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
ẳ
ẳ
Ai I2 ỵ I1 2 V2 ỵ 1 V1
The normal incidence transmissivity in both coordinate systems is
T12 ¼
At
2I1
21 V1
ẳ
ẳ
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
ẳ
ẳ
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
ẳ
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
ẳ
2 VP2 ỵ 1 VP1 IP2 ỵ IP1
1
lnIP2 =IP1 ị
2
2 VS2 1 VS1 IS2 IS1
ẳ
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