Computing the Fourier transform in geophysics with the transform decomposition DFT

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1707-1709
Author(s):  
Michael J. Reed ◽  
Hung V. Nguyen ◽  
Ronald E. Chambers
Keyword(s):  
Fourier Transform ◽  
Phase Shift ◽  
Fast Fourier Transform ◽  
Discrete Fourier Transform ◽  
Large Percentage ◽  
Computationally Efficient ◽  
Operation Count ◽  
The Fourier Transform

The Fourier transform and its computationally efficient discrete implementation, the fast Fourier transform (FFT), are omnipresent in geophysical processing. While a general implementation of the discrete Fourier transform (DFT) will take on the order [Formula: see text] operations to compute the transform of an N point sequence, the FFT algorithm accomplishes the DFT with an operation count proportional to [Formula: see text] When a large percentage of the output coefficients of the transform are not desired, or a majority of the inputs to the transform are zero, it is possible to further reduce the computation required to perform the DFT. Here, we review one possible approach to accomplishing this reduction and indicate its application to phase‐shift migration.

Fast Fourier Transform – Calculation and Interpretation

2008 ◽  
Vol 3 (4) ◽  
pp. 74-86
Author(s):  
Boris A. Knyazev ◽  
Valeriy S. Cherkasskij
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Discrete Fourier Transform ◽  
Signal Theory ◽  
Correct Interpretation ◽  
The Fourier Transform ◽  
Application Programs ◽  
Physics Problems

The article is intended to the students, who make their first steps in the application of the Fourier transform to physics problems. We examine several elementary examples from the signal theory and classic optics to show relation between continuous and discrete Fourier transform. Recipes for correct interpretation of the results of FDFT (Fast Discrete Fourier Transform) obtained with the commonly used application programs (Matlab, Mathcad, Mathematica) are given.

scholarly journals Application of two-dimensional fast Fourier transform algorithm, analog of the Cooley-Tukey algorithm, for 4k fixed format digital image of satellite data in frequency domain processing

2020 ◽  
Vol 149 ◽  
pp. 02010 ◽  
Author(s):  
Mikhail Noskov ◽  
Valeriy Tutatchikov
Keyword(s):  
Fourier Transform ◽  
Numerical Experiment ◽  
Fast Fourier Transform ◽  
Frequency Domain ◽  
Discrete Fourier Transform ◽  
Satellite Data ◽  
Digital Image ◽  
Digital Images ◽  
Two Dimensional ◽  
Fast Fourier Transform Algorithm

Currently, digital images in the format Full HD (1920 * 1080 pixels) and 4K (4096 * 3072) are widespread. This article will consider the option of processing a similar image in the frequency domain. As an example, take a snapshot of the earth's surface. The discrete Fourier transform will be computed using a two-dimensional analogue of the Cooley-Tukey algorithm and in a standard way by rows and columns. Let us compare the required number of operations and the results of a numerical experiment. Consider the examples of image filtering.

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.

Elimination of minimal FFT grid-size limitations

2002 ◽  
Vol 35 (4) ◽  
pp. 505-505 ◽  
Author(s):  
David A. Langs
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Input Data ◽  
Complex Structure ◽  
Grid Size ◽  
Structure Factors ◽  
The Fourier Transform

The fast Fourier transform (FFT) algorithm as normally formulated allows one to compute the Fourier transform of up toNcomplex structure factors,F(h),N/2 ≥h> −N/2, if the transform ρ(r) is computed on anN-point grid. Most crystallographic FFT programs test the ranges of the Miller indices of the input data to ensure that the total number of grid divisions in thex,yandzdirections of the cell is sufficiently large enough to perform the FFT. This note calls attention to a simple remedy whereby an FFT can be used to compute the transform on as coarse a grid as one desires without loss of precision.

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.

scholarly journals How can the Fourier Transform be Used in Radio Astronomy to Perform Spectral Analysis of Pulsars

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Pranesh Kumar ◽  
Arthur Western
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Low Frequency ◽  
Radio Frequency Interference ◽  
Radio Waves ◽  
Folding Algorithm ◽  
Radio Signals ◽  
Multiple Characteristics ◽  
The Fourier Transform

The analysis of pulsars is a complicated procedure due to the influence of background radio waves. Special radio telescopes designed to detect pulsar signals have to employ many techniques to reconstruct interstellar signals and determine if they originated from a pulsating radio source. The Discrete Fourier Transform on its own has allowed astronomers to perform basic spectral analysis of potential pulsar signals. However, Radio Frequency Interference (RFI) makes the process of detecting and analyzing pulsars extremely difficult. This has forced astronomers to be creative in identifying and determining the specific characteristics of these unique rotating neutron stars. Astrophysicists have utilized algorithms such as the Fast Fourier Transform (FFT) to predict the spin period and harmonic frequencies of pulsars. However, FFT-based searches cannot be utilized alone because low-frequency pulsar signals go undetected in the presence of background radio noise. Astrophysicists must stack up pulses using the Fast Folding Algorithm (FFA) and utilize the coherent dedispersion technique to improve FFT sensitivity. The following research paper will discuss how the Discrete Fourier Transform is a useful technique for detecting radio signals and determining the pulsar frequency. It will also discuss how dedispersion and the pulsar frequency are critical for predicting multiple characteristics of pulsars and correcting the influence of the Interstellar Medium (ISM).

Sign in / Sign up

Export Citation Format

Share Document