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- downward extrapolation

The previous section suggests an alternative implementation of mixed-domain 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 split-step correction or higher-order approximation of kz 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 Vl, 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 nx with modulus nx. 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 Mixed-domain Downward
Stanford Exploration Project
5/3/2005