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## Exploding-reflector migration by downward continuation

Recursive downward continuation migration algorithms start with an operator, R, that marches the wavefield down into the earth,
 (61)
with initial conditions determined by the reflection data recorded on the surface,
 (62)
Taking , these equations can be rewritten in matrix form as
 (63) (64)
where is a single frequency component of data recorded at the surface, is a vector containing a single frequency component of the wavefield at depth-step, z, and is the corresponding extrapolation operator.

If the downward-continuation operator, , contains purely a phase-shift, then its adjoint, will fully describe the inverse process of upward-continuation. However, for the amplitudes to be treated accurately, must respect the physics of wave propagation. Stolt and Benson (1986) show that v(z) extrapolators based on WKBJ Green's functions contain an amplitude term as well as a phase term, and for v(x,y,z) earth models this effect is even more pronounced. So while kinematically-correct extrapolators are pseudo-unitary, true-amplitude depth extrapolators are not. In this introductory section, I follow conventional seismic processing methodology, and treat the depth extrapolator as a unitary operator; however, in section , I discuss how to model amplitudes correctly.

The recursion in equations () and () can be rewritten as:
 (65)
or more simply,
 (66)
where now contains for all depth-steps, is a zero-padding operator, and is the extrapolation matrix in equation () that can be inverted rapidly by recursion.

Equation () encapsulates the idea that we can reconstruct the wavefield at every depth-step in the earth from the wavefield at the surface by inverting matrix, .

As a final step, to produce a migrated image we need to invoke an imaging condition. For the case of exploding-reflector (zero-offset) migration [e.g. Claerbout (1995)], we need to extract the image, , corresponding to t=0. In the temporal frequency domain, we can do this by summing over frequency. This stacking process is described by the matrix equation,

or again more simply,
 (67)
where is the reflectivity model, is the total wavefield, is the identity matrix of rank N, and is the (exploding-reflector) imaging operator that sums over frequencies.

The process of imaging by exploding-reflector migration can then be summarized as the chain of composite operators:
 (68)
where

Next: Exploding-reflector modeling by upward Up: Shot-profile migration and modeling Previous: Shot-profile migration and modeling
Stanford Exploration Project
5/27/2001