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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
 
(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 squareroot operators commute.
Thus the same solution is obtained by splitting as by full separation.
EXERCISES:

Migrate a two dimensional data set with velocity v_{1}.
Then migrate the migrated data set with a velocity v_{2}.
Rocca pointed out that this double migration
simulates a migration with a third velocity v_{3}.
Using a method of deduction similar to the Jakubowicz deduction
equations (), (), and ()
find v_{3} in terms of v_{1} and v_{2}.

Consider migration of zerooffset 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.
Next: THE ACOUSTIC WAVE EQUATION
Up: SPLITTING AND SEPARATION APPLICATIONS
Previous: Separability of 3D migration
Stanford Exploration Project
12/26/2000