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The proposition I want to prove is that the vector perpendicular to the
constantoffset isochron bisects the angle between the ray coming
from the source and the one returning to the receiver.
This is the direction of the zerooffset ray, as the incident angle is equal to
the reflected angle and the zerooffset ray bisects both angles.
The gradient to a constantoffset isochron is a vector
where
The constantoffset traveltime field can be decomposed into the sum of
the traveltime from the source (T_{s}) and the traveltime to the receiver
(T_{r}).
The vector can be therefore as the sum of vectors
where T_{sx}, T_{sz}, T_{rx}, T_{rz} are the partial derivatives of the
traveltime from source and receiver, respectively.
If we note
then
which are two vectors: along the ray from the source and
along the ray from the receiver.
The gradient of the constantoffset traveltime field is a sum of the
two vectors and , and because the two vectors
have equal length (from the
eikonal equation), the vector bisects the angle between
the vectors and .
Figure 7:
The gradient of the constantoffset
traveltime field bisects the angle between the ray from the source
and the ray to the receiver.

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Stanford Exploration Project
12/18/1997