Using an upward-traveling coordinate system

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Problem 9.28

Show that a change from one-way traveltime to a new variable ,


(9.28a)

changes from a space-fixed coordinate system to a coordinate system that “rides along on an upcoming wavefront.” This transform is used in migration. Assume that the velocity is constant and is the depth.

Solution

Consider a wavefront that passes through the point at time ; at this instant, . After a further interval , the wavefront has advanced upward the distance , so

Since , , that is, the coordinate system keeps pace with the upcoming wavefront.

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