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
 
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