Refractions and refraction multiples

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Problem 6.16a

Determine the traveltime curve for the refraction and the refraction multiple in Figure 6.16a.


We assume that the velocities are known to three significant figures. Then, using equation (3.1a),

The traveltimes can be obtained either graphically or by calculation. Calculating, we get for the refraction traveltimes

Figure 6.16a.  Refraction multiple.

The critical distance (see equation (6.15a) is

The traveltime curve for SMNTUWPQR is parallel to that for SMNPQR and displaced toward longer time by the amount where

The critical distance for SMNTUWPQR is increased to

The traveltime curves are plotted as curves (a) in Figure 6.16b.

Figure 6.16b.  Traveltime curves. Letters denote curves for respective parts (a), (b), (c).

Problem 6.16b

Determine the traveltime curves when both refractor and reflector dip down to the left, the depths shown in Figure 6.16a now being the slant distances from to the interfaces.


A combined graphical and calculated solution probably provides the easiest solution although Adachi’s method (see problem 11.5) could be used to give greater precision if the data accuracy warranted. A large-scale graph was used to achieve better accuracy; Figure 6.16c is a reduced-scale replica. The traveltime curves are shown in Figure 6.16b labeled (b).

The critical distance for the refraction is km, and

The -layer outcrops at km,

Figure 6.16c.  Geometry and raypaths for dip to the left.

The headwave has a different slope to the right of . To plot the curve in this zone, we use point at the offset . Then,

and the headwave curve is a straight line joining the traveltimes at the points and .

We have two types of reflected refractions: a typical path for the first type is , the reflection occurring at the shallow dipping interface, The second type, , involves reflection at the surface. The first type exists between and , and the curve is parallel to the head-wave curve to the right of . The second type exists to the right of and the curve is parallel to the other reflected refraction. To plot the reflected-refraction curves, we need one point on each curve and then use the refraction-curve slope to the right of . For the first type, we find the coordinates of :

For the second type we find coordinates of :

Problem 6.16c

What happens when the reflector dips to the left and the refractor to the left?

Figure 6.16d.  Geometry and raypaths for and dips to the left.


A summary of the detailed graphical solution is as follows. The traveltime curves are shown in Figure 6.16b labeled (c). The various angles in Figure 6.16d are

The refraction curve is a straight line though and :

The incident angle at is , which is less than the critical angle, so that no refraction will be generated there, only a reflection. However, the refraction that starts at will give rise to upgoing rays which will be reflected, giving a reflected refraction, such as the ray that ends at . The traveltime curve is parallel to the refraction curve and exists beyond whose coordinates are

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