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## Evaluation of the impulse response of the transformation to GOCIGs

The transformation to GOCIG of an image is defined as
 (26)
The transformation to GOCIG of an impulse located at is thus (after inverse Fourier transforms):
 (27)

We now approximate by stationary phase the inner double integral. The phase of this integral is:
 (28)
The stationary path is defined by the solutions of the following system of equations:
 (29) (30)
By moving both and to the right of equations (29) and (30), and then dividing equation (29) by equation (30), we obtain the following relationship between and :
 (31)
Furthermore, by multiplying equation (29) by kz and equation (30) by kx, and then substituting them appropriately in the phase function (28), we can evaluate the phase function along the stationary path as follows:
 (32)
which becomes, by substituting equation (31),
 (33)
Notice that the minus sign comes from the function in expression (23). By substituting expression (33) in equation (27) it is immediate to evaluate the kinematics of the impulse response as follows:
 (34)

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Stanford Exploration Project
7/8/2003