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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 s_{m} in terms of x,y, and z, we
can transform into tau space through
 

 (5) 
 
where is the twoway 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 xdirection, , and one in the ydirection,
,
 
(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 3D 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
(1113),
 
(14) 
We can then take the derivative with respect to the focusing slowness s_{f},
 
(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 s_{f} and evaluate at the
ray segment,
 
(18) 
 (19) 
As a result we now have a linear relation between
and S_{f} in the tau domain.
Next: Data
Up: Clapp: 3D tomography field
Previous: Review
Stanford Exploration Project
9/5/2000