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2 Phase, group, and energy velocities

2 Phase, group, and energy velocities

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84



Seismic wave propagation



of the cophasal surfaces. Born and Wolf (1980) consider the phase velocity to be

devoid of any physical significance because it does not correspond to the velocity of

propagation of any signal and cannot be directly determined experimentally.

Waves encountered in rock physics are rarely perfectly monochromatic but instead

have a finite bandwidth, Do, centered around some mean frequency o. The wave

may be regarded as a superposition of monochromatic waves of different frequencies,

which then gives rise to the concept of wave packets or wave groups. Wave packets,

or modulation on a wave containing a finite band of frequencies, propagate with the

group velocity defined as

1



Vg ẳ 

grad@p=@oịo 

which for plane waves becomes

 

@o

Vg ẳ

@k o

The group velocity may be considered to be the velocity of propagation of the

envelope of a modulated carrier wave. The group velocity can also be expressed in

various equivalent ways as

Vg ẳ Vp l



dVp

dl



Vg ẳ Vp ỵ k



dVp

dk



or

1

1

o dVp



2

Vg Vp Vp do

where l is the wavelength. These equations show that the group velocity is different

from the phase velocity when the phase velocity is frequency dependent, direction

dependent, or both. When the phase velocity is frequency dependent (and hence

different from the group velocity), the medium is said to be dispersive. Dispersion is

termed normal if the group velocity decreases with frequency and anomalous or

inverse if it increases with frequency (Elmore and Heald, 1985; Bourbie´ et al., 1987).

In elastic, isotropic media, dispersion can arise as a result of geometric effects such as

propagation along waveguides. As a rule such geometric dispersion (Rayleigh waves,

waveguides) is normal (i.e., the group velocity decreases with frequency). In a

homogeneous viscoelastic medium, on the other hand, dispersion is anomalous or

inverse and arises owing to intrinsic dissipation.

The energy velocity Ve represents the velocity at which energy propagates and

may be defined as



85



3.2 Phase, group, and energy velocities



kz /w

Ve = Vg



k



Vp = Ve cos

kx /w



anisotropic slowness

surface



Figure 3.2.1 In anisotropic media, energy propagates along Ve, which is always normal to the

slowness surface and in general is deflected by the angle c away from Vp and the wave vector k.



Ve ¼



Pav

Eav



where Pav is the average power flow density and Eav is the average total energy

density.

In isotropic, homogeneous, elastic media all three velocities are the same. In a lossless

homogeneous medium (of arbitrary symmetry), Vg and Ve are identical, and energy

propagates with the group velocity. In this case the energy velocity may be obtained

from the group velocity, which is usually somewhat easier to compute. If the medium is

not strongly dispersive and a wave group can travel a measurable distance without

appreciable “smearing” out, the group velocity may be considered to represent the

velocity at which the energy is propagated (though this is not strictly true in general).

In anisotropic, homogeneous, elastic media, the phase velocity, in general, differs

from the group velocity (which is equal to the energy velocity because the medium is

elastic) except along certain symmetry directions, where they coincide. The direction

in which Ve is deflected away from k (which is also the direction of Vp) is obtained

from the slowness surface (shown in Figure 3.2.1), for Ve (¼ Vg in elastic media) must

always be normal to the slowness surface (Auld, 1990).

The group velocity in anisotropic media may be calculated by differentiation of

the dispersion relation obtained in an implicit form from the Christoffel equation given by

jk2 cijkl nj nl À o2 ik j ẳ ẩo; kx ; ky ; kz ị ¼ 0

where cijkl is the stiffness tensor, ni are the direction cosines of k, r is the density, and

dij is the Kronecker delta function. The group velocity is then evaluated as

Vg ¼ À



rk È

@È=@o



where the gradient is with respect to kx, ky, and kz.



86



Seismic wave propagation



The concept of group velocity is not strictly applicable to attenuating viscoelastic

media, but the energy velocity is still well defined (White, 1983). The energy

propagation velocity in a dissipative medium is neither the group velocity nor the

phase velocity except when

(1) the medium is infinite, homogeneous, linear, and viscoelastic, and

(2) the wave is monochromatic and homogeneous, i.e., planes of equal phase are

parallel to planes of equal amplitude, or, in other words, the real and imaginary

parts of the complex wave vector point in the same direction (in general they do

not), in which case the energy velocity is equal to the phase velocity (BenMenahem and Singh, 1981; Bourbie´ et al., 1987).

For the special case of a Voigt solid (see Section 3.8 on viscoelasticity) the energy

transport velocity is equal to the phase velocity at all frequencies. For wave propagation in dispersive, viscoelastic media, one sometimes defines the limit

V1 ¼ lim Vp ðoÞ

o!1



which describes the propagation of a well-defined wavefront and is referred to as the

signal velocity (Beltzer, 1988).

Sometimes it is not clear which velocities are represented by the recorded travel

times in laboratory ultrasonic core sample measurements, especially when the sample

is anisotropic. For elastic materials, there is no ambiguity for propagation along

symmetry directions because the phase and group velocities are identical. For nonsymmetry directions, the energy does not necessarily propagate straight up the axis of

the core from the transducer to the receiver. Numerical modeling of laboratory

experiments (Dellinger and Vernik, 1992) indicates that, for typical transducer widths

(10 mm), the recorded travel times correspond closely to the phase velocity. Accurate

measurement of group velocity along nonsymmetry directions would essentially

require point transducers of less than 2 mm width.

According to Bourbie´ et al. (1987), the velocity measured by a resonant-bar

standing-wave technique corresponds to the phase velocity.



Assumptions and limitations

In general, phase, group, and energy velocities may differ from each other in both

magnitude and direction. Under certain conditions two or more of them may become

identical. For homogeneous, linear, isotropic, elastic media all three are the same.



3.3



NMO in isotropic and anisotropic media

Synopsis

The two-way seismic travel time, t, of a pure (nonconverted) mode from the surface,

through a homogeneous, isotropic, elastic layer, to a horizontal reflector is hyperbolic



87



3.3 NMO in isotropic and anisotropic media



x3

x1

Surface



x2



e

lin

ke

Dip line



ri

St



θ



Azimuth



Zero-offset ray

Dip f

ctor



Refle



Figure 3.3.1 Schematic showing the geometry for NMO with a dipping reflector.



t2 ẳ t20 ỵ



x2

V2



where x is the offset between the source and the receiver, t0 is the zero-offset,

two-way travel time, and V is the velocity of the layer. The increase in travel time

with distance, specifically the extra travel time relative to t0, is called normal

moveout (or NMO), and the velocity that determines the NMO, is called the

NMO velocity, VNMO. In the case of a homogeneous layer above a horizontal

reflector, VNMO ¼ V.

When the reflector is dipping with angle f from the horizontal (Figure 3.3.1), the

travel time equation in the vertical plane along dip becomes (Levin, 1971)

t2 ẳ t20 ỵ



x2

V= cos ị2



or VNMO ị ẳ V= cos . Again, t0 is the zero-offset, two-way travel time. More

generally, the azimuthally varying NMO velocity is (Levin, 1971)

2

VNMO

ð; ị ẳ



V2

1 cos2  sin2 



where f is the reflector dip and z is the azimuth relative to a horizontal axis in the dip

direction. Note that in the strike direction VNMO ¼ V.



88



Seismic wave propagation



When the Earth consists of horizontal, homogeneous, isotropic layers down to the

reflector, the two-way travel time equation is approximately hyperbolic, with the

approximation being best for small offsets:

t2 % t20 ỵ



x2

2

VNMO



At zero dip, the NMO velocity is often approximated as the root mean squared

(RMS) velocity, VNMO % VRMS , where

2

¼

VRMS



N

X

i¼1



Vi2 ti =



N

X



ti



i¼1



where Vi is the velocity of the ith layer and ti is the two-way, zero-offset travel time of

the ith layer. The summations are over all layers from the surface to the reflector. The

approximation VNMO % VRMS is best at offsets that are small relative to the reflector

depth.

If the NMO velocity is estimated at two horizontal reflectors (e.g., VNMỒðnÞ at the

base of layer n and VNMỒðnÀ1Þ at the base of layer (n À 1)), then the RMS equation

can be inverted to yield the Dix equation (Dix, 1955) for the interval velocity, Vn, of

the nth layer:

!

n

nÀ1

X

X

1

2

2

VNMOÀ

ti À VNMO

ti

Vn2 %

n ị

n1ị

tn

iẳ1

iẳ1

It is important to remember that this model assumes: (1) flat, homogeneous, isotropic

layers; (2) the offsets are small enough for the RMS velocity to be a reasonable

estimate of the moveout velocity; and (3) the estimated interval velocities are

themselves averages over velocities of thinner layers lying below the resolution of

the seismic data.

The equivalent of the Dix equation for anisotropic media is shown below.



NMO in an anisotropic Earth

For a heterogeneous elastic Earth with arbitrary anisotropy and arbitrary dip, the

NMO velocity of pure modes (at offsets generally less than the depth) can be written

as (Grechka and Tsvankin, 1998; Tsvankin, 2001)

1

2

VNMO

 ị



ẳ W11 cos2  ỵ 2W12 sin  cos  ỵ W22 sin2 



where z is the azimuth relative to the x1-axis. The components Wij are elements of a

À

Á

~ and are defined as Wij ¼ 0 @pi =@xj , where pi ¼ @=@xi are

symmetric matrix W

horizontal components of the slowness vector for rays between the zero-offset

Â

Ã

reflection point and the surface location xi ; xj ,  ðx1 ; x2 Þ is the one-way travel time



89



3.3 NMO in isotropic and anisotropic media



from the zero-offset reflection point, and t0 is the one-way travel time to the CMP

(common midpoint) location x1 ¼ x2 ¼ 0. The derivatives are evaluated at the CMP

location. This result assumes a sufficiently smooth reflector and sufficiently smooth

lateral velocity heterogeneity, such that the travel time field exists (e.g., no shadow

zones) at the CMP point and the derivatives can be evaluated. This can be rewritten as

1

2

VNMO

 ị



ẳ l1 cos2  ị ỵ l2 sin2  ị



~ and is the rotation of the eigenvectors of

where l1 and l2 are the eigenvalues of W,

~

W relative to the coordinate system. l1 and l2 are typically positive, in which case

VNMO ð Þ is an ellipse in the horizontal plane. The elliptical form allows the exact

expression for VNMO ð Þ to be determined by only three parameters, the values of the

NMO velocity at the axes of the ellipse, and the orientation of the ellipse relative to

the coordinate axes. The elliptical form simplifies modeling of azimuthally varying

NMO for various geometries and anisotropies, since only VNMO ð Þ along the ellipse

axes needs to be derived. For example, we can write

1

2

 ị

VNMO







cos2 

sin2 



2

2

 ẳ 0ị VNMO

 ¼ =2Þ

VNMO



where the axes of the ellipse are at  ¼ 0 and  ¼ =2.

A simple example of the elliptical form is the azimuthally varying moveout

velocity for a homogeneous, isotropic layer above a dipping reflector, given above,

which can be written as (Grechka and Tsvankin, 1998)

1

sin2 

cos2 





2

V2

VNMO

; ị

V= cos Þ2

where f is the dip and z is the azimuth relative to a horizontal axis in the dip

direction.



VTI symmetry with horizontal reflector

NMO velocities in a VTI medium (vertical symmetry axis) overlying a horizontal

reflector are given for P, SV, and SH modes (Tsvankin, 2001) by

p

VNMO;P 0ị ẳ VP0 1 ỵ 2

p

VNMO;SV 0ị ẳ VS0 1 ỵ 2;

VNMO;SH 0ị ẳ VS0



 2

VP0

ẳ

" ị

VS0



p

1 ỵ 2



where VP0 and VS0 are the vertical P- and S-wave velocities, and e, d, and g are the

Thomsen parameters for VTI media. In this case, the NMO ellipse is a circle, giving



90



Seismic wave propagation



an azimuthally independent NMO velocity. These expressions for NMO velocity hold

for anisotropy of arbitrary strength (Tsvankin, 2001).

In the special case of elliptical anisotropy, with vertical symmetry axis, e ¼ d, the

expressions for NMO velocity become (Tsvankin, 2001):

p

VNMO;P 0ị ẳ VP0 1 ỵ 2"

VNMO;SV 0ị ẳ VS0

VNMO;SH 0ị ẳ VS0



p

1 ỵ 2



Vertical symmetry axis with dipping reflector

A dipping reflector creates an azimuthal dependence for the NMO velocity, even with

a VTI medium. For general VTI symmetry (vertical symmetry axis) and assuming

weak anisotropy, the NMO velocities overlying a dipping reflector are given for

P and SV modes by Tsvankin (2001). In the dip direction,





VP0 1 ỵ ị

1 þ  sin2  þ 3ð" À Þ sin2  2 sin2 

cos 

 2



VS0 1 ỵ ị

VP0

VNMO;SV 0; ị ẳ

" ị

1 5 sin2  ỵ 3 sin4  ;  ẳ

VS0

cos 



VNMO;P 0; ị ẳ



where f is the dip, VP0 and VS0 are the vertical P- and S-wave velocities, and e, d, and

g are the Thomsen parameters for VTI media. In the strike direction with VTI

symmetry:

 







VNMO;P ;  ẳ VP0 1 ỵ ị 1 þ ð" À Þ sin2  2 À sin2 

2

 2

 







VP0

VNMO;SV ;  ẳ VS0 1 ỵ ị 1 À  sin2  2 À sin2  ;  ¼

ð" À Þ

VS0

2

The complete azimuthal dependence can be easily written as an ellipse, where the

strike and dip NMO velocities are the semiaxes.

For the special case of elliptical symmetry (e ¼ d) of arbitrary strength anisotropy,

the dip-direction NMO velocities are (Tsvankin, 2001):

p q

VP0 1 ỵ 2"

VNMO;P 0; ị ẳ

1 ỵ 2" sin2 

cos 

VS0

cos 

p q

VS0 1 ỵ 2

VNMO;SH 0; ị ẳ

1 ỵ 2
sin2 

cos 



VNMO;SV 0; ị ẳ



91



3.3 NMO in isotropic and anisotropic media



where f is the layer dip angle. In the strike direction,

 

pffiffiffiffiffiffiffiffiffiffiffiffiffi

VNMO;P ;  ¼ VP0 1 ỵ 2"

2

 

VNMO;SV ;  ẳ VS0

2

 

p

VNMO;SH ;  ẳ VS0 1 ỵ 2

2



Tilted TI symmetry with dipping reflector

For TI symmetry with symmetry axis tilting an angle n from the vertical and

assuming weak anisotropy, the NMO velocities are given by Tsvankin (2001). In

the dip direction

&





VP0

1 ỵ  ỵ  sin2  nị ỵ 3" ị sin2 ð À nÞ 2 À sin2 ð À nÞ

VNMO;P ð0; ị ẳ

cos 

'



2 sin n sin nị

2

 ỵ 2" ị sin  nị



cos 

&





VS0

1 ỵ  þ  sin2 ð À nÞ À 3 sin2 ð À nÞ 2 À sin2 ð À nÞ

VNMO;SV ð0; Þ ẳ

cos 

'

sin n sin nị cos 2 nị

ỵ 2

cos 



where f is the reflector dip, VP0 and VS0 are the symmetry axis P- and S-wave

velocities, and e, d, and g are the Thomsen parameters for TI media. It is assumed that

the azimuth of the symmetry axis tilt is the same as the azimuth of the reflector dip. In

the strike direction, for weak tilted TI symmetry

 

È

Â

ÃÉ

VNMO;P ;  ẳ VP0 1 ỵ ị 1 ỵ " ị sin2 ð À nÞ 2 À sin2 ð À nÞ

2 







VNMO;SV ;  ẳ VS0 1 ỵ ị 1  sin2 ð À nÞ 2 À sin2 ð À nị

2

 2

VP0

" ị

ẳ

VS0

In the special case of tilted elliptical symmetry with a tilted symmetry axis, the

NMO velocity in the dip direction is

!

q

VP0 p

sin n sin nị 1

2

1 ỵ 2 1 ỵ 2 sin  nị 1 2

VNMO;P 0; ị ẳ

cos 

cos 

VS0

VNMO;SV ị ẳ

cos 

!

q

VS0 p

sin n sin nị 1

VNMO;SH 0; ị ẳ

1 ỵ 2
1 ỵ 2
sin2  nị 1 2

cos 

cos 



92



Seismic wave propagation



In the strike direction for tilted elliptical symmetry

 

VNMO;P ;  ẳ VP0 1 ỵ ị

2

 

VNMO;SV ;  ¼ VS0

2



Orthorhombic symmetry with horizontal reflector

We now consider an orthorhombic layer of arbitrary strength anisotropy over a

horizontal reflector. A symmetry plane of the orthorhombic medium is also horizontal.

The vertically propagating (along the x3-axis) P-wave velocity (VP0) and vertically

propagating S-wave velocities, polarized in the x1 (VS0) and x2 (VS1) directions are

pffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffi

VP0 ¼ c33 =; VS0 ¼ c55 =; VS1 ¼ c44 =

where r is the density. Additional constants necessary for this discussion are

"2ị ẳ



c11 c33

2c33



"1ị ẳ



c22 c33

2c33



2ị ẳ



c13 ỵ c55 ị2 c33 c55 ị2

2c33 c33 c55 ị



1ị ẳ



c23 ỵ c44 ị2 c33 c44 ị2

2c33 c33 c44 ị



2ị ẳ



c66 c44

2c44



1ị ẳ



c66 c55

2c55







2ị



 2 h

i

VP0



"2ị 2ị

VS0







1ị







VP0



VS1



2 h



"1ị 1ị



i



For a CMP line in the x1 direction, the NMO velocities are

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

VNMO;P  ẳ 0;  ẳ 0ị ẳ VP0 1 ỵ 22ị

p

VNMO;SV  ẳ 0;  ẳ 0ị ẳ VS0 1 ỵ 22ị

q

VNMO;SH  ẳ 0;  ẳ 0ị ẳ VS1 1 ỵ 2
2ị

where f is the dip and z is the azimuth measured from the x1-axis. For a line in the x2

direction,





pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



VNMO;P  ¼ ;  ¼ 0 ¼ VP0 1 ỵ 21ị

2





p



VNMO;SV  ẳ ;  ẳ 0 ẳ VS1 1 ỵ 21ị

2

q







VNMO;SH  ẳ ;  ẳ 0 ẳ VS0 1 ỵ 2
1ị

2



93



3.4 Impedance, reflectivity, and transmissivity



NMO in a horizontally layered anisotropic Earth

The effective NMO for a stack of horizontal homogeneous layers above a dipping

reflector can be written as (Tsvankin, 2001)

2

VNMO





N h

i2

1X

iị

VNMO pị ti0 pị

t0 iẳ1



where ti0 pị is the interval travel time in layer i computed along the zero-offset ray,

P

iị

t0 ẳ Niẳ1 ti0 is the total zero-offset time, and VNMO ð pÞ is the interval NMO velocity

for the ray parameter p of the zero-offset ray. The ray parameter for the zero-offset

ray is





sin 

VN ðÞ



where f is the reflector dip, and VN ðÞ is the velocity at angle f from the vertical in

the Nth layer at the reflector.



Synopsis

The expressions for NMO in a layered and/or anisotropic elastic medium generally

work best for small offsets.



3.4



Impedance, reflectivity, and transmissivity

Synopsis

The impedance, I, of an elastic medium is the ratio of the stress to the particle velocity

(Aki and Richards, 1980) and is given by rV, where r is the density and V is the wave

propagation velocity. At a plane interface between two thick, homogeneous, isotropic, elastic layers, the normal incidence reflectivity for waves traveling from

medium 1 to medium 2 is the ratio of the displacement amplitude, Ar , of the reflected

wave to that of the incident wave, Ai, and is given by

R12 ¼

%



Ar I2 À I1 2 V2 1 V1





Ai I2 ỵ I1 2 V2 ỵ 1 V1

1

lnðI2 =I1 Þ

2



The logarithmic approximation is reasonable for |R| < 0.5 (Castagna, 1993).

A normally incident P-wave generates only reflected and transmitted P-waves.

A normally incident S-wave generates only reflected and transmitted S-waves. There

is no mode conversion.



94



Seismic wave propagation



The above expression for the reflection coefficient is obtained when the particle

displacements are measured with respect to the direction of the wave vector (equivalent to the slowness vector or the direction of propagation). A displacement is taken to

be positive when its component along the interface has the same phase (or the same

direction) as the component of the wave vector along the interface. For P-waves, this

means that positive displacement is along the direction of propagation. Thus, a

positive reflection coefficient implies that a compression is reflected as a compression, whereas a negative reflection coefficient implies a phase inversion (Sheriff,

1991). When the displacements are measured with respect to a space-fixed coordinate system, and not with respect to the wave vector, the reflection coefficient is

given by

R12 ¼



Ar I1 I2 1 V1 2 V2





Ai I2 ỵ I1 2 V2 ỵ 1 V1



The normal incidence transmissivity in both coordinate systems is

T12 ẳ



At

2I1

21 V1





Ai I2 ỵ I1 2 V2 þ 1 V1



where At is the displacement amplitude of the transmitted wave. Continuity at the

interface requires

Ai ỵ Ar ẳ At

1ỵRẳT

This choice of signs for Ai and Ar is for a space-fixed coordinate system. Note that

the transmission coefficient for wave amplitudes can be greater than 1. Sometimes the

reflection and transmission coefficients are defined in terms of scaled displacements

A0 , which are proportional to the square root of energy flux (Aki and Richards, 1980;

Kennett, 1983). The scaled displacements are given by

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A0 ¼ A V cos 

where y is the angle between the wave vector and the normal to the interface. The

normal incidence reflection and transmission coefficients in terms of these scaled

displacements are

R012 ¼

0

T12

¼



A0r

¼ R12

A0i



pffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 V2 2 1 V1 2 V2

A0t

p

ffiffiffiffiffiffiffiffiffi



¼

¼

T

12

A0i

1 V1 2 V2 ỵ 1 V1



Reflectivity and transmissivity for energy fluxes, Re and T e, respectively, are given

by the squares of the reflection and transmission coefficients for scaled displacements. For normal incidence they are



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