numerical analysis of a narrow–angle one–way elastic...
TRANSCRIPT
Numerical analysis of a narrow–angle one–way elastic wave
equation and extension to curvilinear coordinates
D.A. Angus ∗ & C.J. Thomson †
Department of Geological Sciences & Geological Engineering, Queen’s University,
Kingston, ON, Canada, K7L 3N6
(March 13, 2006)
Short title: Narrow–angle elastic wave equation
ABSTRACT
In this paper, we review the finite–difference implementation of a narrow–angle
one–way vector wave equation for elastic, three–dimensional media. Extrapolation is
performed in the frequency domain, where the second–order accurate lateral spatial
difference operators are found to be sufficiently accurate for narrow–angle propaga-
tion. We perform a numerical analysis of the finite–difference scheme to highlight
the stability and dispersion characteristics. The von Neumann stability criterion in-
dicates that extracting a reference phase during the extrapolation step noticeably
improves the forward–marching scheme and dispersion analysis shows that numerical
grid anisotropy is minimal for the propagation path lengths, source pulse spectral con-
tent and angular range of forward propagation of interest. Although reasonable, we
observe that the computational efficiency of the algorithm is limited by the second–
order accurate extrapolation step and therefor further improvements to the extrap-
∗Now at: Department of Earth Sciences, University of Bristol, Bristol, UK
†Now at: Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge
1
olation scheme can be made. We extend the Cartesian narrow–angle formulation to
curvilinear coordinates, where the computational grid tracks the true wavefront in a
reference medium and the wavefield derivative normal to the reference wavefront is
evaluated locally using the Cartesian propagator. An example of curvilinear extrap-
olation for a simple model consisting of a high velocity sphere within a homogeneous
background velocity structure shows that the narrow–angle propagator is capable of
modeling frequency–dependent geometrical spreading and diffraction effects in curvi-
linear coordinates.
INTRODUCTION
Improvements in data quality and quantity have stimulated the need for greater
understanding and enhanced modeling of elastic–wave propagation for increasingly
complicated anisotropic and inhomogeneous media. Since there is no general analytic
solution for wave propagation in anisotropic, inhomogeneous elastic media, various
approximate methods are used and these are often based on physically motivated
arguments specific to the problem under study (Carcione et al., 2002). Elastic wave
propagation in anisotropic and heterogeneous media may lead to wavefront folding (or
caustics), frequency–dependent wave coupling and mode conversions. Furthermore,
when orientation variations and averaging of significant fine–scale elastic anisotropy
and heterogeneity are present, the net effect of these variations on longer wavelength
seismic signals can sometimes be difficult to assess.
Thomson (1999) used a displacement–vector formulation to derive a hierarchy
of one–way elastic wave equations which is based on the factorization of the wave
equation itself rather than its solution. This intermediate method simulates one–way
propagation of elastic waves in three–dimensional (3–D) generally anisotropic media
and is closely related to but more generally applicable than many conventional one–
way (e.g., ray based) methods. It is not limited to regions of weak anisotropy or to
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particular directions far from polarization singularities (Crampin and Yedlin, 1981),
and coupling by gradients is included naturally. The wide angle forms of this one–
way wave equation can handle wavefront folding effects (caustics) without special
attention; such folding can arise due to heterogeneities or it can be due to dimples
(indentations) on the slowness sheets of anisotropic materials.
The narrow angle form of the one–way wave equation was implemented via fi-
nite differences and applied to various 3–D wave propagation problems to verify the
method as well as to study frequency–dependent three–component waveform effects.
Angus et al. (2004) simulate waveforms in homogeneous anisotropic media and show
that the propagator is accurate for angles up to ±15◦ to the preferred direction of
propagation and that the propagator is well behaved for the most extreme anisotropic
singularity (i.e. the conical–point singularity). It is shown that the relationship be-
tween waveforms and the underlying elasticity of the medium can be complicated
for curved incident wavefronts and this stresses the importance of considering the
integrated effect of a range of slownesses. Angus (2005) model waveforms in various
deterministic and stochastic isotropic, heterogeneous media, where the length scales
of the heterogeneities span several orders of size relative to the seismic wavelength.
The examples presented in Angus (2005) suggest that the narrow–angle propaga-
tor is an efficient tool for simulating elastic waves for an assortment of 3–D forward
diffraction and scattering problems and is capable of incorporating the frequency–
dependent effects of wave propagation due to smooth variations in the medium down
to the sub–Fresnel scale.
Although there is extensive literature describing the numerical implementation
and characteristics of all types of wave equation (e.g., two–way acoustic, elastic, visco–
elastic, etc.), our vector equation is sufficiently different that it requires revisiting some
of these basic numerical considerations. In this paper, we perform a formal numerical
analysis of the finite–difference implementation of the narrow–angle one–way wave
equation. The numerical analysis will reveal practical limitations of the narrow–
3
angle formulation and highlight possible avenues for improvement. We also extend
the Cartesian formulation to curvilinear coordinates and provide a simple example of
wave extrapolation in curvilinear coordinates.
CARTESIAN NARROW–ANGLE FORMULATION
The frequency–domain narrow–angle one–way wave equation is written
∂u
∂x1= iωP0u + Pα
∂u
∂xα
+1
iωPαβ
∂2u
∂xα∂xβ
, (1)
where u is the three–component displacement vector, ω is angular frequency and
the summation convention is being used. The propagation direction is taken to be
along the x1 axis and the lateral coordinates x2,3 and slownesses p2,3 are denoted with
Greek subscripts (e.g. xα and pα). The sub–propagator matrices P0, Pα and Pαβ are
obtained from the following recursive equations (Thomson, 1999)
P0 =√
ρC−111 ,
P0Pα + PαP0 = −C−111 (C1α + Cα1)P0 ,
P0Pαβ + PαβP0 = −C−111 Cαβ − C−1
11 (C1α + Cα1)Pβ − PαPβ . (2)
The matrices Cjk are submatrices of the elastic tensor given by (Cjk)il = cijkl and ρ
is density.
Finding the optimum finite–difference approach to equation (1) is made compli-
cated by the high dimensionality (three dimensions in space) of the partial differential
equation, the presence of mixed derivatives and the existence of variable coefficients.
Although there are several alternative numerical formulations of equation (1), the
frequency–domain wave equation offers the greatest reduction in dimensionality of
the finite–difference operators and mixed derivatives, and avoids the stability limita-
tions of time–integration schemes (Marfurt, 1984a). That is, the frequency–domain
equation allows direct control over the frequencies modeled (i.e., ability to filter any
4
high frequency numerical noise) and avoids possible parasitic modes which may evolve
in higher–order time domain solutions. There are further computational advantages
to frequency domain methods which are expected to be significant if the narrow–
angle formulation is implemented as a Green’s function or propagator/predictor for
3–D seismic–imaging and inversion algorithms (Muller, 1983; Marfurt, 1984b; Pratt,
1989; Song and Williamson, 1995).
A useful strategy in forward–marching algorithms is to extract a reference phase
during the extrapolation step. Using the diagonal matrix of estimated local Cartesian
x1–components of slowness P1 = (pqP1 , pqS1
1 , pqS2
1 ) for each wavetype, equation (1) is
rewritten
∂u
∂x1= iω (P0 − P1) u + Pα
∂u
∂xα
+1
iωPαβ
∂2u
∂xα∂xβ
. (3)
The extrapolation (or forward propagation) step for the wavefield u is written
u(x1 + ∆x1) = u(x1 − ∆x1) + 2∂u
∂x1∆x1 + O(∆x3
1) (4)
and is second–order accurate in x1 as indicated (Angus et al., 2004). This is a
simple explicit scheme requiring only three x1–planes to be stored during each prop-
agation step: the new or forward plane x1 + ∆x1 to which the wavefield is to be
extrapolated; the current or middle plane x1 upon which the narrow–angle wave
equation (3) is evaluated; and the previous or back plane x1 − ∆x1 which is re-
quired so that the extrapolation step (4) is second–order accurate with respect
to x1. In the extrapolation step (4), the back plane is de–phased according to
u(x1 − ∆x1) = exp [iωP1∆x1]u(x1 − ∆x1). The forward plane is then re–phased
according to u(x1 + ∆x1) = exp [iωP1∆x1]u(x1 + ∆x1).
Numerous approaches can be used in developing an appropriate finite–difference
scheme; see, for example, Tannehill et al. (1997) for the heat equation or Durran
(1999) for the advection equation. In particular, the application of high–order dif-
ference operators (Alford et al., 1974; Dablain, 1986; Fornberg, 1987; Zingg, 2000)
5
and the use of staggered–grids (Madariaga, 1976; Virieux, 1986; Luo and Schuster,
1990; Graves, 1996; Ghrist, 2000; Moczo et al., 2000) have been found to improve
numerical accuracy with minimal loss in computational efficiency. Ristow and Ruhl
(1997) review finite–difference methods used to solve acoustic one–way wave equa-
tions and present implicit 3–D finite–difference schemes using multiway splitting of
the dispersion relation. Although an unconditionally–stable algorithm would be ideal,
the primary goal of our work was to develop a finite–difference scheme that was sim-
ple to code and reasonably efficient computationally. For the interior grid, we use
the following equi–spaced Cartesian formulation of second–order accurate difference
operators for all derivatives (Angus et al., 2004)
u(xi+11 , xj
2, xk3, ω) = u(xi−1
1 , xj2, x
k3, ω) + 2iω∆x1P0u(xi
1, xj2, x
k3, ω)
+∆x1
∆x2P2
[
u(xi1, x
j+12 , xk
3, ω) − u(xi1, x
j−12 , xk
3, ω)]
+∆x1
∆x3P3
[
u(xi1, x
j2, x
k+13 , ω) − u(xi
1, xj2, x
k−13 , ω)
]
−2∆x1(iω
−1)
∆x22 P22
[
u(xi1, x
j+12 , xk
3, ω) − 2u(xi1, x
j2, x
k3, ω)
+u(xi1, x
j−12 , xk
3, ω)]
−2∆x1(iω
−1)
∆x32 P33
[
u(xi1, x
j2, x
k+13 , ω) − 2u(xi
1, xj2, x
k3, ω)
+u(xi1, x
j2, x
k−13 , ω)
]
−∆x1(iω
−1)
2∆x2∆x3
(P23 + P32)[
u(xi1, x
j+12 , xk+1
3 , ω) − u(xi1, x
j+12 , xk−1
3 , ω)
−u(xi1, x
j−12 , xk+1
3 , ω) + u(xi1, x
j−12 , xk−1
3 , ω)]
. (5)
Since numerical errors introduced by interior differencing schemes often dominate
in wave simulation (Zingg, 2000), the following numerical analysis focuses strictly
on the interior finite–difference operators without any formal consideration of the
boundary operators. Specifically, the stability and dispersion characteristics of the
interior difference scheme (5) are examined to understand the limitations of and
possible means of improving the narrow–angle extrapolation.
6
Weak stability – the amplification matrix
Stability analysis provides a measure of the limit or extent to which an initial
discrete function can be amplified during numerical propagation (Richtmeyer and
Morton, 1967). This definition of stability has been shown to be technically equivalent
to the analytic von Neumann stability condition (von Neumann and Richtmeyer,
1949) and, for various wave equations, can be reduced to a simple Courant number
(Marfurt, 1984a; Lines et al., 1999). From early numerical experiments (Angus et al.,
2004) we found the parameter relation
C =∆x1
ω∆xα
≤ K (6)
could be used as a crude estimate of stability, where the constant K is assessed
post–mortem. Although similar to the Courant number discussed by Tannehill et al.
(1997), it is by no means a strict criterion for stability. In fact, it was used prior to the
stability analysis presented in this paper and primarily as a guide in grid refinement
once a general region of stability was obtained. Hereafter, C will be referred to as
the rough Courant number for the narrow–angle difference scheme (5).
In a strict sense, the von Neumann analysis is only valid for problems with constant
coefficients and simple boundary conditions, but it is attractive because it reduces
the technical stability condition to a practical numerical criterion. Fortunately, this
analysis can be applied to more complicated problems (e.g., ones with variable coef-
ficient(s), vector equations and multi–level schemes), though it is then not as exact.
Richtmeyer (1962) gives an excellent account of the von Neumann analysis, high-
lighting the necessary and sufficient conditions for stability and means of adaptation
to more complex problems. The von Neumann analysis provides a local measure of
stability (Press et al., 1992) or continued boundedness of the local error (Durran,
1999) and so gives an estimate of the global error growth rate. This is important be-
cause the growth characteristics of the global error can severely limit the usefulness
of a particular scheme.
7
Consider the three–component plane–wave expressed by
u(x, ω) = u(ω) exp [ik · x] , (7)
where k(k1, k2, k3) is the wavenumber and u(ω) is the frequency–dependent ampli-
tude. Substituting the plane–wave (7) into the standard second–order centered finite–
difference operators for the lateral first and second derivatives yields
∂αu(x, ω) ≈ u(x, ω)
(
i sin (kα∆xα)
∆xα
)
(8)
and
∂α∂βu(x, ω) ≈ u(x, ω)
2(
cos (kα∆xα)−1∆x2
α
)
α = β
−(
sin (kα∆xα) sin (kβ∆xβ)
∆xα∆xβ
)
α 6= β, (9)
respectively. Equations (8) and (9) represent approximations to the lateral derivatives
of the plane–wave (7) sampled on the equi–spaced grid. Substitution of equations (8)–
(9) into the three–level interior–grid extrapolation scheme (5) yields the analogue
u(xi+11 , xj
2, xk3, ω) = A0u(xi
1, xj2, x
k3, ω) + A−1u(xi−1
1 , xj2, x
k3, ω) , (10)
where
A0 = 2∆x1
[
iωP0 + iP2sin (k2∆x2)
∆x2+ iP3
sin (k3∆x3)
∆x3
+2
iω
(
P22cos (k2∆x2) − 1
∆x22
+ P33cos (k3∆x3) − 1
∆x23
−
(
P23 + P32
2
)
sin (k2∆x2) sin (k3∆x3)
∆x2∆x3
)]
(11)
and A−1 = I when a forward reference phase is not extracted. Equation (10) can be
expressed as a two–level extrapolation scheme
u(xi+11 , xj
2, xk3, ω)
u(xi1, x
j2, x
k3, ω)
= A
u(xi1, x
j2, x
k3, ω)
u(xi−11 , xj
2, xk3, ω)
, (12)
where
8
A =
A0 A−1
I 0
. (13)
The 6 × 6 matrix A is called the amplification matrix of the extrapolation operator
(5). When a forward reference phase is extracted, the 6 × 6 amplification matrix is
denoted A, where the submatrices P0 and A−1 in A are replaced by P0 = (P0 −P1)
and A−1 = P−11 .
Stability analysis requires that all coefficients (i.e., the sub–propagator matrices
P0, Pα and Pαβ) of the amplification matrix A be constant, which is not generally
the case for the narrow–angle wave equation. However, the stability of variable–
coefficient equations can be evaluated over a range of expected coefficient values for
the particular problem at hand. This is referred to as freezing of the variable coeffi-
cients and amounts to assuming homogeneity. For the narrow–angle wave equation,
our experience indicates that stability can be achieved for a particular heterogeneous
medium if it is ensured for its homogeneous equivalent. This homogeneous equivalent
or neighbor can be represented by an arithmetic average of the heterogeneous medium
grid elasticities. Another possibility is to examine stability at both the maximum and
minimum elastic limits of the medium (e.g., at the higher and lower wave velocities).
Since the main assumption associated with the narrow–angle approximation is that
the medium be smoothly variable, it is not expected that the medium elasticities will
vary significantly from the homogeneous equivalent. Anisotropy does not limit sta-
bility analysis since it is not expected that anisotropy will exceed ≈ 10% for realistic
media.
Richtmeyer (1962, see pages 59–71) points to four individually sufficient conditions
for stability, only the last and least strict of which can be satisfied by the narrow–
angle scheme. This fourth condition requires all but one of the eigenvalues ℓi of the
amplification matrix A to be within the unit circle. The condition
|ℓmax| ≤ 1 + O(∆x1) (14)
9
must be satisfied by the remaining eigenvalue. Thus for stable propagation equation
(14) requires that the maximum eigenvalue or spectral amplitude ℓmax approach
unity faster than O(∆x1) as ∆x1 → 0.
An obvious implication of condition (14) is that the solution can only be propa-
gated a finite distance before the global errors become significant. For the propagation
path lengths of interest here, our numerical experience indicates that ℓmax ≤ 1.0005
is sufficient for stable and accurate results with the narrow–angle wave equation. It
should be noted that, for these distances and the initial conditions experimented
with thus far, the number of propagation steps fall within the criterion of long–term
stability suggested by Hestholm (2003).
The P– and S–wave velocities in the following (and subsequent) numerical anal-
ysis are 4575 and 2600 m/s, which are the isotropic average velocities of anisotropic
pure halite (Raymer, 2000). The frequencies used (e.g., in evaluating equation (25))
are typical of the waveforms generated in the subsequent numerical example as well
as those in Angus et al. (2004) and Angus (2005). Table 1 presents the stability
results for a plane P–wave at normal incidence on an equi–spaced Cartesian grid. In
this particular example, the indicated frequency ω ≈ 1767 rad/s corresponds to the
dominant period of the waveform pulse (i.e., it is effectively an estimate of the tem-
poral pulse–width). Table 1 demonstrates that by reducing the step size ∆x1, while
keeping the rough Courant factor C (6) constant, ℓmax for the amplification matrix
A satisfies condition (14). It is not surprising to note that in the case of forward
reference–phase extraction, ℓmax for A turns out to be significantly closer to unity
and, in fact, is relatively insensitive to the choice of propagation step ∆x1, at least
for the values tested.
It is important to note that stability is also a function of the propagation angle,
as this affects the off–diagonal components of the matrices A0 and A−1. This is
important in situations when the underlying wavefront is curved or when plane–waves
propagate at angles to the x1–direction. Figure 1 displays the maximum spectral
10
amplitudes ℓmax for a curved incident P–wave, where the curvature is defined by the
wavefront normals varying between 0◦ and 10◦ to the x1–direction at the grid center
and boundary, respectively. The extrapolation scheme for the phased wavefield is
outside the stability regime and so the maximum spectral amplitudes ℓmax appear
insensitive to the wavefront curvature. When a forward reference phase is extracted,
the scheme is within the stability regime and it is apparent that curvature does have an
effect on stability; it is best when the wavefront normal is parallel to the x1–direction
and becomes progressively worse for increasing angles to this direction.
It should be noted that the sampling or numerical resolution (i.e., the number
of extrapolation steps per oscillation period) in the forward or x1–direction is exces-
sive (Durran, 1999). For instance, consider an initial waveform having a temporal
pulse–width of T ≈ 8 ms propagating in an isotropic medium with P– and S–wave
velocities of 4500 and 2600 m/s, respectively. The corresponding P– and S–wave spa-
tial pulse–widths are approximately λP = 36 and λS = 20 m. For stable and accurate
calculations (i.e., ℓmax ≤ 1.0005), the extrapolation increment for an incident P–
and S–wave is limited to ∆x1 ≈ 0.125 m and ∆x1 ≈ 0.0625 m, respectively. Thus,
the P– and S–wave spatial pulse–widths are sampled λ/∆x1 ≈ 300 times. This result
is not unexpected and will be addressed in the section discussing grid dispersion.
Grid dispersion
Stability analysis helps identify if accurate numerical simulations can be achieved
(Claerbout, 1985) and the von Neumann analysis does so by indicating whether a
finite–difference scheme is amplifying, damping or neutral. Even if a scheme is sta-
ble, finite–difference approximations to wave equations nevertheless generate bounded
frequency–dependent error in the wavefield. This phenomenon is commonly referred
to as grid dispersion (Marfurt, 1984a; Etgen, 1988) and it can be especially impor-
tant when studying anisotropic media. Numerical dispersion can introduce a form of
11
anisotropy commonly referred to as grid anisotropy (Virieux, 1986; Igel et al., 1995),
where the grid imposes its own crystallinity. That is, the dispersion relation of the
discrete wave equation differs from that of the continuous wave equation. The errors
introduced by the differing dispersion relations are not only dependent on frequency
and wavenumber, but also on the direction of propagation within the grid. Thus
dispersion analysis determines the cost of achieving accurate numerical simulations.
Dispersion analysis seeks information by examining and comparing the phase evo-
lution properties of the continuous and discrete wave equations. Consider a plane–
wave solution expressed
u = u(ω) exp [ik · x] , (15)
where u(ω) is the three–component displacement vector, k(ω) is the wave vector and
x is the position vector. Substitution of (15) into the continuous narrow–angle wave
equation (1) yields
[
ωP0 + kναPα − kν
1 +1
ωkν
αkνβPαβ
]
u = 0 , (16)
where kνi is the i–th component wavenumber for the wavetype ν.
The determinant of the homogeneous system (16) must equal zero and leads to
a fairly complicated analytic relationship between angular frequency ω and the wave
vector k. For normal incidence (kα = 0) and isotropic homogeneous media, it can be
seen that equation (16) is diagonal and the three solutions of the determinant repre-
sent the k1–component wavenumbers for each wavetype ν. The eigenvalue solution of
this determinant for anisotropic media and propagation angles oblique to the x1–axis
must also represent the k1–component wavenumbers for each wavetype ν. The eigen-
value solution can be evaluated by standard numerical means and expressed in terms
of the x1–component phase velocity vp,ν1 = ω/kν
1 = ω/ℓν, where ℓν is the eigenvalue
for the wavetype ν. The superscript p is used to distinguish the phase velocity vp
from the group velocity vg, which is to be discussed later.
12
Substitution of the vector plane–wave solution (15) into the discrete narrow–
angle wave equation (5) yields
[
ωP0 + kναPα − kν
1 +(
kναkν
β
∣
∣
∣
α6=β− 2kν
αα
)
Pαβ
]
u = 0 , (17)
where
kνi =
sin (kνi ∆xi)
∆xi
and
kναα =
cos (kνα∆xα) − 1
∆x2α
.
As in the continuous case, the eigenvalues ℓν of equation (17) represent the k1–
component wavenumber for each wavetype ν. An expression for the x1–component
phase (or numerical phase) velocity for the discrete wave equation is expressed
vp,ν1 = ω/kν
1 = ω/ℓν, where ℓν is the eigenvalue for the wavetype ν.
Numerical dispersion is strongest for low–order operators and so it is expected
that the second–order difference scheme (5) will require very fine sampling in the x1–
direction to achieve acceptable numerical accuracy. In the following figures, dispersion
curves (i.e., curves of the ratio of ℓν/ℓν traced over all sampled frequencies) for various
grid parameters are presented for plane P– and S–waves over a range of incidence
angles (0◦, 4◦, 8◦ and 12◦) to the x1–axis. In all cases, the P– and S–wave spatial
pulse–widths are approximately 36 and 20 m, where the smallest sampled wavelengths
are approximately 4.5 and 2.5 m.
Figure 2 displays the dispersion curves for P– and S–waves sampled approximately
80 and 50 times per spatial pulse–width (or dominant wavelength) in the x1–direction,
respectively. In this particular example, the phased scheme (i.e., no forward reference
phase is extracted) is dispersive for all wavetypes, although it is particularly poor
for the S–wave. For the de–phased scheme, the dispersion is less and, in fact, much
improved for the S–wave. It is clear that extracting a reference phase reduces numer-
ical dispersion. In Figure 3, the propagation step size ∆x1 is reduced by an order of
13
magnitude from ∆x1 = 0.1 m to 0.01 m. This does improve numerical dispersion,
but at the expense of sampling the P– and S–waves approximately 800 and 500 times
per spatial pulse–width, respectively. It is apparent in both Figures 2 and 3 that grid
anisotropy exists and is illustrated by the greater dispersion with increasing angle to
the x1–axis. As shown in Figure 4, decreasing the lateral grid spacing ∆xα by an
order of magnitude from ∆xα = 10 to 1 m significantly reduces grid anisotropy. In
all three figures, there is no significant improvement in the dispersion ratios of the
phased S–wave, where the phase is being advanced rather than delayed. Figure 5
presents a case where the phased scheme dispersion ratios are much improved for the
S–wave. However, the propagation step length is ∆x1 = 0.0002 m and is on the order
of 50 times smaller than that of Figure 3.
The above results are as expected for a second–order operator applied to forward–
marching schemes, which have been addressed by Fornberg (1987, see Figures 2 and
3) in the context of the pseudo–spectral limit and by Zingg (2000, see Figures 2
and 3) in the context of non–compact/compact (i.e., explicit/implicit) operators.
The sinusoidal oscillations in the x1–direction require small extrapolation steps for
accurate signal estimation. Extracting a reference phase can reduce these oscillations
and improve the accuracy of the second–order extrapolation operator. Sampling in
the x1–direction can be improved further (i.e., made coarser) by increasing the order
of the finite–difference operator, but for the forward extrapolation step this would
require increasing the number of x1–planes that need to be stored (e.g., an n–th order
operator would yield an n–th level extrapolation scheme). An another approach to
coarsen the x1–sampling would be to introduce an implicit operator for the forward
extrapolation step.
Dispersion analysis often involves examining the behavior of not only phase ve-
locity vp but also group velocity vg = ∂ω/∂k. Holberg (1987) demonstrates that
group–velocity error can be an order of magnitude larger than the maximum phase–
velocity error and so potentially more significant. Thus it would seem sufficient to
14
examine only group velocity error. However, an analytic expression for phase velocity
has not been developed and so group velocity would have to be evaluated numerically
(e.g., via finite differencing the phase velocity). Igel et al. (1995) state that phase–
velocity error is responsible for the spatial variation of group–velocity error and so it
may be sufficient to examine the behavior of the phase error to infer the behavior of
group–velocity error.
It is emphasized that for the waveforms obtained by Angus et al. (2004) and
Angus (2005), numerical dispersion is small as a result of the chosen grid parameters,
propagation path lengths and source–pulse spectral content. Our examples fall in
the range 1/G ≈ 0.03 or less on the right–hand side of Figure 4, which is seen to
be highly accurate close to the forward propagation direction (Figure 4, solid line).
In fact, the main source of error stems from phase errors introduced by the narrow–
angle approximation to the wide–angle propagator (Thomson, 1999), with only mild
numerical phase dispersion. There are various approaches to minimize grid dispersion,
some of which are: implementing staggered grids (Virieux, 1986; Moczo et al., 2000);
increasing finite–difference operator size (Dablain, 1986; Holberg, 1987; Min et al.,
2000); including lumped–mass acceleration terms and rotated operators (Cole, 1994;
Jo et al., 1996; Stekl and Pratt, 1998; Min et al., 2000); applying a direct dispersion
correction (Muller et al., 1992) and using compact (or implicit) operators (Zingg,
2000). However, the forward step sizes dictated by the dispersion characteristics (of
the finite–difference narrow–angle propagator) are not computationally prohibitive
for small desktop computer applications.
CURVILINEAR NARROW–ANGLE FORMULATION
The Cartesian narrow–angle formulation is generally only appropriate for homoge-
neous or weakly inhomogeneous media and gently curved initial wavefront conditions
(i.e., non point–sources). When the medium is more heterogeneous, so that steeply
15
dipping and turning waves are possible, a curvilinear formulation is more appropriate.
In this approach, the computational grid (or curvilinear reference frame) attempts to
track a true wavefront.
The curvilinear reference grid is constructed by tracing geometrical rays within
a suitably chosen reference medium. To describe the curvilinear grid it is necessary
to define three coordinate frames; the local (x) and global (X) Cartesian coordinates
and the ray (q) coordinates. Figure 6 is a schematic representation of the curvilinear
reference grid, where the global Cartesian coordinate frame X = (X1, X2, X3) forms
the basis for the other coordinate systems. Ray tracing provides the location of
the reference–frame node points in terms of the ray quantities q = (q1, q2, q3). The
ray quantities q are defined in global Cartesian coordinates, where q1 = T is the
time from the initial data surface q1 = 0 and (q2, q3) = (X02, X03) define the initial
lateral position of a ray in that surface. Note that the subscript 0 specifies the
initial wavefront surface in global Cartesian grid. The ray parameter q1 = T has a
representation in the global Cartesian coordinates q1 = q1(X). The initial data surface
is described by a reference wavefront and is found by tracing rays in the reference
medium from an initial horizontal starting surface using known or estimated incidence
times and lateral slownesses. At each node in the curvilinear grid a local Cartesian
coordinate system x (x1, x2, x3) is introduced, with x1 chosen normal to the wavefront
q1 = T . This local Cartesian frame allows the narrow–angle wave equation (1) to
be used directly rather than a reformulation of the one–way equation in curvilinear
coordinates. We simply interpret x1 in equation (1) as the local x1.
Using the standard relations
∂u
∂qi
=∂xj
∂qi
∂u
∂xj
and∂2u
∂qγ∂qδ
=∂
∂qγ
(
∂xj
∂qδ
∂u
∂xj
)
, (18)
which are evaluated using the ray and geometrical–spreading equations (Cerveny,
2001), equation (1) is recast into the following one–way wave equation in mixed local
Cartesian and curvilinear coordinates
16
∂u
∂x1
= Q−1
[
iωP0u + Pα
∂u
∂qα
+1
iωPαβ
∂2u
∂qα∂qβ
]
, (19)
where
Q =
(
I +1
iωPαβtαγtβδ
∂2x1
∂qγ∂qδ
)
≈ I, (20)
Pα = Pξtξα −1
iωPξβtξγtβδtηα
∂2xη
∂qγ∂qδ
, (21)
and
Pαβ = Pδγtδαtγβ . (22)
The tαβ are the elements of the inverse of the 2×2 matrix of ray geometrical–spreading
quantities ∂xβ/∂qα.
Equation (19) describes the wavefield gradient in the local Cartesian x1–direction.
If the reference medium is not isotropic, then q1 and x1 are not necessarily in the
same direction. For implementation, it is actually preferable to define the wavefield
derivative with respect to the curvilinear coordinate direction q1 via the left–hand
relation in equation (18). The partial derivative ∂u/∂x1 is given by equation (19)
and the ∂u/∂xα are evaluated by second–order finite differences of the wavefield on the
middle x1–plane of the discrete grid. Extrapolation is therefore really performed along
the reference rays rather than the local x1–direction. By the matrix approximation
Q−1 ≈ I and the extraction of a reference phase, equation (19) may be re–written in
mixed local Cartesian x and curvilinear q coordinates
∂ul
∂x1
= iω (P0 − P1) ul + Pα
∂ul
∂qα
+1
iωPαβ
∂2ul
∂qα∂qβ
. (23)
It would appear that the numerical implementation of equation (23) explicitly re-
quires the rotation of the wavefield into the ‘active’ local Cartesian coordinate x.
However, it is not necessary that the wavefield be defined in the local frame; only
the sub–propagator matrices need be evaluated in this frame. Therefore, to reduce
the computational overhead and simplify the algorithm, the wavefield is expressed
17
throughout in terms of the global Cartesian frame. An expression relating the local
and global Cartesian wavefield may be written ul = RUg, where R is the change of
basis tensor from global to local Cartesian coordinates and Ug is the global Cartesian
de–phased wavefield (and similarly ul = RUg for the phased wavefield). Equation
(23) is then re–written
∂Ug
∂x1= iωRT (P0 −P1)RUg + RTPαR
∂Ug
∂qα
+1
iωRTPαβR
∂2Ug
∂qα∂qβ
. (24)
As can be seen in equation (24), only the frequency–independent sub–propagator ma-
trices are rotated and not the frequency–dependent wavefield Ug, which significantly
reduces the computational overhead. The extrapolation step along the ray is then
defined in the global Cartesian coordinate frame by
Ug(q1 + ∆q1) = Ug(q1 − ∆q1) + 2∂Ug
∂q1
∆q1 + O(∆q31) (25)
and is second–order accurate in q1 as indicated. The extrapolation step (25) in the
curvilinear frame also requires the back plane to be de–phased according to Ug(q1 −
∆q1) = RTP1RUg(q1 − ∆q1). The forward plane is then re–phased according to
Ug(q1 + ∆q1) = RTP1RUg(q1 + ∆q1).
An obvious concern is that the evaluation of the curvilinear sub–propagator ma-
trices (20)–(22) will represent additional computational overhead (i.e., rotations be-
tween the local Cartesian and curvilinear coordinates). Fortunately, because these
coefficient matrices are frequency independent and in view of the benefits associated
with the curvilinear formulation (e.g., grid flexibility), the rotation overheads are out-
weighed by the advantages. When constructing the curvilinear coordinate system, it
is important to stress that caustics and ray multi–pathing should be avoided and that
the curvilinear grid should be smooth.
18
Curvilinear amplification matrix
The amplification matrices for curvilinear extrapolation can be derived in a similar
fashion to those for the Cartesian extrapolation. The curvilinear amplification ma-
trices are now also a function of the curvilinear transformation variables, but behave
only slightly differently from those of the Cartesian scheme. This is demonstrated
in Table 2, where the curvilinear grid is constructed to mimic the Cartesian grid in
Table 1. Table 3 demonstrates the effect of grid curvature on the curvilinear amplifi-
cation matrices, where it can be seen that an increase in curvature of the underlying
wavefront results in a general decrease in stability. This statement is purposely vague
and this is because the maximum eigenvalues ℓmax of the amplification matrices
are oscillatory with respect to frequency ω as well as the lateral xα coordinates (as
can be seen in Figure 1). However, when either frequency or the angle of incidence
with respect to the x1–axis increases, the oscillations of the spectral amplitudes ℓmax
are superimposed on a general amplitude increase. When the underlying wavefront
is curved, the curvilinear formulation allows for larger extrapolation step sizes than
the Cartesian formulation and this is because the forward reference phase is extracted
more–or–less in the direction of the wavefront normal rather than strictly in the global
Cartesian x1 direction.
Example
The curvilinear formulation of the narrow–angle wave equation is used to propa-
gate a smoothly curved incident P–wave through a model consisting of a high velocity
sphere embedded within a homogeneous volume. The high velocity sphere is defined
by a smooth analytic velocity function (see Angus (2005), equation 8) with maximum
P– and S–wave velocities of 5030 m/s and 2860 m/s at the center of the sphere and
diameter of approximately 500 m. The curvilinear reference grid is defined by the
19
normals of the incident curved underlying wavefront at the edges of the lateral grid
along the x2– and x3–axes and these normals are inclined at angles up to about 4◦ to
the x1–axis. The reference grid consists of 41×41 lateral node points or rays with ini-
tial lateral spacing ∆qα = 30 m and forward propagation step ∆q1 = ∆T = 0.05 ms.
The center of the initial q1–plane of the curvilinear lateral grid is located directly over
the high velocity sphere at a depth of X1 = 0 m. The reference rays of the curvilinear
grid are traced within a homogeneous reference medium having background isotropic
homogeneous P and S–wave velocities of 4575 m/s and 2600 m/s. A 2–D section of
the curvilinear reference grid is shown in Figure 7 and is superimposed over a 2–D
section of the true velocity profile.
In Figure 8, the q1–component waveforms (i.e. component of displacement along
the ray direction) are plotted for profiles along the q2–direction at the initial q1 = 0
ms plane and two q1–planes at 320 ms and 440 ms. The initially curved underlying q1–
component wavefront on the initial q1 = 0 ms plane panel plot appears planar because
the waveforms are plotted on the curvilinear reference grid, which coincides with the
true wavefront at this stage. Interpolating the initial wavefield onto a Cartesian
grid would yield the curved wavefront. The bottom two frames display the evolved
wavefield on the curvilinear grid at depths of approximately x1 = 1500 and x1 =
2000 m. Similar to the results in Angus (2005) using the Cartesian formulation
and a planar incident wavefront, the central regions of the q2 waveform plots display
reduced amplitudes as a result of geometrical spreading. Also visible are the enhanced
amplitudes and later arriving diffractions along the shoulders due to the funnel–
shaped caustic (Angus, 2005).
DISCUSSION AND CONCLUSIONS
The numerical analysis of the narrow–angle finite–difference propagator indicates
that accurate and stable results can be obtained for reasonable grid parameters, but
20
that extrapolation requires fine numerical resolution in the forward (x1) direction.
Extracting a forward reference phase allows coarsening of the propagation step size,
yet the numerical resolution still remains the limiting factor for efficient and accurate
computations; the x1 sampling needs to be on the order 100 samples per spatial
pulse–width unless the reference phase is extremely close to the true phase of the
wavefield.
Although the second–order accurate explicit difference operators were chosen more
for convenience rather than efficiency, the results suggest that a more stable and less
dispersive implementation is preferable. However, it is important to stress that the
second–order, 3–D extrapolation algorithm tested here is still not computationally
prohibitive for small desktop computational applications. For instance, 3–D calcu-
lations with the Cartesian extrapolator for a homogeneous anisotropic model with
lateral grid dimension of 49× 49 points, 33 frequencies and 10, 000 propagation steps
using Gnu FORTRAN-77 and a 1.8 GHz Athlon processor under a Linux O/S take ap-
proximately 20 minutes (Angus et al., 2004). Wave simulation in 3–D heterogeneous
media with the Cartesian extrapolator for models having similar grid dimensions
as those in Angus et al. (2004) take approximately an hour (Angus, 2005). The
added computation time stems from the algorithm having to read the heterogeneous
elastic model file and evaluating the sub–propagator matrices for each grid point at
each extrapolation step. For curved incident wavefronts the computation times for
heterogeneous media can be reduced using the curvilinear coordinate formulation.
However, there are additional overheads; ray tracing must be performed to evaluate
the curvilinear grid and the elastic model must be specified for each grid point of the
curvilinear grid. The process to set up the curvilinear grid and the elastic model file
for curvilinear coordinates takes approximately one hour.
In terms of stability and dispersion characteristics, implicit methods are generally
considered superior to explicit methods (Claerbout, 1985; Tannehill et al., 1997). In
three dimensions, however, implicit methods may not necessarily lead to the most op-
21
timal numerical scheme. Thus, improvements to the narrow–angle propagator would
likely involve implementing an averaging scheme (e.g., the Dufort–Frankel method)
or an alternating–direction explicit (ADE) method (Tannehill et al., 1997) for the
forward extrapolation step. Either of these two methods would presumably lead to
improved stability and dispersion characteristics. Compared to the forward extrapo-
lation step, the second–order lateral difference operators are not as limiting in terms
of accuracy or grid dispersion. This is primarily because the narrow–angle formula-
tion restricts the range of acceptable lateral wavenumbers and degree of wavefront
curvature (i.e., there is a restriction on initial conditions).
ACKNOWLEDGMENTS
Doug Angus acknowledges Dave Lyness and Bullard Laboratories, Cambridge Uni-
versity for providing access to their computer systems during the initial stages of this
work, as well as Bengt Fornberg and Michelle Ghrist for some helpful discussion with
regard to boundary operators and the Runge phenomenon. Gerhard Pratt is thanked
for the many fruitful discussions about various aspects of finite differences. We thank
Joe Dellinger, two anonymous reviewers, and both the Associate Editor and Editor
for providing thorough reviews. This work was supported by a grant from Imperial
Oil (Canada) and NSERC Individual Research grants to C. J. Thomson. D.A. An-
gus was also supported by scholarships from the Canadian Society of Exploration
Geophysicists and Queen’s University.
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26
TABLES
∆x1(m) ∆xα(m) ω(rad/s) C ℓmax(A) ℓmax(A)
2.00 2000.0 1767.0 5.66 × 10−7 8.37178343 1.00000000
1.00 1000.0 1767.0 5.66 × 10−7 2.84295005 1.00000000
0.50 500.0 1767.0 5.66 × 10−7 1.46072828 1.00000000
0.10 100.0 1767.0 5.66 × 10−7 1.01842897 1.00000000
0.01 10.0 1767.0 5.66 × 10−7 1.00018257 1.00000000
2.00 10.0 1767.0 1.132 × 10−4 8.30270103 1.00000000
TABLE 1. Demonstration of Richtmeyer’s fourth sufficient stability condition for the Cartesian
amplification matrices A and A. The rough Courant number C (6) is held constant by fixing the
angular frequency ω and suitably adjusting ∆x1 and ∆xα. The tabled results are for a plane P–wave
at normal incidence.
27
∆q1(ms) ∆x1(m) ∆qα(m) ω(rad/s) C ℓmax(B) ℓmax(B)
0.2200 1.00 1000.0 1767.0 5.66 × 10−7 2.86740509 1.00000000
0.2200 1.00 100.0 1767.0 5.66 × 10−6 2.86740502 1.00000000
0.0220 0.10 100.0 1767.0 5.66 × 10−7 1.01867405 1.00000000
0.0022 0.01 10.0 1767.0 5.66 × 10−7 1.00018674 1.00000000
0.4400 2.00 10.0 1767.0 1.132 × 10−4 8.46961946 1.00000000
TABLE 2. Demonstration of Richtmeyer’s fourth sufficient stability condition for the
curvilinear amplification matrices B and B. The rough Courant number C (6) is held
constant by fixing the angular frequency ω and suitably adjusting ∆q1 and ∆qα. The results
presented are for a plane P–wave at normal incidence on a curvilinear grid ‘identical’ to
that of the Cartesian grid in Table 1.
28
∆θ/∆x(◦/m) ∆q1 ∆qα ℓmax(B) ℓmax(B) ℓmax(B) ℓmax(B)
(grid center) (grid perimeter)
0.0 × 10−2 0.0022 10.0 1.00018674 1.00000000 1.00018674 1.00000000
5.0 × 10−2 0.0022 10.0 1.02149937 1.01084954 1.02112184 1.00769113
10.0 × 10−2 0.0022 10.0 1.02574406 1.01775906 1.02870654 1.01828778
15.0 × 10−2 0.0022 10.0 1.03185643 1.02557420 1.02761325 1.01735196
30.0 × 10−2 0.0022 10.0 1.03846307 1.03492816 1.03881316 1.03043649
TABLE 3. Effect of grid curvature on the curvilinear amplification matrices B and B.
The curvature (denoted by ∆θ/∆xα in column 1) is specified in terms of the angular rate
of change of the grid normal q1 to the global Cartesian X1–direction with respect to the
global Cartesian lateral Xα–direction.
29
FIGURES
FIGURE 1. Maximum eigenvalue ℓmax plotted as a function of normalized pulse
wavenumber k∗p = ωp∆x1/2πV = kp∆x1/2π = ∆x1/λp = 1/G (where G is the num-
ber of x1 grid points per pulse wavelength λp, ωp is the pulse frequency and V is
the P–wave velocity) and for an incident underlying curved P wavefront at three grid
locations; grid boundary (top panels), point midway between boundary and center
(middle panels) and grid center (bottom panels). Extracting a forward reference–
phase clearly improves the stability regime (left versus right columns). As well, it
is apparent that wavefront curvature also affects stability; at the grid center the
wavefront normal is parallel to the x1–direction and stability is best, whereas at the
boundary the normal is inclined 10◦ and stability is degraded. Note that the lower
panel on the right appears blank because ℓmax is very close to unity.
FIGURE 2. (∆x1 = 0.1 m and ∆xα = 10 m) Dispersion plot for plane P– and
S–waves as a function of the number of grid points per wavelength G. The ratio
of the vertical or x1–component wavenumbers ℓ/ℓ (which is equivalent to the ratio
of x1–component phase velocities vp1/v
p1) at four incidence angles: 0◦ (solid line), 4◦
(large dashes), 8◦ (small dashes) and 12◦ (small and large dashes) to the x1–axis.
The P– and S–waves are sampled approximately 80 and 50 times per spatial pulse–
width, respectively. The ‘phased’ scheme (i.e., when a forward reference–phase is not
extracted) is very dispersive for the S–wave and the phases are advanced rather than
delayed.
FIGURE 3. (∆x1 = 0.01 m and ∆xα = 10 m) Dispersion plot for plane P– and
S–waves (see Figure 2 for details) sampled approximately 800 and 500 times per spa-
tial pulse–width, respectively. The S–wave dispersion curves for the ‘phased’ scheme
have improved in comparison to those of Figure 2. Dispersion is noticeably different
for the various propagation angles and this is indicative of grid–anisotropy in both
30
the ‘phased’ and the ‘de–phased’ schemes, although it is more so for the ‘phased’
S–wave.
FIGURE 4. (∆x1 = 0.1 m and ∆xα = 1 m) Dispersion plot for plane P– and
S–waves (see Figure 2 for details) with the same propagation step length as in Figure
2, but with a lateral grid spacing one order of magnitude smaller. In contrast to
Figures 2 and 3, grid–anisotropy has been reduced significantly.
FIGURE 5. (∆x1 = 0.0002 m and ∆xα = 10 m) Dispersion plot for plane P–
and S–waves (see Figure 2 for details) with propagation step length on the order of
50 times smaller than that of Figure 3. The dispersion curves are very good for both
the ‘phased’ and ‘de–phased’ schemes.
FIGURE 6. The curvilinear reference frame traced in a reference medium. This
figure highlights the various coordinates used in the curvilinear extrapolation: global
Cartesian X, local Cartesian x and ray q coordinates. The solid (black) node points
indicate known values of the wavefield and open (white) node points indicate un-
known quantities.
FIGURE 7. Curvilinear reference grid superimposed over a two–dimensional sec-
tion of the high–velocity sphere model. The grid consists of 41 × 41 node points
or rays with lateral spacing ∆qα = 30 m and forward propagation step ∆q1 = 0.05
ms. The curvilinear reference rays are traced within a reference medium having an
isotropic P–wave velocity of 4575.5 m/s, starting from an approximate initial depth
of x1 = 0 m (i.e. q1 = 0 ms plane) and finishing at an approximate depth of x1 = 2000
m (i.e. q1 = 440 ms plane).
FIGURE 8. Waveforms of the q1–component (i.e. component along the ray direction)
31
of displacement for an incident underlying curved P–wavefront in the high–velocity
sphere model at q1–planes of 0, 320 and 440 ms (approximately equivalent to x1–
planes of 0, 1500 and 2000 m depth), plotted as profiles along the q2–axis. The
indices above the three columns signify the initial lateral position of the profile on
the curvilinear grid defined by 41 × 41 node points. The index iq3 = 20 is located
at x3 = 1970 m and is slightly off the midline, iq3 = 10 is located at x3 = 1670 m
and skirts the side of the sphere, and iq3 = 15 is located x3 = 1820 m and bisects the
other two q3 positions. The region of reduced amplitudes results from geometrical
spreading or de–focussing of the wavefield.
32
1.0004
1.0002
1.0001
1.0000
1.0003
1.0005
1.0004
1.0002
1.0001
1.0000
1.0003
1.0005
1.0004
1.0002
1.0001
1.0000
1.0003
4.2
2.6
1.8
1.0
3.4
5.0
5.0
2.6
1.8
1.0
3.4
4.2
5.0
4.2
2.6
1.8
1.0
3.4
0.100.00 0.02 0.06 0.080.04 0.100.00 0.02 0.06 0.080.04
Grid center
Forward reference−phase
1/G
1.0005
Grid center
Grid midpoint
Grid boundaryGrid boundaryextracted
Grid midpoint
not extractedForward reference−phase
ℓ max
ℓ max
ℓ max
33
FIGURE 1. Maximum eigenvalue ℓmax plotted as a function of normalized pulse
wavenumber k∗p = ωp∆x1/2πV = kp∆x1/2π = ∆x1/λp = 1/G (where G is the number
of x1 grid points per pulse wavelength λp, ωp is the pulse frequency and V is the P–wave
velocity) and for an incident underlying curved P wavefront at three grid locations; grid
boundary (top panels), point midway between boundary and center (middle panels) and
grid center (bottom panels). Extracting a forward reference–phase clearly improves the
stability regime (left versus right columns). As well, it is apparent that wavefront curvature
also affects stability; at the grid center the wavefront normal is parallel to the x1–direction
and stability is best, whereas at the boundary the normal is inclined 10◦ and stability is
degraded. Note that the lower panel on the right appears blank because ℓmax is very close
to unity.
34
1.0
0.9
0.8
1.2
1.1
1.0
0.9
0.8
1.2
0.09 0.12 0.150.00 0.03 0.060.09 0.12 0.150.00 0.03 0.06
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10
1.1
1.0
0.9
0.8
1.2
1.1
1.0
0.9
0.8
1.2
extracted
P−wave
S−wave
Forward reference−phase
1.1
P−wave
not extractedForward reference−phase
(dis
cret
e/co
ntin
uous
)R
atio
of v
ertic
al w
aven
umbe
rs
1/G
S−wave
FIGURE 2. (∆x1 = 0.1 m and ∆xα = 10 m) Dispersion plot for plane P– and S–waves
as a function of the number of grid points per wavelength G. The ratio of the vertical or
x1–component wavenumbers ℓ/ℓ (which is equivalent to the ratio of x1–component phase
velocities vp1/v
p1) at four incidence angles: 0◦ (solid line), 4◦ (large dashes), 8◦ (small dashes)
and 12◦ (small and large dashes) to the x1–axis. The P– and S–waves are sampled approx-
imately 80 and 50 times per spatial pulse–width, respectively. The ‘phased’ scheme (i.e.,
when a forward reference–phase is not extracted) is very dispersive for the S–wave and the
phases are advanced rather than delayed.
35
1.0
0.9
0.8
1.2
1.1
1.0
0.9
0.8
1.2
0.09 0.12 0.150.00 0.03 0.060.09 0.12 0.150.00 0.03 0.06
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10
1.1
1.0
0.9
0.8
1.2
1.1
1.0
0.9
0.8
1.2
1/10G
Forward reference−phaseextracted
1.1
P−wave
not extractedForward reference−phase
(dis
cret
e/co
ntin
uous
)R
atio
of v
ertic
al w
aven
umbe
rs
S−wave
P−wave
S−wave
FIGURE 3. (∆x1 = 0.01 m and ∆xα = 10 m) Dispersion plot for plane P– and S–waves
(see Figure 2 for details) sampled approximately 800 and 500 times per spatial pulse–width,
respectively. The S–wave dispersion curves for the ‘phased’ scheme have improved in com-
parison to those of Figure 2. Dispersion is noticeably different for the various propagation
angles and this is indicative of grid–anisotropy in both the ‘phased’ and the ‘de–phased’
schemes, although it is more so for the ‘phased’ S–wave.
36
1.0
0.9
0.8
1.2
1.1
1.0
0.9
0.8
1.2
0.09 0.12 0.150.00 0.03 0.060.09 0.12 0.150.00 0.03 0.06
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10
1.1
1.0
0.9
0.8
1.2
1.1
1.0
0.9
0.8
1.2
1/G
Forward reference−phaseextracted
1.1
P−wave
not extractedForward reference−phase
(dis
cret
e/co
ntin
uous
)R
atio
of v
ertic
al w
aven
umbe
rs
S−wave
P−wave
S−wave
FIGURE 4. (∆x1 = 0.1 m and ∆xα = 1 m) Dispersion plot for plane P– and S–waves
(see Figure 2 for details) with the same propagation step length as in Figure 2, but with
a lateral grid spacing one order of magnitude smaller. In contrast to Figures 2 and 3,
grid–anisotropy has been reduced significantly.
37
1.0
0.9
0.8
1.2
1.1
1.0
0.9
0.8
1.2
0.09 0.12 0.150.00 0.03 0.060.09 0.12 0.150.00 0.03 0.06
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10
1.1
1.0
0.9
0.8
1.2
1.1
1.0
0.9
0.8
1.2
1/1000G
Forward reference−phaseextracted
1.1
P−wave
not extractedForward reference−phase
(dis
cret
e/co
ntin
uous
)R
atio
of v
ertic
al w
aven
umbe
rs
S−wave
P−wave
S−wave
FIGURE 5. (∆x1 = 0.0002 m and ∆xα = 10 m) Dispersion plot for plane P– and
S–waves (see Figure 2 for details) with propagation step length on the order of 50 times
smaller than that of Figure 3. The dispersion curves are very good for both the ‘phased’
and ‘de–phased’ schemes.
38
q
X α
X 1
Ray
Τ−∆Τ
Τ+∆ΤLocal coordinates
x
Global coordinatesX Curvilinear coordinates
qαqα −∆qα
qαqα +∆
Wav
efro
nt
x
xα
1
T
FIGURE 6. The curvilinear reference frame traced in a reference medium. This figure
highlights the various coordinates used in the curvilinear extrapolation: global Cartesian
X, local Cartesian x and ray q coordinates. The solid (black) node points indicate known
values of the wavefield and open (white) node points indicate unknown quantities.
39
Average isotropic velocity (km/s)
2000.
1750.
1500.
1250.
1000.
750.
500.
250.
0.
0
.
50
0.
100
0.
150
0.
200
0.
250
0.
300
0.
350
0.
400
0.
Dep
th (
m)
Offset (m)
FIGURE 7. Curvilinear reference grid superimposed over a two–dimensional section of
the high–velocity sphere model. The grid consists of 41 × 41 node points or rays with
lateral spacing ∆qα = 30 m and forward propagation step ∆q1 = 0.05 ms. The curvilinear
reference rays are traced within a reference medium having an isotropic P–wave velocity of
4575.5 m/s, starting from an approximate initial depth of x1 = 0 m (i.e. q1 = 0 ms plane)
and finishing at an approximate depth of x1 = 2000 m (i.e. q1 = 440 ms plane).
40
q1=440 ms
q1=320 ms
q1=0 ms
q2
q2
q2
Time
iq3=20
iq3=15
iq3=10
FIGURE 8. Waveforms of the q1–component (i.e. component along the ray direction)
of displacement for an incident underlying curved P–wavefront in the high–velocity sphere
model at q1–planes of 0, 320 and 440 ms (approximately equivalent to x1–planes of 0, 1500
and 2000 m depth), plotted as profiles along the q2–axis. The indices above the three
columns signify the initial lateral position of the profile on the curvilinear grid defined by
41 × 41 node points. The index iq3 = 20 is located at x3 = 1970 m and is slightly off
the midline, iq3 = 10 is located at x3 = 1670 m and skirts the side of the sphere, and
iq3 = 15 is located x3 = 1820 m and bisects the other two q3 positions. The region of
reduced amplitudes results from geometrical spreading or de–focussing of the wavefield.
41