Wave-equation migration is known for its ability to generate accurate structural images in complex geological settings. Recently, imaging principles have been developed that allow for the extraction of amplitude variations as a function of offset ray-parameter or angle (AVP or AVA) from the downward continued wavefield. We propose the least-squares (LS) approach to wave-equation migration in order to generate high quality ray parameter common image gathers (CIGs). As we have previously demonstrated with the Marmousi model, leastsquares imaging with a smoothing constraint on the ray parameter CIGs can mitigate kinematic artifacts. In this paper we study the effect of the smoothing regularization for incomplete and noisy data in more detail. Relatively simple examples based on the full wave-equation allow us to better assess the performance and appropriateness of the LS smoothing regularization in terms of AVP/AVA preservation. The results are promising and suggest that least-squares migration, although computationally expensive, holds benefits for producing high quality AVP/AVA estimates.

Phase‐shift migration techniques that attempt to account for lateral velocity variations make substantial use of the fast Fourier transform (FFT). 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.