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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