Wavefield interpolation in the Fourier wavefield extrapolation

Geophysics ◽  
2004 ◽  
Vol 69 (1) ◽  
pp. 257-264 ◽  
Author(s):  
Li‐Yun Fu
Keyword(s):  
Fourier Transform ◽  
Computational Cost ◽  
Practical Implementation ◽  
Seismic Migration ◽  
Practical Applications ◽  
Depth Migration ◽  
Thick Slab ◽  
Frequency Components ◽  
The Fourier Transform ◽  
Wavefield Extrapolation

The computational cost for seismic migration relies heavily on the methods used for wavefield extrapolation. In general, seismic migration by current industry techniques extrapolates wavefields through a thick slab and then interpolates wavefields in small layers inside the slab. In this paper, I first optimize practical implementation of the Fourier wavefield extrapolation. I then design three interpolation algorithms: Fourier transform, Kirchhoff, and Born‐Kirchhoff for mild, moderate, and large to strong lateral heterogeneities, respectively. The Fourier transform interpolation simultaneously implements wavefield interpolation and imaging without needing to invoke the imaging principle by summing over all frequency components of the interpolated wavefield. The Kirchhoff interpolation is based on the traditional Kirchhoff migration formula and is performed by diffraction summation with a very limited aperture using the average velocity of a laterally heterogeneous slab. The Born‐Kirchhoff interpolation is based on the Lippmann‐Schwinger integral equation. It differs from the Kirchhoff interpolation in that it accounts not only for the obliquity, spherical spreading, and wavelet shaping factors but also for the relative slowness perturbation in a laterally heterogeneous slab. Recursive seismic migration usually accounts for a 20‐ to 40‐ms depth size for wavefield extrapolation in practical applications. Using the above interpolation techniques, Fourier depth migration methods are shown to tolerate a 40‐ to 60‐ms depth size with the SEG/EAGE salt model. Therefore, the Fourier depth migration techniques with thick‐slab extrapolation plus thin‐slab interpolation can be used to image structures with salt‐related complexes.

scholarly journals Computationally Efficient Sources Location Method for Nested Array via Massive Virtual Difference Co-Array

Sensors ◽  
2019 ◽  
Vol 19 (9) ◽  
pp. 1961
Author(s):  
Wei Wu ◽  
Yunfei Wang ◽  
Xiaofei Zhang ◽  
Jianfeng Li
Keyword(s):  
Fourier Transform ◽  
Parametric Method ◽  
Computational Cost ◽  
Dft Method ◽  
Doa Estimation ◽  
Practical Implementation ◽  
Computationally Efficient ◽  
Virtual Sensors ◽  
Phase Rotation ◽  
Array Aperture

In this paper, we derive the discrete Fourier transform (DFT) method for direction of arrival (DOA) estimation by generating the massive virtual difference co-array with the nested array. By contrast with the spatial smoothing (SS) subspace-based methods for nested array, which halve the array aperture, the proposed method can take full advantage of the total array aperture. Since the conventional DFT method is a non-parametric method and is limited by Rayleigh threshold, we perform the phase rotation operation to obtain the fine DOA estimates. Owing to the full utilization of the array aperture and phase rotation operation, the proposed method can achieve better performance than SS subspace-based methods for far-field sources especially with massive virtual difference co-arrays which possess a large number of virtual sensors. Besides, as the fast Fourier transform (FFT) is attractive in practical implementation, the proposed method lowers the computational cost, as compared with the subspace-based methods. Numerical simulation results validate the superiority of the proposed method in both estimation performance and complexity.

scholarly journals A New Application Methodology of the Fourier Transform for Rational Approximation of the Complex Error Function

2016 ◽  
Vol 8 (1) ◽  
pp. 14 ◽  
Author(s):  
S. M. Abrarov ◽  
B. M. Quine
Keyword(s):  
Fourier Transform ◽  
Rational Approximation ◽  
Error Function ◽  
Practical Importance ◽  
Exponential Functions ◽  
New Approach ◽  
Practical Applications ◽  
Input Parameters ◽  
The Fourier Transform

<p>This paper presents a new approach in application of the Fourier transform to the complex error function resulting in an efficient rational approximation. Specifically, the computational test shows that with only $17$ summation terms the obtained rational approximation of the complex error function provides accuracy ${10^{ - 15}}$ over the most domain of practical importance $0 \le x \le 40,000$ and ${10^{ - 4}} \le y \le {10^2}$ required for the HITRAN-based spectroscopic applications. Since the rational approximation does not contain trigonometric or exponential functions dependent upon the input parameters $x$ and $y$, it is rapid in computation. Such an example demonstrates that the considered methodology of the Fourier transform may be advantageous in practical applications.</p>

scholarly journals Review and Simulation of Fixed and Adaptive Hysteresis Current Control Considering Switching Losses and High-Frequency Harmonics

2011 ◽  
Vol 2011 ◽  
pp. 1-6 ◽  
Author(s):  
Hani Vahedi ◽  
Abdolreza Sheikholeslami ◽  
Mohammad Tavakoli Bina ◽  
Mahmood Vahedi
Keyword(s):  
Fourier Transform ◽  
High Frequency ◽  
Current Control ◽  
Switching Frequency ◽  
Hysteresis Current Control ◽  
Switching Losses ◽  
Frequency Components ◽  
Source Current ◽  
Harmonic Components ◽  
The Fourier Transform

Hysteresis Current Control (HCC) is widely used due to its simplicity in implementation, fast and accurate response. However, the main issue is its variable switching frequency which leads to extraswitching losses and injecting high-frequency harmonics into the system current. To solve this problem, adaptive hysteresis current control (AHCC) has been introduced which produces hysteresis bandwidth which instantaneously results in smoother and constant switching frequency. In this paper the instantaneous power theory is used to extract the harmonic components of system current. Then fixed-band hysteresis current control is explained. Because of fixed-band variable frequency disadvantages, the adaptive hysteresis current control is explained that leads to fixing the switching frequency and reducing the high-frequency components in source current waveform. Due to these advantages of AHCC, the switching frequency and switching losses will be diminished appropriately. Some simulations are done in MATLAB/Simulink. The Fourier Transform and THD results of source and load currents and the instantaneous switching frequency diagram are discussed to prove the efficiency of this method. The Fourier Transform and THD results of source and load currents are discussed to prove the validity of this method.

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.

scholarly journals Dynamically Modulating Plasmonic Field by Tuning the Spatial Frequency of Excitation Light

Nanomaterials ◽  
2020 ◽  
Vol 10 (8) ◽  
pp. 1449
Author(s):  
Sen Wang ◽  
Minghua Sun ◽  
Shanqin Wang ◽  
Maixia Fu ◽  
Jingwen He ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Spatial Frequency ◽  
Excitation Light ◽  
Potential Applications ◽  
Frequency Components ◽  
Simulation Results ◽  
The Fourier Transform ◽  
Theoretical Analyses ◽  
Plasmon Polaritons ◽  
Good Agreement

Based on the Fourier transform (FT) of surface plasmon polaritons (SPPs), the relation between the displacement of the plasmonic field and the spatial frequency of the excitation light is theoretically established. The SPPs’ field shifts transversally or longitudinally when the spatial frequency components f x or f y are correspondingly changed. The SPPs’ focus and vortex field can be precisely located at the desired position by choosing the appropriate spatial frequency. Simulation results are in good agreement with the theoretical analyses. Dynamically tailoring the plasmonic field based on the spatial frequency modulation can find potential applications in microparticle manipulation and angular multiplexed SPP focusing and propagation.

scholarly journals Time reversibility in acoustic signals

2004 ◽  
Vol 2 (02) ◽  
Author(s):  
J.M. Velázquez ◽  
A. Ramí­rez ◽  
C. A. Vargas
Keyword(s):  
Fourier Transform ◽  
Green’S Function ◽  
Green's Function ◽  
Acoustic Signals ◽  
Sound Waves ◽  
Time Reversibility ◽  
Practical Applications ◽  
The Fourier Transform

In this work we make a description of the time reversion problem in sound waves. Our objective is to explain the phenomena through the Fourier transform of the Green's function. With this function it is possible to characterize the propagation of the emitted signal. It also can be used to express the time-reversed signal in such a way that we can select precisely the destination site of the signal. Finally, we show some possible practical applications of this problem.

scholarly journals Generalizing the inverse FFT off the unit circle

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Vladimir Sukhoy ◽  
Alexander Stoytchev
Keyword(s):  
Fourier Transform ◽  
Computational Complexity ◽  
Fast Fourier Transform ◽  
Unit Circle ◽  
Parameter Space ◽  
Numerical Stability ◽  
Inverse Fast Fourier Transform ◽  
Numerical Accuracy ◽  
Practical Applications ◽  
Frequency Components

Abstract This paper describes the first algorithm for computing the inverse chirp z-transform (ICZT) in O(n log n) time. This matches the computational complexity of the chirp z-transform (CZT) algorithm that was discovered 50 years ago. Despite multiple previous attempts, an efficient ICZT algorithm remained elusive until now. Because the ICZT can be viewed as a generalization of the inverse fast Fourier transform (IFFT) off the unit circle in the complex plane, it has numerous practical applications in a wide variety of disciplines. This generalization enables exponentially growing or exponentially decaying frequency components, which cannot be done with the IFFT. The ICZT algorithm was derived using the properties of structured matrices and its numerical accuracy was evaluated using automated tests. A modification of the CZT algorithm, which improves its numerical stability for a subset of the parameter space, is also described and evaluated.

Multiscale Terrain Characterization Using Fourier and Wavelet Transforms for Unmanned Ground Vehicles

2009 ◽  
Author(s):  
Jeremy J. Dawkins ◽  
David M. Bevly ◽  
Robert L. Jackson
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Ground Vehicles ◽  
Quarter Car Model ◽  
Power Spectral ◽  
Before And After ◽  
Frequency Components ◽  
The Fourier Transform ◽  
Density Methods

This paper investigates the use of the Fourier transform and Wavelet transform as methods to supplement the more common root mean squared elevation and power spectral density methods of terrain characterization. Two dimensional terrain profiles were generated using the Weierstrass-Mandelbrot fractal equation. The Fourier and Wavelet transforms were used to decompose these terrains into a parameter set. A two degree of freedom quarter car model was used to evaluate the vehicle response before and after the terrain characterization. It was determined that the Fourier transform can be used to reduce the profile into the key frequency components. The Wavelet transform can effectively detect discontinuities of the profile and changes in the roughness of the profile. These two techniques can be added to current methods to yield a more robust terrain characterization.

scholarly journals A Fourier Descriptor of 2D Shapes Based on Multiscale Centroid Contour Distances Used in Object Recognition in Remote Sensing Images

Sensors ◽  
2019 ◽  
Vol 19 (3) ◽  
pp. 486 ◽  
Author(s):  
Yan Zheng ◽  
Baolong Guo ◽  
Zhijie Chen ◽  
Cheng Li
Keyword(s):  
Remote Sensing ◽  
Fourier Transform ◽  
Object Recognition ◽  
Frequency Domain ◽  
Computational Cost ◽  
Shape Descriptor ◽  
Fourier Descriptor ◽  
Remote Sensing Images ◽  
Distance Method ◽  
Practical Applications

A shape descriptor is an effective tool for describing the shape feature of an object in remote sensing images. Researchers have put forward a lot of excellent descriptors. The discriminability of some descriptors is very strong in the experiments, but usually their computational cost is large, which makes them unsuitable to be used in practical applications. This paper proposes a new descriptor-FMSCCD (Fourier descriptor based on multiscale centroid contour distance)—which is a frequency domain descriptor based on the CCD (centroid contour distance) method, multiscale description, and Fourier transform. The principle of FMSCCD is simple, and the computational cost is very low. What is commendable is that its discriminability is still strong, and its compatibility with other features is also great. Experiments on three databases demonstrate its strong discriminability and operational efficiency.

scholarly journals One Technique to Enhance the Resolution of Discrete Fourier Transform

Electronics ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 330 ◽  
Author(s):  
Ivan Kanatov ◽  
Dmitry Kaplun ◽  
Denis Butusov ◽  
Viacheslav Gulvanskii ◽  
Aleksander Sinitca
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Digital Signal ◽  
Analysis Tool ◽  
Signal Phase ◽  
Practical Applications ◽  
Advantages And Disadvantages ◽  
Engine Vibration ◽  
Window Functions ◽  
The Fourier Transform

Discrete Fourier transform (DFT) is a common analysis tool in digital signal processing. This transform is well studied and its shortcomings are known as well. Various window functions (e.g., Hanning, Blackman, Kaiser) are often used to reduce sidelobes and to spread the spectrum. In this paper, we introduce a transformation that allows removing the sidelobes of the Fourier transform and increasing the resolution of the DFT without changing the time sample. The proposed method is based on signal phase analysis. We give the comparison of the proposed approach with known methods based on window functions. The advantages and disadvantages of the proposed technique are explicitly shown. We also give a set of examples illustrating the application of our technique in some practical applications, including engine vibration analysis and a short-range radar system.

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