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In this section, we examine the ODCIG volumes generated by
forwardscattered wavefields. For simplicity, we illustrate these
concepts using planewave S and R wavefields. We
also assume that S and R propagate at constant, though not
necessarily equal, slownesses (i.e., reciprocal of velocity). These
idealizations allow us to generate an analytic surface in ODCIG space
for both PP diffracted and PS converted waves. We specify
planar S and R wavefields in constant velocity media using source
and receiver ray parameter vectors, and , defined by,
 
(3) 
where p_{s} and p_{r} are the source and receiver horizontal ray
parameters, q_{s} and q_{r} are the source and receiver vertical ray parameters, and
s_{s} and s_{r} are the source and receiver wavefield propagation
slownesses, respectively. Also, we use a convention where angles are
defined clockwise positive with respect to the vertical depth axis.
Forwardscattered S and R wavefields must satisfy the causality
arguments illustrated in Figure , which requires
a negative sign in the source and receiver extrapolation operators.
Using the aforementioned assumptions, we generate the following
extrapolated S and R wavefield volumes,
 
(4) 
Applying a Fourier transform over the taxis of both S and R yields,
 
(5) 
Evaluating the imaging condition in Equation (2)
with the wavefields in Equation (5) generates the
following imagespace volume,
 

 (6) 
 
The nonzero function argument,
 
(7) 
represents an analytic forwardscattered ODCIG hyperplane surface.
Importantly, this surface interrelates source and receiver
planewave angles, propagation slownesses and imagespace variables,
and . In the next section, we manipulate this formula
to generate constraint equations that help isolate the receiverside
contribution to the total reflection angle.
Next: From ODCIGs to ADCIGs
Up: Shragge et al.: Forwardscattered
Previous: Shotprofile migration
Stanford Exploration Project
5/3/2005