Geology Reference
In-Depth Information
Δ
x
x
x
Δ
x
5.4 Geometry of refracted ray paths:
irregular (non-planar) interfaces
The assumption of planar refracting interfaces would
often lead to unacceptable error or imprecision in the
interpretation of refraction survey data. For example, a
survey may be carried out to study the form of the con-
cealed bedrock surface beneath a valley fill of alluvium or
glacial drift. Such a surface is unlikely to be modelled ad-
equately by a planar refractor. In such cases the constraint
that refracting interfaces be interpreted as planar must be
dropped and different interpretation methods must be
employed.
The travel-time plot derived from a survey provides a
first test of the prevailing refractor geometry. A layered
sequence of planar refractors gives rise to a travel-time
plot consisting of a series of straight-line segments, each
segment representing a particular refracted phase and
characterized by a particular gradient and intercept time.
Irregular travel-time plots are an indication of irregular
refractors (or of lateral velocity variation within individ-
ual layers — a complication not discussed here). Methods
of interpreting irregular travel-time plots, to determine
the non-planar refractor geometry that gives rise to
them, are based on the concept of
delay time
.
S
1
S
2
D
1
D
2
v
1
z
1
z
2
θ
θ
γ
v
2
>
v
1
Fig. 5.9
Refraction interpretation using the single-ended
profiling method. (After Cunningham 1974.)
and from S
2
to D
2
the travel time is given by
2
z
cos
x
sin
(
+
)
+
q
qg
2
t
=
(5.14)
2
v
v
1
1
where
z
1
and
z
2
are the perpendicular depths to the re-
fractor under shot points
S
1
and
S
2
, respectively. Now,
-=
\=+
zz x
zz x
sin
sin
D
g
2
1
(5.15)
D
g
2
1
Substituting equation (5.15) in (5.14) and then subtract-
ing equation (5.13) from (5.14) yields
5.4.1 Delay time
Consider a horizontal refractor separating upper and
lower layers of velocity
v
1
and
v
2
(>
v
1
), respectively (Fig.
5.1). The travel time of a head wave arriving at an offset
distance
x
is given (see equation (5.3)) by
x
D
t
-= =
t
t
2
sin
cos
)
D
(
g
q
21
v
1
x
sin
+
x
sin
-
D
(
qg
)
-
D
(
qg
)
=
v
v
1
1
x
v
t
=+
2
t
Substituting equations (5.11) and (5.12) in the above
equation and rearranging terms
i
The intercept time
t
i
can be considered as composed
of two delay times resulting from the presence of the
top layer at each end of the ray path. Referring to
Fig. 5.10(a), the
delay time
(or
time term
)
d
t
is defined as the
time difference between the slant path AB through the
top layer and the time that would be required for a ray to
travel along BC. The equation above clearly shows that
the total travel time can be considered as the time a wave
would take to travel the whole distance
x
at refractor ve-
locity
v
2
, plus additional time
t
i
taken for the wave to
travel down to the refractor at the shot point, and
back up to the receiver.These two extra components of
time are the
delay times
at the shot and receiver. Each
t
xv
11
D
D
=-
v
2
d
2
u
where
v
2u
and
v
2d
are the updip and downdip apparent
velocities, respectively. In the case considered
v
2d
is
derived from the single-ended travel-time curves, hence
v
2u
can be calculated from the difference in travel time
of refracted rays from adjacent shots recorded at the same
offset distance
x
. With both apparent velocities calcu-
lated, interpretation proceeds by the standard methods
for conventional reversed profiles discussed in Section
5.2.4.
