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4 Impedance, reflectivity, and transmissivity

4 Impedance, reflectivity, and transmissivity

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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 À I2 1 V1 2 V2





Ai I2 ỵ I1 2 V2 ỵ 1 V1



The normal incidence transmissivity in both coordinate systems is

T12 ¼



At

2I1

21 V1





Ai I2 ỵ I1 2 V2 ỵ 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









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



95



3.4 Impedance, reflectivity, and transmissivity



Re12 ¼



Er À 0 Á2 ð1 V1 À 2 V2 ị2

ẳ R12 ẳ

Ei

2 V2 ỵ 1 V1 ị2



e

T12





Et 0 2

41 V1 2 V2

ẳ T12 ẳ

Ei

2 V2 ỵ 1 V1 Þ2



where Ei, Er , and Et are the incident, reflected, and transmitted energy fluxes,

respectively. Energy reflection coefficients were first given by Knott (1899). Conservation of energy at an interface where no trapping of energy occurs requires that

Ei ¼ Er þ Et

1 ¼ Re þ T e

The reflection and transmission coefficients for energy fluxes can never be greater

than 1.



Simple band-limited inverse of reflectivity time series

Consider a flat-layered Earth, with an impedance time series In ẳ ntịV ntị,

where r is density, V is velocity (either P or S), and t ¼ nDt, where n ¼ 0,1,2, . . . ,N is

the normal incidence two-way travel time, equally sampled in intervals Át. In these

expressions, nDt is the argument of r (·) and V(·), not a factor multiplying r and V.

The reflectivity time series can be approximated as

Rnỵ1 ẳ



Inỵ1 In

Inỵ1 ỵ In



Solving for Inỵ1 yields a simple recursive algorithm

Inỵ1 ẳ In



1 ỵ Rnỵ1 ị

;

1 Rnỵ1 ị



n ẳ 0: N 1



where I0 , the impedance at time t ¼ 0, must be supplied. For small Át and small Rn ,

this equation can be written as

"

#

n À

X

Á

2 3

2 5

In % I0 exp

2Ri þ 3 Ri þ 5 Ri þ Á Á Á

i¼1



In contrast, a simple running summation of Rn leads to

"

#

n

X

ð2Ri Þ

In % I0 exp

i¼1



which is a low-order approximation of the first algorithm.



96



Seismic wave propagation



Rough surfaces

Random interface roughness at scales smaller than the wavelength causes incoherent

scattering and a decrease in amplitude of the coherent reflected and transmitted

waves. This could be one of the explanations for the observation that amplitudes of

multiples in synthetic seismograms are often larger than the amplitudes of corresponding multiples in the data (Frazer, 1994). Kuperman (1975) gives results that

modify the reflectivity and transmissivity to include scattering losses at the interface.

With the mean-squared departure from planarity of the rough interface denoted by s2,

the modified coefficients are

R~12 ¼ R12 ð1 À 2k12 2 ị ẳ R12 ẵ1 82 =l1 ị2

"

!



2 #

1



T~12 ẳ T12 1 k1 k2 ị2 2 ẳ T12 1 22 l2 l1 ị2

2

l 1 l2

where k1 ¼ o/V1, k2 ¼ o/V2 are the wavenumbers, and l1 and l2 are the wavelengths

in media 1 and 2, respectively.



Uses

The equations presented in this section can be used for the following purposes:

 to calculate amplitudes and energy fluxes of reflected and transmitted waves at

interfaces in elastic media;

 to estimate the decrease in wave amplitude caused by scattering losses during

reflection and transmission at rough interfaces.



Assumptions and limitations

The equations presented in this section apply only under the following conditions:

 normal incidence, plane-wave, time-harmonic propagation in isotropic, linear, elastic media with a single interface, or multiple interfaces well separated by thick layers

with thickness much greater than the wavelength;

 no energy losses or trapping at the interface; and

 rough surface results are valid for small deviations from planarity (small s).



3.5



Reflectivity and amplitude variations with offset (AVO)

in isotropic media

Synopsis

The seismic impedance is the product of velocity and density (see Section 3.4), as

expressed by



97



3.5 Reflectivity and AVO in isotropic media



r1Ј V1

r2Ј V2



Figure 3.5.1 Reflection of a normal-incidence wave at an interface between two thick

homogeneous, isotropic, elastic layers.



IP ¼ VP



IS ¼ VS



where IP , IS are P- and S-wave impedances, VP , VS are P- and S-wave velocities, and

r is density.

At an interface between two thick homogeneous, isotropic, elastic layers, the

normal incidence reflectivity, defined as the ratio of the reflected wave amplitude

to the incident wave amplitude, is

RPP ¼

%

RSS ¼

%



2 VP2 À 1 VP1 IP2 IP1



2 VP2 ỵ 1 VP1 IP2 ỵ IP1

1

lnIP2 =IP1 ị

2

2 VS2 1 VS1 IS2 IS1



2 VS2 ỵ 1 VS1 IS2 ỵ IS1

1

lnIS2 =IS1 ị

2



where RPP is the normal incidence P-to-P reflectivity, RSS is the S-to-S reflectivity,

and the subscripts 1 and 2 refer to the first and second media, respectively

(Figure 3.5.1). 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.



AVO: amplitude variations with offset

For non-normal incidence, the situation is more complicated. An incident P-wave

generates reflected P- and S-waves and transmitted P- and S-waves. The reflection



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