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# Extension to 3-D

To construct we must derive a relationship between dt and . We will begin by defining two different slownesses: focusing and mapping slowness. The focusing slowness is the slowness that best focuses the data. The mapping slowness is the slowness that correctly positions the data.

Starting with mapping slowness sm in terms of x,y, and z, we can transform into tau space through
 (5)
where is the two-way vertical traveltime, x' is our new x coordinate, and y' is our new y coordinate. Using the chain rule we can derive the relationship between the derivatives of our coordinates,
 (6) (7) (8)
We can simplify the above relations by defining two quantities, one in the x-direction, , and one in the y-direction, ,
 (9) (10)
Taking the derivative of both sides of the transform of (5) we can obtain a relation for dz, dx, and dy,
 (11) (12) (13)
To obtain our tomography operator in 3-D we begin by defining the traveltime along a single ray segment (where quantities measured along the ray segment are indicated by ) in tau space using equations (11-13),
 (14)
We can then take the derivative with respect to the focusing slowness sf,
 (15)
We new have an expression in terms of , , and .To get an expression in terms of just we start by taking the partial derivative of with respect to x',
 (16)
and similarly
 (17)
We then take the derivative with respect to sf and evaluate at the ray segment,
 (18) (19)
As a result we now have a linear relation between and Sf in the tau domain.

Next: Data Up: Clapp: 3-D tomography field Previous: Review
Stanford Exploration Project
9/5/2000