My first migrations of reflection seismic data with the wave equation
were based on the *U*/*D* concept.
The first wave-equation migration program was in the frequency domain
and worked on synthetic profiles.
Since people generally ignored such work I resolved to
complete a realistic test on field data.
Frequency-domain methods were deemed ``academic.''
I found
I could use the bilinear transformation of *Z*-transform analysis
to convert the 15 wave equation to the time domain.
As a practical matter,
it was apparent that a profile migration program could be used on a section.
But the theoretical justification was not easy.
At that time I thought of the exploding-reflector concept as a curious
analogy, not as a foundation for the derivation.

The actual procedure by which the first zero-offset section
was migrated with finite differences was more circuitous and
complicated than the procedure
later introduced by Sherwood (Loewenthal et al [1976])
and adopted generally.
The equation for profile migration in moveout-corrected coordinates
has many terms.
Neglecting all those with offset as a coefficient
(since you are trying to migrate a zero-offset section),
you are left with an equation that resembles the retarded,
15 extrapolation equation.
But there is one difference.
The term is scaled by a mysterious
coefficient, .This is the equation I used.
As the travel-time depth increases from zero
to the stopping depth *t*',
the mystery coefficient increases slowly from 1/4 to 1.

Unfortunately my derivation was so complicated that few people followed it. (You notice that I do not fully include it here). My 1972 paper includes the derivation but by way of introduction it takes you through a conceptually simpler case, namely, the seismic section that results from a downgoing plane-wave source. This simpler case brings you quickly to the migration equation. But the mystery coefficient is absent. Averaged over depth the mystery coefficient averages to a half. (The coefficient multiplies the second x-derivative and arises from decreasing as geophones descend along a coordinate ray path toward the shot). Sherwood telephoned me one day and challenged me to explain why the coefficient could not be replaced by its average value, 1/2. I could give no practical reason, nor can I today. So he abandoned my convoluted derivation and adopted the exploding-reflector model as an assumption, thereby easily obtaining the required 1/2. I felt more comfortable about the mystery coefficient later when the survey-sinking concept emerged from my work with Doherty, Muir, and Clayton.

My first book, FGDP,
describes how the *U*/*D* concept can be used to deal
with the three problems of migration,
velocity analysis, and multiple suppression.
In only one of these three applications, namely,
zero-offset migration (really CDP-stack migration),
has the wave-equation methodology become a part of routine practice.
None-the-less, the *U*/*D* concept has been generally forgotten
and replaced by Sherwood's exploding-reflector concept.

10/31/1997