Variation of reflection point with offset

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Problem 4.11a

4.11a Equation (4.3a) for an offset geophone can be written


(4.11a)

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


(4.11b)

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,


(4.11c)

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

4.11b Verify the following relations:


(4.11d)


(4.11e)

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


(4.11f)

We now solve equations (4.11c) and (4.11f) as simultaneous equations. Eliminating gives

Using the equations ,  ; this reduces to


(4.11g)


(4.11h)

From equation (4.11f) we get

Since ,

We now have

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