Reflector Maps and Earth Parameter Estimation from Seismic Data: From Ruler and Compass Construction to Gaussian Beam Propagators
Norman Bleistein,  Colorado School of Mines

Hagedoorn created a ruler and compass method for imaging reflectors in the Earth for seismic exploration from pictures of the reflection response in seismic data records. Although it was not apparent at the time (1954), his method anticipated a hierarchy of asymptotic methods for reflector imaging and estimation of geometrical optics or ray-theoretic reflection coefficients via integral processes reminiscent of a Green's function representation of solutions of the wave equation from boundary data. Exploration geophysicists call the basic method ``migration'' and the integral form, ``Kirchhoff migration.'' Here, the data record in space-time has the appearance of reflectors and thus the movement of that data from its temporal location to its spatial location leads to the name ``migration.'' This is the workhorse technique in the exploration geophysics community.

When the processing integral is properly weighted, the peak amplitude of the image on each reflector is in know proportion to a reflection coefficient at a determinable incidence angle. The method is then referred to as ``true-amplitude migration'' or ``inversion,'' although it is only a partial inversion that relies on low wave number information about a background wave speed to propagate the data. This talk describes the basic ideas and the use of ray-theoretic Green's functions to propagate the observed data (and the source) into the Earth. For about twenty years, Gaussian beam representations have been used for Green's functions because of the smoother and more coherent images that they produce, although the cost can be prohibitively and non-competitively expensive.

Hill proposed a method for reducing the number of integrals and, hence, the cost of this Kirchhoff processing with Gaussian beams. Moving that less costly method from migration to true-amplitude migration has only recently been achieved.

We describe this history as time allows.