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## Geometrical derivation

To derive equation (2) we will change the coordinates system in order to find the 2-D case illustrated in Figure -a) and follow the derivation given in Fomel (1996).

 geometry Figure 8 a) 2-D geometry described by Sava and Fomel (2000). b) Common-azimuth formulation of the problem. The derivation of equation (1) relies on the definition of the problem into the propagation plane. Defined this way, the 3-D problem becomes equivalent to the 2-D problem.

Consider the source and the receiver rays constrained to a slanted propagation plane as illustrated in Figure -b). The plane has dip angle .Within the propagation plane, the source ray has a dip angle and the receiver ray a dip angle .We define z' as being a new vertical axis within the propagation plane: the 3-D problem becomes a 2-D problem in this new basis. The aperture angle and the dipping angle of the normal of the interface at the reflection point can be expressed as a function of and :
 (8) (9)
The offset ray parameter in the propagation plane is:
 (10)
We can write a 2-D version of the Double Square Root (DSR) in the propagation plane:
 (11)
The change of variable between the pseudo vertical axis z' and the real vertical axis z is made with the relation:
 (12)
With equations (11) and (12) the DSR expression in the original coordinates system becomes:
 (13)
The quotient of equations (10) and (13) leads to the expected relation 2:
 (14)

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7/8/2003