Historical notes on a mysterious scale factor

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$^\circ$ 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$^\circ$ extrapolation equation. But there is one difference. The $v\, \partial_{gg}$ term is scaled by a mysterious coefficient, $[ t' /( 2t' - \tau )]^2$.This is the equation I used. As the travel-time depth $\tau$ 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 $\Delta x$ 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.