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
ẳ
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
ẳ
cos2
sin2
ỵ
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
ỵ
ẳ
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
ỵ
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
ẩ
ẫ
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
ẳ
"2ị 2ị
VS0
1ị
VP0
ẳ
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
ẳ
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
p¼
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
ẳ
ẳ
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
ẳ
ẳ
Ai I2 ỵ I1 2 V2 ỵ 1 V1
The normal incidence transmissivity in both coordinate systems is
T12 ẳ
At
2I1
21 V1
ẳ
ẳ
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
ﬃ
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