Problem 4.11a
Equation (4.3a) for an offset geophone can be written
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(4.11a)
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where
is the offset and
is the slant depth at the midpoint between the source
and receiver
(see Figure 4.11a). The point of reflection
is displaced updip the distance
from the zero-dip position
. Show that the coordinates of a point
on the line
must satisfy the relation
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(4.11b)
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where
is the image point,
are the direction cosines of
,
is the slant depth at the source, and
is a parameter fixing the location of a point on
.
Solution
Referring to Figure 4.11a,
are the direction cosines of
where
To get the coordinates of
, a point on
, we draw
and
perpendicular to
. Then, using the similar triangles
and
, we have
, that is,
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(4.11c)
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being the horizontal distance from
. If we write
Figure 4.11a. Displacement of reflection point for offset geophone.
we can vary
to get different points on
.
Problem 4.11b
Verify the following relations:
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(4.11d)
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(4.11e)
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Solution
To get
, the point of intersection of
and
, we first find the equation of
; the line
has slope
and passes through
, so the equation is
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(4.11f)
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We now solve equations (4.11c) and (4.11f) as simultaneous equations. Eliminating
gives
Using the equations
,
; this reduces to
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(4.11g)
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(4.11h)
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From equation (4.11f) we get
Since
,
We now have
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