Three‐dimensional modeling and migration of seismic data using Fourier transforms

Programs for 3-D modeling and migration, using 3-D Fourier transforms to solve the scalar wave equation in frequency‐wavenumber space, are developed, implemented, tested, and applied to synthetic and scale‐model data. With microtasking to fully use four CRAY processors in parallel, we can solve a complete [Formula: see text] modeling problem in about 2.5 minutes (elapsed time); of this time, the two 3-D Fourier transforms take 1 minute and the wave‐equation calculations take 1.5 minutes. The corresponding migration also takes 2.5 minutes. Thus, even iterative 3-D processing is now feasible. The two main assumptions in our algorithm are that the earth has a constant velocity and that the data are zero‐offset (or stacked). Tests with model data verify that the algorithm produces the correct results when these assumptions are satisfied. Tests with scale‐model data show that approximate images may still be obtained when the assumptions are not strictly met; but the images contain a variety of distortions, primarily related to undermigration and overmigration, so caution is required in interpretation.