des Instituts fur Allgemeineund Angewandte Geophysik, die eine außerst angenehme Arbeitsatmospharebewirkten.I also thank Malcom Sambridge and Jean Braun for providing their algo-rithm to determine natural neighbours on arbitrary grids.Thanks to Jonathan Shewchuck providing his free 2-D quality mesh genera-tor and Delaunay triangulator.Discussions with Wolfgang Bangerth and Hamish Macintyre were also veryhelpful in improving this work.Mein großter Dank gilt jedoch meiner Familie und meiner Freundin Katja,ohne deren Unterstutzung diese Arbeit sicherlich nicht moglich gewesen ware.

iii

General Introduction

Seismology is a venerable science with a long history. Nearly two thou-sand years ago Chinese scientists invented the first functional seismoscope,a primitive device to register the arrival of seismic waves and even infer thedirection from where they came. The path to logical understanding of natu-ral phenomena like earthquakes was laid in the 17th century by systematicobservations of scientists like Galileo and the discovery and statement of fun-damental physical laws by Newton. However, another 250 years were to passbefore Navier, Cauchy and Poisson completed the foundations of the moderntheory of elasticity.In the first decades of this century analyses of travel times of seismic bodywaves proposed the existence of the Earths core. Several boundaries werediscovered dividing the Earth into an inner and outer core, the mantel, andthe crust. The definition of these and other discontinuities associated withthe deep internal structure of the Earth have since been greatly refined. Es-pecially the techniques of refraction and reflection seismology - developedin the search for hydrocarbon resources - have improved the resolution ofdetailed crustal structures in the 1960s.Knowledge of the Earths structure not only allows us to find hydrocarbons,which still represent our main energy source, but also enables us to studyearthquakes and possibly understanding Earths dynamics. Therefore, it ismost important to improve our techniques to reveal the Earths interior andto understand the measurements carried out at its surface.Within the last few decades a revolution in computer technology pushedseismology one step further. The theories became numerical programs andsynthetic data can now be compared to real data. The numerical solutionsto wave propagation problems enable us to calculate theoretical seismogramsthat are similar to those observed after earthquakes or recorded in seismicexploration experiments. To solve the inverse problem, we try to find theEarth model that correctly explains synthetic data and adjust theoreticalpredictions to reality. The model, that leads to the minimal difference be-tween synthetic data and real seismograms can be considered as the mostlikely, as a unique solution to the seismic inverse problem does not exist.Numerous numerical methods to compute wave propagation have been de-veloped through the years covering the range from global seismology to lab-oratory measurements. Especially the finite-difference technique has turnedout to be a useful and powerful method and has constantly been improved.The numerical calculation of the complete wave field is of uttermost impor-

iv

tance to create datasets, which can be compared to real field data. Methodsbased on ray theory cannot accomplish todays demand of high resolution andcomplete wave field information.In the sense of developing numerical methods other than standard finite-difference techniques this work represents a feasibility study with an alterna-tive approach to compute synthetic data in media with complex geometry.Contrary to standard finite-difference techniques, this work discusses a nu-merical method based on an irregular (arbitrary) discretization of media, inwhich the propagation of seismic waves is simulated.In the following, a brief description of the different chapters is given.

Chapter 1: An introduction to numerical simulation methods and theirneed in modern seismology is given and - with respect to the current stateof the art - the aim of this research is outlined.The elastodynamic equations describing the propagation of a seismic wavefield are shown. Based on this system of equations the fundamental conceptof the numerical simulation algorithm is briefly discussed and the use of astaggered grid scheme is explained.Chapter 2: The advantages of arbitrary, irregular grids to discretize mediawith complex geometries or curved boundaries are shown. The generationof arbitrary grids is discussed with respect to the classification of grids withdifferent degrees of irregularity.Three explicit differential operators are introduced, to compute spatial deriva-tives of a vector field on an arbitrary grid. Their accuracy on irregular grids iscompared to usual finite-difference operators on two regular reference grids.Chapter 3: The differential operators are implemented in a simulation algo-rithm to propagate waves in an acoustic and elastic medium. The simulationsetup is outlined and the implementation of the seismic sources and the re-ceivers is discussed.The accuracies of the synthetic seismograms obtained from the irregular gridsand the regular reference grids are compared to analytical solutions. Prob-lems occurring for different source implementations and grid symmetries areoutlined.Chapter 4: The natural neighbour operator is applied to simulate wavepropagation in different realistic models. Acoustic waves are propagatedthrough a cylindrical model and a model describing a mountainous topogra-phy. In a final elastic model representing a simplified basin structure sim-ulation results are compared to standard finite-difference results. The ap-plication of arbitrary, irregular grids is discussed for the different modelsespecially with respect to the design of possible staggering schemes.

Contents

Acknowledgments ii

General Introduction iii

1 Fundamentals of Numerical Simulations 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Elastodynamic Equations . . . . . . . . . . . . . . . . . . . . . 31.3 Principles of Finite-Difference Algorithms . . . . . . . . . . . . 4

1.3.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Extrapolation . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Discretization on Staggered Grids . . . . . . . . . . . . . . . . 6

2 Explicit Differential Operators 9

2.1 Advantages of Irregular Grids . . . . . . . . . . . . . . . . . . 92.2 Regular Reference Grids . . . . . . . . . . . . . . . . . . . . . 112.3 Irregular Grid Generation . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Grid Perturbation . . . . . . . . . . . . . . . . . . . . . 132.3.2 Grid Quality . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Irregular Staggered Grids . . . . . . . . . . . . . . . . 16

2.4 Difference Weights on Arbitrary Grids . . . . . . . . . . . . . 192.4.1 Natural Neighbour Weights . . . . . . . . . . . . . . . 202.4.2 Finite Volume Weights using all Neighbours . . . . . . 202.4.3 Finite Volume Weights Using Three Neighbours . . . . 202.4.4 Reference Cases . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Accuracy of Space Derivatives . . . . . . . . . . . . . . . . . . 232.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Simulation of Wave Propagation 29

3.1 Initialization of Simulation Parameters . . . . . . . . . . . . . 293.1.1 Geometrical Aspects . . . . . . . . . . . . . . . . . . . 293.1.2 Stability of Simulation Algorithms . . . . . . . . . . . 303.1.3 Source and Receiver Positioning . . . . . . . . . . . . . 34

3.2 Acoustic Wave Propagation . . . . . . . . . . . . . . . . . . . 363.2.1 Influence of Grid Symmetry . . . . . . . . . . . . . . . 37

v

vi CONTENTS

3.2.2 Misfit Energy of Synthetic Seismograms . . . . . . . . 383.2.3 Spatial Variation of Seismogram Accuracy . . . . . . . 393.2.4 Accuracy of Synthetic Seismograms . . . . . . . . . . . 40

3.3 Elastic Wave Propagation . . . . . . . . . . . . . . . . . . . . 453.3.1 Separation of P- and S-Waves . . . . . . . . . . . . . . 453.3.2 Elastic Sources . . . . . . . . . . . . . . . . . . . . . . 463.3.3 Simulation of an Explosive Source . . . . . . . . . . . . 473.3.4 Accuracy of Synthetic Seismograms . . . . . . . . . . . 543.3.5 Simulation of a Rotational Source . . . . . . . . . . . . 583.3.6 Accuracy of Synthetic Seismograms . . . . . . . . . . . 59

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Application to Realistic Models 67

4.1 Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . 674.1.2 Simulation Results . . . . . . . . . . . . . . . . . . . . 69

4.2 Mountain Topography . . . . . . . . . . . . . . . . . . . . . . 714.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . 714.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . 73

4.3 Margin of a Basin . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . 754.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . 77

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

General Conclusions 83

Bibliography 85

A Derivative Weights 91

A.1 Natural Neighbour Weights . . . . . . . . . . . . . . . . . . . 91A.2 Finite Volume Weights . . . . . . . . . . . . . . . . . . . . . . 92

A.2.1 Using all Natural Neighbours . . . . . . . . . . . . . . 92A.2.2 Using Three Neighbours . . . . . . . . . . . . . . . . . 94

B Analytical Solution 95

B.1 Acoustic Medium . . . . . . . . . . . . . . . . . . . . . . . . . 95B.2 Elastic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Chapter 1

Fundamentals of Numerical

Wave Propagation Simulations

1.1 Introduction

Seismic wave propagation is one of the main tools in geophysics for imag-ing the structure of the Earths interior and understanding the correspondinggeodynamic phenomena. In this context, seismic tomography played an im-portant role to provide velocity models of the subsurface. Especially surfacewaves and seismic body waves have been used to uncover the velocity struc-ture of the Earth (e.g. Woodhouse & Dziewonski, 1984; Hara et al., 1993;Dziewonski, 1996).However, to answer the current questions of geodynamics (e.g. the behaviourof subduction zones (e.g. Lay, 1994; Zhong & Gurnis, 1995) or the origin ofhot spots) a structural resolution is necessary, which cannot be accomplishedby seismic tomography, as it is based on ray theory. So far, it is impossi-ble to compute the complete seismic wave field in a three-dimensional Earthwithout severe approximations.The calculation of complete synthetic seismograms in global seismology ismainly based on spherical harmonic functions or normal modes (Dahlen &Tromp, 1998). However, the extension of this approach to general three-dimensional models means an enormous algorithmic effort.An alternative approach is the calculation of synthetic seismograms by di-rectly solving the seismic wave equation using numerical methods (e.g. finite-differences, finite-elements, spectral-elements, finite-volumes, or pseudospec-tral methods). The numerical methods for seismic wave propagation haveintensively been developed in exploration seismology (e.g. Kelly et al., 1976;Virieux, 1986; Carcione et al., 1988; Igel et al., 1995). Hereby, the techno-logical progress in parallel computing plays an important role and enablesthe highly efficient implementation of the seismic wave equation using localfinite-difference operators.

1

2 CHAPTER 1. FUNDAMENTALS OF NUMERICAL SIMULATIONS

Nevertheless, the method was mainly developed and improved for regular,cartesian geometries. Unfortunately, these standard finite-difference tech-niques can not simply be applied to cylindrical or particularly spherical co-ordinates of a complete sphere (r, φ, θ), as for regular gridding the size of thegrid cells decreases towards the axis θ = 0◦ or θ = 180◦. As the stability of afinite-difference algorithm is proportional to the time increment and indirectproportional to the grid spacing, unrealistically small time increments haveto be used to keep the simulation stable. This in turn, leads to very longcomputation times of the wave field, if no multi-domain methods are used.Alternatively, methods have to be developed that operate on arbitrary gridsto avoid such singularities. For example, the spectral-element technique canbe used (Chaljub & Vilotte, 1999). However, the discretization is based oncurved cubic elements leading to an inappropriate description of the Earthsinterior by cubes.Therefore, another numerical method is investigated in this work to approachthe problem of simulating seismic wave propagation on arbitrary grids.

So far, numerical algorithms for the elastic wave propagation in two andthree dimensions are mainly based on methods, that work with more or lessregular grids. These have major disadvantages, when dealing with cylindrical(e.g. borehole core) or spherical (e.g. planets) geometry.The alternative approach discussed in this work is the discretization of two-dimensional, space-dependent variables on arbitrary, irregular grids. There-fore, operators have to be designed to compute spatial derivatives of theseismic wave field in a two-dimensional medium discretized on such irregulargrids.By using the Delaunay triangulation the so-called natural neighbours of eachgrid point are determined. For these points the differential operators have tobe found. Several methods can be used (e.g. finite volume method, naturalneighbour coordinates, etc.). However, these methods are quite inaccuratecompared to regular, finite-difference-operators. The issue is, to find oper-ators for the elastic wave equation, which provide sufficient accuracy in thesense of reducing errors in the calculation of spatial derivatives.Therefore, the operators are tested by investigating the difference of numer-ically and analytically computed derivative values of a test function (two-dimensional sinusoidal functions).In a further step, these operators are implemented in a program to simulateseismic wave propagation on arbitrary grids. This work focuses on the influ-ence of grid irregularity on the performance of the differential operators andtherefore on the accuracy of synthetic seismograms.

1.2. ELASTODYNAMIC EQUATIONS 3

1.2 Elastodynamic Equations

The numerical methods used in this work are based on the theory of elasto-dynamics. In general the wave equation for two-dimensional problems canbe written as

ρ∂2ux∂t2

=∂σxx∂x

+∂σxz∂z

, (1.1)

ρ∂2uz∂t2

=∂σxz∂x

+∂σzz∂z

, (1.2)

σxx = (λ+ 2µ)∂ux∂x

+ λ∂uz∂z

, (1.3)

σzz = (λ+ 2µ)∂uz∂z

+ λ∂ux∂x

, (1.4)

σxz = µ(∂ux∂z

+∂uz∂x

) (1.5)

where ux and uz are the components of the displacement vector and σxx, σzzand σxz) are the elements of the stress tensor. The medium is described bythe density ρ(x, z) and the Lame coefficients λ(x, z) and µ(x, z). This systemcan be transformed into the following first-order hyperbolic system

∂vx∂t

=1

ρ(∂σxx∂x

+∂σxz∂z

), (1.6)

∂vz∂t

=1

ρ(∂σxz∂x

+∂σzz∂z

), (1.7)

∂σxx∂t

= (λ+ 2µ)∂vx∂x

+ λ∂vz∂z

, (1.8)

∂σzz∂t

= (λ+ 2µ)∂vz∂z

+ λ∂vx∂x

, (1.9)

∂σxz∂t

= µ(∂vx∂z

+∂vz∂x

) (1.10)

where vx and vz are the components of the velocity vector. This system iscalled the velocity-stress formulation of the wave equation (Virieux, 1986)and is the basis for all simulation algorithms, which are discussed in laterchapters.


1.3 Principles of Finite-Difference Algorithms

To simulate wave propagation through a medium the computation can bedivided into two main parts. One is the interpolation in space and the secondis the extrapolation in time. The finite-difference method (e.g. Marsal,1989; Heinrich, 1987; Thomas, 1995) can not be described in full detail here.However the fundamental equations clarify, in which part of the simulationalgorithm the irregular grid methods will be implemented. A more detaileddescription of the FD technique applied to wave propagation problems isgiven by Aki & Richards (1980) and Igel (course notes, 1999).

1.3.1 Interpolation

The finite-difference technique is based on the approximation of the Taylor-series

f(x±∆x) = f(x)± f ′(x)∆x+∆x2

2!f ′′(x)± ∆x3

3!f ′′′(x) + ... , (1.11)

which leads to the differential quotients

∂f

∂x≈ f(x)− f(x−∆x)

∆x(1.12)

∂f

∂x≈ f(x+∆x)− f(x)

∆x(1.13)

for backward and forward differences, respectively. Here f can be an arbi-trary function, e.g. the velocities vx and vz or the stresses σxx, σzz or σxz and∆x is the distance of the discrete grid points. These simple one-dimensionalequations show the fundamental concept of calculating the spatial derivativeof a function using the neighbouring function values. This work is concentrat-ing on the investigation of spatial differential operators for two-dimensionalproblems as shown in later chapters.

1.3.2 Extrapolation

Simulation of seismic waves means that we are interested in how the wavepropagates through a medium. Therefore, we have to calculate the completewave field for a series of time steps. This process includes a time extrapolationand is again based on the Taylor-series

f(t+∆t) = f(t) + f ′(t)∆t+∆t2

2!f ′′(t) +

∆x3

3!f ′′′(t) + ... (1.14)

where ∆t is the time increment. To compute the wave field for the nexttime step in the future f(t + ∆t) we again use the truncated series as an

1.3. PRINCIPLES OF FINITE-DIFFERENCE ALGORITHMS 5

σij(x, z, t+∆t/2),S(x, z, t+∆t/2) −→ ∂jσij(x, z, t+∆t/2)

∂jσij(x, z, t+∆t/2), ρ(x, z, t+∆t/2) −→ ∂tvi(x, z, t+∆t/2)

∂tvi(x, z, t+∆t/2), vi(x, z, t) −→ vi(x, z, t+∆t)

vi(x, z, t+∆t) −→ ∂jvi(x, z, t+∆t)

∂jvi(x, z, t+∆t),λ(x, z, t+∆t),µ(x, z, t+∆t) −→ ∂tσij(x, z, t+∆t)

∂tσij(x, z, t+∆t),σij(x, z, t+∆t/2) −→ σij(x, z, t+ 3∆t/2)

Table 1.1: Schematic algorithm to propagate elastic waves as implemented inthe simulation program.

approximation

f(t+∆t) ≈ f(t) +∂f(t)

∂t∆t. (1.15)

This shows that we have to know the wave field at the present f(t) and the

first derivative of the wave field with respect to time ∂f(t)∂t

, which can becalculated using the wave equation in the velocity-stress formulation.The simulation algorithm to propagate elastic waves is implemented as givenin Table 1.1. The values on the left side are used to calculate the values onthe right side.The space and time derivatives are denoted by ∂j and ∂t, respectively. vi arethe components of the velocity vector and σij are the elements of the stresstensor and S is the source term. This scheme is processed for each time stepresulting in a propagating wave field. The system of elastodynamic equationsconnect the interpolation part with the extrapolation part. Therefore thefundamental steps in simulating seismic wave propagation is (1) calculatingspace derivatives of velocities and stresses knowing their values at discretepoints (interpolation part), (2) evaluate the time derivatives by using thewave equation and (3) compute velocity and stress values for each grid pointfor the next time step (extrapolation part). Note, that the scheme implies astaggered scheme in space and time.


vx

σxx σzz

zv

σxz

ijσ

jv

a b

Figure 1.1: (a) Example of a staggered grid scheme for a standard FD gridwith quadratic grid cells (e.g. Virieux 1986). (b) Example of a staggered gridscheme for an irregular grid with triangular grid cells.

1.4 Discretization on Staggered Grids

The basis for numerical solutions to many kinds of time-dependent problemsin geophysics is the discretization of the medium on a spatial grid. Thismeans that data values are only defined on particular grid points or nodes.For example, the velocity-stress formulation of the elastic wave equation,written as a first-order system as shown above, suggests the use of a stag-gered grid scheme. This implies, that the velocities are defined on one grid(primary grid) and the stresses on another (secondary grid). The conceptapplied to elastic wave propagation was first used by Madariaga (1976) andVirieux (1984, 1986). This so-called grid splitting or grid staggering is mainlycaused by the definition of the derivative operators, as the values of thederivatives are located halfway between the function nodes. Staggered gridsgenerally result in improved accuracy compared to non-staggered grids withall fields defined at the same locations due to the antisymmetry of the differ-ential operator. For irregular grids the derivative values cannot in general bedefined right in between two function values as nodes are not aligned alongany coordinate axes. Therefore the question of positioning the secondarygrid with respect to the primary grid to define derivatives is non-trivial (seeChapter 2). Small sections of a standard rectangular staggered grid and anirregular staggered grid are shown in Figure 1.1. Note that in the irregularcase all stress elements are defined at the same grid points (non-staggered).This would allow modeling of general material anisotropy without the needof additional interpolations which decrease the overall accuracy (Igel et al.1995).

1.4. DISCRETIZATION ON STAGGERED GRIDS 7

It is shown, that the wave equation can be written as a hyperbolic systemof equations. This velocity-stress formulation suggests the use of a staggeredgrid scheme in time and space (see Table 1.1). As our aim is to propagateseismic waves through a medium discretized on an irregular spatial grid (Fig-ure 1.1b), operators have to be found, which are capable to compute spatialderivatives of a two-dimensional function on such unstructured grids.

Chapter 2

Explicit Differential Operators

on Arbitrary Grids

In this chapter the advantages of irregular or arbitrary grids are outlined incontrast to commonly used regular grids. A detailed description of gener-ating arbitrary, triangular grids is given, and methods to classify grids withvarying degrees of irregularity are discussed. However, problems of irregulargrids are also shown, especially with respect to their application as staggeredgrid schemes.Furthermore, explicit differential operators for unstructured grids are intro-duced1 and their accuracy is compared to standard two-point FD operatorson quadratic grids and operators used for regular hexagonal grids as intro-duced by Magnier, Mora & Tarantola (1994).The chapter particularly focuses on the question, how grid irregularity influ-ences the accuracy of the space derivatives compared to schemes on regulargrids with equivalent node densities. While Dormy & Tarantola (1995) statethat grid irregularity has only small effects, they did not perform a thoroughquantitative analysis. Therefore the accuracy of the numerical derivativesfor harmonic trigonometric test functions is given here as a function of gridquality - i.e. average triangle quality.

2.1 Advantages of Irregular Grids

So far numerical solutions to the wave equation have been dominated by(quasi-) regular grid methods such as the finite-difference (FD) method (e.g.Kelly et al. 1976) or the pseudospectral (PS) method (e.g. Fornberg 1987,1988; Tessmer, Kosloff & Behle 1992). The numerical techniques capable of

1Main parts of this chapter have been published in the Proceedings of the FourthInternational Conference on Theoretical and Computational Acoustics held in May 1999in Trieste, Italy.

9

10 CHAPTER 2. EXPLICIT DIFFERENTIAL OPERATORS

handling arbitrary grids such as the finite-element (FE) method (e.g. Smith1975; Marfurt 1984), the finite-volume (FV) method (Dormy & Tarantola1995), or the spectral element (SE) method (e.g. Padovani et al. 1994), hadfar less attention, probably because their implementation is more involvedthan regular grid methods.But methods with single domains and (quasi-)regular grids have their lim-itations. For example, when media are to be simulated with large velocitycontrasts, then parts of the model are oversampled, because - for stabilityreasons - the grid has to be adjusted to the smallest velocities resulting infiner gridding.Furthermore, the accurate implementation of boundary conditions often re-quires denser gridding near the boundaries than within the medium. Simu-lations on multidomain regular grids are possible (Jastram & Tessmer 1994;Falk, Tessmer & Gajewski 1996), yet the implementation usually becomesfar more difficult and less flexible.Another difficulty of (quasi-)regular grids arises, when one attempts to solveproblems which suggest the use of curvilinear coordinates, e.g. cylindrical orspherical problems (Figure 2.1a). Here a major problem occurs, as regular

a b

Figure 2.1: (a) Section of a finite-difference grid for modeling in cylindricalcoordinates. Note the necessary depth-dependent grid spacing to comply withthe stability criterion. (b) Section of a triangular grid for modeling a cylin-drical problem in cartesian coordinates. Note the approximately equal gridspacing throughout the entire section.

gridding of cylindrical coordinates leads to decreasing grid spacing towardthe axis r = 0 at the grid center. In time-dependent problems this small gridspacing requires unrealistically small time steps (see Chapter 3).Alternatively, when using irregular grids (e.g. Zhang & Tielin 1999; Zhang1997), the problems can often be solved using the elastodynamic equationsin cartesian coordinates (Figure 2.1b). The boundary conditions in cartesian

2.2. REGULAR REFERENCE GRIDS 11

a b

Figure 2.2: Three layers are divided by curved boundaries. (a) The mediumis discretized by a regular, rectangular grid leading to a blocky nature of theboundaries. (b) The medium is discretized by an irregular, triangular gridadapting much better to the curved boundaries.

coordinates could also be applied on curved boundaries. This is difficult withregular grid methods, because the blocky nature of a curved boundary de-scribed by rectangles leads to strong artifacts. An irregular, triangular gridis more flexible and can be adjusted to curved boundaries or layer interfaceswith high accuracy (Figures 2.2a, b).

2.2 Regular Reference Grids

In general, randomly distributed points sampling a two-dimensional mediumcan be considered as an arbitrary, irregular grid by connecting the grid pointsto a mesh. As the aim of this work is to analyse different numerical methodson arbitrary grids, the main task is to investigate the influence of grid irreg-ularity on the accuracy of the simulation results. Furthermore, the resultsobtained from irregular grid methods are compared to regular grid meth-ods. The two reference grids used in this work are a rectangular, quadraticfinite-difference grid (Figure 2.3b) and a hexagonal grid (Figure 2.3a). Theirregular grids are based on the hexagonal grid and are generated by ran-domly perturbating the hexagonal grid as discussed in the following sections.For further investigations it is important to find a fair method of comparingresults obtained from regular and irregular grids. Usually grids with identi-cal grid spacing are compared. However, this is impossible for this attempt,due to the varying sizes and shapes of irregular grid cells. Therefore, we areusing the average density of nodes as a measure to compare results on equallyspaced regular and irregular grids. The density of nodes is given by

d =N

A, (2.1)


a

a2

3 b

b

0 1 1.07

10.93

z

x

a b c

Figure 2.3: (a) Hexagonal and (b) quadratic, regular grids with equivalentdensity of nodes. (c) Comparison between the unit cells of the two differentbut equally spaced grids.

where A is the area sampled by N grid points. In general, a number of N×Nnodes covers an area of

Ahex = (N − 1)2 · a · a2

√3 (2.2)

in the hexagonal case, and

Aquad = (N − 1)2 · b2 (2.3)

in the quadratic case (Figure 2.3a, b). Assuming an equal density of nodesfor both grid types, leads to the following relation between the grid spacingparameters a and b.

N2

(N − 1)2 · a · a2

√3

=N2

(N − 1)2 · b2 (2.4)

⇒ a =

√

2

3

√3 b (2.5)

b =

√

1

2

√3 a (2.6)

A direct comparison of the unit cells of the two reference grids (Figure 2.3c)displays difference in grid spacing along the coordinate axes. A quadraticgrid spacing of b = 1 is leading to a hexagonal grid spacing of a ≈ 1.07 inx-direction and a

2

√3 ≈ 0.93 in z-direction.

Irregular grids in this work are created by randomly distorting a perfecthexagonal grid (see next section), causing the density of nodes to vary fromone location to another. Therefore the average density of nodes - calculatedfrom the entire grid size - is important.

2.3. IRREGULAR GRID GENERATION 13

2.3 Irregular Grid Generation

2.3.1 Grid Perturbation

To investigate the influence of grid irregularity on the accuracy of derivativeoperators or the results of elastic wave simulations, a method has to be de-veloped to generate irregular grids. Therefore, it is also necessary to find away to control the irregularity of the generated grid.All irregular grids used in this work are based on a random process, thatperturbates the nodes of a perfect hexagonal grid. Each of the nodes of

r

r

a a

a

Figure 2.4: Schematic representation of creating irregular grids by randomlydistorting perfectly hexagonal grids. The empty circles are the initial nodesforming the hexagonal grid, the filled circles are the nodes after randomlymoving them to new locations with in a perturbation box. Note that the sizeof the boxes controls the grid irregularity.

the initial, hexagonal grid is surrounded by a box (Figure 2.4). The ini-tial, hexagonal nodes (empty circles) represent the center of each quadraticbox with side length r. The distortion of the hexagonal grid is achieved byrandomly moving each node to a new location within its corresponding box(filled circles). The perturbed nodes are then connected to form an irregular


mesh using a Delaunay2 triangulation (e.g. Watson 1981; Sibson 1981; De-vijver & Dekesel 1983). In principle, after using the Delaunay algorithm theperturbed nodes are triangulated in a way to achieve triangular grid cells,which are as equilateral as possible.The grid irregularity can be controlled by the size of the box, i.e. by the sidelength r. Small boxes allow only a slight perturbation, whereas large boxeslead to strong grid distortions. The distortion or perturbation of a hexagonalgrid is defined by

p =r2

a· 100 [%] (2.7)

and can be considered as the relation of the side length r of the perturbationboxes to the side length a of the initial, hexagonal unit cells (see Figure 2.4).Therefore the perturbation of a hexagonal grid is always given as a percent-age. Increasing the size of the perturbation boxes leads to overlapping areasof adjacent boxes, if p > 1

4

√3 · 100 ≈ 43%.

The perturbation of a hexagonal grid can only be used as a measure of grid ir-regularity, when using this particular method of irregular grid generation. Toquantify an irregular, triangular grid a more general parameter is necessary.

2.3.2 Grid Quality

Arbitrary, triangular grids can be classified by investigating the shapes ofthe triangular unit cells. Therefore, a parameter is used that describes theparticular shape of a triangle. This parameter is called the triangle qualityfactor q and is defined by

q =4√3A

a2 + b2 + c2, (2.8)

where A is the area and a, b and c are the side lengths of the triangle. Thequality factor q ranges from 0 to 1 with its maximum value for an equilateraltriangle and its minimum value for triangle degenerated to a straight line.The irregularity of a grid can be measured by calculating the triangle quali-ties of all grid cells and determine their average value. The average trianglequality then describes the irregularity of the entire grid. In this case, thequality factor does not depend on the method of how the irregular grid isgenerated, but only considers the final shape of the triangular grid (Figure2.5).For further investigations it is important to know the relation between the

2The Delaunay triangulation was developed in the field of computational geometry (e.g.O’Rouke, 1988) and is based on the concept of Voronoi cells. A brief introduction to thismethod is given in Appendix A, as a detailed description would go beyond the scope ofthis work.


q = 1.00 q = 0.96

q = 0.90 q = 0.78

a b

c d

primary grid

Figure 2.5: Four different sections of grids with equal node density are shown.Increasing grid irregularity leads to decreasing average triangle quality q.Also note the appearance of very low-quality triangles in highly irregular grids.

perturbation p of a hexagonal grid and the average triangle quality of theresulting triangular grid (Figure 2.6). An initial, hexagonal grid with 50×50grid points was distorted using perturbations between 0% and 120%. Appar-ently, the grid size is large enough to obtain at least one perfect equilateraltriangle with q = 1 in the grid, no matter how strong the perturbation of thegrid is chosen. For smaller grid sizes this might not be true, as the grid gen-eration is a random process. A remarkable result is, that the average trianglequality converges to q ≈ 0.7 for increasing grid perturbation. This effect canbe explained by the overlapping perturbation boxes, if p > 1

4

√3 · 100 ≈ 43%.


0 20 40 60 80 100 1200.6

0.7

0.8

0.9

1

tria

ngle

qua

lity

perturbation [%]

Qmax

Qmin

Qaverage

q

qqavemin

max

Figure 2.6: Relation between perturbation of grid points and triangle quality q.The three graphs show the values for the best (qmax) and worst (qmin) triangleof a 50×50 grid and the average triangle quality (qave) for perturbations up to120%. Note that qave is converging to about 0.7 for increasing perturbations.

This value represents the threshold, where the three nodes of an initial, equi-lateral unit cell can be located on a straight line (q = 0). For even greatervalues of the perturbation p, triangles can form just having their vertices indifferent order. The quality factor of the worst triangle tends to become zerowith increasing perturbation. As the Delaunay algorithm triangulates theperturbed nodes in a way to optimize the quality of the triangles, values ofq = 0 are automatically avoided. Nevertheless, the worst triangle will play animportant role in our investigation of local derivative operators on irregulargrids, as it affects the stability conditions in the simulation algorithm (seeChapter 3).

2.3.3 Irregular Staggered Grids

The elastodynamic equations of wave propagation suggest the use of a stag-gered grid scheme as discussed in Chapter 1. The symmetry of the derivativeoperators on a regular, quadratic FD grid causes grid splitting as shown inFigure 1.1a. This means, that the two components of the velocity vector vxand vz and the elements of the stress tensor σxx, σzz and σxz are defined atfour different locations. Using a hexagonal or an irregular grid, all velocitiesare defined on one grid (primary grid), and all stresses are defined on theother (secondary grid) as displayed in Figure 1.1b. The question of how the


primary gridsecondary grid

velocities

stresses

q = 1.00 q = 0.96

q = 0.90 q = 0.78

a b

c d

Figure 2.7: Four sections of staggered grids with different average grid qualityare shown. Note, that a slightly perturbed grid (b) still shows the systematicstructure of a perfect hexagonal grid (a). With increasing perturbation theaverage triangle quality decreases and the grid loses its structure (c) and (d),which makes the initialization of the secondary grid very difficult.

two grids are located with respect to each other can be solved easily in thehexagonal case (Figure 2.7a). The centers of gravity of the triangles of theprimary grid are used as secondary grid nodes. However, there are moreprimary triangles than primary grid points. To obtain the same numberof primary and secondary grid points, we are using only particular trianglecenters of the primary grid as nodes of the secondary grid. Hereby, we are


assuming, that the triangles of the primary grid are numbered from left toright - row by row - and effectively are using every second primary triangle todefine a secondary node at the corresponding triangle center. If the primarygrid is slightly perturbed (Figure 2.7b), the method of using every secondprimary triangle to define a secondary node still works nicely. For strongperturbations the initial hexagonal grid loses its structure (Figure 2.7c and2.7d) and becomes an arbitrary grid. This makes the choice of the secondarygrid points much more difficult. However, the staggered, irregular grids usedin this work are still generated as explained before. This means, that westill assume the triangles of the primary grid to be numbered and use everysecond triangle to define a secondary node. In other words we still assume acertain structure of the primary grid, though it does not exist any more. Asshown in Chapter 3, this problem will affect the stability of the simulationalgorithm.Of course, other techniques of defining secondary grid points must be con-sidered. For example, optimization techniques based on the definition ofVoronoi cells (e.g. Aurenhammer, 1991; Okabe et al. 1992) could be used. Apossible attempt would be to use particular vertices of Voronoi cells of a setof primary nodes to define secondary nodes, as the vertices of Voronoi cellsare somehow located optimally in between the primary grid points.Other methods are using vertices of irregular quadrangles (Zhang & Tielin1999; Zhang 1997) as primary grid points and their centers as secondary gridpoints. This attempt would overcome the problem of choosing only particu-lar centers of the primary grid, as the number of primary points equals thenumber of primary grid cells. Every center of a primary cell can be definedas a secondary node.In general, the question of finding a corresponding secondary grid to an ar-bitrary set of primary nodes is not trivial and deserves further study, butwould go beyond the scope of this work.

2.4. DIFFERENCE WEIGHTS ON ARBITRARY GRIDS 19

2.4 Difference Weights on Arbitrary Grids

As the time-dependent problem of wave propagation suggests the use of stag-gered grids (Figure 1.1) different schemes are investigated for the calculationof space derivatives on two separate grids. The differential operators usedare explicit and local in the sense that they use only information of the func-tion in their nearest neighbourhood, so that no matrix inversion is necessary.Three different ways to calculate partial derivatives are discussed, the nat-ural neighbour derivative (NND), introduced to geophysics by Sambridge etal. (1995) and Braun et al. (1995), the finite volume method using naturalneighbours (FVN) (Dormy & Tarantola, 1995) and the finite volume methodusing only three neighbours (FV3).In general, the derivative ∂if(x0) of a scalar function f(x) will be obtainedby calculating the sum over all neighbouring values f(xj) weighted by somevalue wij according to

∂if(x0) ≈N

∑

j=1

wijf(xj) , (2.9)

where N is the number of neighbouring points (Figure 2.8).

N 1

N 2

N 3

N 4

N 5

Figure 2.8: Calculation of derivatives on a 2-dimensional irregular grid usingfunction values at neighbouring points N1,...,N5. The neighbours are deter-mined through the concept of Voronoi cells (Appendix A).


2.4.1 Natural Neighbour Weights

As mentioned above, the Delaunay triangulation is the key to operate on ir-regular grids. A set of arbitrary points is triangulated by linking adjacent gridpoints in a way to obtain triangles with optimal quality. A secondary gridis obtained by putting grid points into particular centers of these triangles(Figure 2.7). The goal is to evaluate partial derivatives on one grid knowingthe function values on the other. To achieve this we use the concept of natu-ral neighbours (e.g. Sambridge et al. 1995; Watson 1985, 1992). Sambridge’smethod of natural neighbour coordinates determines the neighbouring pointsof each node, for which the partial derivative has to be evaluated. Theseneighbouring points are uniquely determined through the concept of Voronoicells (see Sambridge et al. 1995; Fortune 1992).Once a list of neighbours and their coordinates is obtained, differential weightscan be calculated by simply using the derivative of the interpolation weights.The interpolation weights are computed by the relative contribution of theneighbouring Voronoi cells (Sambridge et al. 1995). The formal expressionsfor the calculation of natural-neighbour-weights are outlined in Appendix A.

2.4.2 Finite Volume Weights using all Neighbours

An alternative to this approach is the use of finite-volume weights (Dormy& Tarantola 1995). The FV method is based on a discretization of Gauss’divergence theorem. For the calculation of the differential weights againthe natural neighbours are used as introduced before. They are connected toform a hull (FV cell) around the point where the derivative is to be evaluated.The length of the cell sides and the components of the corresponding normalvectors are then determined. Finally the derivative computed by summingover all natural neighbour values, weighted by the lengths of the adjacentcell sides and the corresponding components of the normal vectors.Gauss’ theorem implies that the derivative is assumed constant within thecell and is independent of the location of the point where the derivative is tobe evaluated. The formal expression for the calculation of FV weights usingall natural neighbours is given in Appendix A.

2.4.3 Finite Volume Weights Using Three Neighbours

This method is identical to the FV method described above except the num-ber of neighbours used for calculating the derivative is limited to three. Asshown by Magnier et al. (1992; 1994) it is sufficient to use only three pointsfor calculating spatial derivatives on a 2-D grid or four points in the 3-Dcase. In general, the authors call an N-dimensional grid using N + 1 pointsto compute derivatives a minimal grid.While the previous methods of determining natural neighbours on irregular

2.4. DIFFERENCE WEIGHTS ON ARBITRARY GRIDS 21

grids used the Delaunay-triangulation after inserting a secondary grid point,the problem is now, that it is not obvious which three of the natural neigh-bours will lead to the most accurate result. Therefore another method offinding the best three natural neighbours is used. As mentioned before, aprimary grid is initialized, on which the velocities are defined. The Delaunayalgorithm then produces a triangular primary mesh. If a secondary point isinserted into the existing primary grid, this point will be located inside ofone Delaunay triangle of the primary grid. To calculate the spatial deriva-tive at that secondary point using only three neighbours, the best choiceswill be the three primary grid points that are forming the triangle in whichthe secondary point is located (Figure 2.7).A major problem when operating with arbitrary grids is to know which trian-gle contains a given point. For an increasing number of primary grid points itis becoming increasingly inefficient to search through all triangles. Thereforewe use the very efficient walking triangle algorithm as referred to in previouswork (Sambridge et al. 1995; Lawson 1977), which quickly finds the trian-gle containing any given secondary point. Once the list of the three bestneighbours is determined, the FV method can be used for calculating thederivative. But the summation is limited to the three best neighbours.

2.4.4 Reference Cases

The accuracy of the three different irregular grid methods is compared withtwo different regular grid techniques, a rectangular standard FD grid and aregular hexagonal grid (Figure 2.3a and b). As mentioned above, the compar-ison of different grids can only be done by using the average density of nodesas a measure to compare the accuracy of operators on equally spaced regularand irregular grids. Different methods are compared by using the (average)number of grid points per wavelength λ, with which a two-dimensional testfunction is sampled.

FD on Regular Grids

The FD grid used for the comparison consists of square shaped grid cellswhich leads to equal grid spacing in the x- and y-dimensions. We also usea staggered grid scheme (Figure 1.1). The FD operator is second order in∆x. This implies, that the spatial derivatives on the rectangular grid arecalculated by using only two neighbouring grid points in each direction. Asthe numerical derivative is defined between the two function values, numericaland analytical results of the same location are compared in this case. Thederivatives are simply computed by the differential quotients (see Chapter 1)


∂xf(x0) ≈f2 − f1

∆x(2.10)

∂zf(x0) ≈f4 − f3

∆x(2.11)

where ∆x is the grid spacing and fi(i = 1, ..., 4) are the surrounding functionvalues (Figure 2.9a).

f

f

f

f

4

3

21 ox

f

ff1 2

3

px

zp

xo

x

z

a b

∆x x∆

Figure 2.9: (a) FD grid cell (staggered scheme) to compute ∂x,zf(x0) at thecenter of the cell. Note that the numerical derivatives are defined at x0.(b) Triangular grid cell (staggered scheme) of a hexagonal grid to compute∂x,zf(x0) at the center of the cell. Note that here the numerical value of∂xf(x0) is defined at (i.e. interpolated to) point px and ∂zf(x0) at point pz.The analytical values of ∂x,zf(x0) are evaluated at x0, the location of the gridpoint on the secondary grid.

Hexagonal Minimal Grids

The hexagonal grid is equal to a perfectly shaped triangular grid. Thereforethe derivative operator for the staggered scheme uses the three nodes formingthe primary grid cell (an equilateral triangle). In this case the weights canbe calculated explicitly as shown by Magnier et al. (1994). The derivativesare computed by

∂xf(x0) ≈ f2 − f1

∆x(2.12)

∂zf(x0) ≈ f1 + f2 − 2f3

∆x√3

(2.13)

where ∆x is the side length of the triangle and fi(i = 1, 2, 3) are the sur-rounding function values (Figure 2.9b). In the sense of an interpolation the

2.5. ACCURACY OF SPACE DERIVATIVES 23

derivative values are defined in between the corresponding function values,i.e. ∂xf(x0) is defined at location px and ∂zf(x0) at pz. The analytical deriva-tive value of the test function is defined at the triangle center x0, which isthe equivalent location of the grid point on the secondary grid. Thereforeanalytical and numerical derivatives are located at different points in con-trast to rectangular FD schemes.The performance of the hexagonal grid operator could of course be improvedusing additional points and interpolate the derivative value to the corre-sponding location. However, this would impair the simplicity of the methodand is not even necessary due to an interesting effect of error compensation.In other words, when implementing the hexagonal grid operator in a simula-tion algorithm, it is applied twice in each time step. Derivatives of velocitiesare calculated by stresses and vice versa. Therefore, the geometry of theoperator differs for the two cases. This in turn leads to a compensation ofthe errors. A detailed description of this behaviour will be given in Chapter3.

2.5 Accuracy of Space Derivatives

On irregular grids an analytical analysis of the accuracy of derivatives isnot possible. To determine the accuracy of the derivative operators a two-dimensional sinusoidal test function f(x) = sin(πx)sin(πz) is used (Figure2.10). The function is defined in the interval −1 ≤ x ≤ 1 and −1 ≤ z ≤ 1and remains unchanged for all test grids. If this test function is discretizedby using a staggered grid as outlined above (Figure 2.7, discrete functionvalues are defined on the primary (i.e. velocity) nodes. At the secondary

Figure 2.10: Two-dimensional sinusoidal function to test operator accuracyby comparing analytical derivatives with numerically calculated derivative val-ues.


(i.e. stress) nodes of the staggered grid space derivatives can numerically becalculated by applying the NND, FV3 or FVN method. Derivative values ofthe test function can also be analytically computed at all secondary nodes.To compare numerical and analytical results the difference of numerical andanalytical derivative values is determined at each secondary grid point anda mean absolute error ε is calculated. Formally the mean absolute error isgiven by

ε =1

ns

ns∑

i=1

|˜∂fx,z − ∂fx,z

π| (2.14)

where ns is the number of secondary points, at which numerical derivativevalues ( ˜∂fx,z) and analytical derivative values (∂fx,z) are compared. Thedivision by π normalizes the error with respect to the maximum analyticalderivative value.Sampling the test function with grids of varying node densities ranging from 6to 100 grid points per wavelength λ and calculating the corresponding meanabsolute error leads to characteristic distinct error-curves (Figure 2.11).To avoid effects due to the edges of the grid the summation is limited tosecondary grid points with coordinates in the interval −0.7 ≤ x ≤ 0.7 and−0.7 ≤ z ≤ 0.7.After computing these error values for nine different grids with grid-cells ofvarying average triangle qualities (1.000 ≥ q ≥ 0.842), a series of nine graphs(Figure 2.11) is obtained. As mentioned before, the average grid quality inrealistic applications (e.g. a cylindrical problem as shown in Figure 2.1b)was observed to be around q ≈ 0.93.Error-curves obtained by using a rectangular standard FD grid and a hexag-onal grid (Magnier-grid) are also shown as references. Note that even for thehexagonal grid consisting of equilateral triangles the accuracy of the z- andespecially of the x-derivatives is considerably worse than for the rectangularFD grid. Increasing distortion of the hexagonal grid in general leads to de-creasing accuracy of the space derivatives. Particularly the accuracy of thez-derivative is strongly affected by grid irregularity, whereas the accuracy ofthe x-derivative does not change remarkably. The diagrams also show thatthe difference in the accuracy of x- and z-derivatives (as clearly apparent forthe hexagonal case) decreases for increasing grid irregularity, i.e. for highlyirregular grids there is no difference between the accuracy of derivatives inx- and z-direction.Comparing the three described techniques of calculating space derivatives,the NND method provides the most accurate results of the numerical deriva-tive values. For slightly distorted hexagonal grids this method even leadsto higher accuracy of the x-derivative (zoomed section in Figure 2.11) asMagnier’s method applied on perfect hexagonal grids, provided that the ori-entation of the triangles is chosen as shown in Figure 2.5. Surprisingly, the

2.5. ACCURACY OF SPACE DERIVATIVES 25

accuracy obtained by the FV3 method is comparable to NND, though onlythree neighbouring points are used in the computation of space derivatives.The FVN method is less precise than the other two, but does not show aconsiderable difference between the accuracy of x- and z-derivatives.A more direct comparison between the three techniques is given in Figure2.12, as error curves of different methods are shown in a single diagram.Accuracies of x- and z-derivatives are shown for four different grids with de-creasing average triangle qualities of q = 1.000, q = 0.988, q = 0.930 and

20 40 60 80 1000

2

4

6x−derivative

erro

r [%

]

20 40 60 80 1000

2

4

6

erro

r [%

]

20 40 60 80 1000

2

4

6

gridpoints / λ

erro

r [%

]

20 40 60 80 1000

2

4

6z−derivative

20 40 60 80 1000

2

4

6

20 40 60 80 1000

2

4

6

gridpoints / λ

32 34 36 38 40 32 34 36 381.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

FVN

FV3

NNDq=0.988

q=0.842

q=0.842

q=0.842

q=0.842

q=0.842

q=0.842

FD FDMagnierMagnier

Magnier

Figure 2.11: Mean absolute errors ε (as percentage of the maximum valueπ) of the numerically computed x- and z-derivatives of the test function areshown for the three different methods. Graphs are drawn for nine differentgrid qualities ranging from 0.842 ≤ q ≤ 1.000. The accuracies obtained bythe FD grid and the perfect hexagonal grid are also given as references (thicklines).


20 40 60 80 1000

1

2

3

4

5

6

erro

r [%

]

q = 1.000

20 40 60 80 1000

1

2

3

4

5

6q = 0.988

20 40 60 80 1000

1

2

3

4

5

6q = 0.930

erro

r [%

]

gridpoints / λ20 40 60 80 100

0

1

2

3

4

5

6q = 0.842

gridpoints / λ

20 40 60 80 1000

1

2

3

4

5

6

erro

r [%

]

q = 1.000

20 40 60 80 1000

1

2

3

4

5

6q = 0.988

20 40 60 80 1000

1

2

3

4

5

6q = 0.930

erro

r [%

]

gridpoints / λ20 40 60 80 100

0

1

2

3

4

5

6

gridpoints / λ

q = 0.842

a

b

Magnier, NND,FV3,FVN

FD FD

FD FD

FD

FD FD

FD

Magnier,NND,FV3,FVN

NNDFV3

FVN

NNDFV3

FVN

NNDFV3

FVN

FVN

FVN

NNDFV3

NNDFV3

NNDFV3

FVN

Figure 2.12: Errors of the numerically computed x-derivatives (a) and z-derivatives (b) of the test function are shown for four different grids withaverage triangle qualities q = 1.000, q = 0.988, q = 0.930 and q = 0.842.NND clearly is more accurate than FV3 or FVN. Note the decreasing accu-racy with decreasing grid quality q. The accuracies obtained by the FD gridand the perfect hexagonal grid are also given as references.

2.6. DISCUSSION 27

q = 0.842.For the hexagonal grid (q = 1.000) all methods converge to a single line andprovide the same results as Magnier’s method. Again accuracy of the spacederivatives is decreasing as the average triangle quality is decreasing. In theseplots it is clearly shown, that NND always provides more accurate results forcomputing numerical derivatives than the other methods do, independent ofgrid irregularity. It is also remarkable, that there is not very much decreasein accuracy when using the FV3 method with respect to NND. Again notethe considerable difference between the accuracies obtained by using the FDgrid and the hexagonal Magnier-grid. The influence of grid irregularity onthe x-derivative is comparatively small.

2.6 Discussion

The irregular grids used in this work can be considered to be completelyarbitrary, in a sense that the grid generation is based on a random process.This means, we do not deliberately avoid the generation of triangles withvery poor quality. In realistic problems of cylindrical models (Figure 2.1b)the average triangle quality is q ≈ 0.93, and the grid cells have very similarshapes throughout the entire model. This means, that triangles with verypoor quality can be avoided, when using irregular grids in realistic applica-tions. In contrast, the grids obtained by randomly perturbating hexagonalgrids are in a way worst case grids. This is important for further investiga-tions and tests on arbitrary grids, as it means, that the results are obtainedfor the worst case. A deliberate suppression of triangles with low qualitywould lead to a much better performance of irregular grids, though the gridquality (i.e. the average triangle quality) might be the same.The investigation of derivative operators on arbitrary triangular grids hasshown, that the accuracy of space derivatives is dependent on the averagequality of the grid cells. A remarkable result is, that even for undistortedhexagonal grids consisting of equilateral triangles (q = 1) the derivatives areconsiderably less accurate than for rectangular FD grids. This effect canbe explained by the fact that numerical and analytical derivatives in thestaggered hexagonal scheme are not defined at (or interpolated to) the samelocation as it is the case in the staggered FD scheme (Figure 2.9). The effectof deviating locations of derivative values does not appear for plane wavestraveling along the x- or z-axis, i.e. the z-derivative of a plane wave travelingin x-direction is constant in z-direction and the x-derivative of a plane wavetraveling in z-direction is constant in x-direction. With respect to hexagonalgrids, the introduced derivative operators on arbitrary grids yield comparableaccuracy in x-direction and are in contrast to the z-direction relatively lessaffected by increasing grid irregularity. Whereas the accuracy of z-derivatives


is high for hexagonal grids (assuming a triangle orientation as shown in Fig-ure 2.5), the accuracy of the x-derivative is rather low. Therefore additionaldistortion of a hexagonal grid, does rather affect the precise z-derivativesthan the less accurate x-derivatives. For highly irregular grids without anystructure the difference of accuracies of x- and z-derivatives finally vanishes.Eventually, when using explicit local differential operators on arbitrary gridsas introduced here the NND method provides the most accurate results as thelocation of the secondary grid point with respect to the surrounding neigh-bours is considered in the computation algorithm. Contrary, the FVN andthe FV3 method both assume the derivative to be constant within the entireFV-cell, which leads to low accuracy especially when using large cells of theFVN method, i.e. cells, that are defined by more and therefore more distantneighbours. The FV3 method is very similar to the approach of Magnier etal. as it also uses a minimal number of neighbours defining a plane of linearinterpolation, with the extension that the triangle can be arbitrary shapedand oriented. As the FV3 algorithm only uses three neighbours to computespace derivatives on the staggered grid it is very fast, which offers the possi-bility to use denser grids and potentially achieve higher accuracy than NND,without considerably increasing computation time.We have shown, that different local operators can be designed to computespatial derivatives on an arbitrary grid. Furthermore we have clarified, howthe grid irregularity affects the accuracy of the differential operators. Theoperator based on the natural-neighbour-method (NND) provides most accu-rate results independent of grid irregularity. Problems of difference operatorson hexagonal grids as introduced in previous work (Magnier et al. 1994) havebeen pointed out and all results were compared to standard finite-differencemethods on rectangular grids with equivalent node densities.Even though the operators working on hexagonal and irregular grids showconsiderable differences of numerically and analytically calculated deriva-tives, the following chapter will indicate, that these operators are capableto simulate seismic wave propagation. As we are using the velocity-stressformulation of the wave equation, the operators are applied twice within onetime step, which means that velocities are calculated from stresses and viceversa. Using this method the errors obtained from the computation in onedirection (stress −→ velocity) are in a way compensated by the computa-tion in the other direction (velocity −→ stress) and therefore provide resultscomparable to FD.However, the important property of all operators described is, that they arelocal in the sense that they use only information of a function in the nearestneighbourhood of the point where the derivative is to be calculated. There-fore, a potential possibility of these local operators is their application inlarge scale simulation problems as they are well suited for parallelization.

Chapter 3

Simulation of Wave

Propagation on Arbitrary Grids

In this chapter the different derivative operators are implemented in a nu-merical simulation algorithm for acoustic and elastic wave propagation. Ahomogeneous, isotropic model is chosen. The different methods of comput-ing space derivatives as discussed in Chapter 2 are tested and their effecton the accuracy of synthetic seismograms is investigated. The chapter fo-cuses on the analysis of the synthetic seismograms, especially with respect tonumerical anisotropy, grid perturbation and the influence of signal frequency.

3.1 Initialization of Simulation Parameters

3.1.1 Geometrical Aspects

When comparing methods, that are applied on regular, quadratic finite-difference (FD) grids, regular hexagonal grids and completely irregular grids,the first problem is to cover the same area with different grids of equal av-erage density of nodes (see Chapter 2). Therefore, three cases do appear(Table 3.1).(1) In the finite-difference case a grid size of 400 × 400 points is chosen tosample a quadratic area. The corresponding grid spacing is set to b = 5m(Figure 2.3b) leading to a model size of 2000m× 2000m (Figure 3.1a).(2) In the hexagonal case the equivalent average node density d leads to agrid size of 372 × 430 points, which approximately covers the same area1.

Therefore, a grid spacing of a = 5m ·√

23

√3 ≈ 5, 37m is obtained for the

hexagonal case.

1The exact coverage of the same area is not possible due to the geometry of the partic-ular unit grid cells. However, this does not affect the test adversely. Only the equivalentgrid spacing, i.e. the relation of a and b (see Chapter 2) is important.

29

30 CHAPTER 3. SIMULATION OF WAVE PROPAGATION

(3) The irregular grids are obtained by randomly perturbating a hexagonalgrid and also consist of 372 × 430 points. However, the grid points formingthe grid boundary are not affected by the perturbation to keep the outershape unchanged.For all simulations the source is located in the center of the model and anarray of 36 receivers - one every 10◦ - is forming a circle around the source(Figure 3.1a). The distance between source and receivers is set to 500m.A ricker wavelet (Figure 3.1b) is used as the source time function and ismultiplied by a spatial Gauss function (see section 3.1.3) to obtain a seismicsource. The formal expression of the source function is given by

S(x, z, t) = M0 e− 1

α2[(x−x0)2+(z−z0)2] · [−2 1

T 2(t− t0) e

− 1

T2(t−t0)2 ] , (3.1)

where M0 is a scaling factor and x0, z0 and t0 are the source coordinates inspace and time. The value of α determines the width of the spatial Gaussfunction and T represents the period of the source time function.

v = 4000 m/s

v = 2310 m/ss

p

2000 m

2000 m500 m

0 100 200 300 400−1

0

1

1

10

19

28Receivers

Source

Nor

mal

ized

Am

plitu

de

Time (msec)a b

Figure 3.1: (a) The geometrical setup for the numerical tests with 36 receivers(numbered anticlockwise) surrounding a source in the center of the model. (b)The source time function (here shown with a dominant frequency of 20Hz) isinput with a spatial Gaussian (see section 3.1.3) at each time step to producethe seismic source in time and space.

3.1.2 Stability of Simulation Algorithms

To test the fundamental properties of the different explicit operators in asimulation algorithm a homogeneous, isotropic medium is chosen with a P-wave velocity of vp = 4000m

sand an S-wave velocity of vs = 2310m

s. The

density is set to ρ = 2500 kgm3 (Table 3.1).

3.1. INITIALIZATION OF SIMULATION PARAMETERS 31

A major problem of numerical simulations of is to keep the simulation al-gorithm stable. As shown in previous work (e.g. Bamberger et al., 1980;Virieux, 1986), the stability condition can formally be written as

vmax∆t

∆xmin

< γ , (3.2)

where vmax denotes the maximum wave speed, ∆xmin the minimal grid spac-ing and ∆t the allowed time step. The value of γ depends on the geometryand the number of dimensions of the used grid. For the two-dimensional FDproblem with a common grid spacing ∆x in x- and z-direction in an isotropic,homogeneous model the stability criterion (Virieux, 1986) is given by

vp∆tFD

∆xFD

<1√2≈ 0.707 , (3.3)

where the P-wave velocity vp is the maximum wave speed. In the hexagonalcase the stability criterion (Bamberger et al., 1980) is slightly modified andgiven by

vp∆thex∆xhex

<

√3

2≈ 0.866 . (3.4)

Note that the stability criterion required for the hexagonal grid is greaterthan that required for the quadratic FD grid. As we are using grids withequal average node density d (see Chapter 2), the relation of grid spacing isgiven by

∆xhex∆xFD

=a

b=

√

2

3

√3 . (3.5)

Therefore, the relation between the time increments of stable simulations canbe determined by

∆tFD < vp · b ·1√2

(3.6)

∆thex < vp · a ·√3

2= vp · b ·

√

2

3

√3 ·√3

2

< vp · b ·√

1

2

√3 (3.7)

⇒ ∆thex∆tFD

=

√√3 ≈ 1.32 . (3.8)

This means, that in the case of equal node densities the time increment ∆thexcan be about 1.3 times larger than the corresponding time increment ∆tFD.


25 30 35 40 45 50 55 60 65 70 750

1

2

3

4

5

6

gridpoints / λ

erro

r [%

]

99%

90%

80%

70%

diffe

renc

e

Figure 3.2: The difference between analytical and numerical results dependson the time increments ∆t determining the percentage of the stability crite-rion gamma. The graphs show, that the difference is decreasing, the closerthe stability criterion is approached.

Therefore, in the hexagonal case less time steps are necessary to achieve thesame total simulation time.Similar to the grid spacing ∆x the time increment ∆t influences the accuracyof simulation results (Figure 3.2). Different time increments are chosen toobtain 70%, 80%, 90% and 99% of the FD stability criterion 1√

2. The graphs

show, that the difference between simulation results and analytical solutions2

tends to become minimal for the maximum allowed time step. Analogoustests on hexagonal and irregular grids lead to similar results. However, adetailed explanation would go beyond the scope of this work. Generally, re-sults of a simulation are getting more accurate, the more we approach thestability criterion. Therefore, it is necessary to determine the maximum timeincrement ∆t allowed by the stability criterion (equations 3.3, 3.4) for a givengrid spacing ∆x and maximum wave speed vpmax

to achieve the best results.When working on arbitrary grids, the stability criterion for hexagonal gridscan only be an estimation, but no exact definition a stability criterion doesexist. In this work, the smallest diameter of all triangular grid cells is usedas ∆x in equation 3.4 to determine the maximum time increment. Four dif-ferent irregular grids are used for the following simulations (Table 3.2). Thismeans, that each grid has to be searched for the smallest triangle diameterto evaluate the corresponding maximum time increment (Table 3.2). The

2A detailed description of computing the difference of analytical and numerical simu-lation results will be given in section 3.2.2


FD hexagonal irregular

number of grid points N 400× 400 372× 430 372× 430average node density d [m−2] 0.0402 0.0402 0.0402grid spacing ∆x [m] 5.00 5.37 variabletime increment ∆t [ms] 0.875 1.150 variablenumber of time steps nt 400 304 variableP-wave velocity vp [

ms] 4000 4000 4000

S-wave velocity vs [ms] 2310 2310 2310

density ρ [ kgm3 ] 2500 2500 2500

Table 3.1: Overview of the simulation parameters.

Grid 1 Grid 2 Grid 3 Grid 4perturbation p [%] 0 10 20 30average triangle quality q 1.00 0.98 0.95 0.90minimum triangle quality qmin 1.00 0.92 0.70 0.38minimum grid spacing ∆xmin [m] 4.65 3.30 2.00 0.90time increment ∆t [ms] 1.150 0.817 0.495 0.223number of time steps nt 304 428 707 1569

Table 3.2: Simulation parameters for the different irregular grids.

stronger a grid is perturbed, the smaller the worst triangle will be, which inturn also requires smaller time increments.In the following simulations the time increments ∆t are calculated for eachgrid to obtain a stability factor of 99% of the allowed maximum value. Thetotal simulation duration is set to 350ms. This prevents reflections from themodel boundaries to be recorded by the receivers3.To investigate the frequency dependence of the different operators with re-spect to grid irregularity or numerical anisotropy - caused by grid symmetry- the dominant signal period T ranges from T = 0.02s to T = 0.1s, whichequals a frequency band of 10Hz to 50Hz (e.g. Figure 3.1b).No matter the spatial dimensions of a model, the important parameter toconsider is the number of grid points per wavelength. In the following simu-lations this value ranges from 16 to 75 points.

3Boundary effects are not considered in this test simulations.


3.1.3 Source and Receiver Positioning

When using a seismic simulation algorithm for generating synthetic seismo-grams to analyse subsurface models, it is important to provide a flexiblecomputer code, which allows us to place the seismic source and the receiversat arbitrary locations. This means, that the source and receivers should notbe restricted to the locations of particular grid nodes, but can lie somewherein between, although the values of velocity and stress are defined only ondiscrete grid points. Therefore it is necessary to find interpolation schemes,which consider the arbitrary position of the source and the receivers withrespect to the surrounding grid points.Commonly a Gauss function in space is used to solve this problem. Insteadof using a impulsive source defined at a single node, the source is defined bya two-dimensional Gauss function (Figure 3.3a). The maximum of the Gaus-sian represents the source location but must not lie on a particular node.However, the values of the surrounding nodes are initialized by the Gaussian(Figure 3.3b). This method is equivalent to defining several point sources

Figure 3.3: A seismic source is represented by a two-dimensional Gauss func-tion. Note that the maximum of the Gaussian does not have to be located ata particular grid point. The values initialized at the surrounding grid pointscan be considered as several impulsive sources, whose amplitudes are deter-mined by the Gaussian. The resulting signal represents the superposition ofthe signals of these discrete sources.

at discrete grid nodes scaled by the Gaussian. The generated wave is thesuperposition of waves generated at each of these grid points.Another problem arising only on arbitrary grids can be overcome by usingthis technique. As the initialization of a source at a single grid point ispossible for regular grids, it leads to severe amplitude effects on irregulargrids. The amplitude of the outward propagating wave strongly depends onthe local grid irregularity at the particular source location. If the local gridirregularity is high at the source location, the adjacent triangles are stronglydistorted and have rather poor quality. Therefore, large errors in the cal-culations of derivatives (see Chapter 2) already occur within the first time


ba

x4

x1

Rx

x3

x2

x

x

34

12

hx hz

∆ x

x5

x1

x3

2x

x4

Rx

Figure 3.4: (a) Interpolation scheme for receivers locations using naturalneighbours on an triangular irregular grid. (b) Interpolation scheme for re-ceiver locations using a linear interpolation method on a regular, quadraticfinite-difference grid.

step and influence the amplitude of the generated wave field. Using the spa-tial Gauss function as described above, leads to an averaging (smoothing)of triangle qualities over a broad area and therefore prevents the observedamplitude errors. Usually a Gauss functions half width of 2∆x to 5∆x issufficient.Similar to the source location problem, it is required to put receivers atarbitrary locations within the model without being restricted to grid pointlocations. Again the problem is, that the wave field - recorded by the re-ceivers - is only defined on discrete grid points, whereas the receivers canlie somewhere in between. Therefore, an interpolation of the wave field to aparticular receiver location is necessary. In the simulation algorithms usedin this work, two interpolation schemes are applied.

• In the finite-difference algorithm a linear interpolation for two dimen-sions is used (Figure 3.4b). The interpolated value at a particularreceiver location is calculated by

f(xR) ≈ f(x12) +f(x34)− f(x12)

∆xhz with (3.9)

f(x12) ≈ f(x1) +f(x1)− f(x2)

∆xhx (3.10)

f(x34) ≈ f(x4) +f(x4)− f(x3)

∆xhx , (3.11)

where f(xi) are the function values on the discrete grid points xi, ∆x is


the grid spacing and hx and hz determine the receiver location insidea unit cell (Figure 3.4b).

• In the algorithm operating on hexagonal or irregular grids the naturalneighbour coordinates - as introduced by Sambridge et al (1995) - areused (Figure 3.4a). The interpolated value is given by

f(xR) ≈N

∑

j=1

wjf(xj) (3.12)

where N is the number of neighbouring points and wj are the interpola-tion weights. The weights are calculated by considering the overlappingareas of Voronoi cells. For further detail see Appendix A.

Using the two discussed interpolation methods, there are no restrictions tosource or receiver positioning. Both can be put to any location within themedium, no matter if discretized by a regular or irregular grid.

3.2 Acoustic Wave Propagation

In the following sections results of different numerical simulations will be dis-cussed and compared to analytical solutions. A wave field is generated by asource in the center of an acoustic medium and recorded by 36 receivers (Fig-ure 3.1a). The resulting 36 synthetic seismograms are obtained by differentmethods:

• The finite-difference operator (Figure 2.9a) is applied on a regular,quadratic grid (Figure 2.3b).

• The minimal grid operator (Figure 2.9b) is applied on a hexagonal grid(Figure 2.3a).

• The NND operator (Figure A.1) is applied on different irregular grids(Figure 2.5).

• The FVN operator (Figure A.2) is applied on different irregular grids(Figure 2.5).

• The FV3 operator (Figure A.3) is applied on different irregular grids(Figure 2.5).

We simulate acoustic wave propagation in a homogeneous medium for eachoperator on four different irregular grids characterized in Table 3.2 leading to

3.2. ACOUSTIC WAVE PROPAGATION 37

12 simulations. These results are compared to the results of the two referencegrids with equivalent node density. Appendix C includes several snapshots ofthe acoustic wave field obtained on the quadratic FD grid displaying the z-component of the velocity vector ~v. Similar results are obtained, when usingan hexagonal or irregular grid. To uncover the differences of the introducedmethods it is necessary to analyse the recorded seismograms in detail.

3.2.1 Influence of Grid Symmetry

Considering the results of Chapter 2, an interesting question poses itself:How does the difference of operator accuracy in x- and z-direction will affect

100 125 150 175 200

0

50

100

150

200

250

300

350

Pressure Amplitude

Time (msec)

Azim

uth

(deg

)

hig

h a

ccu

racy

alo

ng

gri

d d

iag

on

al

Figure 3.5: Synthetic seismograms showing the amplitude of pressure wavesin an acoustic medium. Simulation parameters are given in Table 3.1. Theseismograms are plotted versus azimuth. Note, that the dispersion is muchstronger for waves traveling along the coordinate axes than for waves travelingalong the diagonal axes. Therefore, a quadratic grid is called anisotropic.

the wave propagation. In other words, it is important to analyse, how thegeometry or orientation of the derivative operators influence the accuracy ofthe synthetic seismograms.In general, the geometry of a grid and their corresponding operators are caus-ing numerical anisotropy. For example, regular FD grids produce numericalanisotropy in a homogeneous, isotropic medium as shown in previous work(e.g. Igel, 1993). This is due to the denser spatial sampling of a wave trav-eling along the diagonal of the quadratic grid compared to the sampling of


an equivalent wave traveling along one of the coordinate axes4. Therefore,the accuracy of synthetic seismograms with respect to the analytical solutionis high along diagonal axes and low along the coordinate axes. This can beshown very impressively by using simulation parameters, that lead to slightdispersion of the propagating wave (Figure 3.5) and plot all 36 synthetic seis-mograms obtained by a simulation on a quadratic finite-difference grid versusazimuth. It can be seen very clearly, how the grid symmetry influences theaccuracy of the seismograms.The same simulation parameters were given for the simulation on a hexag-onal grid with equal node density. The derivative operator introduced byMagnier (1992) is used and all 36 seismograms are plotted versus azimuth(Figure 3.6). All seismograms show quite strong dispersion, but no effect ofthe grid symmetry can be detected. Therefore, hexagonal grids are calledisotropic grids, as they do not produce numerical anisotropy. This is advan-tageous, when modeling physical anisotropy.However, the Figures only show qualitatively, how the geometry of a gridand the corresponding operators may influence the accuracy of seismograms.Therefore, it is desirable to quantitatively analyse the accuracy of syntheticseismograms.

3.2.2 Misfit Energy of Synthetic Seismograms

To quantify the accuracy of synthetic seismograms, they are compared toanalytically calculated solutions 5. The error of such numerically calculatedseismogram will often be called the misfit of the numerical and analyticalseismogram in this work and is defined by the equation

ε = (nt

∑

j=1

(sj − sj)2) / (

nt∑

j=1

s2j) (3.13)

where nt is the number of time samples of a seismogram, sj are the discretenumerical amplitude values and sj the analytical amplitude values. The errorε can also be considered as the relative misfit energy of a numerical seismo-gram with respect to its analytical solution. In other words, the integral ofthe squared difference seismogram is divided by the integral of the squaredanalytical seismogram. In that way, the accuracy of a synthetic seismogramcan be measured quantitatively (Figure 3.7).

4If the grid spacing of a quadratic FD grid is ∆x = 1 along the coordinate axes, theeffective grid spacing along the diagonal axes is ∆xdia =

1

2

√2 ≈ 0.70.

5A detailed description of the calculation of analytical solutions is given in AppendixB.


100 125 150 175 200

0

50

100

150

200

250

300

350

Pressure Amplitude

Time (msec)

Azim

uth

(deg

)

sam

e ac

cura

cy in

all

dir

ecti

on

s

Figure 3.6: Synthetic seismograms showing the amplitude of pressure wavesin an acoustic medium. Simulation parameters are chosen equivalently to theFD case (see Figure 3.5 and Table 3.1). The seismograms are plotted versusazimuth. Note, that the dispersion is the same for all waves and does notdepend on the direction of propagation. Therefore hexagonal grids are calledisotropic.

3.2.3 Spatial Variation of Seismogram Accuracy

An acoustic wave propagation is simulated using the different methods - FD,hexagonal and irregular - as discussed above. The misfit of each of the 36seismograms can be calculated and plotted in a polar coordinate system (Fig-ure 3.8). The distance between the origin of the polar coordinate system andthe graph represents the size of the error ε and is plotted versus azimuth.A strong directional dependency can be seen in the finite-difference case,whereas the hexagonal grid provides perfect isotropic results. The seismo-grams obtained by the irregular grid show a rather unstructured distributionof errors. They do not show any systematic numerical anisotropy, as thewave field propagates through many different triangles in all directions andtherefore an averaging process causes the almost isotropic error distribution.It is notable, that the errors of the hexagonal method lie somewhere in be-tween the maximum and the minimum errors of the finite-difference methodfor all frequencies. However, the corresponding errors of the irregular gridare larger than in the hexagonal or quadratic finite-difference case. For un-realistically high frequencies the error distribution of the hexagonal grid alsoshows the grid symmetry (data not shown), but the effect is still very weak


0 50 100 150 200 250 300 350

difference misfit : 2.50%

Time (msec)

Pre

ssur

e Amp

litude

Figure 3.7: A numerical seismogram (solid) and the corresponding analyticalsolution (dashed) are shown. The difference of the two seismograms is givenas an additional trace below. Dividing the integral of the squared differenceseismogram by the integral of the squared analytical seismogram leads to therelative misfit energy. The corresponding simulation parameters are given inTable 3.1.

compared to the finite-difference case.

3.2.4 Accuracy of Synthetic Seismograms

The relative misfit ε not only varies with azimuth, but also depends on thegrid irregularity and the signal frequency. Examples of synthetic seismo-grams are given (Figure 3.9) to demonstrate, how grid irregularity influencesthe accuracy of the seismograms. With increasing grid perturbation - i.e. de-creasing average triangle quality - the seismograms show increasing artifacts.After the wave has passed the receiver, a considerable level of numerical noisecaused by the grid (operator) geometry can be recognized. Phase errors alsoincrease with increasing grid irregularity.The bare optical comparison of the seismograms clearly shows, that the NNDmethod introduced in Chapter 2 performs better than the other two methods,whereas the FVN method provides the least accurate results. An interest-ing aspect is, that the FV3 method using only the three best neighbours ofall natural neighbours is becoming unstable for higher grid perturbations.The reason can be found in investigating the minimum triangle qualities ofthe distorted grid. (Figure 2.6). As mentioned before, the minimum tri-angle diameter of all unit grid cells has to be found to determine the timeincrement ∆t with the use of the stability criterion for the hexagonal grid


b

c

a

0.02

0.04

0.06

30

210

60

240

90

270

120

300

150

330

180 0

0.02

0.04

0.06

0.08

0.1

30

210

60

240

90

270

120

300

150

330

180 0

2.5

5

7.5

30

210

60

240

90

270

120

300

150

330

180 0

6.5

13

30

210

60

240

90

270

120

300

150

330

180 0

0.02

0.04

0.06

30

210

60

240

90

270

120

300

150

330

180 0

2.5

5

7.5

30

210

60

240

90

270

120

300

150

330

180 0

253545

253545

253545

5565

5565

5565

un

structu

redh

exago

nal

FD

zoom

zoom

error in % error in %

zoom

zoom

error in %7.5

5.0

2.5

0.06

0.04

0.02

error in %

13.0

6.5

1.00

0.02

0.04

0.08

0.06

zoom

zoom

error in %7.5

5.0

2.5

0.06

0.04

0.02

error in %

Figure 3.8: The relative misfit of synthetic pressure seismograms obtainedon different grids is plotted versus azimuth. (a) FD grid, (b) hexagonal gridand (c) irregular grid with average triangle quality of q = 0.98. The dis-tance from a point on the graph to the origin represents the size of the errorε. The zoomed sections show the decrease of the error with decreasing sig-nal frequency, whereas the directional influence of the grid symmetry staysalmost unchanged. On the left side, signal frequencies were chosen to ob-tain sampling rates of 25, 35 and 45 grid points per dominant wavelength.On the right side the sampling rates are 55 and 65 grid points per dominantwavelength.


0 50 100 150 200 250 300 350

misfit : 0.02%

Time (msec)

misfit : 0.02%

misfit : 0.03%

misfit : 0.08%

0 50 100 150 200 250 300 350

misfit : 5.68%

Time (msec)

misfit : 12.54%

misfit : 17.95%

misfit : 23.50%

0 50 100 150 200 250 300 350

misfit : 0.02%

Time (msec)

misfit : 0.02%

0 50 100 150 200 250 300 350

misfit : 5.68%

Time (msec)

misfit : 13.78%

0 50 100 150 200 250 300 350

misfit : 5.68%

Time (msec)

misfit : 19.63%

misfit : 38.14%

misfit : 67.30%

0 50 100 150 200 250 300 350

misfit : 0.17%

Time (msec)

misfit : 1.05%

misfit : 1.67%

misfit : 0.02%1.00

0.98

Grid P

erturbation [%]

10

20

30

0

0.90

0.95

1.00

0.98

Ave

rage

Tria

ngle

Qua

lity

Ave

rage

Tria

ngle

Qua

lity

0.95

1.00

0.90

Grid P

erturbation [%]

10

20

30

0

Grid P

erturbation [%]

10

20

30

0

0.95

0.90

Ave

rage

Tria

ngle

Qua

lity

λ25 grid points /

0.98

NND

FV3

FVN

λ65 grid points /

Figure 3.9: Examples of numerical seismograms (solid), the correspondinganalytical solution (dashed) and the resulting relative misfit ε. The threemethods are shown for two different sampling rates (25 and 65 grid pointsper dominant wavelength). For each method seismograms are displayed fordifferent grid qualities to show the decrease in accuracy with decreasing qual-ity. Simulations for grids with high perturbation become unstable when usingthe FV3 method.


(equation 3.4). However, the stability criterion can only be an estimationfor stable conditions on an irregular grid. In the case of the FV3 method itmight happen, that the triangle formed by the three best neighbours has avery low quality. Therefore, the stability criterion might not be fulfilled anymore. In other words, the spatial derivatives calculated at that particularnode will have extremely low accuracy, which affects the further calculationof the wave field in the following time steps.This problem does not appear for the other methods, as the calculation ofspace derivatives is not based on a single triangle, but on a Voronoi cell (Fig-ure A.1) or a finite-volume cell (Figure A.2), which generally include morethan three points and therefore more than one triangle. It is very unlikely,that all adjacent triangles included in a finite-volume cell have very low qual-ity.

To answer the question of the overall accuracy of seismograms, the relativemisfit ε is calculated for all 36 seismograms (Figure 3.8) and their averagevalue

εave =1

36

36∑

n=1

[(nt

∑

j=1

(sj,n − sj)2) / (

nt∑

j=1

s2j)] (3.14)

is determined for different frequencies (Figure 3.10). The computation of theaverage relative misfit εave is done for the two reference grids and four grids ofdifferent grid perturbations (Table 3.2). The misfit-diagrams of the syntheticseismograms (Figure 3.10) are similar to the error curves (Figure 2.11). Thefinite-difference technique again leads to the most accurate results, althoughthe seismograms recorded along the coordinate axes showed less accuracythan the seismograms obtained from the hexagonal grid. Compared to thenotable difference of operator accuracy on the test function (especially in x-direction) the hexagonal grid provides quite high accuracy in the simulation.For the undistorted hexagonal grid (q = 1.00) all three methods result in onesingle error curve equivalent to Magniers results of minimal grid operators.With decreasing average triangle quality the seismograms clearly show theexpected decrease in accuracy. It is obvious that the NND method performsmuch better than the other two methods. A direct comparison of the differentmethods (Figure 3.11) for an irregular grid with average triangle quality ofq = 0.98 (p = 10%) shows their accuracies with respect to the reference cases.Similar to the investigation of operator accuracy in Chapter 2 the NND andFV3 method are leading to comparable results, whereas the FVN methodperforms worse. Especially for grids of high irregularity, (p > 20%) thespatial sampling would have to be unrealistically high for the FVN methodto obtain seismograms with reasonable accuracy.


25 30 35 40 45 50 55 60 65 70 750

2

4

6

gridpoints / λ

err

or

[%]

25 30 35 40 45 50 55 60 65 70 750

2

4

6

err

or

[%]

25 30 35 40 45 50 55 60 65 70 750

2

4

6

err

or

[%]

FVN

FV3

NNDq=0.90

q=0.98

q=0.98

FDMagnier

FDMagnier

q=0.90

q=0.95

q=0.98

FDMagnier

q=0.95

Figure 3.10: The relation of seismogram accuracy to sampling rate is shownfor the three methods (NND,FV3,FVN) and different grid irregularities. Thesimulation results of the quadratic finite-difference grid and the hexagonalgrid are plotted as a reference. For the regular cases the finite-differencetechnique leads to higher accuracy than Magniers method. Compared to these,the irregular grid methods perform worse. Whereas NND leads to the mostaccurate seismograms and FVN to the least accurate, the FV3 method is inbetween but is only stable for slightly distorted grids.

3.3. ELASTIC WAVE PROPAGATION 45

25 30 35 40 45 50 55 60 65 70 750

1

2

3

4

5

6

gridpoints / λ

erro

r [%

]

q = 0.98

FDMagnier

FV3

NND

FVN

Figure 3.11: The accuracy of seismograms with respect to sampling rate isshown for all three methods (NND,FV3,FVN) applied on a grid of averagetriangle quality q = 0.98. The NND method clearly provides the highestaccuracy.

3.3 Elastic Wave Propagation

In this chapter the derivative operators are implemented in the same numer-ical algorithm as used before, with the only difference that the medium isnow elastic instead of acoustic. Surprisingly, numerical artifacts are observed,when propagating elastic waves through hexagonal and irregular grids, whichdid not appear in the acoustic case. The initialization of two different sources- an explosive source and a vertical force - and the corresponding generationof P- and S-waves are discussed. The origin of the numerical artifacts isexplained on the basis of grid symmetry and the corresponding geometry ofthe differential operators.

3.3.1 Separation of P- and S-Waves

In a homogeneous, isotropic, infinite, elastic medium the seismic wave fieldcan be divided into two parts describing the P-wave and the S-wave prop-agation. To investigate the accuracy of the simulated elastic wave field, itis useful to analyse the P- and S-waves separately. This separation can berealized by computing the divergence ∇·~v and the curl ∇×~v of the velocityfield. For two-dimensional problems the divergence of the seismic velocity


field is given by

∇ · ~v =∂vx∂x

+∂vz∂z

(3.15)

and the curl of the velocity field is given by

∇× ~v =

0∂vx∂z− ∂vz

∂x

0

. (3.16)

The P-wave and S-wave signals are represented by the values of ∇ · ~v and∇ × ~v respectively. Therefore, we directly record the divergence ∇ · ~v andthe curl ∇ × ~v of the velocity field instead of the components vx and vz ofthe velocity vector. The two equations imply, that the divergence and thecurl6 of the velocity field are also defined on discrete grid points, i.e. theirvalues are defined at the same location, where the spatial derivatives of thevelocities (stresses) are defined.The following sections will outline a major problem resulting from the dis-cretization of these values on hexagonal or irregular grids.

3.3.2 Elastic Sources

The system of elastodynamic equations as introduced in Chapter 1, can bewritten in a compact formulation (Aki & Richards, 1980)

ρ∂tvi = ∇j(σij +Mij) + fi (3.17)

σij = cijklεkl (3.18)

∂tεij =1

2(∂ivj + ∂jvi) (3.19)

where vi are the components of the velocity vector, σij and Mij are theelements of the stress and moment tensor, respectively, ρ is the mass density,εij are the elements of the deformation tensor, cijkl are elastic stiffnesses, andfi are volumetric forces, and the summation convention applies.The seismic source is implemented in the simulation algorithm by using aspatial Gauss function as shown in Figure 3.3. Initializing the values of thenormal stresses σxx and σzz on the secondary grid is representing an explosive

6In the two-dimensional case it is sufficient to consider the y-component ∂vx

∂z− ∂vz

∂xof

the curl only, as the x- and z-components are zero.


source. Applying the Gaussian on the primary grid, where the componentsof the velocity vector vx and vz are defined, is simulating an external forceacting in a specific direction. It is shown in previous work (Coutant et al.,1995) that seismic sources can be implemented in a staggered grid schemeby initializing the stress values or the velocity values.In the first operator tests an explosive source is used to obtain an isotropicradiation pattern similar to the acoustic case.

3.3.3 Simulation of an Explosive Source

The simulation setup is identical to the setup shown in Figures 3.1a. Usingan explosive source we would expect similar results to the acoustic case, asonly a circular P-wave is propagating through the medium. In AppendixC snapshots of the wave field are shown for different time steps displayingthe vertical component vz of the velocity vector. The resulting wave fieldobtained on a quadratic FD grid is equivalent to the acoustic case, whereasin the hexagonal or irregular case an S-wave with small amplitude can bedetected.On each grid - finite-difference, hexagonal and irregular - an explosive sourceis initialized on the secondary grid of normal stresses σxx and σzz. As shownbefore, the elastic wave field can be divided into a P-wave and an S-wave partby using the divergence and the curl of the wave field. In the FD simulation(Figure 3.12), all 36 receivers only recorded the P-wave (divergence), but noS-wave (curl).On the hexagonal grid the divergence of the velocity field seems to be similarto the one obtained by FD (Figure 3.12), but the curl of the velocity fieldis showing a considerable difference. An S-wave with systematically varyingamplitude and polarity appears at about 250ms. Theoretically, no S-waveshould be observed. As the S-wave polarity changes six times within the to-tal azimuth of 360◦, an obviously strong correlation consists between S-waveartifacts and the hexagonal grid symmetry.In the third case, shown in Figure 3.12, the simulation is based on the NNDmethod on an irregular grid with an average triangle quality of q = 0.90. Theirregular grid method still shows acceptable results for the divergence, butalso leads to artifacts in the curl of the velocity field. However, comparedto the hexagonal grid simulation, these errors do not show the same system-atic manner, but become more indistinct and also show smaller amplitudes.In fact, the z-component of the velocity field obtained from the irregulargrid shows very small S-wave artifacts compared to the results of the perfecthexagonal grid.It can be observed, that the amplitudes of the S-wave artifacts obtained bythe hexagonal and irregular grid methods depend on frequency and grid irreg-ularity, i.e. the amplitudes of the S-wave artifacts decrease with decreasing


frequency and also decrease for increasing grid irregularity.As mentioned before, the occurrence of the S-wave artifacts seems to coherewith the grid symmetry. Therefore, the origin of the artifacts is expected tobe due to the grid symmetry, i.e. to the geometry of the particular derivativeoperators.

Influence of Operator Symmetry

In this section a detailed description of the hexagonal operator is given toclarify the appearance of the observed numerical S-wave artifacts. In Chapter1 the schematic algorithm (Table 1.1) of propagating an elastic wave througha discretized medium is shown. Unlike regular FD grids using the sameoperator both ways (Figure 2.9a), hexagonal or irregular grids require twodifferent operators for calculating spatial derivatives of velocities or stresses.On the hexagonal grid we use operator A (Figure 3.13a) to compute velocitiesfrom stresses leading to

∂f

∂x≈ f2 − f1

∆x(3.20)

∂f

∂z≈ 2f3 − (f1 + f2)

∆x√3

(3.21)

and operator B (Figure 3.13b) to compute stresses from velocities leading to

∂f

∂x≈ f1 − f2

∆x(3.22)

∂f

∂z≈ (f1 + f2)− 2f3

∆x√3

. (3.23)

In the following, an explosive source is initialized on a hexagonal grid ata single node of the secondary (stress) grid. A section of the grid and thecorresponding elements (Figure 3.14) is demonstrating the situation after thefirst time step. Considering equation 3.17 the initialization of the explosivesource leads to

ρ∂vx∂t

=∂M

∂xand ρ

∂vz∂t

=∂M

∂z, (3.24)

with the moment tensor being M(x, z, t) = δ(x − x0, z − z0, t − t0). Thecorresponding coordinates of the central point of the hexagon - defined bypoints 4 to 9 - is (x0, z0).Assuming ∆x = 1 and M = 1 the values of ρ ∂vx

∂tand ρ∂vz

∂tare calculated at

points 1, 2, 3 in the first time step. The corresponding results are given invector form (ρ∂vx

∂t, ρ∂vz

∂t) for each of the three grid points (see Figure 3.14).


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Azim

uth

(deg

)Az

imut

h (d

eg)

Azim

uth

(deg

)

Time (msec)Time (msec)

Divergence

FD

hexagonal

irregular

Curl

Figure 3.12: The left column of seismograms show the divergence of the nu-merically calculated wave field recorded by the 36 receivers, whereas the rightcolumn shows the curl of the wave field. An explosive source generated byinitializing normal stresses is used. Note that the finite-difference techniquedoes not show any artifacts, whereas for the hexagonal and the irregular casean S-wave occurs.


x

z

f

f f

f2

3 1

3ba f

f2

1

Operator A Operator B

x

x

∆

∆

Figure 3.13: (a) The geometry of the operator used to calculate velocitiesfrom stresses and (b) the geometry of the operator used to calculate stressesfrom velocities are shown with respect to their orientation in the cartesiancoordinate system.

x

z 1

7 6

58

9 4

3 2M

σ ij

v i

32

,32

,

32

, 32

,

32

,32

, 32-2

3

3-1

3-1

32 -2

3

-23

32

32

(-1,1, (

((

((

( (

)

))

) )

)

)

( ,0 )

)

Figure 3.14: A section of a staggered hexagonal grid is shown. Initializing anexplosive source at the central stress node and calculating the derivatives ofthe field variables and the corresponding divergence and curl of the velocityfield for the first time step explains the numerical S-wave artifacts. Values ofthe divergence and curl are given in vector form (∇ · ~a , (∇× ~a)2) for stressnodes 4 to 9, whereas (ρ∂vx

∂t, ρ∂vz

∂t) are given at velocity nodes 1 to 3.


Furthermore, we can calculate the divergence and the curl7 of the velocityfield at locations 4 to 9 from the values just obtained at locations 1 to 3 usingequations (3.22) and (3.23). For clarity reasons we use the substitution

ax = ρ∂vx∂t

(3.25)

az = ρ∂vz∂t

(3.26)

leading to the following expressions for the divergence and the curl of thevelocity field

∇ · ~a =∂ax∂x

+∂az∂z

(3.27)

∇× ~a =

0∂ax∂z− ∂az

∂x

0

. (3.28)

The results of the divergence and the curl8 of the velocity field are againgiven in vector form (∇·~a , (∇×~a)2) for the points at locations 4 to 9. Note,that values at all other locations are zero.Obviously, the values of the divergence are 2

3for all six considered stress

points 4 to 9. This implies, that the radiation pattern of the divergence isisotropic (see Figure 3.8b) in the hexagonal case. Contrary, the curl of thevelocity field does not vanish as required for an explosive source, but showsan anisotropic radiation pattern, which explains the systematically varyingS-wave artifacts observed in the synthetic seismograms (Figure 3.12).This systematic, grid-inherent error seems to decrease with increasing gridirregularity (Figure 3.12). However, a detailed investigation of the behaviourof the irregular grid operator is far more involved and not shown in this work.

Frequency Dependence of S-Wave Artifacts

The reason for the observed S-wave artifacts (Figure 3.12) is discussed indetail in the previous section. It is shown, that the S-wave artifacts are gen-erated by the geometry of the derivative operator used for hexagonal grids.The curl of the velocity field is not vanishing and therefore leads to errors inthe computation of shear stresses.It is observed, that this error seems to affect signals of short wavelengths

7In fact, we will obtain the divergence and curl multiplied by the mass density ρ.However, this is not important, as we only want to check the existence of the curl.

8Only the second component is given, as in two dimensional space the other componentsare zero.


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mut

h (d

eg)

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9

10

grid points / λ

rela

tive

ener

gy o

f S−

wav

e

q=1.00

q=0.95

q=0.98

c

Figure 3.15: An example of the numerically calculated divergence (a) andcurl (b) of the velocity field are shown as obtained by an hexagonal grid. Thetotal energy of the P- and S-wave signals is determined within a time window(red lines) to evaluate the size of the grid-inherent S-wave error. The relativeS-wave energy decreases for increasing wavelengths (c) and depends on gridirregularity.

much more than signals of very long wavelength. In other words, if the sig-nal is sampled with an increasing number of grid points per wavelength, theoperator will more and more approximate the wave field correctly. Therefore,the S-wave artifacts will decrease with increasing wavelength. This can beshown by calculating the relative energy of these pseudo S-waves with respectto the total energy of the P-wave signals (Figure 3.15). The P- and S-waveenergies are computed within a time window including the correspondingsignal and their relation is determined for different signal frequencies.It is also observed, that the relative energy of the pseudo S-wave is de-creasing with increasing grid perturbation, i.e. decreasing average trianglequality. The systematic, grid-inherent error seems to be minimized by theunstructured grid (Figure 3.15c). However, for very low triangle qualities(high perturbations) the numerical noise generated by the P-wave signal dis-turbs the calculation of the energy of the pure S-wave artifacts (data notshown). Therefore, the S-wave error would be minimized, but consequentlythe P-wave accuracy is deteriorated.

Suppression of S-Wave Artifacts

As the amplitudes of the S-wave artifacts are considerable, a hexagonal gridcould not be used for simulating the elastic wave filed with sufficient accuracy,if the source is implemented by initializing the normal stresses. Referring tothe work of Coutant et al. (1995), an explosive source is used in a second sim-ulation test, where the source is implemented by four external forces, acting


a b

Figure 3.16: (a) Four external forces acting as explosive source. (b) Fourexternal forces acting as rotational source.

along the positive and negative coordinate axes (Figure 3.16a). The super-position of the wave fields generated by the four external forces successfullysuppresses the appearance of the S-wave artifacts (Figure 3.17) compared tothe previous results (Figure 3.12).



The methods to investigate the accuracy of numerically calculated seismo-grams was already introduced for the acoustic case. The same techniquesare applied to the elastic case. As an explosive source is used to generate theseismic wave field, the analytical solution consists of a pure P-wave signal inanalogy to the acoustic case. The occurrence of S-wave signals is discussedin the previous sections, so the further investigation focuses on the P-wave(divergence) seismograms. Examples and calculations are shown for simula-tions using the explosive source implemented by normal stresses. Using theexternal forces as an explosive source suppresses the S-wave artifacts, butdoes not change the results of the P-wave signals notably.The synthetic seismograms are compared to the analytical solutions by de-termining the relative misfit energy ε.

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0

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100

150

200

250

300

350

Azim

uth

(deg

)Az

imut

h (d

eg)


hexagonal

irregular

Figure 3.17: The left column of seismograms show the divergence of the nu-merically calculated wave field recorded by the 36 receivers, whereas the rightcolumn shows the curl of the wave field. An explosive source generated byfour external forces is used. Note that the S-wave artifacts are suppressedsuccessfully.


0 50 100 150 200 250 300 350

misfit : 8.63%

Time (msec)

misfit : 14.39%

misfit : 15.49%

misfit : 18.32%

0 50 100 150 200 250 300 350

misfit : 0.04%

Time (msec)

misfit : 0.05%

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misfit : 8.63%

Time (msec)

misfit : 8.94%

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misfit : 0.04%

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misfit : 0.25%

misfit : 2.06%

misfit : 6.39%

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misfit : 8.63%

Time (msec)

misfit : 20.02%

misfit : 55.91%

misfit : 85.75%

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misfit : 0.04%

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misfit : 0.04%

misfit : 0.04%

misfit : 0.06%

Grid P

erturbation [%]

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0.98

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rage

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ngle

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lity

1.00

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rage

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ngle

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lity

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erturbation [%]

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rage

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ngle

Qua

lity

λ

0.95

25 grid points /

Grid P

erturbation [%]

NND

FV3

FVN

λ65 grid points /

misfit : 14.94%

Figure 3.18: Examples of numerical seismograms (solid), the correspondinganalytical solution (dashed) and the resulting relative misfit ε. The threemethods are shown for two different sampling rates (25 and 65 grid pointsper dominant wavelength). For each method seismograms are displayed fordifferent grid qualities to show the decrease in accuracy with decreasing qual-ity. Simulations for grids with high perturbation become unstable when usingthe FV3 method.


Single seismograms (Figure 3.18) obtained on the different irregular grids(Table 3.2) show the influence of the grid irregularity for two different fre-quencies. The relative misfit ε is given for each trace. Similar to the acousticcase, the examples already graphically show, that the NND method performsbetter than FVN and FV3. As already observed in the acoustic case the FV3method is unstable for strongly perturbed grids.In contrast to the results of the acoustic wave simulation, a notable detailcan be detected in the seismograms obtained on the perfect hexagonal grid

25 30 35 40 45 50 55 60 65 70 750

2

4

6

gridpoints / λ

err

or

[%]

25 30 35 40 45 50 55 60 65 70 750

2

4

6

err

or

[%]

25 30 35 40 45 50 55 60 65 70 750

2

4

6

err

or

[%]

FVN

FV3

NND

q=0.98

q=0.98

FDMagnier

FDMagnier

FDMagnier

q=0.98

q=0.95

q=0.90

q=0.95

q=0.90

Figure 3.19: The average misfit energy εave of the elastic simulation resultsare plotted versus sampling rate for the three methods (NND,FV3,FVN) anddifferent grid irregularities. The results obtained on the FD grid and thehexagonal grid are plotted as a reference.


(q = 1) and an explosive source9. A signal with very small amplitude ap-pears at about 250msec. It is caused by the grid-inherent error discussed

25 30 35 40 45 50 55 60 65 70 750

1

2

3

4

5

6

gridpoints / λ

erro

r [%

]

q = 0.98

FDMagnier

FV3

NND

FVN

Figure 3.20: The accuracy of seismograms with respect to sampling rate isshown for the three method (NND,FV3,FVN) for a particular grid of anaverage triangle quality of q = 0.98. It is obvious that the NND operatorleads to better results than the two other methods.

in the previous sections. As the shear stresses σxz are calculated incorrectly,the error influences the further computation of the propagating wave field.Therefore, the grid-inherent error also appears in the divergence of the wavefield.Operating on irregular, arbitrary grids the systematic error of the hexago-nal grid is reduced (Figure 3.12 and 3.18), as the incorrect spatial derivativesobtained at one location of the grid might cancel inaccurate results of deriva-tives at another location. Nevertheless, the accuracy of the actual divergencesignal is decreasing for increasing grid irregularity.In analogy to the acoustic case, the average misfit εave is calculated for all36 receivers and all three irregular grid operators - NND, FV3 and FVN -and the results are compared to the results obtained from the two referencegrids (Figure 3.19). All three methods are leading to increasing errors whengrid irregularity is increasing. The NND method is clearly providing moreaccurate results than the other methods.Compared to the accuracy of seismograms in the acoustic case (Figure 3.10)

9Using the explosive source implemented by four external forces, suppresses this arti-fact, but does not affect the following investigation of seismogram accuracy considerably.


Magnier’s method on perfect hexagonal grids performs worse in the elasticcase (Figure 3.19) due to the systematic error in computing the shear stressσxz.The FVN method performs much worse in the elastic case than in the acous-tic case. Increasing grid irregularity is causing very large errors, as the spatialderivative operator does not include the location, where the derivative hasto be evaluated.In contrast, the NND method seems to become less dependent on grid ir-regularity in the elastic case compared to the acoustic case. The reason isthe suppression of the systematic grid-inherent error with increasing gridperturbation. As the NND method considers the actual location, where thespatial derivative has to be evaluated, it leads to the optimal signal accuracyobtained on irregular grids.The FV3 method behaves similar to the NND method, but again becomesunstable for high grid perturbations. The reason is already discussed for theacoustic case.The direct comparison of the three methods on a irregular grid with an av-erage triangle quality of q = 0.98 clearly shows, that the NND method isleading to the highest accuracy (Figure 3.20).

3.3.5 Simulation of a Rotational Source

Previous test of operator accuracy (Chapter 2) and results of simulationsshown in the sections above have shown, that the operator using the naturalneighbour weights (NND method) provides the most accurate results. Theother derivative operators working on irregular grids (FVN, FV3) always per-form worse. Therefore, in the following investigations only the NND methodis used for simulations on arbitrary grids.The simulations of explosive sources already have shown, that the spatialderivatives ∂vx

∂zand ∂vz

∂xare defined at different locations (Figure 3.14b), if

the medium is discretized on a hexagonal or irregular grid. This leads to thegrid-inherent error, when calculating the shear stress σxz. Therefore, S-waveartifact occur, that are caused by this broken symmetry of the derivativeoperators.The important question is, how these S-wave artifact influence the prop-agation of real S-waves. A source has to be implemented, which initiallygenerates S-waves traveling through the discretized medium. To analyse theaccuracy of S-wave seismograms equivalently to the P-wave investigations, aisotropic radiation pattern of S-waves is desirable.A source producing pure S-waves can be realized by a rotational source(Coutant et al., 1995), which is generated by four external forces (Figure3.16b). In analogy to the (divergence-) seismograms of an explosive source(Figure 3.12 or 3.17) the (curl-) seismograms show an isotropic radiation pat-


tern for the rotational source (Figure 3.21), i.e. each seismogram is recordingthe same S-wave signal. The superposition of the four wave fields suppressesthe appearance of P-wave signals successfully. Though the divergence seis-mograms obtained from the hexagonal and the irregular grids show smallamplitudes, when the S-wave is passing the receivers, no P-wave signal canbe detected. However, in the hexagonal case a slight phase shift of the seis-mograms showing the curl can be observed.Similar to previous investigations, the accuracy of the seismograms showingthe curl of the velocity field can be analysed by comparing them to analyt-ical solutions. Therefore, the influence of grid irregularity on the operatoraccuracy can be analysed for pure S-wave propagation.


The relative misfit energy ε is computed for each seismogram, as shown inprevious sections. The error ε can be plotted versus azimuth, similar to theinvestigation of grid symmetry in the acoustic case (Figure 3.8), to displaythe azimuthal error distribution of all seismograms.In the case of pure S-waves, an interesting error distribution is obtained (Fig-ure 3.22). The result of the FD grid show the typical cross-like error patternwith maximum errors along the coordinate axes and minimum errors alongthe diagonals (Figure 3.22a). In contrast, the hexagonal grid provides an un-expected error distribution (Figure 3.22b). Though the hexagonal symmetrycan be observed, maximum errors occur along 30◦, 150◦ and 270◦ and inter-mediate error occur along 90◦, 210◦ and 330◦. Minimum error are observedin between every 60◦. Furthermore, these minimum errors are very smallcompared to the corresponding errors of the FD grid. The expected isotropyof the hexagonal grid seems to be destroyed for the S-wave propagation.Irregular grids are leading to similar results (Figure 3.22c), whereas the sys-tematic error distribution is more and more deteriorated with increasing gridperturbation.The influence of grid irregularity on seismogram accuracy can graphically beshown by single seismograms (Figure 3.23). It is obvious, that the error isdecreasing for higher sampling rates. However, it might happen, that singleseismograms obtained by a highly perturbed grid are more accurate thanseismograms obtained by a less perturbed grid. Especially, phase errors canstrongly affect the seismogram accuracy, i.e. the relative misfit energy ε (seeleft part of Figure 3.23).Therefore, the calculation of the average relative misfit energy εave of thewhole ensemble of 36 seismograms is affected considerably, as the maximumand minimum errors greatly differ from one direction to another.Consequently, plotting the average relative misfit energy εave versus signalfrequency, i.e. spatial sampling rate, also leads to an unexpected diagram


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imut

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eg)

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)


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FD

hexagonal

irregular

Curl

Figure 3.21: The left column of seismograms show the divergence of the nu-merically calculated wave field recorded by the 36 receivers, whereas the rightcolumn shows the curl of the wave field. A rotational source generated byfour external forces is used.


2.3333

4.6667

7

30

210

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270

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180 0

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7

30

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240

90

270

120

300

150

330

180 0

a b c

error in %error in % error in %

Figure 3.22: The relative misfit of the synthetic seismograms (recording thecurl of the velocity field) is plotted versus azimuth. Result obtained on the FDgrid (a) the hexagonal grid (b) and an irregular grid (c) with average trianglequality of q = 0.98 are shown for a sampling rate of 35 grid points perwavelength. The distance from a point on the graph to the origin representsthe size of the error ε.

(Figure 3.24). The accuracy of the synthetic seismograms obtained on theFD grid does not lead to best results for all frequencies. For low samplingrates the FD operator performs worse than the hexagonal operator, and pro-vides similar results as an irregular grids of perturbation p = 10%. For highersampling rates (> 38 grid points / λ) the FD operator lead to the most ac-curate seismograms.Nevertheless, the seismogram accuracy is still decreasing with increasing gridirregularity.

Simulation of an External Force

The elastic wave field can also be generated by a single external force, i.e.the spatial Gauss function (Figure 3.3) is applied on the grid, where one ofthe components of the velocity (vx or vz) is defined. A source initializing vzis representing a external force acting along the z-axis. Therefore, P- andS- waves are generated initially. Snapshots of the propagating wave field aregiven in Appendix C and display the vertical component vz of the velocityvector.Seismograms recording the divergence and the curl of the velocity field (Fig-ure 3.25) show, that the S-waves do have higher amplitudes than the P-waves. Again, artifacts in the divergence seismograms can be detected, thatare caused by the operator symmetry.As the radiation pattern of P- and S-waves is not isotropic in this case, theinvestigation of seismogram accuracy is difficult. The S-waves signals domi-nate the calculation of the relative misfit because of their large amplitudes.


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misfit : 11.54%

Time (msec)

misfit : 14.88%

misfit : 15.76%

misfit : 9.57%

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misfit : 0.28%

Time (msec)

misfit : 0.40%

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25 grid points / λ

Ave

rage

Tria

ngle

Qua

lity

0.90

0.95

1.00

0.98

Grid P

erturbation [%]

10

20

30

0

λ65 grid points /

NND

Figure 3.23: Examples of numerical seismograms (solid), the correspondinganalytical solution (dashed) and the resulting relative misfit ε. The curl of thewave field is shown for two different sampling rates (25 and 65 grid pointsper dominant wavelength).

25 30 35 40 45 50 55 60 650

1

2

3

4

5

6

gridpoints / λ

erro

r [%

]

Magnier

FD

q=0.98

q=0.95

q=0.90

Figure 3.24: The average misfit energy εave obtained from the pure S-wavesimulations are plotted versus sampling rate for different grid irregularities.The NND operator was used on the irregular grids. The results obtained onthe FD grid and the hexagonal grid are plotted as a reference.

3.4. DISCUSSION 63

Especially, if they travel along the coordinate axes, large errors occur in thesynthetic seismograms obtained by FD due to the numerical anisotropy of aquadratic grid. Generally, the error distribution patterns are rather complexand therefore difficult to interpret. Therefore, behaviour of the graphs show-ing the average relative misfit εave versus sampling rate is not understood,yet. The analysis of such wave fields and the investigation of seismogramaccuracy deserves further study, but would go beyond the scope of this work.

3.4 Discussion

In this chapter the initialization of the simulation parameters is discussedfor a homogeneous, isotropic model and the geometry of simulation setupis shown. Introductory, the stability conditions are investigated for the FDand the hexagonal cases, as they are defined differently. It is outlined, thatseparate time increments ∆thex and ∆tFD) have to be used to approach thethreshold defined by the stability criterion as close as possible. Providedthat the node density (see Chapter 2) is equal, we obtain a ratio of time

increments of ∆thex =√√

3. We have shown, that on irregular grids thetime increment has to be determined with respect to the irregularity of theparticular grid. The diameter of the smallest grid cell seems to be an appro-priate parameter to estimate the maximum time step for the correspondingirregular grid.To make the algorithm as flexible as possible, interpolation methods areused to define sources or receivers at arbitrary locations. In other words,the source and receiver location are not restricted to particular grid nodes.Therefore, a spatial Gauss function is used to implement the source (Figure3.3) and two different interpolation techniques solve the problem of arbitraryreceiver locations on regular or irregular grids. (1) a linear interpolation forthe finite-difference grid (Figure 3.4b) and (2) a interpolation using naturalneighbour coordinates for the hexagonal and irregular grids (Figure 3.4a).The implementation of the different explicit derivative operators in a sim-ulation algorithm shows, that they are capable to propagate seismic wavesthrough an irregular grid. The analysis of synthetic seismograms obtained inan acoustic medium clarifies the influence of grid irregularity, i.e. the influ-ence of the average triangle quality, on seismogram accuracy. As an isotropicradiation pattern of P-waves is used, it can be shown, that the accuracy ofthe synthetic seismograms also strongly depends on the direction, in whicha wave is traveling. Considering the overall accuracy of the whole ensembleof seismograms the natural neighbour method (NND) generally leads to thebest results followed by the FV3 and the FVN methods.The simulation of elastic waves leads to similar results, if an explosive sourceis used. To suppress artifacts - due to the broken symmetry of the derivative


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Figure 3.25: The left column of seismograms show the divergence of the nu-merically calculated wave field recorded by the 36 receivers, whereas the rightcolumn shows the curl of the wave field. A vertical force is used as source.Note that the finite-difference technique does not show any artifacts, whereasfor the hexagonal and the irregular case a P-wave occurs, which seems totravel with the S-wave velocity.

3.4. DISCUSSION 65

operators - on hexagonal or irregular grids, it is necessary to implement thesources by superimposing the wave field of several external forces.An interesting effect occurs for the simulation of pure S-waves, as the simu-lation on a hexagonal grid or a irregular grid with very slight perturbationseems to provide more accurate seismograms for high frequencies than theFD method. The reason is not really understood yet. However, it is no-table, that on hexagonal grids the highest accuracy of S-wave seismogramsis obtained in directions, where the strongest S-wave artifacts occur, whensimulating an explosive source. In other words, the S-wave artifacts due tothe operator symmetry seem to superimpose the real S-waves in a way lead-ing to very high accuracy (even higher than FD). As the S-wave artifactsdecrease with decreasing signal frequency, the influence on the real S-wavesignals also decreases. Therefore, seismograms obtained by FD are gettingmore accurate than the seismograms obtained by hexagonal grids for lowerfrequencies. Nevertheless, a more detailed investigation of the behaviour ofthe differential operators on hexagonal and irregular grids is necessary tofully understand these effects.The investigation of operator accuracy has shown, that they are capable ofpropagating seismic waves through a irregularly discretized medium. Espe-cially the NND operator provides the most accurate results. Though thebehaviour of the NND operator is not fully understood for the elastic case,and boundary conditions are not investigated yet, this method will be appliedto several realistic models in the following.

Chapter 4

Application of the Irregular

Grid Operators to Realistic

Models

In this chapter simulations of acoustic and elastic wave propagation are shownfor different models. The NND operator is applied to the different irregular,triangular grids discretizing the corresponding media in different ways.A simple homogeneous acoustic model with a cylindrical shape is used toclarify, how the singularity problem1 for cylindrical geometry can be over-come.A further homogeneous, acoustic model is shown to demonstrate, that irregu-lar, triangular grids can accurately adapt to model boundaries with arbitraryshape. The problem of finding a optimal staggered scheme is discussed briefly.In a third case study, the NND operator is applied to simulate elastic wavepropagation. The elastic medium consists of two layers with different mate-rial parameters (λ, µ, ρ). The two layers are divided by a curved discontinuity.

4.1 Cylinder

4.1.1 Discretization

In this first example of applying the explicit NND operator to a realisticmodel an acoustic medium with a cylindrical model boundary is used (Fig-ure 4.1a). The medium is discretized by a triangular mesh, based on theDelaunay triangulation as introduced in Chapter 2. The primary (velocity)

1A cylindrical problem would suggest the discretization on a grid using cylindricalcoordinates. However, the center of a cylindrical grid is causing stability problems, as thesize of the grid cells is decreasing towards the center, if no multi-domain grids are used(see Figure 2.1).

67

68 CHAPTER 4. APPLICATION TO REALISTIC MODELS

b c

a

Source

Receivers

2000 m

Figure 4.1: (a) The cylindrical model discretized on an irregular, triangulargrid. (b) The Delaunay triangulation of the primary grid (∗) with the cor-responding staggered grid points (+). (c) The Voronoi cells of the primarygrid points (∗) and the corresponding staggered grid points (+).

grid points are arranged in distinct circular layers (Figure 4.1b). The dis-tance of these different layers approximately equals the distance of points ona particular layer. The Delaunay triangulation of these points leads to a agrid of very high average triangle quality of q ≈ 0.93.The secondary (stress) grid points are ordered in a similar layered system.With respect to the primary grid each layer of secondary points is locatedright in between the layers of primary points. Therefore, the secondary gridalso provides a high average grid quality of q ≈ 0.93. This technique of gener-ating a staggered grid for cylindrical geometries has been used and extendedto the 3-D case of a sphere by Igel et al. (1997).There are three major advantages of discretizing the medium using this stag-gering scheme.

• The size of the unit grid cells is approximately the same throughoutthe entire model and therefore overcomes stability problems at the gridcenter.

• The quality of both grids - velocity and stress - is high.

• The grid considers the layered structure of the Earths major discontinu-

4.1. CYLINDER 69

number of primary points Np 31731number of secondary points Ns 31416average node density d [m−2] 0.01time increment ∆t [ms] 2.228total simulation time t [ms] 500P-wave velocity vp [

ms] 4000

density ρ [ kgm3 ] 2500

dominant frequency [Hz] 20

Table 4.1: Simulation parameters of the cylindrical model.

ities and therefore makes it easy to initialize depth-dependent materialparameters.

As far as the question of designing staggered grid schemes on arbitrary grid isconcerned, we already mentioned the problem of finding the optimal choice ofstaggered grid points (see Chapter 2). In the case of this circular arrangementof nodes it turns out, that the secondary grid points are located very closeto the sides of the Voronoi cells defined by the primary points (Figure 4.1c).Therefore, the choice of particular vertices of Voronoi cells seems to be apromising approach to solve the problem of finding staggered grid points.The following case studies will discuss this topic in further detail.An overview of the simulation parameters for the cylindrical model is givenin Table 4.1.

4.1.2 Simulation Results

The acoustic waves propagated through the cylindrical model are generatedby an explosive source as described in Chapter 3. The source is located inthe bottom part of the model (Figure 4.1a) and the signals of the pressurewave are recorded by an array of 50 receivers located along the diameter ofthe model. The boundaries conditions are representing a free surface, whichis simply realized by setting the values of the field variables to zero.The propagating acoustic wave field is displayed for several time steps (Fig-ure 4.2). As the model is discretized on a grid using cartesian coordinates,the center of the model does not cause any problems, as the size of the gridcells is approximately the same throughout the entire grid. Due to the highquality of the underlying triangular mesh, no notable numerical artifacts canbe detected in the wave field snapshots. The recorded seismograms (Fig-ure 4.3) clearly show the direct wave and the wave reflected at the bottomboundary of the model. An interesting effect is, that the curvatures of themodel boundary and the direct wave lead to a reflected wave propagating


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t=130 ms

t=190 ms

t=250 ms

t=310 ms

t=370 ms

t=430 ms

Figure 4.2: The acoustic wave field generated by an explosive source. Note,that no numerical artifacts are visible by the irregular discretization of themodel with cartesian coordinates. Positive amplitudes are blue, negative arered.

through the medium as an almost plane wave. The occurrence of this plane

4.2. MOUNTAIN TOPOGRAPHY 71

0

0.05

0.1

0.15

0.2

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0.3

0.35

0.4

0.45

Tim

e [s

ec]

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direct wave

reflected wave

Figure 4.3: The pressure seismograms recorded along the receiver line throughthe center of the cylindrical model. Note, that no notable numerical artifactsare visible.

wave does strongly depend on the distance between the source and the re-flecting boundary. However, the seismograms are not affected by notablenumerical artifacts, which would particularly occur after a wave has passedthe receivers. Though the sampling is rather low (≈ 20 grid points per wave-length), the simulation of acoustic waves provides acceptable results in thissimple case.

4.2 Mountain Topography


In this second example the irregular grid operator is applied to an acous-tic medium with a complex boundary geometry. A mountain topographyis represented by a sinusoidal function (Figure 4.4a). Of course, any otherarbitrary topography could be used.The generation of a triangular mesh is more difficult in this case, as thetriangles have to adapt to the topography. Furthermore the grid has to pro-vide a high average quality and all grid cells should have an equivalent sizethroughout the entire model. To solve this problem, a triangular mesh gener-ator developed by Shewchuck (1996) is used to create the primary grid. Themesh nicely adapts to the topography (Figure 4.4b), as the triangle sides arealigned along the curved boundary. Therefore, the corresponding vertices of


Source

a

b c

2000 m

2000 m

Receivers

Figure 4.4: (a) A homogeneous model with curved topography discretized onan irregular, triangular grid. (b) The Delaunay triangulation of the primarygrid (∗) with the corresponding staggered grid points (+). (c) The Voronoicells of the primary grid points (∗) and the corresponding staggered grid points(+).

the triangles of the primary grid are located on the boundary itself2. Theobtained mesh provides an average triangle quality of q ≈ 0.92. However, amajor disadvantage of Shewchuck’s mesh generator is, that the triangles donot converge to perfect equilateral triangles within the homogeneous medium,where the geometry of the boundaries has no influence.The generation of a secondary grid is not trivial in this case, as there is nostructure as in the cylindrical model. Nevertheless, we assume, that the op-timal locations of the secondary nodes are on the edges of the Voronoi cellsof the primary grid. Therefore, we choose the vertices of the Voronoi cells todefine the secondary nodes (Figure 4.4c). Unfortunately, this method leadsto about twice as much secondary grid points as primary grid points. Fur-thermore, the average triangle quality of the secondary grid becomes ratherpoor (q ≈ 0.63), as some of the secondary nodes lie very close together caus-

2The discretization of the space above the free surface is necessary, as the triangulationalgorithm of the simulation program requires the grid points to form a convex hull.

4.2. MOUNTAIN TOPOGRAPHY 73

number of primary points Np 39990number of secondary points Ns 79232average node density d [m−2] 0.01time increment ∆t [ms] 1.683total simulation time t [ms] 500P-wave velocity vp [

ms] 4000

density ρ [ kgm3 ] 2500

dominant frequency [Hz] 20

Table 4.2: Simulation parameters of the mountain topography model.

ing triangles of very low quality.An additional problem arises, as the average number of natural neighboursper grid points is increasing enormously, when using this kind of staggeredscheme. Therefore, the large numbers of secondary grid points and the cor-responding natural neighbours severely affects the computational effort.To overcome this problem a geometrical optimization problem has to besolved. The reduction of secondary nodes to the number of primary nodeswould also reduce the number of natural neighbours for each grid point au-tomatically. The neighbours of a node should also be uniformly distributedwith respect to azimuth. However, a proper solution of this problem wouldgo far beyond the scope of this work.For the following simulation the introduced scheme of using all vertices ofVoronoi cells (Figure 4.4a,b) is applied. An overview of the simulation pa-rameters for the cylindrical model is given in Table 4.2.


In analogy to the cylindrical model the acoustic wave is generated by an ex-plosive source and the boundary conditions of a free surface are realized bysetting the values of the field variables to zero along and above the curvedtopography. The wave field is recorded by an array of 50 receivers along themountain topography (Figure 4.4a).The propagating pressure waves are displayed for several time steps (Figure4.5). To avoid reflections from the sides and the bottom of the model ab-sorbing boundary conditions are implemented. The technique of absorbingboundaries (Cerjan et al., 1985) is based on a sponge function, that simplyreduces the amplitudes of the wave field towards the model boundaries bya taper function. Finding the optimal taper parameters, amplitudes of thereflections can successfully be reduced to only a few percent.The source is located under the central mountain of the sinusoidal topogra-phy. A circular wave front of the initial P-wave (P1) is propagating outwards


(Figure 4.5). Two strong reflections (P2) are caused by by the opposingslopes of the central mountain, whereas the direct wave is diffracted aroundthe two valleys. Therefore, energy is also transported to the summits of theneighbouring mountains. While the direct wave is also reflected (P3) by theslopes of the neighbouring mountains, a strong reflected wave (P4) beneaththe central mountain travels downward. The mountain topography leads toan interesting effect of focusing the energy of this reflected wave.These different phases can also be identified in the numerical pressure seis-

P2

P3 P3

P1

P2P4

t = 110 ms t = 155 ms t = 200 ms

t = 245 ms t = 290 ms

t = 380 ms

t = 335 ms

t = 425 ms t = 470 ms

Figure 4.5: The propagation of the acoustic wave field generated by an ex-plosive source beneath a mountain topography. Note the strong amplitudes ofthe reflected wave due to the focusing effect of the central mountain. Positiveamplitudes are blue, negative are red.

4.3. MARGIN OF A BASIN 75

0

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P1P1

Tim

e [s

ec]

Receivers (1-50)

P2

P3

Figure 4.6: The pressure waves recorded by an array of geophones along thesurface.

mograms (Figure 4.6) recorded along the array of receivers located at thesurface. The amplitudes of the seismograms near the edges of the model areaffected by the absorbing boundary conditions. Therefore, the amplitudes ofthe reflection (P3) at the neighbouring mountain slopes are affected by thetaper and show too small amplitudes. Even though the sampling is ratherlow (≈ 20 grid points per wavelength) and the design of the staggered gridscheme is not optimal, the simulation of acoustic waves provides acceptableresults without notable numerical artifacts. In this example, the irregular,triangular grid represents an elegant way of discretizing acoustic media withcomplex geometries.

4.3 Margin of a Basin


In the third example the irregular grid operator is applied to simulate elasticwave propagation in a layered model. Two layers are divided by a curved dis-continuity separating a medium of low impedance at the top from a mediumof high impedance at the bottom (Figure 4.7a). The model could representthe margin of a basin structure. In the top layer the material parameters λ,µ and ρ are 0.7 times the material parameters of the bottom layer.The model is discretized on an irregular grid created by Shewchuck’s meshgenerator (Figure 4.7b). The obtained grid again provides an average tri-angle quality of q ≈ 0.92. Contrary to the mountain topography model, weare not using the vertices of the Voronoi cells as secondary nodes. In the


a

cb

Receivers

Source

2000 m

2000 m

Figure 4.7: (a) The elastic model with a low velocity layer at the top and ahigh velocity layer underneath. (b) The Delaunay triangulation of the pri-mary grid (∗) with the corresponding staggered grid points (+). (c) TheVoronoi cells of the primary grid points (∗) and the corresponding staggeredgrid points (+).

elastic case, strong artifacts are observed, especially when S-waves propa-gate through the medium due to the low average triangle quality of such asecondary grid. Therefore - as already discussed in Chapter 2 - the centers ofgravity of each primary triangle (Figure 4.7b) are used instead, which leadsto an average triangle quality of the secondary grid of q ≈ 0.76. Again,most of these secondary grid points are located very close to the edges ofthe Voronoi cells of the primary nodes (Figure 4.7b), which confirms ourassumption, that the optimal staggered grid points might also be located onthese edges. The azimuthal distribution of neighbours seems to be more bal-anced compared to the previous scheme in the topography model, althoughthe problem of having twice as much secondary points as primary pointsmaintains. Nevertheless, this staggered grid scheme is leading to less strong


number of primary points Np 160038number of secondary points Ns 318512average node density d [m−2] 0.04time increment ∆t [ms] 0.743total simulation time t [ms] 500P-wave velocity (bottom) vp1 [

ms] 4000

P-wave velocity (top) vp2 [ms] 2800

S-wave velocity (bottom) vs1 [ms] 2310

S-wave velocity (top) vs2 [ms] 1617

density (bottom) ρ [ kgm3 ] 2500

density (top) ρ [ kgm3 ] 1750

dominant frequency [Hz] 10Hz

Table 4.3: Simulation parameters of the basin model.

artifacts and is used for the following simulation of elastic wave propagation.The simulation parameters of the basin model are given in Table 4.3. Note,that the grid is much larger than in the previous examples due to the ne-cessity of a high enough sampling rate of the S-waves in the low velocity layer.


Similar to the previous examples an explosive source is used to generate theelastic wave field. The source is located close to the discontinuity (Figure4.7a) with an array of 50 receivers 200m above the source. Absorbing bound-ary conditions (Cerjan et al., 1985) are used to avoid reflections of the modelboundaries to interfere with the recorded signals.Snapshots of the propagating elastic wave field are given for several timesteps (Figure 4.8 and 4.9). After the initial P-wave (P1) has reached thediscontinuity a part of the wave field energy is transmitted into the highvelocity zone and another part is reflected (P2)3. A converted S-wave (S1)also occurs after the reflection and passes the receivers. Along the disconti-nuity refracted (conical) waves can be observed in the low velocity layer. Aninteresting effect is the interference of the reflected and the refracted S-wave.Similar to the mountain topography model the concave geometry of the dis-continuity focuses the energy of the wave field leading to high amplitudes.

3As the receivers are located above the discontinuity, only the upward traveling wavefield is of interest in this case.


t = 120 ms t = 165 ms t = 210 ms

t = 255 ms t = 300 ms t = 345 ms

t = 390 ms t = 435 ms t = 480 ms

P2

S1

P1

Figure 4.8: The x-component of the velocity vector is shown. The elastic wavefield is generated by an explosive source close to the discontinuity between twomedia with high and low seismic impedance. Note the different wave types andthe energy focusing effect of the concave part of the discontinuity. Positiveamplitudes are blue, negative are red.


t = 120 ms t = 165 ms t = 210 ms

t = 255 ms t = 300 ms t = 345 ms

t = 390 ms t = 435 ms t = 480 ms

P2S1

P1

Figure 4.9: The z-component of the velocity vector is shown. The elastic wavefield is generated by an explosive source close to the discontinuity between twomedia with high and low seismic impedance. Note the different wave types andthe energy focusing effect of the concave part of the discontinuity. Positiveamplitudes are blue, negative are red.


Comparison with FD

The major phases observed and described in the snapshots of the elastic wavefield can be identified in the corresponding numerical seismograms (Figure4.10). Here, simulation results obtained on the discussed irregular grid (Fig-ure 4.10c and d) and results of a FD simulation (Figure 4.10a and b) arecompared.As shown in Chapter 3 the accuracy of synthetic seismograms is decreasingwith decreasing average triangle quality (Figure 3.10 and 3.19). Though onlyhomogeneous media have been considered in these investigations, we will usethe results and extrapolate them as a rough estimation to this case.In the discussed elastic wave simulation the S-wave in the low velocity mediumhas the shortest occurring wavelength of λ ≈ 160m. Considering the givennode density of d = 0.04m−2, this wavelength is assumed to be sampled with32 grid points. To achieve the same accuracy with a standard FD simula-tion, the results of Chapter 3 state, that a sampling of 25 grid points perwavelength is sufficient in the FD case. Therefore, the node density d can bereduced by the factor ( 25

32)2 ≈ 0.61 to dFD = 0.024m−2. The resulting grid

size of the FD grid is 312× 312 grid points. The comparison of the seismo-

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Tim

e [s

ec]

Receivers (1−50)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Tim

e [s

ec]

Receivers (1−50)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Tim

e [s

ec]

Receivers (1−50)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Tim

e [s

ec]

Receivers (1−50)

a

c

b

d

S1

P2

P1

P2

P1

S1

Figure 4.10: Numerical seismograms of the x-component and z-componentof the velocity vector are shown as obtained from the FD (a),(b) and theirregular grid method (c),(d).

4.4. DISCUSSION 81

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Time [sec]

v x am

plitu

de

FD

irregular

P1

P2 S1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

v z am

plitu

de

Time [sec]

FD

irregular

P1

S1 P2

a b

Figure 4.11: Numerical eismograms calculated on the FD and the irregulargrid showing the x-component (a) and z-component (b) of the velocity vectorare compared for a particular receiver. The direct P-wave (P1) the reflectedP-wave (P2) and the reflected S-wave (S1) are labeled.

grams obtained by the FD simulation (Figure 4.10a and b) and the irregulargrid method (Figure 4.10c and d) shows, that similar results are obtained.To analyse the results in more detail, single traces of both simulations areselected (Figure 4.11). Whereas the dominant signals obtained on the irreg-ular grid are very similar to the signals obtained by FD, slight differencesare observed. In particular, small phase shifts can be detected especially forthe S-wave signal (S1). Furthermore, a certain level of numerical noise isvisible after the dominant signals have passed the receiver in the irregularcase. However, the wave forms of the recorded signals correspond quite wellwith the results obtained by FD.

4.4 Discussion

In this chapter the natural neighbour difference (NND) operator has beenapplied to simulate wave propagation in several realistic models discretizedon irregular, triangular grids with different staggering schemes.The acoustic cylindrical model shows, how the singularity problem of regulargrid using cylindrical coordinates can easily be overcome with our approach.As the staggered grid scheme is designed in a way to provide very high av-erage triangle qualities (q ≈ 0.93) for both grids - i.e. velocity and stress- numerical artifacts are very small. The computational effort is kept on aminimum by using a similar number of primary and secondary grid points.The acoustic mountain topography model impressively outlines, how an irreg-ular, triangular grid can elegantly adapt to an arbitrary boundary geometry.


Hereby, a quality mesh generator of (Shewchuck, 1996) is used to create thecorresponding primary grid (q ≈ 0.92). A problem arising when working onsuch completely unstructured grids is the optimal choice of staggered gridpoints. Considering the definition of Voronoi cells (see Appendix A), we in-tuitively assume the optimal locations of secondary nodes to lie on the edgesof the Voronoi cells of the primary nodes. However, the design of an optimalirregular staggered grid scheme is non-trivial and is not solved here. There-fore, all vertices of the Voronoi cells are used as secondary grid points. Theresulting average triangle quality of this secondary grid turns out to be verylow (q ≈ 0.63). Nevertheless, this method is capable to simulate acousticwaves propagation without causing notable artifacts.In the elastic model describing the margin of a basin structure, the sameproblem of finding a staggered grid scheme occurs. Here, the low grid qualityobtained by the use of the vertices of Voronoi cells does lead to considerableartifacts and therefore requires an alternative method of defining secondarynodes. As suggested in Chapter 2, the centers of gravity of the primary trian-gular grid cells are used as secondary grid points resulting in a much higheraverage secondary grid quality (q ≈ 0.76) and far less numerical artifacts.Finally, the obtained numerical seismograms are compared to seismogramsprovided by a standard FD scheme discretizing the same model. The compar-ison confirms the results of Chapter 3, where the influence of grid irregular-ity on the accuracy of numerical seismograms has been investigated. Slight,residual artifacts, observed in the simulation results of the irregular gridmethod are mainly due to the choice of the grid design and the correspond-ing staggering scheme. As mentioned before, Shewchuck’s mesh generatordoes not provide perfect equilateral triangles in regions of the model, wherethe geometry of boundaries has no influence. The results could definitely beimproved by a more ingenious technique of generating grids, that use equi-lateral triangles in major regions as well as adapt to curved boundaries bydistorting the hexagonal structure as slightly as possible.Nevertheless, the discussed examples show the feasibility of propagating seis-mic waves through completely unstructured grids using an explicit derivativeoperator.

General Conclusions

The aim of this research has been to investigate the accuracy of explicitderivative operator on irregular grids. As the development of an algorithmto compute the complete seismic wave field in a three-dimensional Earth isof uttermost importance, but the extension of existing regular grid methodsfor this purpose is difficult, an alternative technique is discussed capable ofhandling arbitrary grids.The investigation of the explicit differential operators has shown, how gridirregularity influences the accuracy of the computation of spatial derivatives.Whereas operators working on regular grids in general are more accurate,their application on complex geometries with curvelinear boundaries is inel-egant or even inappropriate (e.g. for cylindrical or spherical geometries).Alternatively, distorted, irregular grids can be applied to arbitrary geometry.Therefore, operators have been tested, that are capable of handling irregular,triangular grids based on Delaunay triangulations.The velocity-stress formulation of the elastic wave equation suggests the useof a staggered grid scheme defining velocities on one grid and stresses on theother.Using the derivative operator based on natural neighbour differences hasturned out to be the most accurate method followed by the finite-volumeoperators. The investigation of operator accuracy and it’s application toseismic wave propagation simulations in realistic models have shown, thatthe grid irregularity and the design of the staggered scheme are most sig-nificant. By improving the grid quality of both grids and optimizing theirlocations with respect to each other, numerical artifacts can be reduced andthe computational effort can be minimized.The comparison to regular reference grid - i.e. a quadratic FD grid anda hexagonal grid - has clarified, how errors in the computation of spatialderivatives on irregular grids can be minimized by using grids of higher nodedensity. Consequently, the major drawback is the increase of computationtime. Additionally to the greater number of necessary grid points, the num-ber of necessary time steps increases. Due to the irregularity of the grid,small grid cells may appear, that for stability reasons require smaller timesteps than regular cases.It could also be shown, that a considerable difference between simulations ofacoustic and elastic waves exists, as the accurate calculation of shear stressesturns out to be a substantial problem on irregular grids, and even on hexag-onal grids.

83

However, this research representing a feasibility study of the discussed ex-plicit operators has pointed out, that they are capable of propagating seis-mic wave through irregular grids. Especially, the local character of elasticbehaviour of materials suggests the use of such explicit local operators. Con-trary to finite-elements, which also can handle irregular grids and definitelyhave to be considered as an alternative method, the introduced explicit lo-cal operators only use their nearest neighbourhood. Therefore, they are wellsuited for parallel algorithms, which become increasingly important for large-scale problems (e.g. global seismology).The main interest in the future will be, how the investigated explicit opera-tors can be extended to the 3-D case. The actual aim is to use these explicitoperators on grids with complex geometries (e.g. the sphere) and inhomoge-neous media. This implies dense gridding in low velocity zones and coarsemeshes in regions of high velocities. Therefore, the ingenious generation ofhigh quality grids serving this purpose seems to be the greatest challenge.

84

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88

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89

Appendix A

Methods to Compute Weights

of Space Derivative Operators

A.1 Natural Neighbour Weights

A detailed description of the calculation of the natural neighbour weights isgiven by Sambridge et al. (1995). The method was originally developed forsolving interpolation problems of irregularly distributed data.The method is based on the theory of Voronoi cells and the Delaunay trian-gulation algorithm. In 2-D a set of irregular distributed nodes divides theplane into a corresponding set of regions, such that any point in a particularregion is closer to that region’s note than to any other one. These regionsare called Voronoi cells. The cells are unique and space filling and are themost fundamental geometrical constructs, when working with an irregularlyspaced set of points. Voronoi cells also uniquely define so-called Delaunaytriangles, which are formed by connecting the nodes, whose Voronoi cellsshare a common side. Delaunay triangles have the useful property, that theygenerate a mesh with optimal triangular grid cells, i.e. the grid cells are asequilateral as possible. Delaunay tessellations in turn lead to the definitionof the natural neighbours of a node, which are those grid points, to which theparticular node is connected by the sides of Delaunay triangles.As mentioned above natural neighbours were originally used to interpolateirregular data (Sambridge et al., 1995). The interpolation weights are com-puted by the relative contribution of second-order Voronoi cells as shown inFigure A.1 and are formaly given by

f(x0) ≈1

A

N∑

j=1

aj(x0)f(xj) (A.1)

where N is the number of natural neighbours, aj are the areas of second-order Voronoi cells and A is the total area of the actual Voronoi cell of the

91

point x0, which is

A =N

∑

j=1

aj(x0) . (A.2)

Extending this method to the problem of calculating derivatives is realized bydifferentiating equation (A.1) with respect to the different directions leadingto

∂if(x0) ≈1

A

N∑

j=1

∂iaj(x0)f(xj)−1

A2∂iA

N∑

j=1

aj(x0)f(xj) (A.3)

with (i = 1, 2) for the 2-D case. An extension to 3-D is straight forwardand terms of areas or triangles can be replaced by volumes or tetrahedra (seeBraun et al. 1995).

��

��

a1

2

3

4 5x x

x

x

x

1

x0A

Figure A.1: After inserting a secondary point (x0) and re-meshing the gridpoints, a new Voronoi cell (shaded) with an area A appears. It is subdividedinto second-order Voronoi cells of size aj (j = 1, ..., 5). The interpolationweight of point x1 is given by the ratio a1/A.

A.2 Finite Volume Weights

A.2.1 Using all Natural Neighbours

The finite volume method is based on Gauss’ divergence theorem. To ap-proximately estimate the gradient of a vector field f at a particular point x0

92

knowing the values of the field at the surrounding points Gauss’ divergencetheorem can be used in the 2-D formulation

∫

S

dS∂if =

∫

L

dLnif (A.4)

where f is a vector field, ni the outward normal vector and L the boundaryof the cell surface S (Figure A.2b). A detailed description of the calculationof finite volume weights using this method is given by Dormy et al. (1995).In this work we are using the natural neighbours as the vertices of the finitevolume cell as shown in Figure A.2. To calculate the derivative of the vector

a b

x0

4x x5

x1

x0

x5

x1n1

n5

n 2n

2n3

n4

1n

n5

x2

x3 x3

2x

x4

3n

n4

Figure A.2: (a) After inserting a secondary point (x0) and re-meshing thegrid points, a new Voronoi cell as shown in Figure A.1 appears. The sides ofa finite volume cell connect the surrounding neighbours and define a polygonsurrounding x0. Note that the finite volume cell is different from the shapeof the corresponding Voronoi cell. (b) The finite volume cell is shown withrespect to the triangular mesh obtained by Delaunay triangulation.

field f at the location x0 the neighbouring values are weighted by half ofthe length of each adjacent cell side. The formal expression of the discreteformulation of Gauss’ theorem is given by

∂if(x0) ≈1

S[

N∑

j=1

(∆Lj

2nj,i +

∆Lj+1

2nj+1,i)f(xj) ] (A.5)

where ∆Lj is the length of a side defined between xj and xj−1. However,this method does not consider the position of the inserted grid point x0 inthe calculation of the space derivative, as the polygon is only defined by thelocations of the neighbouring grid points xi (i = 1, ..., 5).

93

A.2.2 Using Three Neighbours

This method is identical to the method described above, with the only dif-ference that the number of neighbours is limited to three. Equation A.5 canbe used to calculate the space derivatives by substituting N = 3. The three

a b

x

x3

2x

1x

n

n n

1

2 3x0

2xx1

x 3

0

Figure A.3: After inserting the secondary node x0 a re-meshing of the gridpoints is not necessary. The problem is to find the three best neighboursdefining the triangle, which includes the grid point x0.

best neighbours, that are chosen out of the whole set of neighouring points,have to define the triangle, in which the inserted grid point x0 is located(Figure A.3). The problem of searching an unstructured grid - in this casesearching through all irregular triangles - to find the inserted node x0 insidea particular grid cell is non-trivial. Searching the grid triangle by trianglecan become very ineffective for large grids. The problem can be overcomeusing the walking triangle algorithm introduced in previous work (Sambridgeet al. 1995; Lawson 1977) and also has been extented to the 3-D case.

94

Appendix B

Calculation of Analytical

Solutions Using Green’s

Functions

To verify the results of the numerical simulations and to investigate theaccuracy of the corresponding seismograms, it is necessary to compare themto analytical solutions. The computation of analytical solutions is based onGreen’s functions.

B.1 Acoustic Medium

The Green’s function for a two-dimensional, homogeneous acoustic mediumis found by solving the scalar wave equation. Using an impulsive source inspace and time, the solution for the displacement vector is given by (e.g.Carcione & Kosloff, 1988; Morse & Feshbach, 1953)

u(r, t) = F ·H(t− r

v)(t2 − r2

v2)−1/2 , (B.1)

where F is a scaling factor of the source amplitude, r =√

x21 + x2

2 is thedistance from the receiver to the source, v is the velocity of the compressionalwave and H is the Heaviside function.

B.2 Elastic Medium

The calculation of the elastic wave field generated by an impulsive verticalforce in an homogeneous elastic medium is also based on Green’s functions(e.g. Carcione & Kosloff, 1988; Eason, Fulton & Sneddon, 1956). The solu-tion for the displacement vector (u1, u2) of an impulsive force acting in the

95

positive x2 direction can be expressed by

u1(r, t) =F

2πρ· x1, x2

r2(G1(r, t) +G2(r, t)) (B.2)

u2(r, t) =F

2πρ· 1r2(x2

2G1(r, t)− x21G2(r, t)) , (B.3)

where F is a scaling factor for the force amplitude, r =√

x21 + x2

2 is thedistance from the receiver to the source and

G1(r, t) =1

v21

(t2 − τ 21 )

−1/2H(t− τ1) (B.4)

+1

r2(t2 − τ 2

1 )1/2H(t− τ1)−

1

r2(t2 − τ 2

2 )1/2H(t− τ2) (B.5)

G1(r, t) = − 1

v22

(t2 − τ 22 )

−1/2H(t− τ2) (B.6)

+1

r2(t2 − τ 2

1 )1/2H(t− τ1)−

1

r2(t2 − τ 2

2 )1/2H(t− τ2) , (B.7)

where τν = rvν, (ν = 1, 2) with v1 and v2 being the compressional and shear

wave velocities respectively. H represents the Heaviside function.

96

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