Geoscience Reference
In-Depth Information
where the reflected P-wave travel direction angle, β
P
R
, equals the incident P-wave travel direction
angel, β
P
I
; the reflected SV-wave travel direction angle is β
SV
R
; the transmitted P-wave travel direc-
tion angle is β
P
T
; the transmitted SV-wave travel direction angle is β
SV
T
; and all other quantities have
been previously defined. With respect to an incident SV-wave, Snell's law gives
( )
=
()
=
( )
=
1
()
=
( )
2
sin
β
sin
β
sin
β
sin
β
sin β
SV
P
SV
P
SV
I
R
R
T
T
(8.5)
V
V
V
V
V
S
P
1
S
1
P
S
2
where the reflected SV-wave angle, β
SV
R
, equals the incident SV-wave angle, β
SV
I
. Finally, in regard
to an incident SH-wave, Snell's law gives:
( )
=
( )
=
( )
1
sin
β
sin
β
sin
β
SH
SH
SH
I
R
T
(8.6)
V
V
V
S
S
1
S
2
where the reflected SH-wave angle, β
SH
R
, equals the incident SH-wave angle, β
SH
I
, and the transmit-
ted SH-wave travel direction angle is β
SH
T
.
When the transmitted angle of a seismic wave equals 90°, a process called refraction occurs in
which the seismic wave travels along the interface at the V
2
velocity while continually causing oscil-
lations at the interface that transmit derivative seismic waves back into Layer 1. A refracted wave is
often called the head wave. The incident seismic wave angle that produces a refracted wave (or head
wave) is referred to as the critical angle. For a seismic wave initially traveling downward, this refrac-
tion phenomenon can occur only at an interface where the seismic wave velocity in the layer below
the interface (V
2
) is greater than the seismic wave velocity in the layer above the interface (V
1
).
Incident P-waves that are normal (perpendicular) to the interface (β
P
I
= 0°) produce only P
R
and
P
T
waves (no S-waves) with travel paths that are also perpendicular to the interface (β
P
R
= β
P
T
= 0°).
Incident SV-waves that are normal to the interface (β
SV
I
= 0°) produce only SVT
R
and SV
T
waves (no
P-waves) with travel paths normal to the interface (β
SV
R
= β
SV
T
= 0°). Incident SH-waves that are nor-
mal to the interface (β
SH
I
= 0°) still produce only SHT
R
and SH
T
waves (Figure 8.8c), with travel paths
normal to the interface (β
SH
R
= β
SH
T
= 0°). The ratio of the reflected wave amplitude to the incident
wave amplitude is called the reflection coefficient. For incident P-waves normal to the interface, the
reflection coefficient,
R
C-P
, is expressed as follows:
ρ
V
−
+
ρ
V
R
−
=
22
P
11
P
(8.7)
CP
ρ
V
ρ
V
11
P
22
P
The reflection coefficient given an incident S-wave (SV or SH) normal to the interface is
ρ
V
−
+
ρ
V
11
S
22
S
R
−
=
(8.8)
CS
ρ
V
ρ
V
11
S
22
S
(Note the difference in form of the numerators for Equation (8.7) and Equation (8.8). Negative val-
ues of
R
C-P
or
R
C-S
simply imply that the polarity of the reflected seismic wave has been reversed
with respect to the incident seismic wave. The transmission coefficient,
T
C
, is the ratio of the trans-
mitted wave amplitude to the incident wave amplitude, and given an incident seismic wave normal
to the interface (P, SV, and SH), is quantified using the following relationship:
