1975-1988: Explosive Algorithm Development

Regardless of what some of the Geophysicists of the day thought, wave equations, digital processing, and seismic imaging was here to stay. It should not be a surprise that a hyperbolic partial differential equation was the fundamental basis. Within the short-offset approximation, hyperbolicity is almost guaranteed. Once accepted, the drive to produce computationally efficient algorithms became a race. Figure 28 shows just how explosive this process was. Although not developed in this order, two-way reverse time propagation topped the list in accuracy while the beam method of Sherwood (yes, its the one from Chevron) and Stolt's Fourier based methodology were and still are clearly the most efficient. Note that the vast majority of algorithms in Figure 28 are below the "one-way" line. The reason for this is clear. To get a one-way equation one must first factor the full two-way equation. Every mathematician will tell you this is really not possible. Every practitioner will argue that "it works." Yes, it works, but the factorization process introduces many problems along the way that are at best very difficult to resolve. For example, when one factors the second order equation as if its just a second order quadratic, one loses all wave propagation phenomenon associated with lateral propagation. As a result, the amplitudes of these one-way equations are not correct and any kind of "true" amplitude processing is not possible without some kind of ``fixup''. Many such issues are being researched today and new solutions appear frequently, so be patient, there is still a lot to come and a lot to do.

Hierarchy Figure 28. A modern migration hierarchy.

There were many contributors to the development of efficient algorithms. In 1971 Schneider's 1971 () "Developments in seismic data processing and analysis" tied diffraction stacking and Kirchhoff migration together. In 1974 and 1975, French 1975 (); 1974 (), and Gardner 1974 () clarified this even more. In 1978, Schneider's 1978 () "integral formulation" of migration put our diffraction schemes on firm theoretical foundations. Gazdag 1978 () entered the fray in 1978 with an adaptation to Stolt's original algorithm that was one of the first to begin the removal of Stolt's constant velocity assumption and appeared almost simultaneously with Bob's. Gazdag's 1978 phase-shift method was modified to "phase-shift plus interpolation" 1984 () and followed shortly thereafter by Stoffa's 1990 () split-step method that would foretell the "phase-screen" methods of Wu (see below). In 1980, Berkhout 1980 () published a detailed description of a general framework for seismic migration. This and the later book 1985 () are certainly classics in this effort. While it never received the acceptance it deserved, the end of this period saw Gardner and Forel 1988 () (see also 1999 ()) develop a completely velocity independent migration technique. It sounded impossible, but both Shell and Amerada Hess proved that it worked just fine in practice.

In 1982, Dan Whitmore at AMOCO along with co-workers (AMOCO U at work) imaged the overturned flanks of the Hackberry Dome in Louisiana by the use of reverse-time migration. According to Larry Lines, 2004 () this "surprised everyone at the 1982 SEG Workshop on migration. In the fall of that year, R. G. Keys, working for me at Cities Services, did the same thing for a dome in Southeast Texas. Keys fed the reversed-time seismic data into a finite-element modeling program written by Kurt Marfurt under John Kuo's direction at Columbia University. I should like to note that when I presented Keys' results to management they basically told me you could not do depth migration. They all new that "model-in meant model-out." Thus, even with all these developments the wave-equation was still not fully accepted. During the course of a presentation entitled "A Discrete Look at 1-D Inverse Scattering" at Cities Service in Tulsa in the Spring of 1982 I discretized the one-dimensional version of the wave equation and automatically came up with Goupillaud's method. One of the listeners in the back of the room virtually screamed "Goupillaud's method has nothing to do with this nuclear equation you have written." I'll come back to this notion of inverse scattering a bit later, but for now suffice it to say that the ideas here provide yet another approach to imaging the Earth's interior (or at least the first 30 to 40 thousand feet).

In 1983, the cat came out of the bag in nearly three simultaneously published papers on this new two-way solution to migration. Whitmore 1983 (), McMechan 1983 (), and Baysal, Sherwood, and Kosloff 1983 () authored these papers. McMechan's paper was rejected by Geophysics, but later appeared in Geophysical Prospecting. It is one of the clearest descriptions of finite-difference methods in back propagation to date. Its a great place to start a research effort into seismic migration.

In 1987, Bleistein 1987 () published what became one of the defining articles on Kirchhoff migration (he would say inversion) using a vastly improved approach to amplitudes and phases (traveltimes). Seismic migration could now be done in the space-time , frequency-space , wavenumber-space , or almost any combination of these domains. Both Whitmore's and Keys' images were poststack depth migrations so the move from time to depth was on the way.

John F. Kennedy's space initiative had provided the nation with a vast supply of young physicists, mathematicians, engineers, and computer scientists. Sadly, I fear this supply is dwindling at an alarming rate today. The space-program also added impetus to the development of powerful vector processors (most notably Cray Research) that made development and application of advanced imaging algorithms possible. By 1979 an Apple II fit on one's desk and was 1000 times more powerful than the first computer I ever programmed. In 1981/1982 the Cray-1 was 1400 times more powerful than a VAX 780. Today almost any PC on the market is vastly more powerful than that original Cray. Unfortunately the computers of this period were still not powerful enough to do prestack migration with any of the more advanced and accurate algorithms. John Sherwood's algorithm could have been used this way (and may have been), but full downward continuation using something like Claerbout's algorithm was prohibitively computationally expensive. In 1986 Gerald Neale and I at Amerada Hess tried to do prestack reverse time migration on a small 2D line. The attempt was a total failure. Even on an IBM 3090-200J imaging that 2D line would have taken months. Prestack reverse time was impractical then and may have only limited use today.