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The previous section suggests an alternative implementation of
mixeddomain migration. To see this more clearly, assume that, at each
depth step, we downward continue the wavefield with the true velocities
at that depth. In other words, compute wavefields, each one
corresponding to the model velocity at each spatial location. No
splitstep correction or higherorder approximation of k_{z} would then
be required. The wavefield interpolation in  domain
reduces to a simple selection of the appropriate wavefield, operation that
can be expressed as:
 
(2) 
where is the row of the array of wavefields extrapolated
with the velocity V_{l}, and is the Kronecker delta that
selects from that row the corresponding j=l component. Notice that, since
we are extrapolating as many wavefields as there are spatial positions
(traces), . Figure shows a schematic of the
velocity selection.
bin_vels1
Figure 1 Diagram illustrating velocity selection when there are
as many velocities as spatial locations.

 
In the  domain, Equation (2) becomes:
 
(3) 
where and we are using a single index to
represent the spatial axis in order to simplify the notation. The symbol
represents circular convolution.
Notice that Equation (3) was derived without any approximation.
Let's make the computations more explicit in order to gain a better
appreciation for what it means:
where means that the summation is over the range n_{x}
with modulus n_{x}. That is,
Let and exchange the order of summation:
This equation shows that in order to compute the jth component of the
extrapolated wavefield in the  domain, we need to
take the dot product of the wavefield at the previous depth step with
a vector that contains all the velocity and interpolation information.
That is,
 
(4) 
where is the vector given by
 
(5) 
Next: Practical Implementation
Up: Alvarez and Artman: Wavefield
Previous: Overview of Mixeddomain Downward
Stanford Exploration Project
5/3/2005