Geology Reference
In-Depth Information
Incident ray,
amplitude
A
0
Reflected ray,
amplitude
A
1
v
1
,
ρ
1
Transmitted ray,
amplitude
A
2
v
2
,
ρ
2
ρ
2
v
2
=
/
ρ
1
v
1
Fig. 3.8
Reflected and transmitted rays associated with a
ray normally incident on an interface of acoustic
impedance contrast.
where
r
1
,
v
1
,
Z
1
and
r
2
,
v
2
,
Z
2
are the density, P-wave
velocity and acoustic impedance values in the first and
second layers, respectively. From this equation it follows
that -1 £
R
£+1. A negative value of
R
signifies a phase
change of
p
(180°) in the reflected ray.
The
transmission coefficient T
is the ratio of the ampli-
tude
A
2
of the transmitted ray to the amplitude
A
0
of the
incident ray
interface, and more energy is reflected the greater the
contrast. From common experience with sound, the best
echoes come from rock or brick walls. In terms of physical
theory, acoustic impedance is closely analogous to elec-
trical impedance and, just as the maximum transmission
of electrical energy requires a matching of electrical im-
pedances, so the maximum transmission of seismic ener-
gy requires a matching of acoustic impedances.
The
reflection coefficient R
is a numerical measure of the
effect of an interface on wave propagation, and is calcu-
lated as the ratio of the amplitude
A
1
of the reflected ray
to the amplitude
A
0
of the incident ray
TAA
=
20
For a normally incident ray this is given, from solution of
Zoeppritz's equations, by
RAA
=
10
Z
ZZ
2
1
T
=
To relate this simple measure to the physical properties
of the materials at the interface is a complex problem. As
we have already seen, the propagation of a P-wave de-
pends on the bulk and shear elastic moduli, as well as the
density of the material. At the boundary the stress and
strain in the two materials must be considered. Since the
materials are different, the relations between stress and
strain in each will be different. The orientation of stress
and strain to the interface also becomes important. The
formal solution of this physical problem was derived
early in the 20
th
century, and the resulting equations are
named the Zoeppritz equations (Zoeppritz 1919; and
for explanation of derivations see Sheriff & Geldart
1982). Here, the solutions of these equations will be
accepted. For a normally incident ray the relationships
are fairly simple, giving:
+
2
1
Reflection and transmission coefficients are some-
times expressed in terms of energy rather than wave am-
plitude. If energy intensity
I
is defined as the amount of
energy flowing through a unit area normal to the direc-
tion of wave propagation in unit time, so that
I
0
,
I
1
and
I
2
are the intensities of the incident, reflected and trans-
mitted rays respectively, then
2
I
I
ZZ
ZZ
-
+
È
Í
˘
˙
1
0
2
1
R
¢=
=
2
1
and
I
I
ZZ
ZZ
4
v
-
+
v
ZZ
ZZ
-
+
r
r
2
1
12
T
¢=
=
22
11
2
1
R
=
=
2
v
v
(
+
)
r
r
22
11
2
1
2
1
