Separability in shot-geophone space

Reflection seismic data gathering is done on the earth's surface. One can imagine the appearance of the data that would result if the data were generated and recorded at depth, that is, with deeply buried shots and geophones. Such buried data could be synthesized from surface data by first downward extrapolating the geophones, then using the reciprocal principle to interchange sources and receivers, and finally downward extrapolating the surface shots (now the receivers). A second, equivalent approach would be to march downward in steps, alternating between shots and geophones. The result is simply stated by the equation  

$${\partial U \over \partial z} \eq \left( \ \sqrt{ {{(-\,i... ... )^2}\ -\ {\partial^2 \ \over \partial g^2} } \ \right) \ U$$ (83)

The equivalence of the two approaches has a mathematical consequence. The shot coordinate s and the geophone coordinate g are independent variables, so the two square-root operators commute. Thus the same solution is obtained by splitting as by full separation.

EXERCISES:

1. Migrate a two dimensional data set with velocity v₁. Then migrate the migrated data set with a velocity v₂. Rocca pointed out that this double migration simulates a migration with a third velocity v₃. Using a method of deduction similar to the Jakubowicz deduction equations (), (), and () find v₃ in terms of v₁ and v₂.
2. Consider migration of zero-offset data P(x,y,t) recorded in an area of the earth's surface plane. Assume a computer with a random access memory (RAM) large enough to hold several planes (any orientation) from the data volume. (The entire volume resides in slow memory devices). Define a migration algorithm by means of a program sketch (such as in IEI section 1.3). Your method should allow velocity to vary with depth.