Problem
The Blondeau method of making weathering corrections (Musgrave and Bratton, 1967, 231−246) is useful in areas where, because of appreciable compaction within the low-velocity layer, the velocity is given approximately by the equation
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(8.21a)
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and
being constants,
The Blondeau method starts with a curve of first breaks versus offset [Figure 8.21a(ii)] plotted on log-log graph paper; the curve is approximately a straight line with slope
(see Musgrave and Bratton, 1967, 244).
To remove the effect of a surface layer of thickness
, we find
, a tabulated function of
. Then
. Next we use the
plot to find the corresponding
. Finally,
, the vertical traveltime to the depth
, is given by
.
Verify the Blondeau procedure by deriving these relations:
-
, where
is the angle of incidence measured with respect to the vertical at depth
;
-
, where
function of
, hence also of
;
-
, where
;
[Note that
can be obtained from the table for
by writing
, finding for
, and multiplying the value by
-
, the horizontal component of the apparent velocity at any point
of the trajectory, is
;
-
.
Solution
i) Note that the quantities
and
are the offset and traveltime for the point of emergence. For intermediate points we write
except for the deepest point, where we have
,
,
.
Solving equation (8.21a) for
gives
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(8.21b)
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From Snell’s law we have
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(8.21c)
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Thus, using equations (8.21b,c) we get
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(8.21d)
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ii) From equations (4.17d), we get
where we have differentiated equation (8.21d) to replace
with
. If we integrate from 0 to
and multiply by 2, the result is
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(8.21e)
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iii) From Figure 4.17a we have
, so
where equation (8.21d) was used to replace
with
. Using equation (8.21c) to eliminate
, we get
Integration from 0 to
gives
, so
Since
from equations (8.21a) and
from equation (8.21e), this result can be written
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(8.21f)
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iv) Solving equation (8.21f) for
gives
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(8.2lg)
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Equation (8.21f) can be used to eliminate
, so equation (8.21g) becomes
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(8.21h)
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The angle between the wavefront and a horizontal line equals the angle between a ray and the vertical, both angles being the angle of incidence (see Figure 4.2c), so
[compare with equation (4.2d)]. Therefore,
using Snell’s law. Thus the apparent velocity
; since this is a constant, it holds at every point of the trajectory, including the point of emergence. Therefore, from equation (8.21h),
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(8.21i)
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v) We have
time to travel vertically from the surface to the depth
, Thus,
where we have used equation (8.21a) in the first and last steps. From equations (8.21e) and (8.21i) we have
and
, so
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(8.21j)
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Also in this chapter
External links
find literature about Blondeau weathering corrections
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