4 Random spherical grain packings: contact models and effective moduli
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130
Seismic wave propagation
S-waves, s is the shear traction across each interface, and w is the tangential
component of the particle velocity. Each layer matrix Ak has the form
2
3
odk
k
V
sin
cos od
i
k k
Vk
Vk 7
6
5
Ak ¼ 4
odk
i
k
cos od
k Vk sin Vk
Vk
where o is the angular frequency.
Kennett (1974, 1983) used the invariant imbedding method to generate the
response of a layered medium recursively by adding one layer at a time (Figure
^ D and T
^ D , respectively,
3.10.1). The overall reflection and transmission matrices, R
for downgoing waves through a stack of layers are given by the following recursion
relations:
h
i1
^ kị ẳ Rkị ỵ Tkị Ekị R
^ kỵ1ị Ekị I Rkị Ekị R
^ kỵ1ị Ekị Tkị
R
D
D
U
D
D
D
U
D
D
D
D
^ kỵ1ị Ekị I Rkị Ekị R
^ kị ẳ T
^ kỵ1ị Ekị
T
D
U
D
D
D
D
D
kị
kị
kị
kị
!1
kị
TD
where RD ; TD ; RU ; and TU are just the single-interface downward and upward
reflection and transmission matrices for the kth interface:
2
3
#" VPðkÀ1Þ cos kÀ1 1=2
#"
SP
PP
6
7
VSk1ị cos k1
6
7
kị
7
RD ẳ6
6
7
1=2
#"
4 #" VSk1ị cos k1
5
PS
SS
VPk1ị cos kÀ1
2
1=2 ##
1=2 3
##
k VPðkÞ cos k
k VPðkÞ cos k
SP
6 PP
7
6
7
k1 VPk1ị cos k1
k1 VSk1ị cos k1
6
7
kị
7
TD ẳ6
##
6
1=2
1=2 7
6 ##
7
##
k VSðkÞ cos k
k VSðkÞ cos k
4
5
PS
SS
kÀ1 VPðkÀ1Þ cos kÀ1
kÀ1 VSðkÀ1Þ cos kÀ1
2
3
"# VPðkÞ cos k 1=2
"#
SP
PP
6
7
VSðkÞ cos k
6
7
kị
7
RU ẳ6
6
7
1=2
"#
4 "# VSkị cos k
5
PS
SS
VPkị cos k
2
3
"" kÀ1 VPðkÀ1Þ cos kÀ1 1=2
"" kÀ1 VPðkÀ1Þ cos kÀ1 1=2
SP
6 PP
7
6
7
k VPkị cos k
k VSkị cos k
6
7
kị
6
7
TU ẳ6
##
1=2
1=2 7
6 ""
7
"" kÀ1 VSðkÀ1Þ cos kÀ1
kÀ1 VSðkÀ1Þ cos kÀ1
4
5
PS
SS
k VPðkÞ cos k
k VSðkÞ cos k
131
3.10 Full-waveform synthetic seismograms
Homogeneous half-space
Reflected
Incident
Interface 1
Interface 2
Layer 1
Layer 2
(1)
RD
(k)
Interface k
Layer k
Interface n − 1
Interface n
Interface n + 1
Layer n − 1
Layer n
Transmitted
RD
(n)
RD
(n + 1)
RD
Homogeneous half-space
Figure 3.10.1 Recursively determined transfer functions in a layered medium.
with
0#" #" "" ""1
B PP SP PP SP C
B#" #" "" ""C
B PS SS PS SS C
À1
B
C
B## ## "# "#C¼M N
B PP SP PP SP C
@
A
##
##
"#
"#
PS SS PS SS
2
À sin kÀ1
6
6
cos kÀ1
6
M ¼6
6 2ISðkÀ1Þ sin kÀ1 cos kÀ1
4
ÀIPðkÀ1Þ ð1 À 2 sin2 kÀ1 Þ
2
sin kÀ1
6
cos kÀ1
6
N¼6
6 2I
4 SðkÀ1Þ sin kÀ1 cos kÀ1
IPðkÀ1Þ ð1 À 2 sin2 kÀ1 Þ
À cos kÀ1
sin k
À sin kÀ1
cos k
ISðkÀ1Þ ð1 À 2 sin2 kÀ1 Þ
2ISðkÞ sin k cos k
ISðkÀ1Þ sin 2kÀ1
IPðkÞ ð1 À 2 sin2 k Þ
cos k
3
7
7
7
7
2
ISðkÞ ð1 À 2 sin k Þ 7
5
À sin k
ÀISðkÞ sin 2k
cos kÀ1
À sin k
À cos k
À sin kÀ1
cos k
À sin k
3
7
7
7
ISðkÀ1Þ ð1 À 2 sin2 kÀ1 Þ
2ISðkÞ sin k cos k
ISðkÞ ð1 À 2 sin2 k Þ 7
5
2
ÀISðkÀ1Þ sin 2kÀ1
ÀIPðkÞ ð1 À 2 sin k Þ
ISðkÞ sin 2k
where IP(k) = rkVP(k) and IS(k) = rkVS(k) are the P and S impedances, respectively,
of the kth layer, and yk and fk are the angles made by the P- and S-wave vectors
with the normal to the kth interface. The elements of the reflection and transmisðkÞ
ðkÞ
ðkÞ
ðkÞ
sion matrices RD ; TD ; RU ; and TU are the reflection and transmission
coefficients for scaled displacements, which are proportional to the square root
of the energy flux. The scaled displacement u0 is related to the displacement u by
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u0 ¼ u V cos :
132
Seismic wave propagation
For normal-incidence wave propagation with no mode conversions, the reflection
and transmission matrices reduce to the scalar coefficients:
kị
RD ẳ
kị
TU
k1 Vk1 k Vk
;
k1 Vk1 ỵ k Vk
p
2 k1 Vk1 k Vk
ẳ
;
k1 Vk1 ỵ k Vk
kị
kị
RU ¼ ÀRD
ðkÞ
ðkÞ
TU ¼ TD
ðkÞ
The phase shift operator for propagation across each new layer is given by ED
ðkÞ
ED
expðiodk cos k =VPðkÞ Þ
¼
0
0
expðiodk cos k =VSðkÞ Þ
!
where yk and fk are the angles between the normal to the layers and the directions of
^ D and T
^ D are functions of o
propagation of P- and S-waves, respectively. The terms R
and represent the overall transfer functions of the layered medium in the frequency
domain. Time-domain seismograms are obtained by multiplying the overall transfer
function by the Fourier transform of the source wavelet and then performing an
inverse transform. The recursion starts at the base of the layering at interface n ỵ 1
nỵ1ị
^ nỵ1ị ẳ 0 and Tnỵ1ị ẳ T
^ nỵ1ị ẳ I simulates a
(Figure 3.10.1). Setting RD
ẳR
D
D
D
stack of layers overlying a semi-infinite homogeneous half-space with properties
equal to those of the last layer, layer n. The recursion relations are stepped up through
^ ð1Þ and T
^ ð1Þ , the overall reflection and
the stack of layers one at a time to finally give R
D
D
transmission response for the whole stack.
Calculate the P-wave normal-incidence overall reflection and transmission functions
ð1Þ
ð1Þ
R^D and T^D recursively for a three-layered medium with layer properties as follows:
VP1ị ẳ 4000 m=s;
1 ẳ 2300 kg=m3 ;
d1 ẳ 100 m
VP2ị ẳ 3000 m=s;
2 ẳ 2100 kg=m ;
d2 ẳ 50 m
VP3ị ẳ 5000 m=s;
3 ẳ 2500 kg=m ;
d3 ẳ 200 m
3
3
4ị
4ị
The recursion starts with R^D = 0 and T^D = 1. The normal-incidence reflection
and transmission coefficients at interface 3 are
ð3Þ
2 VPð2Þ À 3 VP3ị
3ị
3ị
ẳ 0:33; RU ẳ RD
2 VP2ị ỵ 3 VP3ị
p
2 2 VP2ị 3 VP3ị
3ị
3ị
ẳ
ẳ 0:94; TU ẳ TD
2 VP2ị ỵ 3 VP3ị
RD ẳ
3ị
TD
and the phase factor for propagation across layer 3 is
133
3.10 Full-waveform synthetic seismograms
3ị
ED ẳ expi2fd3 =VP3ị ị ẳ expði2f 200=5000Þ
where f is the frequency. The recursion relations give
ð3Þ
ð3Þ
R^D ẳ RD ỵ
3ị
T^D ẳ
3ị 3ị 4ị 3ị
TU ED R^D ED
3ị
T
3ị ð3Þ ^ð4Þ ð3Þ D
1 À RU ED RD ED
ð4Þ ð3Þ
T^D ED
ð3Þ
T
ð3Þ ð3Þ ^ð4Þ ð3Þ D
1 À RU ED RD ED
ð1Þ
The recursion is continued in a similar manner until finally we obtain R^D and
1ị
T^D .
The matrix inverse
kị kị ^ kỵ1ị kị À1
ED
½I À RU ED R
D
is referred to as the reverberation operator and includes the response caused by all
internal reverberations. In the series expansion of the matrix inverse
ðkÞ ðkÞ ^ kỵ1ị kị 1
kị kị ^ kỵ1ị kị
ED ẳ I ỵ RU ED R
ED
ẵI RU ED R
D
D
kị kị ^ kỵ1ị kị kị kị ^ kỵ1ị kị
ỵ RU ED R
ED RU ED RD ED ỵ
D
the first term represents the primaries and each successive term corresponds to
higher-order multiples. Truncating the expansion to m ỵ 1 terms includes m internal
multiples in the approximation. The full multiple sequence is included with the exact
matrix inverse.
Uses
The methods described in this section can be used to compute full-wave seismograms, which include the effects of multiples for wave propagation in layered media.
Assumptions and limitations
The algorithms described in this section assume the following:
layered medium with no lateral heterogeneities;
layers are isotropic, linear, elastic; and
plane-wave, time-harmonic propagation.
134
Seismic wave propagation
3.11
Waves in layered media: stratigraphic filtering
and velocity dispersion
Synopsis
Waves in layered media undergo attenuation and velocity dispersion caused by multiple
scattering at the layer interfaces. The effective phase slowness of normally incident
waves through layered media depends on the relative scales of the wavelength and layer
thicknesses and may be written as Seff ẳ Srt ỵ Sst. The term Srt is the ray theory slowness
of the direct ray that does not undergo any reflections and is just the thickness-weighted
average of the individual layer slownesses. The individual slownesses may be complex
to account for intrinsic attenuation. The excess slowness Sst (sometimes called the
stratigraphic slowness) arises because of multiple scattering within the layers.
A flexible approach to calculating the effective slowness and travel time follows from
Kennett’s (1974) invariant imbedding formulation for the transfer function of a layered
medium. The layered medium, of total thickness L, consists of layers with velocities
(inverse slownesses), densities, and thicknesses, Vj, rj, and lj, respectively.
The complex stratigraphic slowness is frequency dependent and can be calculated
recursively (Frazer, 1994) by
!
n
tj
1 X
Sst ¼
ln
ioL j¼1
1 Rj 2j rj
As each new layer j ỵ 1 is added to the stack of j layers, R is updated according to
Rjỵ1 ẳ rjỵ1 ỵ
Rj 2jỵ1 t2jỵ1
1 Rj 2jỵ1 rjỵ1
(with R0 ẳ 0) and the term
lnẵtjỵ1 1 Rjỵ1 2jỵ1 rjỵ1 ị1
is accumulated in the sum. In the above expressions, tj and rj are the transmission and
reflection coefficients defined as
p
2 j Vj jỵ1 Vjỵ1
tj ẳ
j Vj ỵ jỵ1 Vjỵ1
rj ẳ
jỵ1 Vjỵ1 j Vj
j Vj ỵ jỵ1 Vjỵ1
whereas yj ẳ exp(iolj/Vj) is the phase shift for propagation across layer j and o is the
angular frequency. The total travel time is T ẳ Trt ỵ Tst, where Trt is the ray theory
travel time given by
135
3.11 Stratigraphic filtering and velocity dispersion
Trt ¼
n
X
lj
V
j¼1 j
and Tst is given by
"
n
tj
1 X
Tst ¼ Re
ln
io j¼1
1 À Rj 2j rj
!#
The deterministic results given above are not restricted to small perturbations in the
material properties or statistically stationary geology.
Calculate the excess stratigraphic travel time caused by multiple scattering for a
normally incident P-wave traveling through a three-layered medium with layer
properties as follows:
VP1ị ẳ 4000 m=s;
1 ẳ 2300 kg=m3 ;
l1 ẳ 100 m
VP2ị ẳ 3000 m=s;
2 ẳ 2100 kg=m3 ;
l2 ẳ 50 m
VP3ị ẳ 5000 m=s;
3 ẳ 2500 kg=m3 ;
l3 ¼ 200 m
The excess travel time is given by
"
!#
n
tj
1 X
Tst ¼ Re
ln
io j¼1
1 À Rj 2j rj
The recursion begins with R0 ẳ 0,
R1 ẳ r1 ỵ
R0 21 t21
1 À R0 21 r1
where
t1 ¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2 1 V1 2 V2
¼ 0:98
1 V1 ỵ 2 V2
r1 ẳ
2 V2 1 V1
ẳ 0:19
1 V1 ỵ 2 V2
y1 ẳ exp(i2pfl1/V1) ẳ exp(i2pf100/4000) with f as the frequency.
The recursion is continued to obtain R2 and R3. Setting t3 ¼ 1 and r3 ¼ 0
simulates an impedance-matching homogeneous infinite half-space beneath layer 3.
136
Seismic wave propagation
Finally, the excess travel time, which is a function of the frequency, is obtained by
taking the real part of the sum as follows:
&
!'
1
t1
t2
t3
Tst ẳ Re
ỵ ln
ỵ ln
ln
i2f
1 R1 21 r1
1 R2 22 r2
1 À R3 23 r3
The effect of the layering can be thought of as a filter that attenuates the input
wavelet and introduces a delay. The function
Aoị ẳ expioxSst Þ ¼ expðioTrt Sst =Srt Þ
(where Srt is assumed to be real in the absence of any intrinsic attenuation) is
sometimes called the stratigraphic filter.
The O’Doherty–Anstey formula (O’Doherty and Anstey, 1971; Banik et al., 1985)
^
jAðoÞj % expðÀRðoÞT
rt Þ
approximately relates the amplitude of the stratigraphic filter to the power spectrum
^
RðoÞ
of the reflection coefficient time series r() where
Z x
xị ẳ
dx0 =Vx0 ị
0
is the one-way travel time. Initially the O’Doherty–Anstey formula was obtained by a
heuristic approach (O’Doherty and Anstey, 1971). Later, various authors substantiated the result using statistical ensemble averages of wavefields (Banik et al., 1985),
deterministic formulations (Resnick et al., 1986), and the concepts of self-averaged
values and wave localization (Shapiro and Zien, 1993). Resnick et al. (1986) showed
that the O’Doherty–Anstey formula is obtained as an approximation from the
exact frequency-domain theory of Resnick et al. by neglecting quadratic terms in
the Riccatti equation of Resnick et al. Another equivalent way of expressing the
O’Doherty–Anstey relation is
^
RðoÞ
ImðSst Þ
1
1 ^
ẳ oM2oị
%
%
o
Srt
2Q
2
^
Here 1/Q is the scattering attenuation caused by the multiples, and MðoÞ
is the power
spectrum of the logarithmic impedance fluctuations of the medium,
lnẵịVị hlnẵịVịi, where hi denotes a stochastic ensemble average.
Because the filter is minimum phase, o Re(Sst) and o Im(Sst) are a Hilbert transform pair,
^
ReðSst Þ t HfRðoÞg
%
%
Srt
Trt
o
where H{·} denotes the Hilbert transform and dt is the excess travel caused by
multiple reverberations.
137
3.11 Stratigraphic filtering and velocity dispersion
Shapiro and Zien (1993) generalized the O’Doherty–Anstey formula for nonnormal incidence. The derivation is based on a small perturbation analysis and
requires the fluctuations of material parameters to be small (<30%). The generalized
formula for plane pressure (scalar) waves in an acoustic medium incident at an
angle y with respect to the layer normal is
!
^ cos Þ
Rðo
jAðoÞj % exp À
Trt
cos4
whereas
"
^ cos Þ
ð2 cos2 À 1Þ2 Rðo
Trt
jAðoÞj % exp À
cos4
#
for SH-waves in an elastic medium (Shapiro et al., 1994).
For a perfectly periodic stratified medium made up of two constituents with phase
velocities V1, V2; densities r1, r2; and thicknesses l1, l2, the velocity dispersion relation
may be obtained from the Floquet solution (Christensen, 1991) for periodic media:
cos
!
ol1 ỵ l2 ị
ol1
ol2
ol1
ol2
cos
sin
sin
ẳ cos
V
V1
V2
V1
V2
ẳ
1 V1 ị2 ỵ 2 V2 ị2
21 2 V1 V2
The Floquet solution is valid for arbitrary contrasts in the layer properties. If the
spatial period (l1 ỵ l2) is an integer multiple of one-half wavelength, multiple reflections are in phase and add constructively, resulting in a large total accumulated
reflection. The frequency at which this Bragg scattering condition is satisfied is called
the Bragg frequency. Waves cannot propagate within a stop-band around the Bragg
frequency.
Uses
The results described in this section can be used to estimate velocity dispersion and
attenuation caused by scattering for normal-incidence wave propagation in layered
media.
Assumptions and limitations
The methods described in this section apply under the following conditions:
layers are isotropic, linear elastic with no lateral variation;
propagation is normal to the layers except for the generalized O’Doherty–Anstey
formula; and
plane-wave, time-harmonic propagation is assumed.
138
Seismic wave propagation
3.12
Waves in layered media: frequency-dependent anisotropy,
dispersion, and attenuation
Synopsis
Waves in layered media undergo attenuation and velocity dispersion caused by
multiple scattering at the layer interfaces. Thinly layered media also give rise to
velocity anisotropy. At low frequencies this phenomenon is usually described by the
Backus average. Velocity anisotropy and dispersion in a multilayered medium are
two aspects of the same phenomenon and are related to the frequency- and angledependent transmissivity resulting from multiple scattering in the medium. Shapiro
et al. (1994) and Shapiro and Hubral (1995, 1996, 1999) have presented a wholefrequency-range statistical theory for the angle-dependent transmissivity of layered
media for scalar waves (pressure waves in fluids) and elastic waves. The theory
encompasses the Backus average in the low-frequency limit and ray theory in the
high-frequency limit. The formulation avoids the problem of ensemble averaging
versus measurements for a single realization by working with parameters that are
averaged by the wave-propagation process itself for sufficiently long propagation
paths. The results are obtained in the limit when the path length tends to infinity.
Practically, this means the results are applicable when path lengths are very much
longer than the characteristic correlation lengths of the medium.
The slowness (s) and density (r) distributions of the stack of layers (or a continuous inhomogeneous one-dimensional medium) are assumed to be realizations of
random stationary processes. The fluctuations of the physical parameters are small
(<30%) compared with their constant mean values (denoted by subscripts 0):
1
s2 ðzÞ ẳ 2 ẵ1 ỵ "s zị
c0
zị ẳ 0 ẵ1 ỵ " ðzÞ
where the fluctuating parts es(z) (the squared slowness fluctuation) and er (z) (the
density fluctuation) have zero means by definition. The depth coordinate is denoted
by z, and the x- and y-axes lie in the plane of the layers. The velocity
À1=2
c0 ¼ s2
corresponds to the average squared slowness of the medium. Instead of the squared
slowness fluctuations, the random medium may also be characterized by the P- and
S-velocity fluctuations, a and b, respectively, as follows:
zị ẳ hiẵ1 ỵ " zị ẳ 0 ẵ1 ỵ " zị
zị ẳ hiẵ1 ỵ "