Geoscience Reference
In-Depth Information
Figure 4.38.
Waves
generated at an interface
between two elastic
media by an incident
P-wave. The incident
P-wave has amplitude
A
0
and angle of incidence
i
The reflected P- and
SV-waves have angles of
reflection
e
1
and
f
1
and
amplitudes
A
1
and
B
1
,
respectively. The
transmitted P- and
SV-waves have angles of
refraction
e
2
and
f
2
and
amplitudes
A
2
and
B
2
,
respectively.
B
1
A
0
A
1
SV
P
P
e
1
f
1
Medium 1
i
Interface
Medium 2
f
2
e
2
P
A
2
SV
B
2
The angles made by the various rays with the normal to the interface are defined
by Snell's law. The constant along each ray path, sin
i
/velocity, is often called
p
,
the
ray parameter
.For the case of the incident P-wave (Fig. 4.38), the angles for
the reflected and transmitted P- and SV-waves are therefore determined from
sin
i
α
1
=
sin
e
1
α
1
sin
f
1
β
1
sin
e
2
α
2
sin
f
2
β
2
=
=
=
=
p
(4.55)
Clearly, for P-waves in the first layer, the angle of incidence is equal to the angle
of reflection (
i
e
1
), and the other rays bend according to the seismic velocities
of the media and the angle of incidence.
To determine the relative amplitudes of the reflected and transmitted waves, it
is necessary to calculate the displacements and stresses resulting from the wave
fields and to equate these values at the interface. Displacement and stress must
both be continuous across the interface or else the two layers would move relative
to each other. The following equations, which relate the amplitudes of the various
waves illustrated in Fig. 4.38, are called the
Zoeppritz equations
:
=
A
1
cos
e
1
−
B
1
sin
f
1
+
A
2
cos
e
2
+
B
2
sin
f
2
=
A
0
cos
i
(4.56)
A
1
sin
e
1
+
B
1
cos
f
1
−
A
2
sin
e
2
+
B
2
cos
f
2
=−
A
0
sin
i
(4.57)
A
1
Z
1
cos(2
f
1
)
−
B
1
W
1
sin(2
f
1
)
−
A
2
Z
2
cos(2
f
2
)
−
B
2
W
2
sin(2
f
2
)
=−
A
0
Z
1
cos(2
f
1
)
(4.58)
A
1
γ
1
W
1
sin(2
e
1
)
+
B
1
W
1
cos(2
f
1
)
+
A
2
γ
2
W
2
sin(2
e
2
)
−
B
2
W
2
cos(2
f
2
)
=
A
0
γ
1
W
1
sin(2
i
)
(4.59)
