Theory of differential offset continuation |

From the offset continuation characteristic equation
(4), we can conclude that the first-order traveltime
derivative with respect to offset decreases with decreasing
offset. The derivative equals zero at the zero offset, as predicted by the
principle of reciprocity (the reflection traveltime has to be an * even* function of offset). Neglecting
in (4) leads to the
characteristic equation

Comparing equations (13) and (4), we can note that approximation (13) is valid only if

To find the geometric constraints implied by inequality (15), we can express the traveltime derivatives in geometric terms. As follows from expressions (10) and (11),

Expression (9) allows transforming equations (16) and (17) to the form

Without loss of generality, we can assume to be positive. Consider a plane tangent to a true reflector at the reflection point (Figure 2). The traveltime of a wave, reflected from the plane, has the known explicit expression

where is the length of the normal ray from the midpoint. As follows from combining (20) and (9),

We can now combine equations (21), (18), and (19) to transform inequality (15) to the form

where is the depth of the plane reflector under the midpoint. For example, for a dip of 45 degrees, equation (14) will be satisfied only for offsets that are much smaller than the depth of the reflector.

ocobol
Reflection rays and
tangent to the reflector in a constant velocity medium (a scheme).
Figure 2. | |
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Theory of differential offset continuation |

2014-03-26