Geology Reference
In-Depth Information
velocity is by considering the increase of reflected
travel time with offset distance, the
moveout
, as discussed
below.
Equation (4.3) can also be rearranged
Note that NMO is a function of offset, velocity and re-
flector depth
z
(since
z
=
Vt
0
/2).The concept of move-
out is fundamental to the recognition, correlation and
enhancement of reflection events, and to the calculation
of velocities using reflection data. It is used explicitly or
implicitly at many stages in the processing and interpre-
tation of reflection data.
As an important example of its use, consider the
T
-
D
T
method of velocity analysis. Rearranging the
terms of equation (4.7) yields
12
12
2
2
È
˘
È
˘
z
V
x
z
x
Vt
2
Ê
Ë
ˆ
¯
+
Ê
Ë
ˆ
¯
t
=
1
=
t
1
Í
˙
Í
˙
(4.5)
0
2
Î
˚
Î
˚
0
This form of the equation is useful since it indicates
clearly that the travel time at any offset
x
will be the
vertical travel time plus an additional amount which
increases as
x
increases,
V
and
t
0
being constants.
This relationship can be reduced to an even simpler form
with a little more rearrangement. Using the standard
binomial expansion of equation (4.5) gives
x
tT
V
ª
(4.8)
12
2
0
)
(
D
Using this relationship, the velocity
V
above the reflector
can be computed from knowledge of the zero-offset
reflection time (
t
0
) and the NMO (
D
T
) at a particular
offset
x
. In practice, such velocity values are obtained by
computer analysis which produces a statistical estimate
based upon many such calculations using large numbers
of reflected ray paths (see Section 4.7). Once the
velocity has been derived, it can be used in conjunction
with
t
0
to compute the depth
z
to the reflector using
z
=
Vt
0
/2.
2
4
È
Í
˘
˙
1
2
x
Vt
1
8
x
Vt
=+
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
t
t
1
-
+
...
0
0
0
Remembering that
t
0
= 2
z
/
V
, the term
x
/
Vt
0
can be
written as
x
/2
z
. If
x
=
z
, the second term in this series
becomes 1/8 of (1/2)
4
, i.e. 0.0078, which is less than a
1% change in the value of
t
. For small offset/depth ratios
(i.e.
x
/
z
<< 1), the normal case in reflection surveying,
this equation may be truncated after the first term to
obtain the approximation
4.2.2 Sequence of horizontal reflectors
In a multilayered ground, inclined rays reflected from
the
n
th interface undergo refraction at all higher inter-
faces to produce a complex travel path (Fig. 4.3(a)). At
offset distances that are small compared to reflector
depths, the travel-time curve is still essentially hyper-
bolic but the homogeneous top layer velocity
V
in e
qu
a-
tions (4.1) and (4.7) is replaced by the
average velocity
or,
to a closer approximation (Dix 1955), the
root-mean-
square velocity V
rms
of the layers overlying the reflector.As
the offset increases, the departure of the actual travel-
time curve from a hyperbola becomes more marked
(Fig. 4.3(b)).
The root-mean-square velocity of the section of
ground down to the
n
th interface is given by
2
È
Í
1
2
x
Vt
˘
˙
ª+
x
Vt
2
2
ª+
Ê
Ë
ˆ
¯
t
t
1
t
(4.6)
0
0
2
0
0
This is the most convenient form of the time-distance
equation for reflected rays and it is used extensively in the
processing and interpretation of reflection data.
Moveout
is defined as the difference between the
travel times
t
1
and
t
2
of reflected-ray arrivals recorded at
two offset distances
x
1
and
x
2
. Substituting
t
1
,
x
1
and
t
2
,
x
2
in equation (4.6), and subtracting the resulting equations
gives
V
2
2
xx
Vt
-
2
1
t
-ª
t
21
2
2
0
12
n
n
=
È
˘
˙
Normal moveout
(NMO) at an offset distance
x
is the
difference in travel time
D
T
between reflected arrivals at
x
and at zero offset (see Fig. 4.2)
ÂÂ
V
v
2
t
t
Í
rms,
n
i
i
i
i
=
1
i
=
1
where
v
i
is the interval velocity of the
i
th layer and
t
i
is the
one-way travel time of the reflected ray through the
i
th layer.
x
Vt
2
2
D
Tt
=-ª
t
(4.7)
0
2
0
