numerical modeling of seismic wave propagation in selected...
TRANSCRIPT
1
Numerical modeling of seismic wave propagation in selected
anisotropic media
by., Danek, T., Lesniak, A, Pieta, A.
2
Preface
This study is the first step toward creation of a computer system which allows efficient
modeling of seismic wave field in all kinds of 3D media. At the beginning, this study was a
part of a project of Oil and Gas Institute, Cracow, Poland, where solutions were created and
tested under the supervision of Prof. Halina Jędrzejowska-Tyczkowska. Then the study was
continued as a part of Statutory Research of a Department of Geoinformatics and Applied
Computer Science at AGH University of Science and Technology, Cracow, Poland,
no.11.11.140.561. Of course this kind of research and performed computation would have
never been possible without appropriate computer infrastructure which was financed by
special donation from the Polish Ministry of Science and Higher Education.
Summary
This book reports a comprehensive study of numerical modeling of wave propagation in 3D
anisotropic media. Finite-difference methods were used for efficient modeling of wave
propagation in such a media. After a short introduction, the book presents the theory of
anisotropic model that describes real geological media and its approximation using horizontal,
vertical, titled transverse isotropic and orthorhombic media in the second chapter. Also, the
construction of efficient and universal wave field modeling algorithm that is deduced from the
basic ones using stiffness matrix transformation by Bond matrices is shown. The third
chapter presents numerical solution of the elastic wave equation in various types of
anisotropic media. The second order in space and time finite-difference method was used to
obtain a detailed solution for all considered types of anisotropic media. Stability criteria for all
discussed algorithms are also presented. Fourth order in time finite-difference method is
presented additionally as an example of alternative numerical solution. In the next chapter
implementation of the finite-difference schemas including initial and border conditions and
computational environment is discussed. In the last chapter example results of wave
propagation modeling for different types of anisotropy and for various seismic sources are
presented and discussed.
3
Contents
1. Introduction (ALeśniak, TDanek, APięta)...........................................................................5
2. Description of anisotropic models of real geological medium and its approximation (VTI,
HTI, TTI, orthorhombic media)(ALeśniak).............................................................................6
2.1. Introduction: Analogies to crystallography...................................................................6
2.2. Symmetry of stiffness tensor........................................................................................8
2.3. Anisotropy types of rock media....................................................................................9
2.4. Transformations of stiffness tensor.............................................................................12
2.5. Effective parameters for VTI medium........................................................................17
2.6. Thomsen parameters for VTI medium........................................................................22
2.7. Thomsen parameters for orthorhombic medium .........................................................24
3. Numerical solution of the elastic wave equation (APięta) .................................................26
3.1. Rhombic, transversely isotropic and isotropic media ..................................................27
3.2. Tilted transverse anisotropy media .............................................................................29
3.2.1 Rotation of vertical transverse isotropy medium around x1 axis by angle θ ........................ 29 3.2.2 Rotation of vertical transverse isotropy medium around x2 axis by angle θ ........................ 30 3.2.3 Rotation of horizontal transverse isotropy medium around x3 axis by angle θ .................... 31
3.3. Numerical solution of equation of motion using finite-difference method...................33
3.4. The stability criterion of second-order finite-difference schema .................................35
3.4.1 The stability criterion for vertical transverse isotropy medium............................................. 35 3.4.2 The stability criterion for horizontal transverse isotropy medium ........................................ 36 3.4.3 The stability criterion for rhombic isotropy medium ............................................................. 36 3.4.4. The stability criterion for tilted transverse anisotropy media - rotation of vertical transverse
isotropy medium around x1 axis by angle θ .................................................................................. 37 3.4.5. The stability criterion for tilted transverse anisotropy medium - rotation of vertical
transverse isotropy media around x2 axis by angle θ ................................................................... 38 3.4.6. The stability criterion for tilted transverse anisotropy medium - rotation of horizontal
transverse isotropy media around x3 axis by angle θ ................................................................... 39 4. Implementation of the finite-difference schemas (TDanek) ...............................................41
4.1. Initial and border conditions.......................................................................................41
4.2 Seismic wave propagation, finite-difference algorithm................................................41
4.3 Computational environment........................................................................................42
5. Example results of wave propagation modeling in anisotropic media(ALeśniak, TDanek,
APięta) .................................................................................................................................43
4
5.1 Sources .......................................................................................................................45
5.1.1 Explosive source.................................................................................................................. 45 5.1.2 Double couple source .......................................................................................................... 45 5.1.3 Single force source .............................................................................................................. 45
5.2 Selected results and discussion....................................................................................45
5.2.1 Isotropic medium.................................................................................................................. 45 5.2.2 HTI medium ......................................................................................................................... 46 5.2.3 VTI medium.......................................................................................................................... 47 5.2.4 Orthorhombic medium ......................................................................................................... 48 5.2.5 Heterogeneous orthorhombic medium ................................................................................ 49
6. Summary(ALeśniak, TDanek, APięta)..............................................................................53
7. References........................................................................................................................54
Appendix 1. Matrix notation for constitutive relations ..........................................................56
Appendix 2.The finite-difference schema for horizontal transverse isotropy media...............57
Appendix 3. The finite-difference schema for tilted transverse anisotropy media - rotation
around x1 axis .......................................................................................................................58
Appendix 4. The finite-difference schema for tilted transverse anisotropy media - rotation
around x2 axis .......................................................................................................................59
Appendix 5. The finite-difference schema for tilted transverse anisotropy media - rotation
around x3 axis .......................................................................................................................60
Appendix 6. The fourth-order in time and second-order in space finite-difference schema for
transversely isotropic media .................................................................................................61
Appendix 7. Main computational kernel loop .......................................................................64
Appendix 8. Modeling results...............................................................................................66
5
1. Introduction
Full-wave field modelling is a powerful tool of seismic exploration and seismology. It can be
used for various earthquake-related analyses and during many stages of seismic
investigations. Recent rapid progress in full-wave form seismic inversion methods has made
these simulations even more important.
The seismic wave field propagation is studied in simplified geological media. The heterogenic
medium is one of the common models of the real, complicated subsurface structure of earth. It
consists of homogeneous parts of different shapes and sizes. In most cases they are considered
as isotropic ones; e.g., mechanical parameters inside each part are constant and independent
of directions. But in an anisotropic medium, parameters that are measured depend on the
direction. The anisotropic and heterogeneous medium can be nowadays regarded as a
sufficient approximation of the real medium of seismic wave propagation.
Generally, the more complicated analyzed medium is, the more important correct
understanding of anisotropy is necessary. In the case of three dimensional (3D) anisotropic
media, abundance of possible waves and interactions between them can make wave form
analysis very complicated even if there is lack of any heterogeneity (e.g. layer borders).
Analysis of synthetic seismograms and modeled snapshots can be very interesting not only for
theoretical studies but also for solving real, “production” problems (Jędrzejowska-
Tyczkowska, 2006). Sometimes one synthetic seismogram or snapshot can reveal the true
nature and/or genesis of observed complicated wave forms. In complicated anisotropic media,
various wave forms can be created by simple wave propagation through homogeneous
medium. In a detailed analysis anisotropy can no longer be treated as a simple variation of
seismic wave speed with direction. Various waves, sometimes surprisingly hard to interpret,
can be observed in the case of even very limited anisotropy. On the other hand very subtle
distortion of wave speed symmetry can be crucial for final seismic or seismological
interpretation. Proper understanding of all effects of seismic anisotropy seems to be one of the
most important problems of modern geophysics.
The beginning of these studies can be traced back to 2004 when a series of papers concerning
wave filed modeling in complex and/or anisotropic media were published by authors (Leśniak
& Danek 2004; Danek & Franczyk 2004a, 2004b, Danek 2004). One year later these works
were continued as a part of European Union program – HPC Europa (Danek 2005a, 2005b).
Obtained results were a background to more mature studies which were conducted later (e.g.
Leśniak & Danek 2006, Pietsh et al. 2007). Finally during the second HPC Europe
cooperation (Danek 2007) additional efficiency studies for big, 3D models were carried out.
These results led to some complicated, full scale simulation (e.g. Danek et al. 2008, Pięta et
al. 2009) and to studies concerning alternative ways of wave filed modeling (Danek 2009).
All this experience in wave field propagation theory, computational methods and parallel
computing gave a proper background for the most ambitious task – full scale 3D modeling in
anisotropic media.
6
2. Description of anisotropic models of real geological medium and its
approximation (VTI, HTI, TTI, orthorhombic media)
2.1. Introduction: Analogies to crystallography
At microscopic scale, anisotropy is related with crystals. Anisotropic properties of crystals
reflect their periodic structure at atomic scale. The terminology and basic models of
anisotropy that are used in geophysics are based mainly on information and names from
crystallography. Hence, there is a direct reference to the crystallography in the first chapter of
this publication.
Properties of anisotropic media can be represented by tensors. In the case of anisotropic media
measured physical properties depend on direction. A response of a crystal to the applied force
will not depend on force amplitude alone but on its direction as well. The manner in which a
given crystal will change its properties along with changes of the direction depends on its
symmetry. If a 3D crystal has defined planes of symmetry, its form will not change after the
symmetrical reflection about those planes.
Likewise, physical parameters measured in that direction do not change as a result of such
transformations. In other words, it is symmetry planes that determine the directions in which
physical parameters of a given crystal do not change. If stresses are applied in those
directions, the deformations will be the same. This also means that if a medium is
symmetrical about given transformation, the transformation does not change the stiffness
matrix that describes the medium.
The 3D space periodic structures (like crystals) can be constructed only in 32 ways: so-called
32 classes of symmetry that are grouped in seven systems that are listed below (e.g. Auld,
1973):
- triclinic
- monoclinic
- rhombic
- hexagonal
- rhombohedral (trigonal)
- tetragonal
- regular
The symmetry planes for basic classes of symmetry are shown in Fig.1.
7
Fig.2.1. Symmetry planes for basic symmetry systems; a) monoclinic, b) tetragonal, c) rhombic, d) hexagonal, e)
rhombohedral, f) cubic (modified after Crampin 1984)
Rys. 2.1. Płaszczyzny symetrii podstawowych systemów symetrii. a) jednoskośny b) tetragonalny c) rombowy d)
heksagonalny e) romboedryczny f) kubiczny(Crampin 1984, zmodyfikowany)
Generally, more than 32 symmetry classes can be distinguished for a rock medium
(Winterstein, 1990). For instance, a plane-parallel medium is isotropic in any horizontal
direction. Any vertical plane is the symmetry plane for such a medium. This does not happen
in crystals. The same type of anisotropy is observed for a homogeneous medium with a
system of cracks with horizontal boundaries. Such media are anisotropic with so-called
vertical transverse isotropy (VTI). Adding two systems of parallel cracks with plane
boundaries into an anisotropic medium gives anisotropy that simultaneously belongs to
triclinic, rhombic, and tetragonal systems. Three-crack systems in a homogeneous medium
give anisotropy which is simultaneously included to triclinic, rhombohedric, hexagonal, and
regular systems.
Of greatest importance to geophysicists are media with vertical transverse isotropy; however,
other types of anisotropic media can be employed in seismics. We shall discuss them later in
this study.
8
2.2. Symmetry of stiffness tensor
Anisotropic properties in geophysics are analyzed through studying the symmetry of stiffness
tensor that links stress with deformation:
3,2,1,,, == lkjic klijklij εσ
It is known that anisotropic systems can be classified based on the mutual position of their
symmetry planes (Crampin 1984). One can prove a theorem saying that ‘a plane 0=px is
the symmetry plane of an anisotropic system with stiffness tensor ijklc if and only if one or
three indexes i,j,k,l are equal to p’ (Crampin 1984).
To illustrate how this theorem acts, the stiffness tensor must be somewhat simplified.
Theoretically, it has 34
= 81components; however, they are not mutually independent.
Because stress and deformation are symmetric
lkkljiij εεσσ == (2.1)
we get the following identities
jilkijlkjiklijkl cccc === (2.2)
which reduce the number of independent components of the stiffness tensor to 36. In turn, the
identity
klijijkl cc = (2.3)
reduces that number to 21. The components can be written as a 6 x 6 symmetrical matrix (see
Appendix 1).
αβ
1111111
11
11
1111111111111111
ijkl C
cccccc
cccccc
cccccc
cccccc
cccccc
cccccc
cccccc
cccccc
cccccc
cccccc
cccccc
cccccc
C =
=
=
665646362616
565545352515
464544342414
363534332313
262524232212
65432
121212131223123312221211
131213131323133313221311
231223132323233323222311
3312331333233333332233
2212221322232233222222
23233322
(2.4)
Let a rhombic system with symmetry planes 01 =x , 02 =x and 03 =x be an example. Using
the Crampin theorem we conclude that the stiffness matrix for rhombic system has the form
9
=
66
33313
31
13111
αβ
c00000
0c0000
00c000
000ccc
000ccc
000ccc
C
55
44
2
2222
2
(2.5)
with nine independent elements.
2.3. Anisotropy types of rock media
Throughout this book appropriate stiffness matrices are used in computer programs for wave-
field modeling. Each anisotropic medium has a specific stiffness matrix. What is important,
the number of independent matrix elements decreases when symmetry of the medium
increases.
The simplest medium we can consider is homogeneous and isotropic: only two components of
elastic tensor Cijkl are mutually independent. A symmetric tensor has the following form:
−−
−−
−−
=
44
44
44
44334433
4433334433
4433443333
c00000
0c0000
00c000
000c2cc2cc
0002ccc2cc
0002cc2ccc
C33
αβ
(2.6)
The Lame constants are given as: c44 = µ and c33 = λ+2µ.
From a practical point of view of the seismic exploration, three types of anisotropic media are
distinguished:
1. With hexagonal symmetry (transverse isotropy) in which physical properties do not change
perpendicular to the symmetry axis but they change parallel to it. Four types of anisotropy are
distinguished in these media:
• Vertical transverse isotropy (VTI) - for almost horizontal plane-parallel finely layered
media; the symmetry axis is vertical.
• Horizontal transverse isotropy (HTI) – for almost vertical plane-parallel finely layered
media; the symmetry axis is horizontal.
10
Fig. 2.2. Models of VTI and HTI finely layered media with symmetry axis shown.
Rys. 2.2 Model ośrodka VTI i HTI zbudowane z ośrodków cienkowarstwowanych z zaznaczonymi osiami symetrii.
• Extensive dilatancy anisotropy (EDA) – caused by cracks with orientation different
from horizontal; the symmetry axis is different from vertical.
• Tilted transverse anisotropy (TTI) – for tilted plane-horizontal finely layered media;
the symmetry axis is different from vertical.
There are five independent components of stiffness tensors for these types of media. They
have the forms
− for VTI anisotropy:
−
−
=
66
331313
13116611
13661111
c00000
0c0000
00c000
000ccc
000cc2cc
000c2ccc
C
55
55
)(VTI
αβ
(2.7)
− for HTI anisotropy:
(2.8)
To explicitly obtain the stiffness matrix for TTI medium, the matrix for VTI medium
must be rotated around the corresponding horizontal axis.
−
−
=
55
55
44
4433
443333
)(
c00000
0c0000
00c000
000c2ccc
0002cccc
000ccc
C3313
13
131311
HTI
αβ
11
2. Second, there is orthorhombic (rhombic) anisotropy that is a superposition of VTI and
EDA. It occurs for a vertical system of cracks in horizontally stratified media (e.g.
limestones). Usually VTI >> EDA.
Fig. 2.3. A scheme of orthorhombic system – plane-parallel medium with a system of vertical cracks (modified after
Tsvankin, 2001).
Rys. 2.3 Schematyczny model układu ortoromboidalnego - ośrodek płasko -równoległy wraz z nałoŜonym układem
pionowych szczelin (Tsvankin, 2001, zmodyfikowany).
The stiffness matrix for orthorhombic anisotropy has the form below.
=
66
33313
31
13111
αβ
c00000
0c0000
00c000
000ccc
000ccc
000ccc
C
55
44
2
2222
2
(2.9)
As it has been mentioned earlier, there are nine independent components of the stiffness
matrix.
3. Third, there is a monoclinic anisotropy that is observed if a system of non-vertical cracks
occurs in horizontally stratified systems. The monoclinic anisotropy also exists in media
where two, nonorthogonal systems of cracks exist.
12
Fig. 2.4. Two systems of parallel cracks forming monoclinic anisotropy. There, the symmetry plane is horizontal
(modified after Tsvankin, 2001).
Rys. 2.4 Dwa systemy równoległych szczelin tworzących układ jednoskośny. Płaszczyzną symetrii jest w tym wypadku
płaszczyzna pozioma (Tsvankin, 2001, zmodyfikowany).
The stiffness tensor has the following form:
=
661
33313
31
113111
αβ
c00ccc
0cc000
0cc000
c00ccc
c00ccc
c00ccc
C
36266
5545
4544
362
262222
62
(2.10)
An interesting case occurs when 0
21 90=+φφ . Then the above shown system reduces to the
rhombic system.
Theoretically, at least a dozen or so other symmetry systems can be distinguished for the
elastic tensor, and hence the same number of anisotropy types. However, only the three
above-mentioned types were found during seismic field survey. Maybe other types of
anisotropy do not appear in nature, possibly because of physical reasons.
2.4. Transformations of stiffness tensor
The more complicated anisotropy models can be evaluated from the basic ones (described
above) through application of the specific transformation of the stiffness tensor. This stiffness
tensor transformation is achieved when a specific rotation is made around a given symmetry
axis. Since rotations belong to orthogonal transformations, they can be characterized by
matrix { } 3,2,1, == jiaija that satisfies conditionsTaa =−1
, Iaa =Tand ( ) 1det =a for
rotation and ( ) 1det −=a for symmetry. Equations that transform stress tensors and
13
deformation tensors to a new (‘) coordinate system (the Einstein’s summation convention is
consistently applied) are
kljlikij aa εε =′
kljlikij aa σσ =′ (2.11)
The equations have the following form in matrix notation:
Nεε =′ (2.12)
Mσσ =′
where N and M are transformation matrices from one coordinate system to the other. The
explicit forms of matrices N and M are as follows:
+++
+++
+++=
211222112311211322132312231322212111
311232113311311332133312133312321131
322131223123331232233322332332223121
323131333332
2
33
2
32
2
31
222121232322
2
23
2
22
2
21
121111131312
2
13
2
12
2
11
222
222
222
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
N
+++
+++
+++=
211222112311211322132312231322212111
311232113311311332133312133312321131
322131223123331232233322332332223121
323131333332
2
33
2
32
2
31
222121232322
2
23
2
22
2
21
121111131312
2
13
2
12
2
11
222
222
222
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
M
(2.13)
The N matrix and M matrix are known as ‘Bond matrices’. Using formula (2.1) and because
matrix a is orthogonal, one can easily prove that T1 MN =− . It can be concluded that
εCσ ′′=′
1MCNC −=′ (2.14)
and finally:
TMCMC =′ (2.15)
For rotation around the vertical axis, the matrix of rotation by angle θ has the form
14
−
=
100
0cossin
0sincos
θθ
θθ
a
(2.16)
Similarly, for rotation around x1 and x2 axis there is
−=
θθ
θθ
cossin0
sincos0
001
a
−
=
θθ
θθ
cos0sin
010
sin0cos
a
(2.17)
The two above given formulas enable the explicit expressions for stiffness matrix for specific
anisotropy models to be obtained.
As the first example, let us consider a TTI medium (a VTI medium rotated by angle θ around
x1 or x2 axis). First, we must determine the Bond matrices for those rotations. For rotation
around X1 axis the Bond matrix is
−
−
−
=
θθ
θθ
θθθ
θθθ
θθθ
cossin0000
sincos0000
002cos2sin2sin0
002sincossin0
002sinsincos0
000001
22
22
M
(2.18)
whereas for rotation around X2 axis it is
−
−−
=
θθ
θθθθθ
θθ
θθθ
θθθ
cos0sin000
02cos0cossin0cossin
sin0cos000
02sin0cos0sin
000010
02sin0sin0cos
22
22
M
(2.19)
To obtain explicit expressions for stiffness matrix components for TTI (rotation around x1
axis), we must perform pretty arduous matrix multiplication:
−
−
−
θθ
θθ
θθθ
θθθ
θθθ
cossin0000
sincos0000
002cos2sin2sin0
002sincossin0
002sinsincos0
000001
22
22
−
−
66
55
55
331313
13116611
13661111
00000
00000
00000
000
0002
0002
c
c
c
ccc
cccc
cccc
−
−
−
θθ
θθ
θθθ
θθθ
θθθ
cossin0000
sincos0000
002cos2sin2sin0
002sincossin0
002sinsincos0
000001
22
22
As a result we get the following expressions for stiffness tensor coefficients for TTI:
c'11= 11c
15
c'21= c'12= ( ) θθ 2
13
2
6611 sincos2 ccc +−
c'31= c'13= ( ) θθ 2
13
2
6611 cossin2 ccc +−
c'41= c'41= ( ) θθ 2sin2sin2 136611 ccc −−
c'22= θθθθθ 2sinsincos2sincos2
55
22
13
4
33
4
11 cccc +++
c'23= c'32= ( ) ( ) θθθθθ 2sincossincossin 2
55
22
1133
44
13 cccc −+++
c'24= c'42= θθθθθθθθ 2cos2sin2sinsin2cossin2sincos 55
2
33
2
13
2
11 cccc −−−
c'33= θθθθθ 2sincoscossin2sin 2
55
4
33
22
13
4
11 cccc +++
c'43= c'34= θθθθθθθ 2sin2sincos2sin2cos2sinsin2
55
2
3313
2
11 cccc −−+
c'44= ( ) θθ 2cos2sin2 2
55
2
331311 cccc ++−
c'55= θθ sincos 6655 cc +
c'65= c'56= ( ) θθ cossin5566 cc −
c'66= θθ cossin 6655 cc + (2.20)
Other coefficients of the stiffness matrix are equal to zero. Stiffness matrix coefficients for
rotation around x2 axis can be calculated in the same way. The values are as follows:
θθθθθθθ cossin2sinsinsincos2cos 55
4
33
22
13
4
11
'
11 ccccc +++=
( ) θθ 2
13
2
6611
'
21
'
12 sincos2 ccccc +−==
( ) ( ) θθθθθθθ cossin2sincossinsincos 55
44
13
22
3311
'
13 ccccc −+++=
( ) θθθθ 2sin2cossincos 55
2
3313
2
11
'
15 ccccc −++=
11
'
22 cc =
( ) θθ 2
13
2
6611
'
32
'
23 cossin2 ccccc +−==
( ) θ2sin2 136611
'
25 cccc +−=
( ) ( ) θθθθθθθ sincos2sinsincossincos 55
44
13
22
3311
'
31 ccccc ++++=
θθθθθθθ cossin2sincossincos2sin 55
4
33
22
13
4
11
'
33 ccccc −++=
16
( ) θθθθ 2sin2coscossin 55
2
3313
2
11
'
35 ccccc −++= (2.21)
θθ 2
66
2
55
'
44 sincos ccc −=
( ) θθ sincos6655
'
46 ccc +=
( )( )θθθθθθθ 2cossincossincoscossin 55
2
33
22
13
2
11
'
51 ccccc −−−+=
( ) θθ cossin2 136611
'
52 cccc −−=
( )( )θθθθθθθ 2coscossincossincossin 55
2
33
22
13
2
11
'
53 ccccc +−−+=
( ) θθθθ 2cos2sincossin 2
553311
'
55 cccc +−=
( ) θθ sincos6655
'
46
'
64 cccc +−==
θθ 2
66
2
55
'
66 cossin ccc +−=
Let the stiffness matrix transformation around x1 axis by angle θ for HTI be the third
example.
−
−−
=
−
−
=
001
sincos0
cossin0
001
010
100
100
0cossin
0sincos
θθ
θθ
θθ
θθ
a
(2.22)
The Bond matrix has the form:
−−
−−
−
−
=
002cos2sin2
12sin2
10
sincos0000
cossin0001
000000
002sinsincos0
002sincossin022
22
θθθ
θθ
θθ
θθθ
θθθ
M
(2.23)
To obtain an expression for the stiffness matrix for HTI medium rotated by θ around the
vertical axis, we must perform the transformation TMCMC =′ , where C is the stiffness
matrix for VTI medium. As a result we get the following expressions for stiffness tensor
coefficients for TTI medium:
c'11= ( )( ) θθθ 4
11
2
5513
4
33 sin2sin221cos cccc +++
17
c'21= c'12= ( ) ( )[ ]θ4cos424681 5533131155331311 cccccccc −+−−−++
c'31= c'13= θθ 2
13
2
12 cossin cc +
c'61= c'16= ( ) ( )[ ] θθ 2sin2cos4241 553313113311 cccccc −+−++−
c'22= ( )( ) θθθ 2sin221sincos 2
5513
4
33
4
11 cccc +++
c'23= c'32= θθ 2
13
2
12 sincos cc +
c'26= c'62= ( ) ( )[ ] θθ 2sin2cos4241 553313113311 cccccc −+−−+−
c'33= 11c
c'63= c'36= ( )( ) θ2sin21 1213 cc −
c'44= ( )( ) θθ 2
55
2
1211 sincos21 ccc +−
c'55= ( )( ) θθ 2
55
2
1211 cossin21 ccc +−
c'45= c'54= ( )( ) θ2sin241 551211 ccc ++−
c'66= ( ) ( )[ ]θ4cos424281 5533131155331311 cccccccc −+−−++− (2.24)
Using the above given expressions and examples one can relatively easily determine stiffness
matrix coefficients for plane-parallel media with different orientations of the symmetry axis,
for example from VTI through TTI to HTI media.
2.5. Effective parameters for VTI medium
Sedimentary structures often take cyclic forms with plane-parallel boundaries, as it is shown
in the figure below.
18
2.5. Cyclic form of sedimentary structure
Rys. 2.5. Forma cykliczna struktury sedymentacyjnej.
For long-enough wave lengths, the structure shown in the figure will behave as a
homogeneous anisotropic structure.
Fig.2.6. Schematic image of a stratified structure that behaves as an anisotropic medium
Rys. 2.6 Schematyczny obraz struktury warstwowanej zachowującej się jak ośrodek anizotropowy.
At a macroscopic scale, seismic anisotropy in stratified media occurs when the wavelength, λλλλ,
is much bigger than layer thickness, hi. It can be assumed that the following condition must be
satisfied:
19
max8h≥λ (2.25)
Let us assume that a VTI structure is periodic and built of homogeneous and isotropic layers
with thickness 1h and 2h . Each layer is described by the stiffness matrix having two
parameters, λ and µ (Lame constants). On the other hand, an equivalent VTI medium is
characterized by the stiffness matrix with five independent coefficients (6655331311 ,,,, ccccc ). It
should be remembered that 661112 2ccc −= . One can prove that stiffness matrix coefficients
for such a structure (Postma, 1955) take the following form:
c11= ( ) ( )( ) ( ) ( ) ( )[ ]{ }22121212211
2
21 224221
µλµλµµµλµλ +−+−++++ hhhhL
c21= ( ) ( )( ){ }2112221121
2
21 21
hhhhhhL
µµλλλλ ++++
c31= ( ) ( ) ( )[ ]{ }1122221121 221
µλλµλλ ++++ hhhhL
c33= ( ) ( )( ){ }2211
2
21 221
µλµλ +++ hhL
c55=( )
1221
2121
µµ
µµ
hh
hh
+
+
c66=( )
( )21
2211
hh
hh
+
+ µµ
(2.26)
where ( ) ( ) ( )[ ]11222121 22 µλµλ ++++= hhhhL .
A different situation is observed for non-periodic media. We assume that a plane-parallel
medium is built of L single layers whose thickness is much smaller than the dominating
seismic wave length; in this case transverse isotropy rather than isotropy is assumed, with the
symmetry axis perpendicular to stratification. Moreover, the stationarity of layer sequence is
assumed. This means that the percent contribution of individual components is stable for
packets whose thickness is much smaller than the wavelength. In such a case, stiffness matrix
coefficients are a combination of average coefficient values for individual layers (Backus,
1962):
c11=2
1
3313
11
33
1
33
2
1311
−−
−− +− cccccc
c21= 2
1
3313
11
33
1
33
2
1312
−−−− +− cccccc
c31= 1
3313
11
33
−−−ccc
20
c33=1
1
33
−−c
c55=1
1
55
−−c
c66= 66c (2.27)
where the value in cone brackets is the weighted mean of corresponding parameters:
∑=
=L
i
iiawa1
(2.28)
where the weighting value iw depends on layers’ thicknesses.
The results shown above were extended to sequences of arbitrary, thin (as compared to
seismic wavelength) anisotropic layers (Schoenberg, Muir, 1989). Let us assume such a
sequence satisfies the stationary conditions. Let )(n
ijσ be a stress tensor for n-th layer, )(n
klε be a
deformation tensor for that layer, and )(n
ijklc be its stiffness tensor. Changing from fourth-order
tensor notation to second-order tensors (as explained above and in Appendix 1), the
generalized Hooke’s law for n-th layer can be written:
6,5,4,3,2,1,)()()( == kjcn
k
n
jk
n
j εσ (2.29)
We assume that all stress components that act tangentially to stratification are identical for
each layer. A similar assumption is taken for deformation components parallel to
stratification. Other components of both stress and deformation can change from layer to
layer.
Let us define the following vectors:
- tangent stress vector [ ]Tnnnn
TS)(
6
)(
2
)(
1
)( ,, σσσ=
- normal stress vector [ ]TNS 543 ,, σσσ=
- tangent strain vector [ ]TTE 621 ,, εεε=
- normal strain vector [ ]Tnnnn
NE)(
5
)(
4
)(
3
)( ,, εεε= .
Using that notation we can write the Hooke’s law in the following form:
)()()()( n
N
n
TNT
n
TT
n
T ECECS +=
)()()( n
N
n
NNT
n
NTN ECECS += (2.30)
where:
21
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
=nnn
nnn
nnn
n
TT
ccc
ccc
ccc
C
662616
262212
161211
)(
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
=nnn
nnn
nnn
n
TN
ccc
ccc
ccc
C
564636
252423
151413
)(
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
=nnn
nnn
nnn
n
NN
ccc
ccc
ccc
C
554535
454434
353433
)(
while )(n
NTC is a transposition of )(n
TNC .
Coefficients of an equivalent homogeneous medium are obtained as a result of averaging
procedure. Solving the above equations for )(n
TS and )(n
NE we get:
( ) N
n
NN
n
TNT
n
NT
n
NN
n
TN
n
TT
n
T SCCECCCCS )(1)()()(1)()()( −− +−=
N
n
NNT
n
NT
n
NN
n
N SCECCE)(1)()(1)( −− +−= .
(2.31)
The first equation gives values of tangent stresses in each layer. The average of those stresses
i.e. the sum over all layers divided by the total layer thickness gives the mean force tangent to
stratification of the medium. Through averaging procedure the second equation gives the
mean thickness change of the layer packet.
Through weighted averaging (the weighting coefficients are equal to thicknesses of the layers)
we get
( ) NNNTNTNTNNTNTTT SCCECCCCS 11 −− +−=
NNNTNTNNN SCECCE 11 −− +−= (2.32)
where ⋅ stands for weighted averaging procedure.
Solving the second equation for NS and substituting the result into the first equation we get:
NTNTTTT ECECS +=
NNNTNTN ECECS += (2.33)
where
22
11
−−= NNNN CC
NNNNTNTN CCCC 1−=
NTNNNNNNTNNTNNTNTTTT CCCCCCCCCC 111 −−− +−=
The above given equations describe relationships between stress and strain in a homogeneous
medium that is equivalent to a medium consisting of N anisotropic layers (arbitrary
anisotropy).
2.6. Thomsen parameters for VTI medium
There are several disadvantages of the conventional notation for a VTI medium:
• inconvenient evaluation of a strength of anisotropy (particularly for small offsets that
are used in seismic surveys); no possibility to evaluate anisotropy of P waves
propagating in near-vertical direction;
• subordination of P-wave and SV-wave propagation to four parameters (c11, c33, c55,
c13); in the so-called “Thomsen notation”, the number of parameters responsible for P-
wave and SV-wave propagation can be reduced to three;
• due to the relationship between c55 and c13, the inversion of P section does not allow
the above parameters to be reproduced ;
• expressions for kinematic corrections are very complicated; they can be simplified
using the Thomsen parameters.
For VTI medium, Thomsen (1986) proposed substituting five independent material constants
with combination of them so that new constants could better describe seismic wave
propagation. If we take that
ρ33
0
cVP =
ρ55
0
cVS =
33
3311
2c
cc −=ε
( ) ( )( )553333
2
5533
2
5513
2 ccc
cccc
−
−−+=δ
55
5566
2c
cc −=γ
(2.34)
where ρ is medium density, then P-wave and SV-wave propagation will be described by first
four constants while SH-wave propagation will depend on parameters VS0 and γ alone. If the
last three parameters take small values, we can get approximated expressions for longitudinal
and shear wave propagation depending on θ angle (Thomsen, 1986):
( ) ( )
( ) ( )
( ) ( )θγθ
θθδεθ
θεθθδθ
2
0
222
0
2
00
422
0
sin1
cossin1
sincossin1
+=
−+=
++=
SSH
s
pSSV
PP
VV
V
VVV
VV
(2.35)
23
Constants VP0 and VS0 are, respectively, P-wave and S-wave propagation velocities in
direction parallel to the symmetry axis of VTI medium (perpendicular to stratification).
Analyzing expressions for P-wave and S-wave propagation velocities one can find that
coefficient ε controls P-wave velocity for near horizontal propagation (big values of θ angle).
In turn, for small θ angles (almost vertical propagation that is a frequent case in practice) it is
coefficient δ that controls anisotropic properties for P-wave propagation.
Fig. 2.7. P-wave rays (black lines) and wave fronts (white solid lines) in VTI medium with parameters εεεε = 0.1 and δδδδ =
-0.1. P-wave front for isotropic medium is marked in white dashed line (Tsvankin, 2001).
Rys 2.7 Promienie fali podłuŜnej (czarne linie proste) i fronty falowe (linie białe ciągłe) w ośrodku VTI o parametrach
εεεε=0.1 oraz δδδδ= -0.1. Front falowy fali podłuŜnej dla ośrodka izotropowego zaznaczono białą linią przerywaną
(Tsvankin, 2001).
Phase velocities obtained from Thomsen relations are shown in polar diagrams that illustrate a
dependence of P-wave and SV-wave phase velocity on phase angle θ for two anisotropic
media (Fig. 2.8). Values for components of stiffness tensor and corresponding Thomsen
parameters for both media are given in Table 2.1.
Table 2.1. Parameters of two sample anisotropic media of VTI type
ρ C11 C13 C33 C55 C66 VP0 VS0 ε δ γ
Layer 1 2000 18 6.4 12.0 5.5 4.2 2.45 1.66 0.25 0.64 -0.12
Layer 2 2100 40 13.0 33.0 12.0 8.0 3.96 2.39 0.106 0.13 -0.167
24
Fig. 2.8. Direction distribution of P-wave (solid line) and SV-wave (dashed line) phase velocity depending on phase
angle θθθθ for two anisotropic media with parameters listed in Table 1. A – layer 1, B- layer 2.
Rys 2.8 Rozkład kierunkowy prędkości fazowej propagacji fal sejsmicznych P (linia ciągła) i SV (linia przerywana) w
zaleŜności od kąta fazowego θ dla dwóch ośrodków anizotropowych scharakteryzowanych parametrami
przedstawionymi w Tabeli 1. A – warstwa 1, B – warstwa 2.
2.7. Thomsen parameters for orthorhombic medium
A HTI model is merely the first approximation of an anisotropic medium in which the
velocity depends on the azimuth. A more viable model of real anisotropic medium is an
above-defined orthorhombic (rhombic) medium that has two vertical symmetry planes shown
in Fig. 2.3 as [x1,x3] i [x2,x3].
Thomsen parameters for a rhombic medium are defined as follows:
1) equivalent of parameter ε for symmetry plane [x1,x3]
( )
33
33112
2c
cc −=ε
(2.36a)
2) equivalent of parameter δ for symmetry plane [x1,x3]
( ) ( ) ( )( )553333
2
5533
2
55132
2 ccc
cccc
−
−−+=δ
(2.36b)
3) equivalent of parameter γ for symmetry plane [x1,x3]
( )
44
44662
2c
cc −=γ
(2.36c)
4) equivalent of parameter ε for symmetry plane [x2,x3]
( )
33
33221
2c
cc −=ε
(2.36d)
25
5) equivalent of parameter δ for symmetry plane [x2,x3]
( ) ( ) ( )( )443333
2
4433
2
44231
2 ccc
cccc
−
−−+=δ
(2.36e)
6) equivalent of parameter γ for symmetry plane [x2,x3]
( )
55
55661
2c
cc −=γ
(2.36f)
7) equivalent of parameter δ for symmetry plane [x1,x2]
( ) ( ) ( )( )661111
2
6611
2
66123
2 ccc
cccc
−
−−+=δ
(2.36g)
8) vertical velocity for P wave
ρ33
0
cVP =
(2.36h)
9) vertical velocity for S wave polarized in x2 axis direction
ρ44
0
cVS =
(2.36i)
Both VTI medium and HTI medium are special cases of rhombic anisotropy. It reduces to
VTI medium when parameters of the medium are identical around symmetry planes [x1,x3]
and [x2,x3] and P-wave and S-wave velocity in [x1,x2] plane are constant.
26
3. Numerical solution of the elastic wave equation
In this chapter numerical solution of equation of motion in anisotropic medium is presented.
The second-order in space and time finite-difference method was used to obtain a detailed
solution for vertical, horizontal and rhombic transverse isotropy and tilted transverse
anisotropy media. Stability criteria for all discussed algorithms are also presented. As an
example of alternative numerical solution the fourth order in time finite-difference method is
presented additionally.
The equation of motion for perfectly elastic medium can be written as
zyxt
w
zyxt
v
zyxt
u
zzzyzx
zyyyyx
zxyxxx
∂
∂+
∂
∂+
∂
∂=
∂
∂
∂
∂+
∂
∂+
∂
∂=
∂
∂
∂
∂+
∂
∂+
∂
∂=
∂
∂
σσσρ
σσσρ
σσσρ
2
2
2
2
2
2
(3.1)
where ρ is density,
( )wvuu ,,r
is displacement vector and { }zyxjiij ,,,; ∈σ is stress tensor.
After applying relationship between stress and strain (3.2) to the equation (3.1)
εσ C= (3.2)
where
[ ]Tzxyzxyzzyyxx σσσσσσσ = is stress vector;
[ ]Tzxyzxyzzyyxx εεεεεεε 222= is strain vector;
=
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
cccccc
cccccc
cccccc
cccccc
cccccc
cccccc
C is stiffness matrix.
The general form of the equation of motion in anisotropic medium takes the following form:
27
[ ]
[ ]
[ ]zxyzxyzzyyxx
zxyzxyzzyyxx
zxyzxyzzyyxx
ccccccz
ccccccy
ccccccxt
u
εεεεεε
εεεεεε
εεεεεερ
666564636261
464544434241
1615141312112
2
222
222
222
+++++∂
∂
++++++∂
∂
++++++∂
∂=
∂
∂
[ ]
[ ]
[ ]zxyzxyzzyyxx
zxyzxyzzyyxx
zxyzxyzzyyxx
ccccccz
ccccccy
ccccccxt
v
εεεεεε
εεεεεε
εεεεεερ
565554535251
262524232221
4645444342412
2
222
222
222
+++++∂
∂
++++++∂
∂
++++++∂
∂=
∂
∂
(3.3)
[ ]
[ ]
[ ]zxyzxyzzyyxx
zxyzxyzzyyxx
zxyzxyzzyyxx
ccccccz
ccccccy
ccccccxt
w
εεεεεε
εεεεεε
εεεεεερ
363534333231
565554535251
6665646362612
2
222
222
222
+++++∂
∂
++++++∂
∂
++++++∂
∂=
∂
∂
Components of the strain tensor in (3.3) can be eliminated using the strain displacement
relation (3.4) (Igel et al., 1995). In the following chapters detailed forms of equation of
motion which describes wave propagation in vertical and horizontal transversely isotropic
media and tilted transverse anisotropy media will be presented.
∂
∂+
∂
∂=
i
u
j
u jiij
2
1ε (3.4)
3.1. Rhombic, transversely isotropic and isotropic media
In rhombic medium the stiffness matrix takes the form
=
66
55
44
333231
232221
131211
00000
00000
00000
000
000
000
c
c
c
ccc
ccc
ccc
C (3.5)
Hence after applying the strain displacement relation (3.4), equation of motion (3.3) can be
rewritten as
28
( ) ( )zx
wcc
yx
vcc
z
uc
y
uc
x
uc
t
u
∂∂
∂++
∂∂
∂++
∂
∂+
∂
∂+
∂
∂=
∂
∂ 2
6613
2
44122
2
662
2
442
2
112
2
ρ
( ) ( )zy
wcc
yx
ucc
z
vc
y
vc
x
vc
t
v
∂∂
∂++
∂∂
∂++
∂
∂+
∂
∂+
∂
∂=
∂
∂ 2
5523
2
44212
2
552
2
222
2
442
2
ρ (3.6)
( ) ( )zy
vcc
zx
ucc
z
wc
y
wc
x
wc
t
w
∂∂
∂++
∂∂
∂++
∂
∂+
∂
∂+
∂
∂=
∂
∂ 2
5532
2
66312
2
332
2
552
2
662
2
ρ
For rhombic medium nine elastic constants are independent whereas for transversely isotropic
medium only five elastic constants are independent and for isotropic only two (Thomsen,
1986). This is due to symmetry of stiffness matrix and dependences among elastic constants
discussed in detail in Chapter 2.
For rhombic and isotropic media these dependences can be written as: (3.7a) and (3.7b)
respectively.
3223
3113
2112
cc
cc
cc
=
=
=
(3.7a)
4433322331132112
446655
332211
2cccccccc
ccc
ccc
−======
==
==
(3.7b)
For transversely isotropic media with a vertical axis of symmetry (VTI - vertical transverse
isotropy ) and horizontal axis of symmetry (HTI - horizontal transverse isotropy) these
dependences take the form (3.8a and 3.8b respectively):
5544
133223
1122
3113
66112112 2
cc
ccc
cc
cc
cccc
=
==
=
=
−==
(3.8a)
5566
44333223
3322
3113
132112
2
cc
cccc
cc
cc
ccc
=
−==
=
=
==
(3.8b)
29
3.2. Tilted transverse anisotropy media
3.2.1 Rotation of vertical transverse isotropy medium around x1 axis by angle θθθθ
Fig. 3.1. Rotation of vertical transverse isotropy medium around x1 axis by angle θθθθ (the view after rotation)
Rys. 31. Rotacja ośrodka z anizotropią typu VTI względem osi x1 o kąt θθθθ (widok po rotacji).
After rotation of VTI medium around x1 axis the stiffness matrix takes the form
'
66
'
56
'
56
'
55
'
44
'
34
'
24
'
14
'
34
'
33
'
23
'
13
'
24
'
23
'
22
'
12
'
14
'
13
'
12
'
11
0000
0000
00
00
00
00
cc
cc
cccc
cccc
cccc
cccc
(3.9)
where elastic constants are linear combination of unrotated elastic constants and trigonometric
values of rotation angles described in previous chapter (equation 2.20)
Equation of motion for this kind of anisotropy is reduced to
xz
wc
yz
wc
zy
wc
zx
wc
z
vc
y
vc
yx
vc
xy
vc
x
vc
z
uc
y
uc
yx
uc
yx
uc
x
uc
t
u
∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂
+∂
∂+
∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂
+∂
∂+
∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂=
∂
∂
2'
66
2'
56
2'
34
2'
13
2
2'
562
2'
24
2'
12
2'
442
2'
14
2
2'
662
2'
44
2'
14
2'
142
2'
112
2
ρ
30
xz
wc
yz
wc
zy
wc
zx
wc
z
vc
y
vc
yx
vc
xy
vc
x
vc
z
uc
y
uc
yx
uc
yx
uc
x
uc
t
v
∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂
+∂
∂+
∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂
+∂
∂+
∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂=
∂
∂
2'
56
2'
55
2'
23
2'
34
2
2'
552
2'
22
2'
24
2'
242
2'
44
2
2'
562
2'
24
2'
12
2'
442
2'
142
2
ρ
(3.10)
2
2'
332
2'
55
2'
56
2'
562
2'
66
2'
23
2'
34
2'
55
2'
56
2'
13
2'
34
2'
56
2'
662
2
z
wc
y
wc
yx
wc
yx
wc
x
wc
yz
vc
xz
vc
zy
vc
zx
vc
xz
uc
yz
uc
zy
uc
zx
uc
t
w
∂
∂+
∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂
∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂
∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂=
∂
∂ρ
3.2.2 Rotation of vertical transverse isotropy medium around x2 axis by angle θθθθ
Fig. 3.2. Rotation of vertical transverse isotropy medium around x2 axis by angle θθθθ (the view after rotation)
Rys. 3.2. Rotacja ośrodka z anizotropią typu VTI względem osi x2 o kąt θθθθ (widok po rotacji).
Stiffness matrix for VTI medium rotated around x2 axis takes the form:
31
'
66
'
64
'
55
'
53
'
52
'
51
'
46
'
44
'
35
'
33
'
32
'
31
'
25
'
23
'
22
'
21
'
15
'
13
'
12
'
11
0000
00
0000
00
00
00
cc
cccc
cc
cccc
cccc
cccc
(3.11)
Elastic constants from equation (3.11) are calculated using relation described in previous the
chapter (equation 2.21).
For this kind of anisotropy, equation of motion can be written as
xz
wc
xy
wc
zx
wc
yx
wc
xz
vc
xy
vc
zx
vc
yx
vc
z
uc
yz
uc
zy
uc
y
uc
x
uc
t
u
∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂
+∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂
+∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂+
∂
∂=
∂
∂
2'
66
2'
46
2'
13
2'
15
2'
64
2'
44
2'
15
2'
12
2
2'
66
2'
64
2'
462
2'
442
2'
112
2
ρ
2
2'
53
2'
55
2'
232
2'
252
2'
46
2
2'
55
2'
52
2'
252
2'
222
2'
44
2'
51
2'
21
2'
46
2'
442
2
z
wc
yz
wc
zy
wc
y
wc
x
wc
z
vc
yz
vc
zy
vc
y
vc
x
vc
xz
uc
xy
uc
zx
uc
yx
uc
t
v
∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂+
∂
∂
+∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂+
∂
∂
+∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂=
∂
∂ρ
(3.12)
2
2'
33
2'
35
2'
532
2'
552
2'
66
2
2'
35
2'
32
2'
552
2'
522
2'
64
2'
31
2'
51
2'
66
2'
642
2
z
wc
yz
wc
zy
wc
y
wc
x
wc
z
vc
yz
vc
zy
vc
y
vc
x
vc
xz
uc
xy
uc
zx
uc
yx
uc
t
w
∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂+
∂
∂
+∂
∂+
∂∂
∂+
∂∂
∂+
∂
∂+
∂
∂
+∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂=
∂
∂ρ
3.2.3 Rotation of horizontal transverse isotropy medium around x3 axis by angle θθθθ
32
Fig. 3.3. Rotation of horizontal transverse isotropy medium around x3 axis by angle θθθθ (the view after rotation)
Rys. 3.3. Rotacja ośrodka z anizotropią typu HTI względem osi x3 o kąt θθθθ (widok po rotacji).
Stiffness matrix for horizontal isotropic medium rotated around x3 axis can be written as
'
66
'
36
'
26
'
16
'
55
'
45
'
45
'
44
'
36
'
33
'
23
'
13
'
26
'
23
'
22
'
12
'
16
'
13
'
12
'
11
00
0000
0000
00
00
00
cccc
cc
cc
cccc
cccc
cccc
(3.13)
where elastic constants are calculated using relation described in the previous chapter
(equation 2.24)
and equation of motion for this kind of anisotropy is reduced to the form of
2
2'
36
2'
662
2'
45
2'
132
2'
16
2'
26
2'
45
2'
44
2'
12
2
2'
66
2'
162
2'
44
2'
162
2'
112
2
z
wc
xz
wc
y
wc
zx
wc
x
wc
yz
vc
zy
vc
xy
vc
yx
vc
z
uc
xz
uc
y
uc
zx
uc
x
uc
t
u
∂
∂+
∂∂
∂+
∂
∂+
∂∂
∂+
∂
∂
∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂
+∂
∂+
∂∂
∂+
∂
∂+
∂∂
∂+
∂
∂=
∂
∂ρ
33
yz
wc
zy
wc
xy
wc
yx
wc
z
vc
xz
vc
y
vc
zx
vc
x
vc
yz
uc
zy
uc
xy
uc
yx
uc
t
v
∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂
∂
∂+
∂∂
∂+
∂
∂+
∂∂
∂+
∂
∂
∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂=
∂
∂
2'
55
2'
23
2'
26
2'
45
2
2'
55
2'
452
2'
22
2'
452
2'
44
2'
45
2'
26
2'
12
2'
442
2
ρ
(3.14)
2
2'
33
2'
362
2'
55
2'
362
2'
66
2'
23
2'
55
2'
45
2'
26
2
2'
36
2'
132
2'
45
2'
662
2'
162
2
z
wc
xz
wc
y
wc
zx
wc
x
wc
yz
vc
zy
vc
xy
vc
yx
vc
z
uc
xz
uc
y
uc
zx
uc
x
uc
t
w
∂
∂+
∂∂
∂+
∂
∂+
∂∂
∂+
∂
∂
∂∂
∂+
∂∂
∂+
∂∂
∂+
∂∂
∂
+∂
∂+
∂∂
∂+
∂
∂+
∂∂
∂+
∂
∂=
∂
∂ρ
3.3. Numerical solution of equation of motion using finite-difference method
Finite-difference methods are widely used to model seismic wave propagation in elastic
media (Alford, 1974; Kelly i in., 1976, Reynolds 1987). Numerical solution of the equation of
motion can be obtained using various formulations of finite-difference schemes and these
differ from one another by accuracy, efficiency and computational time. In this study the
explicate schemes computationally simpler than implicit ones were used. In this schemes
motion at a given spatial grid point and time level is calculated only from motion at a previous
time level (or levels) and adjacent grid points.
Second-order in space and time explicate finite-difference scheme can be obtained by
adopting the central difference (3.15) to approximate partial derivative of equation of motion
( )( )[ ]2
2
,1,,1
,
2
2 2xO
x
uuu
x
u jijiji
ji
∆+∆
+−=
∂
∂ −+ (3.15)
( ) ( )[ ]221,11,11,11,1
,
2
,222
1yxO
y
uu
y
uu
xyx
u jijijiji
ji
∆∆+
∆
−−
∆
−
∆=
∂∂
∂ −−+−−+++
where ∆x, ∆y denote the spatial increments in x and y directions, respectively,
( )[ ]2xO ∆ and ( ) ( )[ ]22
, yxO ∆∆ are approximation errors.
The discrete form of equations of motion obtained by adopting central difference to second-
order partial derivatives and mixed partial derivatives (3.15) is presented in Appendix 2. In
order to obtain a detailed solution for isotropic, rhombic, vertical transverse isotropy and
horizontal transverse isotropy media adequate dependences (3.7a, 3.7b, 3.8a or 3.8b) must be
satisfied. In Appendices 3, 4, 5 the discrete form of equations of motion for TTI media is
presented.
34
Finite-difference method often suffers from grid dispersion and unphysical oscillations which
appear when the computational grid is too coarse. Although finer spatial grids and time-
sampling can be use to eliminate the numerical dispersion they result in large computational
costs and large amount of memory necessary to store data. The gain in computational
efficiency can be made by using higher-order approximation for derivatives of equation of
motion.
Fourth-order in time finite-difference schema is an example of such approximation. The
discrete form of equations of motion obtained by using fourth order in time schema can be
written as
( ) ( ) ( )
( )
( )[ ] ( )[ ]
( )( ) ( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( ) ( )( ) ( )( )[ ]
∂∂∂
∂++++++
∂∂
∂+++
∂∂
∂++
+∂∂∂
∂++++++
∂∂
∂+++
∂∂
∂++
+∂∂
∂+
∂∂
∂+++
∂∂
∂+++
∂
∂+
∂
∂+
∂
∂
∆+
+
∂∂
∂++
∂∂
∂++
∂
∂+
∂
∂+
∂
∂∆+−= −+
zyx
wcccccccc
zx
wcccc
zx
wcccc
zyx
vcccccccc
yx
vcccc
yx
vcccc
zy
ucc
zx
ucccc
yx
ucccc
z
uc
y
uc
x
uc
t
zx
wcc
yx
vcc
z
uc
y
uc
x
uc
tuuu
nnn
nnn
nnnnnn
nnnnnnnn
2
4
55234412554466133
4
663366133
4
11666613
2
4
55236613665544123
4
442244123
4
44124411
22
4
664422
42
6613661122
42
441244114
42
664
42
444
42
11
2
4
2
6613
2
44122
2
662
2
442
2
11
2
11
222
12
2
ρ
ρ
( ) ( ) ( )
( )
( )[ ] ( )[ ]
( )( ) ( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( ) ( )( ) ( )( )[ ]
∂∂∂
∂++++++
∂∂
∂+++
∂∂
∂++
+∂∂∂
∂++++++
∂∂
∂+++
∂∂
∂++
+∂∂
∂+
∂∂
∂+++
∂∂
∂+++
∂
∂+
∂
∂+
∂
∂
∆+
+
∂∂
∂++
∂∂
∂++
∂
∂+
∂
∂+
∂
∂∆+−= −+
zyx
wcccccccc
zy
wcccc
zy
wcccc
zyx
ucccccccc
yx
ucccc
yx
ucccc
zx
vcc
zy
vcccc
yx
vcccc
z
vc
y
vc
x
vc
t
zy
wcc
yx
ucc
z
vc
y
vc
x
vc
tvvv
nnn
nnn
nnnnnn
nnnnnnnn
2
4
66134412664455233
4
553355233
4
22555523
2
4
66135523556644123
4
224444123
4
44114412
22
4
554422
42
5523552222
42
441222444
42
554
42
224
42
44
2
4
2
5523
2
44122
2
552
2
222
2
44
2
11
222
12
2
ρ
ρ
(3.16)
( ) ( ) ( )
( )
( )[ ] ( )[ ]
( )( ) ( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( ) ( )( ) ( )( )[ ]
∂∂∂
∂++++++
∂∂
∂+++
∂∂
∂++
+∂∂∂
∂++++++
∂∂
∂+++
∂∂
∂++
+∂∂
∂+
∂∂
∂+++
∂∂
∂+++
∂
∂+
∂
∂+
∂
∂
∆+
+
∂∂
∂++
∂∂
∂++
∂
∂+
∂
∂+
∂
∂∆+−= −+
zyx
vcccccccc
zy
vcccc
zy
vcccc
zyx
ucccccccc
zx
ucccc
zx
ucccc
yx
wcc
zy
wcccc
zx
wcccc
z
wc
y
wc
x
wc
t
zy
vcc
zx
ucc
z
wc
y
wc
x
wc
twww
nnn
nnn
nnnnnn
nnnnnnnn
2
4
66134412664455233
4
335555233
4
55225523
2
4
44125523445566133
4
663366133
4
11666613
22
4
556622
42
5523335522
42
661333664
42
334
42
554
42
66
2
4
2
5523
2
66132
2
332
2
552
2
66
2
11
222
12
2
ρ
ρ
The discrete form of the above equation in isotropic, rhombic or transversely isotropic media
obtained after adopting central difference to second- and fourth-order partial derivatives and
mixed partial derivatives is presented in Appendix 6. Detailed solution for isotropic, rhombic,
vertical transverse isotropy or horizontal transverse isotropy media can be obtained after
taking into account equations 3.7a, 3.7b, 3.8a and 3.8b, respectively.
Fourth order in time and second order in space finite-difference schema presented above is
more accurate but also much more computationally complex than the second order in space
and time finite-difference schema. It also suffers for numerical dispersion, thus second-order
in space and time finite-difference schema is presented only for more complicated, tilted
transverse anisotropy media.
35
3.4. The stability criterion of second-order finite-difference schema
General form of equation of motion can be written as (Yang et al., 2002)
uz
Iy
Hx
Gz
uz
Fy
Ex
Dy
uz
Cy
Bx
Axt
u
∂
∂+
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂+
∂
∂
∂
∂+
∂
∂+
∂
∂+
∂
∂
∂
∂=
∂
∂2
2
ρ (3.17)
Applying central differences one can receive
( )( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( ) ( )
∆
+−+
∆
−−
∆
−
∆+
∆
−−
∆
−
∆
∆
∆
−−
∆
−
∆+
∆
+−+
∆
−−
∆
−
∆
∆
∆
−−
∆
−
∆+
∆
−−
∆
−
∆+
∆
+−∆
=−−
−+−−−++−++−−−++−++
−−+−−+++−+−−−++−++
−−+−−+++−−+−−+++−+
−+
2
1,,,,1,,1,1,1,1,1,1,1,1,1,,11,,11,,11,,12
1,1,1,1,1,1,1,1,
2
,1,,,,1,,1,1,1,1,1,1,1,12
1,,11,,11,,11,,1,1,1,1,1,1,1,1,1
2
,,1,,,,12
1
,,,,
1
,,
2
222
1
222
1
222
12
222
1
222
1
222
12
2
z
uuuI
y
uu
y
uu
zH
x
uu
x
uu
zG
t
z
uu
z
uu
yF
y
uuuE
x
uu
x
uu
yD
t
z
uu
z
uu
xC
y
uu
y
uu
xB
x
uuuA
t
uuu
kjil
kjil
kjill
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kjikjil
kjil
kjill
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kji
l
kjikjil
kjil
kjil
l
kji
l
kji
l
kji
ρ
ρ
ρ
(3.18)
Application of von Neumann stability method (Tannehill, 1997) and substituting into the
difference equation a term of the form
zikyikxikatl
kjizyx eeeeu =,,
After a series of mathematical operations, the following stability criterion of the finite-
difference schema is obtained:
( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )
1444
2
maxmaxmax
2
maxmaxmaxmaxmax
2
max
2
≤
∆+
∆∆
++
∆+
∆∆
++
∆∆
++
∆
∆
zzyyzxyxx
tIHFEGCDBA σσσσσσσσσ
ρ (3.19)
where A
maxσ is the largest eigenvalue of A matrix , B
maxσ is the largest eigenvalue of B matrix
etc.
The A, B, C, D, E, F, G, H and I matrices for each type of anisotropy have different form. In
the next sections a detailed account of these matrices and stability criterion for VTI, HTI,
rhombic and TTI media is presented.
3.4.1 The stability criterion for vertical transverse isotropy medium
In this case matrices from equation (3.17) take the form
=
66
55
11
00
00
00
c
c
c
A
−
=
000
00
020
55
6611
c
cc
B
=
00
000
00
66
13
c
c
C
−
=
000
00
020
55
6611
c
cc
D
=
55
11
55
00
00
00
c
c
c
E
=
00
00
000
55
13
c
cF
36
=
00
000
00
66
13
c
c
G
=
00
00
000
55
13
c
cH
=
33
55
66
00
00
00
c
c
c
I
0maxmaxmaxmaxmaxmax ====== HFGCBD σσσσσσ .
Thus stability criterion of finite-difference schema for VTI media can be written as:
( )( ) ( ) ( )
12
max
2
max
2
max
2
≤
∆+
∆+
∆
∆
zyx
tIEA σσσ
ρ,
( ) ( ) ( )2
max
2
max
2
max
zyx
tIEA
∆+
∆+
∆
≤∆σσσ
ρ
where: { }665511max ,,max ccc
A =σ , { }1155max ,max cc
E =σ , { }335566max ,,max ccc
I =σ
3.4.2 The stability criterion for horizontal transverse isotropy medium
In this case matrices from equation (3.17) take the form
=
55
44
11
00
00
00
c
c
c
A
=
000
00
00
44
13
c
c
B
=
00
000
00
55
13
c
c
C
=
000
00
00
44
13
c
c
D
=
55
33
44
00
00
00
c
c
c
E
−=
00
200
000
55
4433
c
ccF
=
00
000
00
55
13
c
c
G
−=
00
200
000
55
4433
c
ccH
=
33
55
55
00
00
00
c
c
c
I
0maxmaxmaxmaxmaxmax ====== HFGCBD σσσσσσ .
Thus stability criterion of finite-difference schema for HTI media can be written as
( )( ) ( ) ( )
12
max
2
max
2
max
2
≤
∆+
∆+
∆
∆
zyx
tIEA σσσ
ρ,
( ) ( ) ( )2
max
2
max
2
max
zyx
tIEA
∆+
∆+
∆
≤∆σσσ
ρ
where { }554411max ,,max cccA =σ , { }553344max ,,max ccc
E =σ , { }3355max ,max ccI =σ .
3.4.3 The stability criterion for rhombic isotropy medium
In this case matrices from equation (3.17) take the form
37
=
66
44
11
00
00
00
c
c
c
A
=
000
00
00
44
12
c
c
B
=
00
000
00
66
13
c
c
C
=
000
00
00
44
12
c
c
D
=
55
22
44
00
00
00
c
c
c
E
=
00
00
000
55
23
c
cF
=
00
000
00
66
13
c
c
G
=
00
00
000
55
23
c
cH
=
33
55
66
00
00
00
c
c
c
I
0maxmaxmaxmaxmaxmax ====== HFGCBD σσσσσσ .
Thus stability criterion of finite-difference schema for rhombic media can be written as
( )( ) ( ) ( )
12
max
2
max
2
max
2
≤
∆+
∆+
∆
∆
zyx
tIEA σσσ
ρ,
( ) ( ) ( )2
max
2
max
2
max
zyx
tIEA
∆+
∆+
∆
≤∆σσσ
ρ
where { }664411max ,,max ccc
A =σ , { }552244max ,,max ccc
E =σ , { }335566max ,,max ccc
I =σ .
3.4.4. The stability criterion for tilted transverse anisotropy media - rotation of vertical
transverse isotropy medium around x1 axis by angle θθθθ
In this case matrices from equation (3.17) take the form
='
66
'
44
'
14
'
14
'
11
00
0
0
c
cc
cc
A
='
56
'
24
'
44
'
12
'
14
00
0
0
c
cc
cc
B
=
0
00
00
'
56
'
66
'
34
'
13
cc
c
c
C
='
56
'
24
'
12
'
44
'
14
00
0
0
c
cc
cc
D
='
55
'
22
'
24
'
24
'
44
00
0
0
c
cc
cc
E
=
0
00
00
'
55
'
56
'
23
'
34
cc
c
c
F
=
0
00
00
'
34
'
13
'
56
'
66
cc
c
c
G
=
0
00
00
'
23
'
34
'
55
'
56
cc
c
c
H
='
33
'
55
'
56
'
56
'
66
00
0
0
c
cc
cc
I
because
( )2'
14
'
44
'
11
'
66)det( ccccA −=
38
( )'
44
'
12
'
24
'
14
'
56)det()det( cccccDB −==
( )2'
24
'
22
'
44
'
55)det( ccccE −=
( )2'
56
'
55
'
66
'
33)det( ccccI −=
0)det()det()det()det( ==== FHGC
0maxmaxmaxmax ==== HFGC σσσσ .
The stability criterion of finite-difference schema for this kind of anisotropy can be written as
( )( ) ( )( ) ( ) ( )
14
2
max
2
maxmaxmax
2
max
2
≤
∆+
∆+
∆∆
++
∆
∆
zyyxx
tIEDBA σσσσσ
ρ
( ) ( ) ( ) ( )( )yxzyx
tBIEA
∆∆+
∆+
∆+
∆
≤∆
2
max
2
max
2
max
2
max σσσσ
ρ
where { }'
44
'
11
'
66max ,max cccA =σ , { }'
22
'
44
'
55max ,max cccE =σ , { }'
55
'
66
'
33max ,max cccI =σ ,
{ }'
44
'
12
'
24
'
14
'
56maxmax ,max cccccDB −== σσ .
3.4.5. The stability criterion for tilted transverse anisotropy medium - rotation of
vertical transverse isotropy media around x2 axis by angle θθθθ
In this case matrices from equation (3.17) take the form
='
66
'
64
'
46
'
44
'
11
0
0
00
cc
cc
c
A
=
00
00
0
'
64
'
44
'
15
'
12
c
c
cc
B
=
00
00
0
'
66
'
46
'
13
'
15
c
c
cc
C
=
00
00
0
'
51
'
21
'
46
'
44
c
c
cc
D
='
55
'
52
'
25
'
22
'
44
0
0
00
cc
cc
c
E
='
53
'
55
'
23
'
25
'
46
0
0
00
cc
cc
c
F
=
00
00
0
'
31
'
51
'
66
'
64
c
c
cc
G
='
35
'
32
'
55
'
52
'
64
0
0
00
cc
cc
c
H
='
33
'
35
'
53
'
55
'
66
0
0
00
cc
cc
c
I
because
( )'
46
'
64
'
44
'
66
'
11)det( cccccA −=
39
( )'
52
'
25
'
22
'
55
'
44)det( cccccE −=
( )'
23
'
55
'
53
'
25
'
46)det( cccccF −=
( )'
32
'
55
'
35
'
52
'
64)det( cccccH −=
( )'
35
'
53
'
55
'
33
'
66)det( cccccI −=
0)det()det()det()det( ==== GDCB
0maxmaxmaxmax ==== GDCB σσσσ .
The stability criterion of finite-difference schema for this kind of anisotropy can be written as
( )( ) ( ) ( )( ) ( )
14
2
maxmaxmax
2
max
2
max
2
≤
∆+
∆∆
++
∆+
∆
∆
zzyyx
tIHFEA σσσσσ
ρ
( ) ( ) ( ) ( )( )yxzyx
tHFIEA
∆∆
++
∆+
∆+
∆
≤∆
4
maxmax
2
max
2
max
2
max σσσσσ
ρ
where { }'
64
'
46
'
66
'
44
'
11max ,max cccccA −=σ , { }'
52
'
25
'
55
'
22
'
44max ,max cccccE −=σ ,
{ }'
35
'
53
'
33
'
55
'
66max ,max cccccI −=σ , { }'
23
'
55
'
53
'
25
'
46max ,max cccccF −=σ , { }'
32
'
55
'
35
'
52
'
64max ,max cccccH −=σ .
3.4.6. The stability criterion for tilted transverse anisotropy medium - rotation of
horizontal transverse isotropy media around x3 axis by angle θθθθ
In this case matrices from equation (3.17) take the form
='
66
'
16
'
44
'
16
'
11
0
00
0
cc
c
cc
A
=
00
0
00
'
26
'
45
'
44
'
12
c
cc
c
B
='
36
'
66
'
45
'
13
'
16
0
00
0
cc
c
cc
C
=
00
0
00
'
45
'
26
'
12
'
44
c
cc
c
D
='
55
'
45
'
22
'
45
'
44
0
00
0
cc
c
cc
E
=
00
0
00
'
55
'
23
'
26
'
45
c
cc
c
F
='
36
'
13
'
45
'
66
'
16
0
00
0
cc
c
cc
G
=
00
0
00
'
23
'
55
'
45
'
26
c
cc
c
H
='
33
'
36
'
55
'
36
'
66
0
00
0
cc
c
cc
I
because
( )2'
16
'
66
'
11
'
44)det( ccccA −=
40
( )'
13
'
66
'
36
'
16
'
45)det()det( cccccGC −==
( )2'
45
'
55
'
44
'
22)det( ccccE −=
( )2'
36
'
33
'
66
'
55)det( ccccI −=
0)det()det()det()det( ==== HFDB
0maxmaxmaxmax ==== HFDB σσσσ .
The stability criterion of finite-difference schema for kind of anisotropy can be written as
( )( ) ( )( ) ( ) ( )
14
2
max
2
maxmaxmax
2
max
2
≤
∆+
∆+
∆∆
++
∆
∆
zyzxx
tIEGCA σσσσσ
ρ
( ) ( ) ( ) ( )( )zxzyx
tCIEA
∆∆+
∆+
∆+
∆
≤∆
2
max
2
max
2
max
2
max σσσσ
ρ
where { }'
66
'
11
'
44max ,max cccA =σ , { }'
13
'
66
'
36
'
16
'
45maxmax ,max cccccGC −== σσ , { }'
55
'
44
'
22max ,max cccE =σ
{ }'
33
'
66
'
55max ,max cccI =σ .
41
4. Implementation of the finite-difference schemas
4.1. Initial and border conditions
Numerical modeling of seismic wave field is a computational problem in which continuous
space-time has to be simulated in discrete mesh with a defined time step. To do this, various
previously mentioned time and space stability conditions have to be fulfilled. But even if
stability of the solution is achieved there are other limitations of this discrete representation.
First of all real waves propagate in continuous and practically unlimited medium. The first
problem with discrete representation is “instant” reaction of neighboring points when wave is
passing through the computational grid. It means that modeled waves seem to have a couple
of percent higher velocities then those defined in modeling parameters. The other, more
important problem is definition of border conditions. Of course 3D computational cube has
borders which limit the propagation. There are various ways to define and then calculate
border conditions. It is possible to add additional “artificial” set of grid points “outside” a
cube, redefine computational schemes for border planes or add dumping zones around a cube.
Of course, all these solutions require additional time and/or memory consuming operations.
Usually the most efficient is dumping solution but it has one important disadvantage:
additional “dumping” cube has to be allocated which is memory consuming. Fortunately in
tests presented below time of computation was short and only wave field snapshots were
taken into consideration.
Initial conditions for this kind of computations are very simple. Motion in each point of
computational grid in time t+1 is calculated using motions of the examined point and
neighboring points in time t and t-1. It means that at the beginning all cubes have to be zeroed
in all points except source point or area. In all later time steps all computational cubes for
time t-1 are recalculated for time t+1 and all other cubes are shifted one time space back.
4.2 Seismic wave propagation, finite-difference algorithm
Modeling of seismic wave propagation in the anisotropic media can be described by the
following four steps:
1. data input – providing values of geological model, location and type of the source of
seismic wave, establishing of proper values of stiffness matrix;
2. providing initial values for 0
,, kjiU , 0
,, kjiV , 0
,, kjiW ;
3. utilizing the finite-difference scheme described in Appendix 2,3,4 or 5 to compute 1
,,
+n
kjiU , 1
,,
+n
kjiV , 1
,,
+n
kjiW , where 0≥n ;
4. providing boundary condition.
The steps 3 and 4 have to be repeated until simulation time is reached.
42
4.3 Computational environment
Full wave form numerical modeling in 3D media is a very complicated time and hardware
resources-consuming task. Even in the case of a simple isotropic, acoustic 3D modeling the
memory consumption is enormous. Single 100 x 100 x 100 cube of 8 byte floating point
numbers needs more than 22 MB of memory and in this simplest case at least four cubes are
needed (without consideration of matrices or cubes for storing results). In case of HTI, VTI or
orthorhombic models at least 19 cubes (elastic constraints and 3D displacements in three
continuous time steps) are needed, which gives about 0.5 GB of memory for every 1,000,000
grid points. The other problem is time of computations. In case of modern CPU one average
scale anisotropic model is usually calculated in 10 to 20 hours. Fortunately parallelization of
this kind of computations is very easy.
All proposed solutions were evaluated on IBM Blade Linux cluster*. One BladeCenter
chassis with 11 HC21 and 3 HC10 Blade servers was used. 88 Intel Xeon E5405 2.00 GHz
cores and 6 Intel Core2 6700 2.66 GHz cores with 200 GB of total RAM were available. This
environment was previously successfully used as a hardware platform for many other projects
connected with seismic wave filed modeling in complex media (e.g. Danek et al. 2008, Kowal
et al. 2008, Pięta et al. 2009, Danek 2009).
All computer codes were written in C language. The most important part of the code –
computational kernel loop is presented in Appendix 7.
* The cluster is located in Department of Geosciences and Applied Computer Science, Faculty of Geology,
Geophysics and Environmental Protection, AGH University of Science and Technology, Kraków, Poland.
43
5. Example results of wave propagation modeling in anisotropic media
In this chapter wave propagation in HTI, VTI and orthorhombic homogeneous media is
presented. Also, propagation in simple isotropic models is calculated for comparison
purposes. In all cases three kinds of seismic sources were used: explosive, unidirectional
vertical force and double couple force. Double couple source was always localized in X2
(“cross line”) plane. For all computations the Ricker signal, which is very often used in
seismic simulations, was chosen. All parameters used in modeling are presented in Table 5.1
and were exactly the same in all analyzed cases of this research. Results of modeling can be
presented in three perpendicular planes (X1, X2, X3) and three components of the modeled
wave field (x1, x2, x3) can be exhibited. It means that up to 9 combinations of motion pictures
(three planes, in each three different directions of wave motions) can be evaluated and
analyzed in every time step. In this study results for planes crossing source area and planes
shifted 20 m away from the source were selected for further, detailed analysis (Fig. 5.1).
Fig.5.1. Illustration of X1 observation planes: (A) plane crossing source area, (B) plane shifted 20 m away from the
source
Rys.5.1. Płaszczyzna obserwacyjna X1: (A) przechodząca przez obszar źródłowy, (B) odsunięta o 20 m od źródła.
Table 5.1. Parameters of seismic propagation modeling
Name Value
Number of points in all directions 250
Distance between grid points 1 m
Time steps of computations 0.05 s
Source point localization (125,125,125)
Peek frequency of Ricker signal 90 Hz
44
Elastic constraints used in this experiment were calculated in a very special manner. First of
all constraints for the orthorhombic anisotropy were chosen. Then, using the method
described in Chapter 3 this medium was reduced to HTI, VTI and finally isotropic media. All
parameters used in numerical experiments are presented below:
Stiffness matrix for:
- isotropic medium:
0.300000
00.30000
000.3000
0003.103.43.4
0003.43.103.4
0003.43.43.10
- horizontal transverse isotropy medium HTI:
0.300000
00.30000
003.3000
0003.107.35.6
0007.33.105.6
0005.65.66.16
- vertical transverse isotropy medium VTI:
9.300000
00.30000
000.3000
0003.105.65.6
0005.66.168.8
0005.68.86.16
- orthorhombic medium:
9.300000
00.30000
003.3000
0003.104.65.6
0004.65.154.7
0005.64.76.16
45
5.1 Sources
5.1.1 Explosive source
Explosive source is one of the most commonly used sources in seismic exploration. It can be
also used in the case of wave field modeling of mine tremors and explosions induced by
mining activity. From numerical point of view the source was defined as a 3 x 3 x 3 grid point
cube and in each point force pointing outside the cube was attached.
5.1.2 Double couple source
Double couple mechanism is one of the most commonly used source models in seismology.
Wave propagation caused by earthquakes related with rock mass movements along faults is
very similar to wave propagation excited by double couple mechanism. In this study a simple
double couple combination of forces was assumed to simulate the vertical fault lying in the X2
plane. This orientation of fault was chosen because it allows simple comparison with a 2D
case. Additionally this strict set of directions of wave propagation results in a very clear
image of waves propagating from the source. In the case of source-crossing planes waves
propagate in all planes (as it was expected) but particles move only in very well-determined
directions (empty frames in figures in Appendix 8 mean no motion at all).
5.1.3 Single force source
Single force source model can be used to simulate vibrations caused by unidirectional force.
In case shallow seismic exploration hammer or hammer source can be approximated by such
a source. But even if this kind of source is not very common in seismic exploration or
seismology analysis its unidirectional nature can be useful because waves generated by it are
very well separated and its polarization is clear. In these studies vertically directed force was
used in all cases.
5.2 Selected results and discussion
5.2.1 Isotropic medium
Propagation in simple isotropic models was calculated for comparison purposes. All
results should be easy to analyze and interpret and then be used as a background for
interpretations in case of more complicated media. All modeling results for this and the rest of
the media are presented in Appendix 8. As mentioned above, three kinds of sources were
used: explosive, double couple and single force. It is clearly visible that in the case of
explosive source and isotopic homogeneous medium wave-form propagation is very regular
and stable (fig. A8.1). As expected, almost all energy is propagating as a primary wave. Very
limited, almost invisible amounts of shear wave energy can be observed only in shifted planes
(fig. A8.2). It is due to the discrete character of computational grid and limited size of the
source. In case of planes crossing directly through the center point of the source these
energies are invisible because all the source forces are lying inside planes. This is also the
reason why there are almost no displacements in directions normal to presented planes (e.g.
plane X1 and x1 direction in figure A8.1). It is also important to mention that the relation
between point of evaluation and source location determines the sign of displacement value. In
46
case of motion directions other than direction of the planes an apparent phase change can be
visible (e.g. plane X1 and x2 direction). When motion direction is normal to plane (e.g. X1
and x1) there is no sign change and this effect is not visible.
In the case of double couple source, P- and S-wave fronts have purely circular shapes
(fig. A8.9-10). Used source mechanism makes S wave stronger and better visible. Typical
changes of wave phases with direction of propagation can also be noticed. Results obtained
for X2 plane are very similar to 2D ones. There is no “cross line” component but both P and S
waves are visible in “in line” and vertical directions. In X1 and X3 planes, motions of shear
wave-related particles are clearly visible in a direction normal to planes.
Results obtained for single force source (fig. A8.17-18) were exactly as expected. P
and S waves have circular shapes and only one shear wave propagates through the media. In
source crossing X3 plane, almost all of the energy is propagating as SV wave with no
polarization changes. In other source crossing X3 planes, P and S waves are visible and their
polarizations are consistent with the direction of the source.
5.2.2 HTI medium
In horizontally transverse isotropic medium which simulates rocks with vertical, parallel
fractures waves should propagate symmetrically along these structures (fig A8.3-4, A8.11-12,
A8.19-20). In analyzed model fractures are parallel to X1 plane. This is why in this plane
primary wave has circular shape whereas shear wave has ellipsoidal shape. It is due to motion
directions of particles involved in S-wave propagation. These directions are of course
perpendicular to X1 plane. In other words, if a particle moves parallel to the plane of isotropy
its motion will generate isotropic wave fronts. The other interesting phenomenon to discuss is
shear waves propagating from the exploding source (fig A8.3-4). This is easy to explain. In
the case of anisotropic media, even as simple as HTI, there is no clear differentiation between
P and S waves. More complicated relation between elastic constraints makes simple
exploding source, which produce only P wave, no longer possible. It is due to the fact that
some of source energy is radiated in directions perpendicular to the direction of motion. In
other words some of P-wave energy is converted into S waves in a source point even without
the presence of any media discontinuity.
In HTI media, a symmetrical propagation of double couple source results in a plane of
isotropy can be observed but in other planes relatively small amounts of - wave energy
propagates with angle dependent velocities (fig. A8.11-12). In shifted planes the shear wave
splitting phenomenon is visible. In X2 and X3 planes in pictures presenting x2 component of
motion two S-wave fronts are separated from each other. In this particular combination of
source mechanism and kind of anisotropy this effect is very easy to observe because most of
energy is propagating as shear waves, and phase variations make both modes better contrasted
(fig 5.2).
Combination of directly polarized source mechanism and HTI medium makes velocity
differences between S waves polarized in different directions clearly visible (fig. A8.19-20).
The other interesting phenomenon is the limited but visible amount of P-wave energy in
47
source crossing X3 plane. More complicated relations between elastic constraints makes P
wave visible in this plane even if no direct “compress” energy is radiated from the source
along x1 and x2 directions.
Figure 5.2. Zoom of S-wave results for double couple source in HTI media. Displacements in x2 direction and shifted
X2 (left) and X3 (right) planes are presented.
Rys 5.2. Powiększenie wyników propagacji fali S dla źródła opisywanego za pomocą podwójnej pary sił w ośrodku
typu HTI. Zaprezentowano przemieszczenia w kierunku x2 w odsuniętej płaszczyźnie obserwacji X2 (lewy obraz) i
odsuniętej płaszczyźnie obserwacji X3 (prawy obraz).
5.2.3 VTI medium
In the case of VTI media and explosive source, results (fig. A8.5-6) are comparable to those
of HTI (fig. A8.3-4). It is because these two media are very similar to each other. When real
media are concerned the most important difference between them is the direction of parallel
fractures which are horizontally oriented in the case of VTI. This orientation of symmetry axis
is clearly visible in snapshot figures. In X1 and X2 planes P-wave fronts have ellipsoidal shape
whereas in X3 (horizontal) plane this front is circular. Shear wave splitting effect is better
visible in this case. In both source-crossing and shifted horizontal planes, it is possible to
separate two wave fronts which are crossing each other. This phenomenon is observed here
because the polarization of S waves varies with angle and the more energy is radiated with
particle vibrations along faster direction the faster the whole wave front is.
48
Fig 5.3 Wave propagation velocity in symmetry plane [X1, X3]
Rys 5.3 Prędkości propagacji fali w płaszczyźnie [X1,X3]
Double couple source generates results in VTI media similar to those observed for real fault-
related earthquakes in horizontally layered rocks. In “in line” cross section P wave
propagates faster along layers (fig. A8.13-14). Phase changes of the S waves determine
possible orientation of fault. In X1 plane S-wave front for x1 component is circular because in
this case all particles involved in shear wave propagation moves in the isotropic (“cross line”)
direction.
In the case of VTI media, results for single force source (fig. A8.21-22) are a very good
background for comparison with double couple source (fig. 5.4). Different orientations of the
symmetry axis are clearly visible (e.g. X2 plane x2 direction). Another interesting and
expected result is that the slightly more energy is radiated from source as P wave in the case
of single force. It is due more to the direct nature of this source.
Figure 5.4. VTI medium, X2 shifted plane, x2 direction results of modeling in the case of double couple (left) and single
force (right) sources
Rys. 5.4. Wyniki modelowań dla ośrodka VTI w odsuniętej płaszczyźnie obserwacji X2. Przemieszczenia w kierunku
x2 dla źródła definiowanego za pomocą podwójnej pary sił (obraz po lewej) i pojedynczej siły (obraz po prawej)
5.2.4 Orthorhombic medium
Orthorhombic anisotropy is the most complicated case analyzed in this study. Results are
very similar to those of VTI case but some subtle but very important differences can be
spotted. The most important one is the expected rotation around X1 axis. It can be observed
that in the case of explosive source (fig. A8.7-8) there are almost no differences of P-wave
ellipsoid in VTI and orthorhombic case in X2 plane. In X3 plane, primary wave shape is no
longer a circle. It is an ellipsoid with a little bit longer horizontal axis (“flattened circle shape”
–fig 5.5). In X1 plane the longer axis of ellipsoid is shorter than that in VTI medium. The
second important fact is better separation of splitted shear waves propagating from source
point. At the same time sharper shapes of wave fronts (especially S waves) are observed. It is
49
probably a purely numerical phenomenon. As the possible explanation we can regard the fact
that in orthorhombic media axis of symmetry are no longer identical with axis of
computational frame.
Double couple source mechanism and additional axis of rotation make separation of shear
waves better visible in some plane-direction combinations (like X3-x1 for example in fig.
A8.15-16). But in the case of X3 plane and x2 (“cross line”), S waves are better separated in
results for VTI model. It could let us make a conclusion that in the case of double couple
source, anisotropic effects connected with S waves are easier to observe on a surface in case
of simpler, VTI model. The other important thing which this detailed study revealed is that
when a full analysis of anisotropy is necessary, tree component surface (on one plane)
recordings are insufficient.
Orthorhombic anisotropy and singe force source is the last analyzed case in this study (fig.
A8.23-24). All previously described phenomena like shear wave splitting and subtle
asymmetry of P-wave “circle” in X3 plane connected with this kind of anisotropy are visible.
The interesting thing to note is the similarity between these results and those of exploding
source. Of course in the case of explosion much more energy propagating as a P wave but
other characteristics are similar. It can be concluded that in numerical experiment focused on
P wave, explosion source should be used but when more energy of S wave is needed single
force source is better.
Figure 5.5. Orthorobic medium, X3 shifted plane, x1 direction results of modeling in exploding source. Inner ellipse
shows actual shape of P wave front. Outer circle represents the shape of the wave front in the case of isotropic solution
Rys. 5.5. Wyniki modelowań dla składowej ruchu w kierunku x1 dla ośrodka o anizotropii ortorombowej, rzut na
płaszczyznę odsuniętą X3. Wewnętrzna elipsa przedstawia bieŜący front falowy fali P, zewnętrzna reprezentuje ten
sam front dla ośrodka izotropowego.
5.2.5 Heterogeneous orthorhombic medium
Orthorhombic medium was also a subject for further studies for bigger and more complicated
models. This time modeling of seismic wave propagation was made for a three-layer
model. The model dimension was 800 x 800 x 400 m. Layers’ borders were on 200
and 300 m depths. Two variants of the model were calculated. In the first one, the
50
central layer was orthorhombic whereas upper and lower were isotropic. In the second
one, all layers were isotropic but the central one was recalculated from orthorhombic
according to the previously mentioned rule. All parameters are presented in Table 1.
All test runs were done for 2 m grid point interval. Wave motion was modeled for 0.8 second.
As a source function a 40 Hz Ricker signal was used. All results were stored with 2 ms time
step. It is also important to mention that for models of this size and complication, 32 bit
computational architecture is no longer sufficient.
Orthorhombic
c11 c12 c13 c22 c23 c33 c44 c55 c66
L1 4.5 0.5 0.5 4.5 0.5 4.5 2.0 2.0 2.0
L2 16.6 7.4 6.5 15.5 6.4 10.3 3.3 3.0 3.9
L3 4.5 0.5 0.5 4.5 0.5 4.5 2.0 2.0 2.0
Isotropic
c11 c12 c13 c22 c23 c33 c44 c55 c66
L1 4.5 0.5 0.5 4.5 0.5 4.5 2.0 2.0 2.0
L2 10.3 4.3 4.3 10.3 4.3 10.3 3.0 3.0 3.0
L3 4.5 0.5 0.5 4.5 0.5 4.5 2.0 2.0 2.0
Table 1 Elastic constraints used in numerical modeling
Results for all combinations of directions and plane orientation were stored. Selected wave
form snapshots and synthetic seismograms for vertical component of motion are presented in
consecutive figures.
51
Fig. 5.6 Snapshots of elastic wave propagation in the case of isotropic (a) and orthorhombic model: vertical
component of motion in X1 (“in-line”) (b) and in X2 (“cross-line”) (c) directions and differences between isotropic and
orthorhombic (d) and orthorhombic models in and cross line directions (e). Amplitudes on pictures (d) and (e) are
normalized.
Fig. 5.6. Migawki wyników propagacji fail elastycznej dla ośrodka izotropowego (a) i ortorombowego: składowej
pionowej ruchu w płaszczyźnie X1 („in line”)(b) i w płaszczyźnie X2 („cross line”)(c) oraz róŜnice pomiędzy
wynikami dla ośrodka izotropowego i ortorombowego (d) ortorombowego w kierunku „cross line”(d). Amplitudy na
rysunkach d i e zostały znormalizowane.
52
Fig. 5.7 Synthetic seismograms obtained for isotropic (a) and anisotropic (b) model, (c) differences between isotropic
and anisotropic model. Vertical component of motion in cross-line plane.
Rys. 5.7. Sejsmogramy syntetyczne obliczone dla modelu izotropowego (a) i anizotropowego (b) oraz sejsmogram
róŜnicowy (c) obliczony dla sejsmogramów (a) i (b). Wszystkie rysunki przedstawiają składową ruchu w płaszczyźnie
„cross-line”.
As it was expected differences between two analyzed models are clearly visible mostly
because of faster wave propagation in the horizontal direction in the anisotropic layer.
The additional phenomenon is a very sharp refracted wave in the upper layer. There
are also visible differences between in-line and cross-line directions in the case of
anisotropic model.
But in the case of ground recordings, the situation is a little bit different. Of course, these are
differences between iso- and anisotropic results but rather minor and hard to spot on synthetic
section. So in this case a difference seismogram is much more informative (figure 5.7.c). It is
clearly visible that in case of all three presented wave forms (top reflected, bottom reflected
and top converted) expected amplitude and time changes in the function of offset are present.
53
6. Summary
The basic aim of the presented study was to build, test and apply the effective parallel
algorithms to evaluate the seismic wave propagation in 3D anisotropic media. The further aim
was to find how important is the impact of the anisotropy on wave propagation for typical
anisotropy parameters and common seismic sources. Contrary to numerous theoretical studies
we have only used the direct method of evaluation – a numerical modeling.
Numerical algorithms used here for full-wave-field propagation modeling use stiffness tensor
of VTI and orthorhombic media. The more complicated models of anisotropy (like HTI or
TTI media) are deduced from the basic ones using the Bond matrices. It allows the
construction of efficient and universal programs that are in a position to evaluate wave fields
in the anisotropic medium (e.g. the same program can be used to model wave propagation in
numerous TTI media for different dip angles).
The results of modeling presented here are encouraging and interesting. The evaluated results
perfectly agree with theoretical models and prove the constructed software is correct, reliable
and well suited to modeling tasks. The important limitation of a numerical modeling is the
size of the model and number of iterations. The 3D anisotropic wave field modeling is
extremely time consuming and unrealizable in a single-node environment. The parallelization
of the computing process is the only effective way to face the real size models of the
geological medium and large offsets.
We can summarize the modeling results from the point of view of practical seismic
prospecting. We focus only on the impact of anisotropy on the shape of wave fronts and
propagation times. The results of performed modeling for different seismic sources show
relatively small differences between related isotropic and real anisotropic media. On the other
hand the most distinctive differences can be observed for heterogeneous, orthorhombic
medium for large offsets.
Another important result that can be drawn from the study is about the order of the differential
scheme used in computation. As it was expected the second order of time and space
expansion is the best compromise between speed and precision of modeling process. The
results of modeling with higher order differential schemes are not beneficial enough to
explain costs of additional computation time.
Further progress of a large scale and massive full-wave field propagation modeling strongly
depends on the progress in computer science especially in hardware development and new
methods of efficient computation. One of the promising techniques is utilization of GPU
instead of CPU, particularly in a multinode version. The experiences show that in this case
several dozens of acceleration can be achieved compared to standard CPU processing. In the
near future GPU processing will increase in importance allowing evaluation of the 3D
modeling with reasonable effort and in reasonable time.
54
7. References
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acoustic wave equation, Geophysics 1974, Vol. 39, No. 6, 834-842
Auld, B., A., 1973, Acoustic fields and waves in solids, vol. 1 , John Wiley & Sons, Inc, New
York.
Backus, G., E., 1962. Long-wave elastic anisotropy produced by horizontal layering: J.
Geophys. Res., 67,4427-4440.
Crampin, S., 1984, An introduction to wave propagation in anisotropic media, Geophys. J.
Roy. Astr. Soc. 76, 17-28.
Danek T., Elastic wave-field modeling in seismic exploration. Science and supercomputing
in Europe : report 2005, HPC-Europa, Pan-European Research Infrastructure on High
Performance Computing
Danek T., Franczyk A., Parallel and distributed seismic wave-field modeling, TASK
Quarterly, 2004 vol. 8 no. 4, 573–581
Danek T., Franczyk A., PC clusters in numerical modeling of seismic wave field,
Proceedings of XI KK KOWBAN '2004 (in Polish)
Danek T., Parallel computing and PC clusters in seismic wave field modeling /
Geoinformatica Polonica, 2005, Vol 7 25–34 (in Polish)
Danek T., Parallel computing in numerical modeling of seismic wave field. GEOPETROL
2004, (in Polish)
Danek T., Seismic wave field modeling as a tool in reservoir evaluation, Science and
supercomputing in Europe : raport 2007 HPC-Europa Pan-European Research Infrastructure
on High Performance Computing.
Danek T., Seismic wave field modeling with graphics processing units, Lecture Notes in
Computer Sciences 2009, 435–442.
Danek T., Pięta A., Leśniak A., Simulation of seismic waveforms from “Rudna” copper
mine, Poland, using staggered grid, Proceedings of 31st General Assembly of the European
Seismological Commission, Hersonissos, Crete, 2008
Igel H., Mora P., Riollet B., Anisotropic wave propagation through finite-difference grids,
Geophysics, 1995, Vol. 60 No 4, 1203-1216
Jędrzejowska-Tyczkowska H., 2006. Integrated Studies of the Fractures Induced Anisotropy
of Devonian Carbonates in the South Ekstern Poland. The 12th International Workshop on
Seismic Anisotropy. 22-27 October 2006, Beijing, China. Expanded abstracts. p 153.
55
Kelly K. R., Ward R. W., Sven Treitel, Alford R., M., Synthetic seismograms: a finite
difference approach, Geophysics, 1976, Vol. 41 No 2, 2-27
Kowal A., Piórkowski A., Danek T., Pięta A., Analysis of selected component technologies
efficiency for parallel and distributed seismic wave field modeling. CISSE 2008: international
joint conference on Computer, Information, and Systems Sciences, and Engineering.
Leśniak A., Danek T., Analysis of elastic wave field in simple anisotropic media with
applications in 3C seismic. Proceedings of GEOPETROL 2004. (in Polish)
Leśniak A., Danek T., Efficiency of Linux clusters in multi-component elastic wave field
modeling in anisotropic media, Proceedings of 68th EAGE conference & exhibition, Vienna
2006
Pięta A., Danek T., Leśniak A., Numerical modeling of ground vibration caused by
underground tremors in LGOM mining area. Mineral Resources Management, 2009, 261–
271 (in Polish)
Pietsh K., Marzec P., Kobylarski M., Danek T., Leśniak A., Tatarata A., Gruszczyk E.,
Identification of seismic anomalies caused by gas saturation on the basis of theoretical P and
PS wavefield in the Carpathian Foredeep, SE Poland. Acta Geophysica 2007 vol. 55 no. 2.
191–208
Postma, G., W., 1955, Wave propagation in stratified medium, Geophysics, 20, 780-806.
Reynolds A. C., Boundary conditions for the numerical solution of wave propagation
problems, Geophysics, 1978, Vol. 43, 1099–1110.
Schoenberg, M., Muir, F., 1989, A calculus for finely layered anisotropic media, Geophysics,
54, 581-589.
Tannehill J. C., Anderson D.A., Pletcher R.H., Computational Fluid Mechanics and Heat
Transfer, Second Eddition, Taylor & Francis, Washington 1997
Thomsen, L., 1986, Weak elastic anisotropy, Geophysics, vol. 51.
Tsvankin, I., 1997, Reflection moveout and parameter estimation for horizontal transverse
isotropy, Geophysics, 62, 614–629.
Tsvankin, I., 2001, Seismic signatures and analysis of reflection data in anisotropic media,
Pergamon, UK.
Winterstein, D., F., 1990, Velocity anisotropy terminology for geophysicists, Geophysics, 55,
1070-1088.
Yang D. H., Liu E., Zhang Z. J., Teng J., Finite-difference modelling in two-dimensional
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320-328
56
Appendix 1. Matrix notation for constitutive relations
Because of the following symmetry of the stiffness tensor,
jilkijlkjiklijkl cccc ===
we can use two indices instead of four, i.e.
6,,1,3,2,1,,, K=== JIlkjicc IJijkl.
There is the following conversion rule:
Tensor notation Matrix notation
ij or
kl i or
j
11 1
22 2
33 3
23 or 32 4
13 or 31 5
12 or 21 6
In shortened notation, a relationship between stress and deformation is
6,5,4,3,2,1, == JIc JIJI εσ
where
( ) ( )TT
121323332211654321 ,,,,,,,,,, σσσσσσσσσσσσ ==σ
( ) ( )TT
121323332211654321 2,2,2,,,,,,,, εεεεεεεεεεεε ==ε
57
Appendix 2.The finite-difference schema for horizontal transverse isotropy media
( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
∆
−−
∆
−
∆+
∆
+
∆
−−
∆
−
∆+
∆
+
∆
+−+
∆
+−+
∆
+−∆+−=
−−+−−+++
−−+−−+++
−+−+−+−+
z
ww
z
ww
xcc
t
y
vv
y
vv
xcc
t
z
uuuc
y
uuuc
x
uuuc
tuuu
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
222
1
222
1
2222
1,,11,,11,,11,,166
,,
13
,,
2
,1,1,1,1,1,1,1,144
,,
12
,,
2
2
1,,,,1,,66
,,2
,1,,,,1,44
,,2
,,1,,,,111
,,
2
1
,,,,
1
,,
ρ
ρ
ρ
( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
∆
−−
∆
−
∆+
∆
∆
−−
∆
−
∆+
∆
∆
+−+
∆
+−+
∆
+−∆+−=
−−+−−+++
−−+−−+++
−+−+−+−+
z
ww
z
ww
ycc
t
y
uu
y
uu
xcc
t
z
vvvc
y
vvvc
x
vvvc
tvvv
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
222
1
222
1
2222
1,1,1,1,1,1,1,1,55
,,
23
,,
2
,1,1,1,1,1,1,1,144
,,
21
,,
2
2
1,,,,1,,55
,,2
,1,,,,1,22
,,2
,,1,,,,144
,,
2
1
,,,,
1
,,
ρ
ρ
ρ
( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
∆
−−
∆
−
∆+
∆
∆
−−
∆
−
∆+
∆
∆
+−+
∆
+−+
∆
+−∆+−=
−−+−−+++
−−+−−+++
−+−+−+−+
z
vv
z
vv
ycc
t
z
uu
z
uu
xcc
t
z
wwwc
y
wwwc
x
wwwc
twww
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
222
1
222
1
2222
1,1,1,1,1,1,1,1,55
,,
32
,,
2
1,,11,,11,,11,,166
,,
31
,,
2
2
1,,,,1,,33
,,2
,1,,,,1,55
,,2
,,1,,,,166
,,
2
1
,,,,
1
,,
ρ
ρ
ρ
58
Appendix 3. The finite-difference schema for tilted transverse anisotropy media - rotation around x1 axis
( )( ) ( ) ( ) ( ) ( ) ( )
( )( )
( )( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+
∆
+−+
∆
+−+
∆
−−
∆
−
∆++
∆
+−∆
+
∆
+−+
∆
+−+
∆
−−
∆
−
∆+
∆
+−∆
+−=
−−−++−++−−+−−+++
−+−+−−+−−+++−+
−+−+−−+−−+++−+
−+
y
ww
y
ww
zcc
z
ww
z
ww
xcc
t
z
vvvc
y
vvvc
y
vv
y
vv
xcc
x
vvvc
t
z
uuuc
y
uuuc
y
uu
y
uu
xc
x
uuuc
t
uuu
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kjikji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
222
1
222
1
22
222
12
22
222
12
2
2
1,1,1,1,1,1,1,1,34'
,,
56'
,,
1,,11,,11,,11,,113'
,,
66'
,,
2
2
1,,,,1,,56'
,,2
,1,,,,1,24'
,,
,1,1,1,1,1,1,1,112'
,,
44'
,,2
,,1,,,,114'
,,
2
2
1,,,,1,,66'
,,2
,1,,,,1,44'
,,
,1,1,1,1,1,1,1,114'
,,2
,,1,,,,111'
,,
2
1
,,,,
1
,,
ρ
ρ
ρ
( )( )
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+
∆
+−+
∆
+−+
∆
−−
∆
−
∆+
∆
+−∆
+
∆
+−+
∆
+−+
∆
−−
∆
−
∆++
∆
+−∆
+−=
−−−++−++−−+−−+++
−+−+−−+−−+++−+
−+−+−−+−−+++−+
−+
y
ww
y
ww
zcc
z
ww
z
ww
xcc
t
z
vvvc
y
vvvc
y
vv
y
vv
xc
x
vvvc
t
z
uuuc
y
uuuc
y
uu
y
uu
xcc
x
uuuc
t
vvv
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kjikji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
222
1
222
1
22
222
12
2
22
222
12
2
1,1,1,1,1,1,1,1,23'
,,
55'
,,
1,,11,,11,,11,,134'
,,
56'
,,
2
2
1,,,,1,,55'
,,2
,1,,,,1,22'
,,
,1,1,1,1,1,1,1,124'
,,2
,,1,,,,144'
,,
2
2
1,,,,1,,56'
,,2
,1,,,,1,24'
,,
,1,1,1,1,1,1,1,112'
,,
44'
,,2
,,1,,,,114'
,,
2
1
,,,,
1
,,
ρ
ρ
ρ
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( )
∆
+−+
∆
+−+
∆
−−
∆
−
∆+
∆
+−∆
+
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+−=
−+−+−−+−−+++−+
−−−++−++−−+−−+++
−−−++−++−−+−−+++
−+
2
1,,,,1,,33'
,,2
,1,,,,1,55'
,,
,1,1,1,1,1,1,1,156'
,,2
,,1,,,,166'
,,
2
1,1,1,1,1,1,1,1,23'
,,
55'
,,
1,,11,,11,,11,,134'
,,
56'
,,
2
1,1,1,1,1,1,1,1,34'
,,
56'
,,
1,,11,,11,,11,,113'
,,
66'
,,
2
1
,,,,
1
,,
22
222
12
2
222
1
222
1
222
1
222
1
2
z
wwwc
y
wwwc
y
ww
y
ww
xc
x
wwwc
t
y
vv
y
vv
zcc
z
vv
z
vv
xcc
t
y
uu
y
uu
zcc
z
uu
z
uu
xcc
t
www
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
ρ
ρ
ρ
59
Appendix 4. The finite-difference schema for tilted transverse anisotropy media - rotation around x2 axis
( )( ) ( )
( )( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+
∆
+−+
∆
−−
∆
−
∆++
∆
+−+
∆
+−∆
+−=
−−+−−+++−−+−−+++
−−+−−+++−−+−−+++
−+−−−++−++−+−+
−+
z
ww
z
ww
xcc
y
ww
y
ww
xcc
t
y
vv
y
vv
xcc
z
vv
z
vv
xcc
t
z
uuuc
y
uu
y
uu
zcc
y
uuuc
x
uuuc
t
uuu
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
222
1
222
1
222
1
222
1
2
222
122
2
1,,11,,11,,11,,113'
,,
66'
,,
,1,1,1,1,11,1,146'
,,
15'
,,
2
,1,1,1,1,1,1,1,112'
,,
44'
,,
1,,11,,11,,11,,164'
,,
15'
,,
2
2
1,,,,1,,66'
,,
1,1,1,1,1,1,1,1,64'
,,
46'
,,2
,1,,,,1,44'
,,2
,,1,,,,111'
,,
2
1
,,,,
1
,,
ρ
ρ
ρ
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( )
( )( ) ( ) ( ) ( )
( )( ) ( )
( )( ) ( ) ( ) ( )
∆
+−+
∆
−−
∆
−
∆++
∆
+−+
∆
+−∆
+
∆
+−+
∆
−−
∆
−
∆++
∆
+−+
∆
+−∆
+
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+−=
−+−−−++−++−+−+
−+−−−++−++−+−+
−−+−−+++−−+−−+++
−+
2
1,,,,1,,53'
,,
1,1,1,1,1,1,1,1,23'
,,
55'
,,2
,1,,,,1,25'
,,2
,,1,,,,146'
,,
2
2
1,,,,1,,55'
,,
1,1,1,1,1,1,1,1,52'
,,
25'
,,2
,1,,,,1,22'
,,2
,,1,,,,144'
,,
2
1,,11,,11,,11,,151'
,,
46'
,,
,1,1,1,1,1,1,1,121'
,,
44'
,,
2
1
,,,,
1
,,
2
222
122
2
222
122
222
1
222
1
2
z
wwwc
y
ww
y
ww
zcc
y
wwwc
x
wwwc
t
z
vvvc
y
vv
y
vv
zcc
y
vvvc
x
vvvc
t
z
uu
z
uu
xcc
y
uu
y
uu
xcc
t
vvv
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
ρ
ρ
ρ
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( )
( )( ) ( ) ( ) ( )
( )( ) ( )
( )( ) ( ) ( ) ( )
∆
+−+
∆
−−
∆
−
∆++
∆
+−+
∆
+−∆
+
∆
+−+
∆
−−
∆
−
∆++
∆
+−+
∆
+−∆
+
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+−=
−+−−−++−++−+−+
−+−−−++−++−+−+
−−+−−+++−−+−−+++
−+
2
1,,,,1,,33'
,,
1,1,1,1,1,1,1,1,53'
,,
35'
,,2
,1,,,,1,55'
,,2
,,1,,,,166'
,,
2
2
1,,,,1,,35'
,,
1,1,1,1,1,1,1,1,32'
,,
55'
,,2
,1,,,,1,52'
,,2
,,1,,,,164'
,,
2
1,,11,,11,,11,,131'
,,
66'
,,
,1,1,1,1,1,1,1,151'
,,
64'
,,
2
1
,,,,
1
,,
2
222
122
2
222
122
222
1
222
1
2
z
wwwc
y
ww
y
ww
zcc
y
wwwc
x
wwwc
t
z
vvvc
y
vv
y
vv
zcc
y
vvvc
x
vvvc
t
z
uu
z
uu
xcc
y
uu
y
uu
xcc
t
www
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
ρ
ρ
ρ
60
Appendix 5. The finite-difference schema for tilted transverse anisotropy media - rotation around x3 axis
( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
( )( )
( )( ) ( ) ( ) ( ) ( )
∆
+−+
∆
+−+
∆
−−
∆
−
∆++
∆
+−∆
+
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+
∆
+−+
∆
+−+
∆
−−
∆
−
∆+
∆
+−∆
+−=
−+−+−−−++−++−+
−−+−−+++−−+−−+++
−+−+−−−++−++−+
−+
2
1,,,,1,,36'
,,2
,1,,,,1,45'
,,
1,,11,,11,,11,,113'
,,
66'
,,2
,,1,,,,116'
,,
2
1,1,1,1,1,1,1,1,26'
,,
45'
,,
,1,1,1,1,1,1,1,112'
,,
44'
,,
2
2
1,,,,1,,66'
,,2
,1,,,,1,44'
,,
1,,11,,11,,11,,116'
,,2
,,1,,,,111'
,,
2
1
,,,,
1
,,
22
222
12
222
1
222
1
22
222
12
2
2
z
wwwc
y
wwwc
x
ww
x
ww
zcc
x
wwwc
t
z
vv
z
vv
ycc
y
vv
y
vv
xcc
t
z
uuuc
y
uuuc
x
uu
x
uu
zc
x
uuuc
t
uuu
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kjikji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
ρ
ρ
ρ
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+
∆
+−+
∆
+−+
∆
−−
∆
−
∆+
∆
+−∆
+
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+−=
−−+−−+++−−+−−+++
−+−+−−−++−++−+
−−+−−+++−−+−−+++
−+
z
ww
z
ww
ycc
y
ww
y
ww
xcc
t
z
vvvc
y
vvvc
x
vv
x
vv
zc
x
vvvc
t
z
uu
z
uu
ycc
y
uu
y
uu
xcc
t
vvv
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
222
1
222
1
22
222
12
2
222
1
222
1
2
1,1,1,1,1,1,1,1,23'
,,
55'
,,
,1,1,1,1,1,1,1,126'
,,
45'
,,
2
2
1,,,,1,,55'
,,2
,1,,,,1,22'
,,
1,,11,,11,,11,,145'
,,2
,,1,,,,144'
,,
2
1,1,1,1,1,1,1,1,45'
,,
26'
,,
,1,1,1,1,1,1,1,112'
,,
44'
,,
2
1
,,,,
1
,,
ρ
ρ
ρ
( )( )
( )( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( )
∆
+−+
∆
+−+
∆
−−
∆
−
∆+
∆
+−∆
+
∆
−−
∆
−
∆++
∆
−−
∆
−
∆+
∆
+
∆
+−+
∆
+−+
∆
−−
∆
−
∆++
∆
+−∆
+−=
−+−+−−−++−++−+
−−+−−+++−−+−−+++
−+−+−−−++−++−+
−+
2
1,,,,1,,33'
,,2
,1,,,,1,55'
,,
1,,11,,11,,11,,136'
,,2
,,1,,,,166'
,,
2
1,1,1,1,1,1,1,1,23'
,,
55'
,,
,1,1,1,1,1,1,1,126'
,,
45'
,,
2
2
1,,,,1,,36'
,,2
,1,,,,1,45'
,,
1,,11,,11,,11,,116'
,,
66'
,,2
,,1,,,,116'
,,
2
1
,,,,
1
,,
22
222
12
2
222
1
222
1
22
222
12
2
z
wwwc
y
wwwc
x
ww
x
ww
zc
x
wwwc
t
z
vv
z
vv
ycc
y
vv
y
vv
xcc
t
z
uuuc
y
uuuc
x
uu
x
uu
zcc
x
uuuc
t
www
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
l
kji
kjikji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
ρ
ρ
ρ
61
Appendix 6. The fourth-order in time and second-order in space finite-difference schema for transversely isotropic media
( ) ( ) ( ) ( )
( ) ( )
( )
( )( )
( )( )
( )( )
( )[ ]( ) ( )
( )[ ]( ) ( )
( ) ( )
( )( )( )
( )( )( )
( )( ) ( )( )[ ]
( )
( )( )( )
( )( )( )
( )( ) ( )( )[ ]
( )
∆∆∆
+−−+−++−+−−
++++++
+∆∆
+−−++−−+++
+∆∆
+−−++−−+++
+∆∆∆
+−−+−++−+−−
++++++
+∆∆
+−−++−−+++
+∆∆
+−−++−−+++
+∆∆
+−+−+−+−+
+∆∆
+−+−+−+−+++
+∆∆
+−+−+−+−+++
+∆
+−+−+
∆
+−+−+
∆
+−+−
∆
+
∆∆
+−−++
∆∆
+−−+
+∆
+−+
∆
+−+
∆
+−
∆+−=
−−−−−++−−+−+−−−++−++−+−−++++−+++
−−−+−−−++−+++−++
−−+−−−+−−+++−+++
−−−−−+−+−−++−−−++−+++−−+−+++−+++
−−−+−−−++−+++−++
−−+−−−+−−+++−+++
−−−+−−+−++++
−−−−+−++−+++
−−−−+−++−+++
−−++−−++−−++
−−+−−+++−−+−−+++
−+−+−+
−+
2
1,1,11,1,11,1,11,1,11,,11,,11,,11,,11,1,11,1,11,1,11,1,1
55
,,
23
,,
44
,,
12
,,
55
,,
44
,,
66
,,
13
,,
3
2,,12,,11,,11,,11,,11,,12,,12,,166
,,
33
,,
66
,,
13
,,
3
1,,21,,21,,11,,11,,11,,11,,21,,211
,,
66
,,
66
,,
13
,,
2
1,1,11,1,11,1,11,1,1,1,1,1,1,1,1,1,11,1,11,1,11,1,11,1,1
55
,,
23
,,
66
,,
13
,,
66
,,
55
,,
44
,,
12
,,
3
,2,1,2,1,1,1,1,1,1,1,1,1,2,1,2,144
,,
22
,,
44
,,
12
,,
3
,1,2,1,2,1,1,1,1,1,1,1,1,1,2,1,244
,,
12
,,
44
,,
11
,,
22
1,1,,1,1,1,1,,,,1,,1,1,,1,1,1,66
,,
44
,,
22
1,,11,,1,,1,,1,,,,11,,11,,1,,1266
,,
13
,,
66
,,
11
,,
22
,1,1,1,,1,1,,1,,,,1,1,1,1,,1,1244
,,
12
,,
44
,,
11
,,
4
2,,1,,,,1,,2,,266
,,4
,2,,1,,,,1,,2,244
,,4
,,2,,1,,,,1,,2211
,,
2
4
1,,11,,11,,11,,166
,,
13
,,
,1,1,1,1,1,1,1,144
,,
12
,,
2
1,,,,1,,66
,,2
,1,,,,1,44
,,2
,,1,,,,111
,,2
1
,,,,
1
,,
4
2222*
*
4
2222
4
2222
4
2222*
*
4
2222
4
2222
224222
224222
224222
464464464
12
44
222
2
yzx
wwwwwwwwwwww
cccccccc
zx
wwwwwwwwcccc
xz
wwwwwwwwcccc
zyx
vvvvvvvvvvvv
cccccccc
yx
vvvvvvvvcccc
xy
vvvvvvvvcccc
yz
uuuuuuuuucc
zx
uuuuuuuuucccc
yx
uuuuuuuuucccc
z
uuuuuc
y
uuuuuc
x
uuuuuc
t
xz
wwwwcc
yx
vvvvcc
z
uuuc
y
uuuc
x
uuuc
tuuu
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikjikjikjikjikjikjikji
kjil
kji
lkji
lkji
lkji
lkji
lkji
lkji
l
kjikjikjikji
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikjikjikji
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikjikjikjikjikjikjikji
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikjikjikji
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikjikjikji
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikji
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikjikjikji
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikjikjikji
kjil
kjil
kjil
kjil
kjil
kji
kjil
kjil
kjil
kjil
kjil
kji
kjil
kjil
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
l
kji
kjikji
l
kji
l
kji
l
kji
kji
l
kji
l
kji
l
kji
kji
kjil
kjil
kjil
kji
l
kji
l
kji
l
kji
ρ
ρ
62
( ) ( ) ( ) ( )
( ) ( )
( )
( )( )
( )( )
( )( )
( )[ ]( ) ( )
( )[ ]( ) ( )
( ) ( )
( )( )( )
( )( )( )
( )( ) ( )( )[ ]
( )
( )( )( )
( )( )( )
( )( ) ( )( )[ ]
( )
∆∆∆
+−−+−++−+−−
++++++
+∆∆
+−−++−−+++
+∆∆
+−−++−−+++
+∆∆∆
+−−+−++−+−−
++++++
+∆∆
+−−++−−+++
+∆∆
+−−++−−+++
+∆∆
+−+−+−+−+
+∆∆
+−+−+−+−+++
+∆∆
+−+−+−+−+++
+∆
+−+−+
∆
+−+−+
∆
+−+−
∆
+
∆∆
+−−++
∆∆
+−−+
+∆
+−+
∆
+−+
∆
+−
∆+−=
−−−−+−+−−++−−−−++−++−−+−+++−++++
−−−+−−−++−+++−++
−−+−−−+−−+++−+++
−−−−−+−+−−++−−−++−+++−−+−+++−+++
−−−+−−−++−+++−++
−−+−−−+−−+++−+++
−−−−+−++−+++
−−−+−−+−++++
−−−−+−++−+++
−−++−−++−−++
−−+−−+++−−+−−+++
−+−+−+
−+
2
1,1,11,1,11,1,11,1,11,1,1,1,1,1,1,1,1,1,11,1,11,1,11,1,1
66
,,
13
,,
44
,,
12
,,
66
,,
44
,,
55
,,
23
,,
3
2,1,2,1,1,1,1,1,1,1,1,1,2,1,2,1,55
,,
33
,,
55
,,
23
,,
3
1,2,1,2,1,1,1,1,1,1,1,1,1,2,1,2,22
,,
55
,,
55
,,
23
,,
2
1,1,11,1,11,1,11,1,1,1,1,1,1,1,1,1,11,1,11,1,11,1,11,1,1
66
,,
13
,,
55
,,
23
,,
55
,,
66
,,
44
,,
12
,,
3
,2,1,2,1,1,1,1,1,1,1,1,1,2,1,2,122
,,
44
,,
44
,,
12
,,
3
,1,2,1,2,1,1,1,1,1,1,1,1,1,2,1,244
,,
11
,,
44
,,
12
,,
22
1,,11,,1,,1,,1,,,,11,,11,,1,,155
,,
44
,,
22
1,1,,1,1,1,1,,,,1,,1,1,,1,1,1,255
,,
23
,,
55
,,
22
,,
22
,1,1,1,,1,1,,1,,,,1,1,1,1,,1,1244
,,
12
,,
22
,,
44
,,
4
2,,1,,,,1,,2,,255
,,4
,2,,1,,,,1,,2,222
,,4
,,2,,1,,,,1,,2244
,,
2
4
1,1,1,1,1,1,1,1,55
,,
23
,,
,1,1,1,1,1,1,1,144
,,
12
,,
2
1,,,,1,,55
,,2
,1,,,,1,22
,,2
,,1,,,,144
,,2
1
,,,,
1
,,
4
2222
*
4
2222
4
2222
4
2222*
*
4
2222
4
2222
224222
224222
224222
464464464
12
44
222
2
xzy
wwwwwwwwwwww
cccccccc
zy
wwwwwwwwcccc
yz
wwwwwwwwcccc
zyx
uuuuuuuuuuuu
cccccccc
yx
uuuuuuuucccc
xy
uuuuuuuucccc
zx
vvvvvvvvvcc
yz
vvvvvvvvvcccc
yx
vvvvvvvvvcccc
z
vvvvvc
y
vvvvvc
x
vvvvvc
t
yz
wwwwcc
yx
uuuucc
z
vvvc
y
vvvc
x
vvvc
tvvv
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikjikjikjikjikjikjikji
kjil
kji
lkji
lkji
lkji
lkji
lkji
lkji
l
kjikjikjikji
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjikjikjikji
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
kjil
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64
Appendix 7. Main computational kernel loop
u[i][j][k]=2.0*pu[i][j][k]-ppu[i][j][k]+((dtr*dtr)/(rho[i][j][k]))*
(
c11[i][j][k]*((pu[i+1][j][k]-2.0*pu[i][j][k]+pu[i-1][j][k])/(ds*ds))
+c44[i][j][k]*
((pu[i][j+1][k]-2.0*pu[i][j][k]+pu[i][j-1][k])/(ds*ds))
+c66[i][j][k]*
((pu[i][j][k+1]-2.0*pu[i][j][k]+pu[i][j][k-1])/(ds*ds))
+((c12[i][j][k]+c44[i][j][k])/(4.0*ds*ds))*
((
pv[i+1][j+1][k]-pv[i+1][j-1][k]-pv[i-1][j+1][k]+pv[i-1][j-1][k]
))
+((c13[i][j][k]+c66[i][j][k])/(4.0*ds*ds))*
((
pw[i+1][j][k+1]-pw[i+1][j][k-1]-pw[i-1][j][k+1]+pw[i-1][j][k-1]
))
);
v[i][j][k]=2.0*pv[i][j][k]-ppv[i][j][k]+((dtr*dtr)/(rho[i][j][k]))*
(
c44[i][j][k]*((pv[i+1][j][k]-2.0*pv[i][j][k]+pv[i-1][j][k])/(ds*ds))
+c22[i][j][k]*
((pv[i][j+1][k]-2.0*pv[i][j][k]+pv[i][j-1][k])/(ds*ds))
+c55[i][j][k]*
((pv[i][j][k+1]-2.0*pv[i][j][k]+pv[i][j][k-1])/(ds*ds))
+((c12[i][j][k]+c44[i][j][k])/(4.0*(ds*ds)))*
((
pu[i+1][j+1][k]-pu[i+1][j-1][k]-pu[i-1][j+1][k]+pu[i-1][j-1][k]
))
65
+((c23[i][j][k]+c55[i][j][k])/(4.0*(ds*ds)))*
((
pw[i][j+1][k+1]-pw[i][j+1][k-1]-pw[i][j-1][k+1]+pw[i][j-1][k-1]
))
);
w[i][j][k]=2.0*pw[i][j][k]-ppw[i][j][k]+((dtr*dtr)/(rho[i][j][k]))*
(
c66[i][j][k]*((pw[i+1][j][k]-2.0*pw[i][j][k]+pw[i-1][j][k])/(ds*ds))
+c55[i][j][k]*
((pw[i][j+1][k]-2.0*pw[i][j][k]+pw[i][j-1][k])/(ds*ds))
+c33[i][j][k]*
((pw[i][j][k+1]-2.0*pw[i][j][k]+pw[i][j][k-1])/(ds*ds))
+((c13[i][j][k]+c66[i][j][k])/(4.0*(ds*ds)))*
((
pu[i+1][j][k+1]-pu[i+1][j][k-1]-pu[i-1][j][k+1]+pu[i-1][j][k-1]
))
+((c23[i][j][k]+c55[i][j][k])/(4.0*(ds*ds)))*
((
pv[i][j+1][k+1]-pv[i][j+1][k-1]-pv[i][j-1][k+1]+pv[i][j-1][k-1]
))
);
Listing A7.1. Computational kernel; u, w, v denote displacements in two horizontal and vertical directions in time t+1;
prefixes p and pp mean time step t and t-1 respectively; c?? represents elastic constraints and rho is density; i, j, k are
discrete coordinates of computational cubes.
Wykaz A7.1. Główna pętla obliczeniowa: u, w, v oznacza przemieszczenie w kierunku pionowym i obu poziomych w
czasie t+1; przedrostki p i pp oznaczają odpowiednio t i t-1 krok czasowy; c?? oznacza składowe macierzy sztywności,
rho gęstość; i, j, k współrzędne dyskretne siatki obliczeniowej.
66
Appendix 8. Modeling results
Fig A8.1. Results for exploding source and isotropic medium. Capital letters denote planes; small letters denote
directions of motion. All planes are source crossing
Rys A8.1 Wyniki modelowań uzyskane dla źródła wybuchowego dla ośrodka izotropowego. DuŜe litery oznaczają
płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez obszar źródłowy.
Fig A8.2 Results for exploding source and isotropic medium. Capital letters denote planes; small letters denote
directions of motion. All planes are shifted 20 from the source.
Rys A8.2 Wyniki modelowań uzyskane dla źródła wybuchowego dla ośrodka izotropowego. DuŜe litery oznaczają
płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od obszaru źródłowego o 20 m.
67
Fig A8.3 Results for exploding source and HTI medium. Capital letters denote planes; small letters denote directions
of motion. All planes are source crossing.
Rys A8.3 Wyniki modelowań uzyskane dla źródła wybuchowego dla ośrodka z anizotropią typu HTI. DuŜe litery
oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez obszar źródłowy.
Fig A8.4 Results for exploding source and HTI medium. Capital letters denote planes; small letters denote directions
of motion. All planes are shifted 20 from the source.
Rys A8.4 Wyniki modelowań uzyskane dla źródła wybuchowego dla ośrodka z anizotropią typu HTI. DuŜe litery
oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od obszaru
źródłowego o 20 m.
68
Fig A8.5. Results for exploding source and VTI medium. Capital letters denote planes; small letters denote directions
of motion. All planes are source crossing.
Rys A8.5 Wyniki modelowań uzyskane dla źródła wybuchowego dla ośrodka z anizotropią typu VTI. DuŜe litery
oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez obszar źródłowy.
Fig A8.6 Results for exploding source and VTI medium. Capital letters denote planes; small letters denote directions
of motion. All planes are shifted 20 from the source.
Rys A8.6 Wyniki modelowań uzyskane dla źródła wybuchowego dla ośrodka z anizotropią typu VTI. DuŜe litery
oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od obszaru
źródłowego o 20 m.
69
Fig A8.7. Results for exploding source and orthorhombic medium. Capital letters denote planes; small letters denote
directions of motion. All planes are source crossing
Rys A8.7 Wyniki modelowań uzyskane dla źródła wybuchowego dla ośrodka ortorombowego. DuŜe litery oznaczają
płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez obszar źródłowy.
Fig A8.8. Results for exploding source and orthorhombic medium. Capital letters denote planes; small letters denote
directions of motion. All planes are shifted 20 from the source.
Rys A8.8 Wyniki modelowań uzyskane dla źródła wybuchowego dla ośrodka ortorombowego. DuŜe litery oznaczają
płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od obszaru źródłowego o 20 m.
70
Fig A8.9. Results for double couple source and isotropic medium. Capital letters denote planes; small letters denote
directions of motion. All planes are source crossing
Rys A8.9 Wyniki modelowań uzyskane dla źródła przybliŜanego podwójną parą sił dla ośrodka izotropowego. DuŜe
litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez obszar
źródłowy.
Fig A8.10. Results for double couple source and isotropic medium. Capital letters denote planes; small letters denote
directions of motion. All planes are shifted 20 from the source.
Rys A8.10 Wyniki modelowań uzyskane dla źródła przybliŜanego podwójną parą sił dla ośrodka ortorombowego.
DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od obszaru
źródłowego o 20 m.
71
Fig A8.11. Results for double couple source and HTI medium. Capital letters denote planes; small letters denote
directions of motion. All planes are source crossing
Rys A8.11 Wyniki modelowań uzyskane dla źródła przybliŜanego podwójną parą sił dla ośrodka z anizotropią typu
HTI. DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez
obszar źródłowy.
Fig A8.12. Results for double couple source and HTI medium. Capital letters denote planes; small letters denote
directions of motion. All planes are shifted 20 from the source.
Rys A8.12 Wyniki modelowań uzyskane dla źródła przybliŜanego podwójną parą sił dla ośrodka z anizotropią typu
HTI. DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od
obszaru źródłowego o 20 m.
72
Fig A8.13. Results for double couple source and VTI medium. Capital letters denote planes; small letters denote
direction of motion. All planes are source crossing
Rys A8.13 Wyniki modelowań uzyskane dla źródła przybliŜanego podwójną parą sił dla ośrodka z anizotropią typu
VTI. DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez
obszar źródłowy.
Fig A8.14. Results for double couple source and VTI medium. Capital letters denote planes; small letters denote
direction of motion. All planes are shifted 20 from the source.
Rys A8.14. Wyniki modelowań uzyskane dla źródła przybliŜanego podwójną parą sił dla ośrodka z anizotropią typu
VTI. DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od
obszaru źródłowego o 20 m.
73
Fig A8.15. Results for double couple source and orthorhombic medium. Capital letters denote planes; small letters
denote direction of motion. All planes are source crossing
Rys A8.15. Wyniki modelowań uzyskane dla źródła przybliŜanego podwójną parą sił dla ośrodka ortorombowego.
DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez
obszar źródłowy.
Fig A8.16. Results for double couple source and orthorhombic medium. Capital letters denote planes; small letters
denote direction of motion. All planes are shifted 20 from the source.
Rys A8.16. Wyniki modelowań uzyskane dla źródła przybliŜanego podwójną parą sił dla ośrodka ortorombowego.
DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od obszaru
źródłowego o 20 m.
74
Fig A8.17.Results for single force source and isotropic medium. Capital letters denote planes; small letters denote
direction of motion. All planes are source crossing
Rys A8.17. Wyniki modelowań uzyskane dla źródła przybliŜanego pojedynczą siłą dla ośrodka izotropowego. DuŜe
litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez obszar
źródłowy.
Fig A8.18. Results for singe force source and isotropic medium. Capital letters denote planes; small letters denote
direction of motion. All planes are shifted 20 from the source.
Rys A8.18. Wyniki modelowań uzyskane dla źródła przybliŜanego pojedynczą siłą dla ośrodka izotropowego. DuŜe
litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od obszaru
źródłowego o 20 m.
75
Fig A8.19.Results for single force source and HTI medium. Capital letters denote planes; small letters denote
direction of motion. All planes are source crossing
Rys A8.19. Wyniki modelowań uzyskane dla źródła przybliŜanego pojedynczą siłą dla ośrodka z anizotropią typu
HTI. DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez
obszar źródłowy.
Fig A8.20. Results for singe force source and HTI medium. Capital letters denote planes; small letters denote direction
of motion. All planes are shifted 20 from the source.
Rys A8.20. Wyniki modelowań uzyskane dla źródła przybliŜanego pojedynczą siłą dla ośrodka z anizotropią typu
HTI. DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od
obszaru źródłowego o 20 m.
76
Fig A8.21. Results for single force source and VTI medium. Capital letters denote planes; small letters denote
direction of motion. All planes are source crossing
Rys A8.21. Wyniki modelowań uzyskane dla źródła przybliŜanego pojedynczą siłą dla ośrodka z anizotropią typu
VTI. DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez
obszar źródłowy.
Fig A8.22. Results for singe force source and VTI medium. Capital letters denote planes; small letters denote direction
of motion. All planes are shifted 20 from the source.
Rys A8.22. Wyniki modelowań uzyskane dla źródła przybliŜanego pojedynczą siłą dla ośrodka z anizotropią typu
VTI. DuŜe litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od
obszaru źródłowego o 20 m.
77
Fig A8.23.Results for single force source and orthorhombic medium. Capital letters denote planes; small letters
denote direction of motion. All planes are source crossing
Rys A8.23. Wyniki modelowań uzyskane dla źródła przybliŜanego pojedynczą siłą dla ortorombowego. DuŜe litery
oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji przechodzą przez obszar źródłowy.
Fig A8.24. Results for singe force source and orthorhombic medium. Capital letters denote planes; small letters
denote direction of motion. All planes are shifted 20 from the source.
Rys A8.24. Wyniki modelowań uzyskane dla źródła przybliŜanego pojedynczą siłą dla ośrodka ortorombowego. DuŜe
litery oznaczają płaszczyzny obserwacji, małe – kierunek ruchu. Płaszczyzny obserwacji są odsunięte od obszaru
źródłowego o 20 m.