Geology Reference
In-Depth Information
where
R
¢ and
T
¢ are the reflection and transmission coef-
ficients expressed in terms of energy.
If
R
or
R
¢=0, all the incident energy is transmitted.
This is the case when there is no contrast of acoustic im-
pedance across an interface, even if the density and ve-
locity values are different in the two layers (i.e.
Z
1
=
Z
2
).
If
R
or
R
¢=+1 or -1, all the incident energy is reflected.
A good approximation to this situation occurs at the free
surface of a water layer: rays travelling upwards from an
explosion in a water layer are almost totally reflected
back from the water surface with a phase change of
p
(
R
=-0.9995).
Values of reflection coefficient
R
for interfaces be-
tween different rock types rarely exceed ±0.5 and are
typically much less than ±0.2.Thus, normally the bulk of
seismic energy incident on a rock interface is transmitted
and only a small proportion is reflected. By use of an em-
pirical relationship between velocity and density (see
also Section 6.9), it is possible to estimate the reflection
coefficient from velocity information alone (Gardner
et
al
. 1974, Meckel & Nath 1977):
Reflected S
Incident P
Reflected P
θ
v
1
v
2
> v
1
Refracted P
Refracted S
Fig. 3.9
Reflected and refracted P- and S-wave rays generated by
a P-wave ray obliquely incident on an interface of acoustic
impedance contrast.
θ
1
θ
1
Incident P
Reflected P
(
)
v
1
v
2
> v
1
R
= 0.625 ln
v
1
v
2
Such relationships can be useful, but must be applied
with caution since rock lithologies are highly variable and
laterally heterogeneous as pointed out in Section 3.4.
Refracted P
θ
2
3.6.2 Reflection and refraction of
obliquely incident rays
When a
P-wave ray
is obliquely incident on an inter-
face of acoustic impedance contrast, reflected and trans-
mitted P-wave rays are generated as in the case of normal
incidence. Additionally, some of the incident com-
pressional energy is converted into reflected and trans-
mitted S-wave rays (Fig. 3.9) that are polarized in a
vertical plane. Zoeppritz's equations show that the am-
plitudes of the four phases are a function of the angle of
incidence
q
. The converted rays may attain a significant
magnitude at large angles of incidence. Detection and
identification of converted waves can be difficult in seis-
mic surveys, but they do have potential to provide more
constraints on the physical properties of the media at
the interface. Here consideration will be confined to the
P-waves.
In the case of oblique incidence, the transmitted P-
wave ray travels through the lower layer with a changed
direction of propagation (Fig. 3.10) and is referred to as a
refracted ray
.The situation is directly analogous to the be-
Fig. 3.10
Reflected and refracted P-wave rays associated with a
P-wave rays obliquely incident on an interface of acoustic
impedance contrast.
haviour of a light ray obliquely incident on the boundary
between, say, air and water and
Snell's Law of Refraction
applies equally to the optical and seismic cases. Snell de-
fined the
ray parameter p
= sin
i
/
v
, where
i
is the angle of
inclination of the ray in a layer in which it is travelling
with a velocity
v.
The generalized form of Snell's Law
states that, along any one ray, the ray parameter remains a
constant.
For the refracted P-wave ray shown in Fig. 3.10,
therefore
sin
sin
q
q
1
2
=
v
v
1
2
or