Geoscience Reference
In-Depth Information
the same as the travel time for the reflected wave (Fig. 4.34) since at this range
the distance AB is zero, and these two ray paths are identical. The slope of the
reflection hyperbola (Eq. (4.31)) is
d
t
d
x
=
1
α
1
4
z
1
+
x
2
d
d
x
x
α
1
1
4
z
1
+
x
2
=
(4.37)
At the critical distance (
x
c
=
2
z
1
tan
i
c
), this slope is
x
=
x
c
=
d
t
d
x
2
z
1
tan
i
c
α
1
1
4
z
1
+
4
z
1
tan
2
i
c
sin
i
c
α
1
=
1
α
2
=
(4.38)
So, at the critical distance, the head wave is the tangent to the reflection hyperbola.
Crossover distance
The crossover distance
x
cross
is the range at which the direct wave and the head
wave have the same travel time. It can be obtai
ned fro
m Eqs. (4.28) and (4.35):
x
cross
α
1
1
−
α
x
cross
α
2
2
z
1
α
1
1
=
+
(4.39)
α
2
On rearranging, we obtain
x
cross
=
2
z
1
α
2
+
α
1
α
2
−
α
1
(4.40)
Time-distance graph
Figure 4.34 shows the travel time versus distance graph for this simple two-layer
model. At short ranges, the first arrival is the direct wave, followed by the reflected
wave;atlong ranges, the first arrival is the head wave, followed by the direct wave,
followed by the reflected wave.
To determine an initial velocity-depth structure from a refraction experiment,
it is necessary to display the data on a time-distance graph. If we had a record
section or first-arrival travel times from a seismic experiment shot over this model,
we would determine
α
1
,
α
2
and
z
1
,inthat order, as follows.
1.
α
1
is determined as the inverse of the slope of the direct-wave time-distance plot for
distances less than
x
cross
.
2.
α
2
is determined as the inverse of the slope of the head-wave time-distance plot for
distances greater than
x
cross
.
3.
z
1
is determined from the intercept of the head-wave time-distance line on the
t
axis
(Eq. (4.35));
z
1
could also be calculated from the crossover distance (Eq. (4.40)), but
this distance is not usually well enough defined to give an accurate value for
z
1
.
