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

2

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