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To derive equation (2) we will change the coordinates system
in order to find the 2D case illustrated in Figure a) and
follow the derivation given in Fomel (1996).
geometry
Figure 8 a) 2D geometry described by Sava and Fomel (2000). b) Commonazimuth formulation
of the problem. The derivation of equation (1) relies on the
definition of the problem into the propagation plane. Defined this way,
the 3D problem becomes equivalent to the 2D 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 3D problem becomes a 2D 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 2D 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) 
Next: Analytical derivation
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Stanford Exploration Project
7/8/2003