The Hartley transform in seismic imaging

Phase-shift migration techniques that attempt to account for lateral velocity variations make substantial use of the fast Fourier transform (FIT). Generally, the Hermitian symmetry of the complex-valued Fourier transform causes computational redundancies in terms of the number of operations and memory requirements. In practice a combination of the FFT with the well-known real-to-complex Fourier transform is often used to avoid such complications. As an alternative means to the Fourier transform, we introduce the inherently real-valued, non-symmetric Hartley transform into phase-shift migration techniques. By this we automatically avoid the Hermitian symmetry resulting in an optimized algorithm that is comparable in efficiency to algorithms based on the real-to-complex FFT. We derive the phase-shift operator in the Hartley domain for migration in two and three dimensions and formulate phase shift plus interpolation, split-step migration, and split-step double-square-root prestack migration in terms of the Hartley transform as examples. We test the Hartley phase-shift operator for poststack and prestack migration using the SEG/EAGE salt model and the Marmousi data set, respectively.