The Hartley transform in seismic imaging

Geophysics ◽  
2001 ◽  
Vol 66 (4) ◽  
pp. 1251-1257 ◽  
Author(s):  
Henning Kühl ◽  
Maurico D. Sacchi ◽  
Jürgen Fertig
Keyword(s):  
Fourier Transform ◽  
Phase Shift ◽  
Shift Operator ◽  
Three Dimensions ◽  
Lateral Velocity ◽  
Data Set ◽  
Prestack Migration ◽  
Hartley Transform ◽  
Alternative Means ◽  
The Fourier Transform

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.

Seismic depth imaging with the Gabor transform

Geophysics ◽  
2008 ◽  
Vol 73 (3) ◽  
pp. S91-S97 ◽  
Author(s):  
Yongwang Ma ◽  
Gary F. Margrave
Keyword(s):  
Fourier Transform ◽  
Phase Shift ◽  
Gabor Transform ◽  
Positioning Error ◽  
Lateral Velocity ◽  
Data Set ◽  
Depth Migration ◽  
Velocity Variations ◽  
Spatial Positioning ◽  
Wavefield Extrapolation

Wavefield extrapolation in depth, a vital component of wave-equation depth migration, is accomplished by repeatedly applying a mathematical operator that propagates the wavefield across a single depth step, thus creating a depth marching scheme. The phase-shift method of wavefield extrapolation is fast and stable; however, it can be cumbersome to adapt to lateral velocity variations. We address the extension of phase-shift extrapolation to lateral velocity variations by using a spatial Gabor transform instead of the normal Fourier transform. The Gabor transform, also known as the windowed Fourier transform, is applied to the lateral spatial coordinates as a windowed discrete Fourier transform where the entire set of windows is required to sum to unity. Within each window, a split-step Fourier phase shift is applied. The most novel element of our algorithm is an adaptive partitioning scheme that relates window width to lateral velocity gradient such that the estimated spatial positioning error is bounded below a threshold. The spatial positioning error is estimated by comparing the Gabor method to its mathematical limit, called the locally homogeneous approximation — a frequency-wavenumber-dependent phase shift that changes according to the local velocity at each position. The assumption of local homogeneity means this position-error estimate may not hold strictly for large scattering angles in strongly heterogeneous media. The performance of our algorithm is illustrated with imaging results from prestack depth migration of the Marmousi data set. With respect to a comparable space-frequency domain imaging method, the proposed method improves images while requiring roughly 50% more computing time.

Prestack migration by split‐step DSR

Geophysics ◽  
1996 ◽  
Vol 61 (5) ◽  
pp. 1412-1416 ◽  
Author(s):  
Alexander Mihai Popovici
Keyword(s):  
Phase Shift ◽  
Square Root ◽  
Variable Velocity ◽  
Step Method ◽  
Lateral Velocity ◽  
Data Set ◽  
Prestack Migration ◽  
Imaging Results ◽  
Velocity Variations ◽  
Migration Equation

The double‐square‐root (DSR) prestack migration equation, though defined for depth variable velocity, can be used to image media with strong velocity variations using a phase‐shift plus interpolation (PSPI) or split‐step correction. The split‐step method is based on applying a phase‐shift correction to the extrapolated wavefield—a correction that attempts to compensate for the lateral velocity variations. I show how to extend DSR prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the Marmousi data set. The split‐step DSR migration is very fast and offers excellent imaging results.

Diffraction by crystals

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Three Dimensional ◽  
Incident Beam ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Dimensional Structure ◽  
Angle Of Incidence ◽  
X Ray ◽  
Max Von Laue ◽  
The Fourier Transform

In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.

True-amplitude one-way wave equation migration in the mixed domain

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S199-S209 ◽  
Author(s):  
Flor A. Vivas ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Numerical Experiments ◽  
Wave Equations ◽  
Laplacian Operator ◽  
Single Shot ◽  
Lateral Velocity ◽  
Data Set ◽  
Classical Formulation ◽  
Imaging Tool

One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium,the extrapolation of the wavefields with the true-amplitude one-way wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain true-amplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.

The use of the Hartley transform in geophysical applications

Geophysics ◽  
1990 ◽  
Vol 55 (11) ◽  
pp. 1488-1495 ◽  
Author(s):  
R. Saatcilar ◽  
S. Ergintav ◽  
N. Canitez
Keyword(s):  
Fourier Transform ◽  
Seismic Data ◽  
Integral Transform ◽  
Complex Multiplication ◽  
One Dimensional ◽  
Hartley Transform ◽  
And Migration ◽  
The Fourier Transform ◽  
Processing Techniques ◽  
Phase Spectra

The Hartley transform (HT) is an integral transform similar to the Fourier transform (FT). It has most of the characteristics of the FT. Several authors have shown that fast algorithms can be constructed for the fast Hartley transform (FHT) using the same structures as for the fast Fourier transform. However, the HT is a real transform and for this reason, since one complex multiplication requires four real multiplications, the discrete HT (DHT) is computationally faster than the discrete FT (DFT). Consequently, any process requiring the DFT (such as amplitude and phase spectra) can be performed faster by using the DHT. The general properties of the DHT are reviewed first, and then an attempt is made to use the FHT in some seismic data processing techniques such as one‐dimensional filtering, forward seismic modeling, and migration. The experiments show that the Hartley transform is two times faster than the Fourier transform.

scholarly journals Сдвиговый спекл-интерферометр с квадролинзой

2020 ◽  
Vol 128 (10) ◽  
pp. 1577
Author(s):  
Г.Н. Вишняков ◽  
В.Л. Минаев ◽  
А.Д. Иванов ◽  
Ф.Ю. Виноградов
Keyword(s):  
Fourier Transform ◽  
Phase Shift ◽  
Optical Axis ◽  
Optical Element ◽  
Shift Method ◽  
Speckle Interferometer ◽  
Speckle Interferogram ◽  
The Fourier Transform ◽  
Phase Shift Method ◽  
Spatial Phase Shift

The paper proposes a new optical element – Quad lens, which is used as part of a shearing speckle interferometer (sherograph) to provide measurements of stress-strain states of objects simultaneously in two mutually perpendicular directions. Quad lens consists of four identical sections cut from the original round lenses and spaced from each other to form a gap. Quad lens creates four images of an object that are offset relative to the optical axis by a distance that depends on the size of the gaps between the sectors. Phase recovery from a single speckle interferogram is performed using the spatial phase shift method based on the Fourier transform. To increase the contrast of interference bands, an aperture diaphragm with four holes is installed in front of quad lens sectors, and polarizing channel isolation can be used to separate channels and reduce the influence of cross interference. Experimental results of using a speckle interferometer with quad lens for the study of microdeformation of a round membrane are presented.

An accurate formulation of log‐stretch dip moveout in the frequency‐wavenumber domain

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 815-820 ◽  
Author(s):  
Binzhong Zhou ◽  
Iain M. Mason ◽  
Stewart A. Greenhalgh
Keyword(s):  
Fourier Transform ◽  
Impulse Response ◽  
Fourier Transforms ◽  
Computationally Efficient ◽  
Prestack Migration ◽  
Industry Standard ◽  
Computational Speed ◽  
Simple Phase ◽  
The Fourier Transform ◽  
Approximate Version

Dip moveout (DMO) processing is a partial prestack migration procedure that has been widely used in seismic data processing. The DMO process has been described in Deregowski (1986), Hale (1991) and Liner (1990). Many different DMO algorithms have been developed over the past decade. These algorithms have been designed to improve either the accuracy or the computational speed of the DMO process. Hale (1984) developed a method for performing DMO via Fourier transforms that is accurate for all reflector dips (assuming constant velocity). Hale’s method is computationally expensive because his DMO operator is temporally nonstationary, but its accuracy and simplicity have made it an industry standard. It has become a benchmark by which results from other DMO algorithms are judged. Of all the methods used to make the frequency‐domain DMO computationally efficient, the technique of logarithmic time stretching, first suggested in Bolondi et al. (1982), is widely used. After logarithmic stretching of the time axis, the DMO operator becomes temporally stationary which enables replacement of the slow temporal Fourier integration with a fast Fourier transform combined with a simple phase shift. Bale and Jakubowicz (1987) presented a log‐stretch DMO operator (hereafter referred to as Bale’s DMO) in the frequency‐wavenumber (F-K) domain without approximations, while Notfors and Godfrey (1987) suggested an approximate version of log‐stretch DMO operator (hereafter referred to as Notfors’s DMO). Surprisingly, Bale’s full log‐stretch DMO operator produces a less satisfactory impulse response than Notfors’s approximate log‐stretch DMO scheme (see Liner, 1990). Liner (1990) attributed this characteristic to the fact that Bale’s DMO derivation implicitly assumes that the Fourier transform frequency in the log‐stretch domain is not time‐dependant. He presented an exact log‐stretch DMO operator (hereafter referred to as Liner’s DMO) which was derived by transforming the time log‐stretched Hale’s (t, x) DMO impulse response into the Fourier domain. Its derivation is relatively complicated, but Liner has shown that his DMO does generate good DMO impulse responses.

scholarly journals Phase Extraction from Single Interferogram Including Closed-Fringe Using Deep Learning

2019 ◽  
Vol 9 (17) ◽  
pp. 3529 ◽  
Author(s):  
Daichi Kando ◽  
Satoshi Tomioka ◽  
Naoki Miyamoto ◽  
Ryosuke Ueda
Keyword(s):  
Fourier Transform ◽  
Deep Learning ◽  
Phase Shift ◽  
Optical Measurement ◽  
Phase Image ◽  
Transform Method ◽  
Optical Components ◽  
Fourier Transform Method ◽  
The Fourier Transform ◽  
Trained Network

In an optical measurement system using an interferometer, a phase extracting technique from interferogram is the key issue. When the object is varying in time, the Fourier-transform method is commonly used since this method can extract a phase image from a single interferogram. However, there is a limitation, that an interferogram including closed-fringes cannot be applied. The closed-fringes appear when intervals of the background fringes are long. In some experimental setups, which need to change the alignments of optical components such as a 3-D optical tomographic system, the interval of the fringes cannot be controlled. To extract the phase from the interferogram including the closed-fringes we propose the use of deep learning. A large amount of the pairs of the interferograms and phase-shift images are prepared, and the trained network, the input for which is an interferogram and the output a corresponding phase-shift image, is obtained using supervised learning. From comparisons of the extracted phase, we can demonstrate that the accuracy of the trained network is superior to that of the Fourier-transform method. Furthermore, the trained network can be applicable to the interferogram including the closed-fringes, which is impossible with the Fourier transform method.

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