Geology Reference
In-Depth Information
13.1.2 Critical refraction and the head wave
Snell's Law (see Section 11.1.5) implies that if, in Figure 11.3, sin
i
=
V
1
/
V
2
,
which is possible only if
V
2
is greater than
V
1
, then the refracted ray will
travel parallel to the interface at velocity
V
2
.
After critical refraction, some energy will return to the ground surface
as a
head wave,
represented by rays that leave the interface at the critical
angle. This planar wavefront travels through the upper layer at velocity
V
1
but, because of its inclination, appears to move across the ground at the
V
2
velocity with which the wavefront expands below the interface. It will
therefore eventually overtake the direct wave, despite the longer travel path.
The
crossover
or
critical
distance for which the travel times of the direct
and refracted waves are equal is:
x
c
=
2
d
√
[(
V
2
+
V
1
)
/
(
V
2
−
V
1
)]
This distance can be estimated from a plot of arrival time against distance
(T-D plot). It is always more than twice the interface depth and is large if
the depth is large or the difference in velocities is small. Simple methods
of refraction interpretation use either this distance directly or the
crossover
time,
which is equal to the crossover distance divided by the direct-wave
velocity.
The term 'critical distance' is also sometimes used for the minimum
distance at which refractions return to the surface, i.e. the distance from
the shot-point at which energy arrives after reflection at the critical angle.
This usage is not common amongst field crews because at this point, and for
some distance beyond it, the refractions arrive after the direct wave and are
difficult to observe. Use of the alternative 'crossover' terminology avoids
this ambiguity.
If more than one interface is involved (Figure 13.1), multiple head waves
are generated. The head wave from the
n
th interface makes an angle
i
n
with
the ground surface given by:
sin
i
n
=
V
1
/
V
n
This angle, which is also the angle at which a ray must leave the source
in order to be critically refracted at the
n
th interface, depends only on the
velocities in the uppermost and lowermost layers and not on the velocities
in between. For each interface, there is a corresponding crossover distance,
but crossovers are difficult to locate precisely if several layers are involved
and the Generalised Reciprocal (intercept-time) method discussed in Sec-
tion 13.2 is preferred.
