The Hartley transform in seismic imaging

Abstract

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.

On discretization of tori of compact simple Lie groups: II

“…Since introduction of the discrete version of the Hartley transform in [3], both continuous and discrete Hartley transforms form fully equivalent real-valued variants of the standard Fourier transforms [30]. As alternatives to complex Fourier transforms, these transforms together with their 2D and 3D versions found applications in many fields including signal processing [28,29], pattern recognition [4], geophysics [23], measurement [31] and optics [24]. In the context of Weyl-orbit functions and their corresponding transforms, the Hartley transforms have not yet been studied.…”

3-D acoustic modeling by a Hartley method

“…Later in 1991, Saatcilar and Ergintav solved the 2-D elastic wave equation with the Hartley transform. Kühl et al (2001) introduced the Hartley transform into seismic imaging. There are also some other techniques to improve the efficiency of the Fourier method.…”

Hartley transform and the use of the Whitened Hartley spectrum as a tool for phase spectral processing

“…Bracewell observed that the real spectrum derived via the Hartley transform from a real signal, contained phase information (as well as magnitude information) and showed that analogue phase measurement was possible with suitable laboratory apparatus [3–7]. Published work related to the Hartley transform in the area of signal processing can be found in [8–12]; the Hartley transform has also found application in diverse areas such as geophysics [13, 14], electrical power engineering [15] and pattern recognition [16–18].…”