Geoscience Reference
In-Depth Information
t
0
then define the model. In the example of Fig. 4.41,itisobvious that curve 2 is
correct, but with real data it is not sufficient to rely on the eye alone to determine
velocities - a numerical criterion must be used. In practice, the reflections are
stacked using a range of values for
α
1
and
t
0
. The power (or some similar entity)
in the stacked signal is calculated for each value of
α
1
and
t
0
.For each value of
t
0
the maximum value of the power is then used to determine the best velocity. A
plot of power against both velocity and time, as in Fig. 4.41(b),isusually called
a
velocity-spectrum
display.
Once the records have been stacked, the common-depth-point record section
shows the travel times as if shots and receivers were coincident. The stacking
process necessarily involves some averaging over fairly short horizontal distances,
but it has the considerable advantage t
h
at the signal-to-noise ratio of the stacked
traces is increased by a factor of
√
n
over the signal-to-noise ratio of the
n
individual traces.
4.4.3 A multilayered model
Atwo-layer model is obviously not a realistic approximation to a pile of sediments
or to the Earth's crust, but it serves to illustrate the principle of the reflection
method. A multiple stack of layers is a much better model than a single layer.
Travel times through a stack of multiple layers are calculated in the same way
as for two layers, with the additional constraint that Snell's law (sin
i
p
,a
constant for each ray) must be applied at each interface (Eq. (4.55)). The travel
times and distances for a model with
n
layers, each with thickness
z
j
and velocity
α
j
, are best expressed in the parametric form
/α
=
n
z
j
p
α
j
x
=
2
1
−
ρ
α
j
2
j
=
1
(4.75)
n
z
j
α
j
1
−
ρ
t
=
2
2
α
j
j
=
1
sin
i
j
/α
j
.Itisunfortunately not usually possible to eliminate
p
from
these equations in order to express the distance curve as one equation. In this
multilayered case, the time-distance curve is
not
ahyperbola as it is when
n
where
p
=
1
(Eq. (4.65)). However, it can be shown that the square of the travel time,
t
2
, can
be expressed as an infinite series in
x
2
:
t
2
=
=
c
0
+
c
1
x
2
+
c
2
x
4
+
c
3
x
6
+ ···
(4.76)
where the coefficients
c
0
,
c
1
,
c
2
,...areconstants dependent on the layer thick-
nesses
z
j
and velocities
α
j
.Inpractice, it has been shown that use of just the first
two terms of Eq. (4.76)(
c
0
and
c
1
)gives travel times to an accuracy of within
about 2%, which is good enough for most seismic-reflection work. This means
that Eq. (4.76) can be simplified to
t
2
=
c
0
+
c
1
x
2
(4.77)
