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4 Random spherical grain packings: contact models and effective moduli

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



;

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 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 Š

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 " ðzފ

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 h iẵ1 ỵ " zị ẳ 0 ẵ1 ỵ " zị

zị ẳ h iẵ1 ỵ "

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