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1
Huang, Xingguo, Morten Jakobsen, and Ru-Shan Wu. "On the applicability of a renormalized Born series for seismic wavefield modelling in strongly scattering media." Journal of Geophysics and Engineering 17, no. 2 (2019): 277–99. http://dx.doi.org/10.1093/jge/gxz105.
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Abstract Scattering theory is the basis for various seismic modeling and inversion methods. Conventionally, the Born series suffers from an assumption of a weak scattering and may face a convergence problem. We present an application of a modified Born series, referred to as the convergent Born series (CBS), to frequency-domain seismic wave modeling. The renormalization interpretation of the CBS from the renormalization group prospective is described. Further, we present comparisons of frequency-domain wavefields using the reference full integral equation method with that using the convergent Born series, proving that both of the convergent Born series can converge absolutely in strongly scattering media. Another attractive feature is that the Fast Fourier Transform is employed for efficient implementations of matrix–vector multiplication, which is practical for large-scale seismic problems. By comparing it with the full integral equation method, we have verified that the CBS can provide reliable and accurate results in strongly scattering media.
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Fontara, I. K., F. Wuttke, S. Parvanova, and P. Dineva. "Wave Propagation Due to an Embedded Seismic Source in a Graded Half-Plane with Relief Peculiarities Part I: Mechanical Model and Computational Technique." Journal of Theoretical and Applied Mechanics 45, no. 1 (2015): 87–98. http://dx.doi.org/10.1515/jtam-2015-0006.
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Abstract This work addresses the evaluation of the seismic wave field in a graded half-plane with free-surface and/or sub-surface relief subjected to shear horizontally (SH)-polarized wave, radiating from an embedded seismic source. The considered boundary value problem is transformed into a system of boundary integral equations (BIEs) along the boundaries of the free-surface and of any sub-surface relief, using an analytically derived frequency-dependent Green’s function for a quadratically inhomogeneous in depth half-plane. The numerical solution yields synthetic seismic signals at any point of the half-plane in both frequency and time domain following application of Fast Fourier Transform (FFT). Finally, in the companion paper, the verification and numerical simulation studies demonstrate the accuracy and efficiency of the present computational approach. The proposed BIE tool possesses the potential to reveal the sensitivity of the seismic signal to the type and properties of the seismic source, to the existence and type of the material gradient and to the lateral inhomogeneity, due to the free-surface and/or sub-surface relief peculiarities.
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Pirooz, Roohollah M., and Peyman Homami. "An improvement over Fourier transform to enhance its performance for frequency content evaluation of seismic signals." Computers & Structures 243 (January 2021): 106422. http://dx.doi.org/10.1016/j.compstruc.2020.106422.
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Krylov, A. A. "THE EMPIRICAL GREEN’S FUNCTION TECHNIQUE MODIFICATION FOR ACCELEROGRAM SYNTHESIS IN LOWSEISMICITY REGIONS." Engineering survey 12, no. 5-6 (2018): 72–80. http://dx.doi.org/10.25296/1997-8650-2018-12-5-6-72-80.
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In the absence of strong motion records at the future construction sites, different theoretical and semi-empirical approaches are used to estimate the initial seismic vibrations of the soil. If there are records of weak earthquakes on the site and the parameters of the fault that generates the calculated earthquake are known, then the empirical Green’s function can be used. Initially, the empirical Green’s function method in the formulation of Irikura was applied for main shock record modelling using its aftershocks under the following conditions: the magnitude of the weak event is only 1–2 units smaller than the magnitude of the main shock; the focus of the weak event is localized in the focal region of a strong event, hearth, and it should be the same for both events. However, short-termed local instrumental seismological investigation, especially on seafloor, results usually with weak microearthquakes recordings. The magnitude of the observed micro-earthquakes is much lower than of the modeling event (more than 2). To test whether the method of the empirical Green’s function can be applied under these conditions, the accelerograms of the main shock of the earthquake in L'Aquila (6.04.09) with a magnitude Mw = 6.3 were modelled. The microearthquake with ML = 3,3 (21.05.2011) and unknown origin mechanism located in mainshock’s epicentral zone was used as the empirical Green’s function. It was concluded that the empirical Green’s function is to be preprocessed. The complex Fourier spectrum smoothing by moving average was suggested. After the smoothing the inverses Fourier transform results with new Green’s function. Thus, not only the amplitude spectrum is smoothed out, but also the phase spectrum. After such preliminary processing, the spectra of the calculated accelerograms and recorded correspond to each other much better. The modelling demonstrate good results within frequency range 0,1–10 Hz, considered usually for engineering seismological studies.
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Xu, Sheng, Yu Zhang, Don Pham, and Gilles Lambaré. "Antileakage Fourier transform for seismic data regularization." GEOPHYSICS 70, no. 4 (2005): V87—V95. http://dx.doi.org/10.1190/1.1993713.
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Seismic data regularization, which spatially transforms irregularly sampled acquired data to regularly sampled data, is a long-standing problem in seismic data processing. Data regularization can be implemented using Fourier theory by using a method that estimates the spatial frequency content on an irregularly sampled grid. The data can then be reconstructed on any desired grid. Difficulties arise from the nonorthogonality of the global Fourier basis functions on an irregular grid, which results in the problem of “spectral leakage”: energy from one Fourier coefficient leaks onto others. We investigate the nonorthogonality of the Fourier basis on an irregularly sampled grid and propose a technique called “antileakage Fourier transform” to overcome the spectral leakage. In the antileakage Fourier transform, we first solve for the most energetic Fourier coefficient, assuming that it causes the most severe leakage. To attenuate all aliases and the leakage of this component onto other Fourier coefficients, the data component corresponding to this most energetic Fourier coefficient is subtracted from the original input on the irregular grid. We then use this new input to solve for the next Fourier coefficient, repeating the procedure until all Fourier coefficients are estimated. This procedure is equivalent to “reorthogonalizing” the global Fourier basis on an irregularly sampled grid. We demonstrate the robustness and effectiveness of this technique with successful applications to both synthetic and real data examples.
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Kühl, Henning, Maurico D. Sacchi, and Jürgen Fertig. "The Hartley transform in seismic imaging." GEOPHYSICS 66, no. 4 (2001): 1251–57. http://dx.doi.org/10.1190/1.1487072.
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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.
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Wu, Lin, and John Castagna. "S-transform and Fourier transform frequency spectra of broadband seismic signals." GEOPHYSICS 82, no. 5 (2017): O71—O81. http://dx.doi.org/10.1190/geo2016-0679.1.
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The S-transform is one way to transform a 1D seismogram into a 2D time-frequency analysis. We have investigated its use to compute seismic interpretive attributes, such as peak frequency and bandwidth. The S-transform normalizes a frequency-dependent Gaussian window by a factor proportional to the absolute value of frequency. This normalization biases spectral amplitudes toward higher frequency. At a given time, the S-transform spectrum has similar characteristics to the Fourier spectrum of the derivative of the waveform. For narrowband signals, this has little impact on the peak frequency of the time-frequency analysis. However, for broadband seismic signals, such as a Ricker wavelet, the S-transform peak frequency is significantly higher than the Fourier peak frequency and can thus be misleading. Numerical comparisons of spectra from a variety of waveforms support the general rule that S-transform peak frequencies are equal to or greater than Fourier-transform peak frequencies. Comparisons on real seismic data suggest that this effect should be considered when interpreting S-transform spectral decompositions. One solution is to define the unscaled S-transform by removing the normalization factor. Tests comparing the unscaled S-transform with the S-transform and the short-windowed Fourier transform indicate that removing the scale factor improves the time-frequency analysis on reflection seismic data. This improvement is most relevant for quantitative applications.
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Malik, Umairia, Dennis Ling Chuan Ching, and Hanita Daud. "Seismic Shear Energy Reflection By Radon-Fourier Transform." MATEC Web of Conferences 38 (2016): 01004. http://dx.doi.org/10.1051/matecconf/20163801004.
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Gülünay, Necati. "Seismic trace interpolation in the Fourier transform domain." GEOPHYSICS 68, no. 1 (2003): 355–69. http://dx.doi.org/10.1190/1.1543221.
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A data adaptive interpolation method is designed and applied in the Fourier transform domain (f‐k or f‐kx‐ky for spatially aliased data. The method makes use of fast Fourier transforms and their cyclic properties, thereby offering a significant cost advantage over other techniques that interpolate aliased data. The algorithm designs and applies interpolation operators in the f‐k (or f‐kx‐ky domain to fill zero traces inserted in the data in the t‐x (or t‐x‐y) domain at locations where interpolated traces are needed. The interpolation operator is designed by manipulating the lower frequency components of the stretched transforms of the original data. This operator is derived assuming that it is the same operator that fills periodically zeroed traces of the original data but at the lower frequencies, and corresponds to the f‐k (or f‐kx‐ky domain version of the well‐known f‐x (or f‐x‐y) domain trace interpolators. The method is applicable to 2D and 3D data recorded sparsely in a horizontal plane. The most common prestack applications of the algorithm are common‐mid‐point domain shot interpolation, source‐receiver domain shot interpolation, and cable interpolation.
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Ma, Yongwang, and Gary F. Margrave. "Seismic depth imaging with the Gabor transform." GEOPHYSICS 73, no. 3 (2008): S91—S97. http://dx.doi.org/10.1190/1.2903821.
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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.
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Puryear, Charles I., Oleg N. Portniaguine, Carlos M. Cobos, and John P. Castagna. "Constrained least-squares spectral analysis: Application to seismic data." GEOPHYSICS 77, no. 5 (2012): V143—V167. http://dx.doi.org/10.1190/geo2011-0210.1.
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An inversion-based algorithm for computing the time-frequency analysis of reflection seismograms using constrained least-squares spectral analysis is formulated and applied to modeled seismic waveforms and real seismic data. The Fourier series coefficients are computed as a function of time directly by inverting a basis of truncated sinusoidal kernels for a moving time window. The method resulted in spectra that have reduced window smearing for a given window length relative to the discrete Fourier transform irrespective of window shape, and a time-frequency analysis with a combination of time and frequency resolution that is superior to the short time Fourier transform and the continuous wavelet transform. The reduction in spectral smoothing enables better determination of the spectral characteristics of interfering reflections within a short window. The degree of resolution improvement relative to the short time Fourier transform increases as window length decreases. As compared with the continuous wavelet transform, the method has greatly improved temporal resolution, particularly at low frequencies.
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Deighan, Andrew J., and Doyle R. Watts. "Ground‐roll suppression using the wavelet transform." GEOPHYSICS 62, no. 6 (1997): 1896–903. http://dx.doi.org/10.1190/1.1444290.
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Low‐frequency, high‐amplitude ground roll is an old problem in land‐based seismic field records. Current processing techniques aimed at ground‐roll suppression, such as frequency filtering, f-k filtering, and f-k filtering with time‐offset windowing, use the Fourier transform, a technique that assumes that the basic seismic signal is stationary. A new alternative to the Fourier transform is the wavelet transform, which decomposes a function using basis functions that, unlike the Fourier transform, have finite extent in both frequency and time. Application of a filter based on the wavelet transform to land seismic shot records suppresses ground roll in a time‐frequency sense; unlike the Fourier filter, this filter does not assume that the signal is stationary. The wavelet transform technique also allows more effective time‐frequency analysis and filtering than current processing techniques and can be implemented using an algorithm as computationally efficient as the fast Fourier transform. This new filtering technique leads to the improvement of shot records and considerably improves the final stack quality.
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Xu, Sheng, Yu Zhang, and Gilles Lambaré. "Antileakage Fourier transform for seismic data regularization in higher dimensions." GEOPHYSICS 75, no. 6 (2010): WB113—WB120. http://dx.doi.org/10.1190/1.3507248.
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Wide-azimuth seismic data sets are generally acquired more sparsely than narrow-azimuth seismic data sets. This brings new challenges to seismic data regularization algorithms, which aim to reconstruct seismic data for regularly sampled acquisition geometries from seismic data recorded from irregularly sampled acquisition geometries. The Fourier-based seismic data regularization algorithm first estimates the spatial frequency content on an irregularly sampled input grid. Then, it reconstructs the seismic data on any desired grid. Three main difficulties arise in this process: the “spectral leakage” problem, the accurate estimation of Fourier components, and the effective antialiasing scheme used inside the algorithm. The antileakage Fourier transform algorithm can overcome the spectral leakage problem and handles aliased data. To generalize it to higher dimensions, we propose an area weighting scheme to accurately estimate the Fourier components. However, the computational cost dramatically increases with the sampling dimensions. A windowed Fourier transform reduces the computational cost in high-dimension applications but causes undersampling in wavenumber domain and introduces some artifacts, known as Gibbs phenomena. As a solution, we propose a wavenumber domain oversampling inversion scheme. The robustness and effectiveness of the proposed algorithm are demonstrated with some applications to both synthetic and real data examples.
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Naghizadeh, Mostafa, and Kristopher A. Innanen. "Seismic data interpolation using a fast generalized Fourier transform." GEOPHYSICS 76, no. 1 (2011): V1—V10. http://dx.doi.org/10.1190/1.3511525.
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We have found a fast and efficient method for the interpolation of nonstationary seismic data. The method uses the fast generalized Fourier transform (FGFT) to identify the space-wavenumber evolution of nonstationary spatial signals at each temporal frequency. The nonredundant nature of FGFT renders a big computational advantage to this interpolation method. A least-squares fitting scheme is used next to retrieve the optimal FGFT coefficients representative of the ideal interpolated data. For randomly sampled data on a regular grid, we seek a sparse representation of FGFT coefficients to retrieve the missing samples. In addition, to interpolate the regularly sampled seismic data at a given frequency, we use a mask function derived from the FGFT coefficients of the low frequencies. Synthetic and real data examples can be used to examine the performance of the method.
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Wu, Guoning, and Yatong Zhou. "Seismic data analysis using synchrosqueezing short time Fourier transform." Journal of Geophysics and Engineering 15, no. 4 (2018): 1663–72. http://dx.doi.org/10.1088/1742-2140/aabf1d.
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Lu, Wenkai, and Fangyu Li. "Seismic spectral decomposition using deconvolutive short-time Fourier transform spectrogram." GEOPHYSICS 78, no. 2 (2013): V43—V51. http://dx.doi.org/10.1190/geo2012-0125.1.
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The spectral decomposition technique plays an important role in reservoir characterization, for which the time-frequency distribution method is essential. The deconvolutive short-time Fourier transform (DSTFT) method achieves a superior time-frequency resolution by applying a 2D deconvolution operation on the short-time Fourier transform (STFT) spectrogram. For seismic spectral decomposition, to reduce the computation burden caused by the 2D deconvolution operation in the DSTFT, the 2D STFT spectrogram is cropped into a smaller area, which includes the positive frequencies fallen in the seismic signal bandwidth only. In general, because the low-frequency components of a seismic signal are dominant, the removal of the negative frequencies may introduce a sharp edge at the zero frequency, which would produce artifacts in the DSTFT spectrogram. To avoid this problem, we used the analytic signal, which is obtained by applying the Hilbert transform on the original real seismic signal, to calculate the STFT spectrogram in our method. Synthetic and real seismic data examples were evaluated to demonstrate the performance of the proposed method.
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Saatcilar, R., S. Ergintav, and N. Canitez. "The use of the Hartley transform in geophysical applications." GEOPHYSICS 55, no. 11 (1990): 1488–95. http://dx.doi.org/10.1190/1.1442796.
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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.
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Tian, Lin. "Seismic spectral decomposition using short-time fractional Fourier transform spectrograms." Journal of Applied Geophysics 192 (September 2021): 104400. http://dx.doi.org/10.1016/j.jappgeo.2021.104400.
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Chen, Yangkang. "Nonstationary local time-frequency transform." GEOPHYSICS 86, no. 3 (2021): V245—V254. http://dx.doi.org/10.1190/geo2020-0298.1.
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Time-frequency analysis is a fundamental approach to many seismic problems. Time-frequency decomposition transforms input seismic data from the time domain to the time-frequency domain, offering a new dimension to probe the hidden information inside the data. Considering the nonstationary nature of seismic data, time-frequency spectra can be obtained by applying a local time-frequency transform (LTFT) method that matches the input data by fitting the Fourier basis with nonstationary Fourier coefficients in the shaping regularization framework. The key part of LTFT is the temporal smoother with a fixed smoothing radius that guarantees the stability of the nonstationary least-squares fitting. We have developed a new LTFT method to handle the nonstationarity in all time, frequency, and space ( x and y) directions of the input seismic data by extending fixed-radius temporal smoothing to nonstationary smoothing with a variable radius in all physical dimensions. The resulting time-frequency transform is referred to as the nonstationary LTFT method, which could significantly increase the resolution and antinoise ability of time-frequency transformation. There are two meanings of nonstationarity, i.e., coping with the nonstationarity in the data by LTFT and dealing with the nonstationarity in the model by nonstationary smoothing. We evaluate the performance of our nonstationary LTFT method in several standard seismic applications via synthetic and field data sets, e.g., arrival picking, quality factor estimation, low-frequency shadow detection, channel detection, and multicomponent data registration, and we benchmark the results with the traditional stationary LTFT method.
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Serdyukov, Aleksander, Anton Azarov, and Alexandr Yablokov. "S-TRANSFORM RIDGE FILTERING." Interexpo GEO-Siberia 2, no. 3 (2019): 168–73. http://dx.doi.org/10.33764/2618-981x-2019-2-3-168-173.
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The problem of time-frequency filtering of seismic data on the basis of S-conversion is considered. S-transform provides a frequency-dependent resolution, while maintaining a direct connection with the Fourier spectrum. S-conversion is widely used in seismic processing. The standard filtering method based on S-conversion is based on its reversibility. From the point of view of temporal localization, this method is not optimal, since the calculation of the inverse S-transform includes time averaging. We propose an alternative filtering method based on signal recovery from S-transform peaks.
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Hu, Liang‐Zie, and George A. McMechan. "Wave‐field transformations of vertical seismic profiles." GEOPHYSICS 52, no. 3 (1987): 307–21. http://dx.doi.org/10.1190/1.1442305.
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Vertical seismic profile (VSP) data may be partitioned in a variety of ways by application of wave‐field transformations. These transformations provide insights into the nature of the data and aid in the design of processing operations. Transformations are implemented in a reversible sequence that takes the observed VSP data from the depth‐time (z-t) domain through the slowness‐time intercept (p-τ) domain (by a slant stack), to the slowness‐frequency (p-ω) domain (by a 1-D Fourier transform over τ), to the wavenumber‐frequency (k-ω) domain (by resampling using the Fourier central‐slice theorem), and finally back to the z-t domain (by an inverse 2-D Fourier transform). Multidimensional wave‐field transformations, combined with k-ω, p-ω, and p-τ filtering, can be applied to wave‐field resampling, interpolation, and extrapolation; separation of P-waves and S-waves; separation of upgoing and downgoing waves; and wave‐field decomposition for isolation, identification, and analysis of arrivals.
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Ouyang, Fang, Jianguo Zhao, Shikun Dai, Longwei Chen, and Shangxu Wang. "Shape-function-based nonuniform Fourier transforms for seismic modeling with irregular grids." GEOPHYSICS 86, no. 4 (2021): T165—T178. http://dx.doi.org/10.1190/geo2020-0575.1.
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Multidimensional Fourier transform on an irregular grid is a useful tool for various seismic forward problems caused by complex media and wavefield distributions. Using a shape-function-based strategy, we have developed four different algorithms for 1D and 2D nonuniform Fourier transforms, such as two high-accuracy Fourier transforms (the linear shape-function-based Fourier transform [LSF-FT] and the quadratic shape-function-based Fourier transform [QSF-FT]) and two nonuniform fast Fourier transforms (NUFFTs) (the linear shape-function-based NUFFT [LSF-NUFFT] and the quadratic shape-function-based NUFFT [QSF-NUFFT]), respectively, based on their linear and quadratic shape functions. The main advantage of incorporating shape functions into the Fourier transform is that triangular elements can be used to mesh any complex wavefield distribution in the 2D case. Therefore, these algorithms can be used in conjunction with any irregular sampling strategies. The accuracy and efficiency of the four nonuniform Fourier transforms are investigated and compared by applying them in frequency-domain seismic wave modeling. All of the algorithms are compared with the exact solutions. Numerical tests indicate that the quadratic shape-function-based algorithms are more accurate than those based on the linear shape function. Moreover, LSF-FT/QSF-FT exhibits higher accuracy but much slower calculation speed, whereas LSF-NUFFT/QSF-NUFFT is highly efficient but has lower accuracy at near-source points. In contrast, a combination of these algorithms, by using QSF-FT at near-source points and LSF-NUFFT/QSF-NUFFT at others, achieves satisfactory efficiency and high accuracy at all points. Although our tests are restricted to seismic models, these improved NUFFT algorithms may also have potential applications in other geophysical problems, such as forward modeling in complex gravity and magnetic models.
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Hargreaves, N. D., and A. J. Calvert. "Inverse Q filtering by Fourier transform." GEOPHYSICS 56, no. 4 (1991): 519–27. http://dx.doi.org/10.1190/1.1443067.
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Although some attention has been paid to the idea that seismic migration is equivalent to a type of deconvolution (of the spatial wavelet), less thought has been given to the opposite perspective: that deconvolution (of the earth Q filter) might itself be equivalent to a form of migration. The key point raised in this paper is that a dispersive 1-D backward propagation can form the basis of a number of different algorithms for inverse Q filtering, each of which is akin to a particular migration algorithm. An especially efficient algorithm can be derived by means of a coordinate transformation equivalent to that in the Stolt frequency‐wavenumber migration. This fast algorithm, valid for Q constant with depth, can be extended to accommodate depth‐variable Q by cascading a series of constant Q compensations, as in cascaded migration. By combining a cascaded phase compensation with a windowed approach to amplitude compensation, we obtain an algorithm that is sufficiently efficient to be used routinely for prestack data processing. Data examples compare the results of conventional processing with the more stable phase treatment that can be obtained by including prestack inverse Q filtering in the processing.
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Liu, Shu Cong, Er Gen Gao, and Chen Xun. "Seismic Data Denoising Simulation Research Based on Wavelet Transform." Applied Mechanics and Materials 490-491 (January 2014): 1356–60. http://dx.doi.org/10.4028/www.scientific.net/amm.490-491.1356.
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The wavelet packet transform is a new time-frequency analysis method, and is superior to the traditional wavelet transform and Fourier transform, which can finely do time-frequency dividion on seismic data. A series of simulation experiments on analog seismic signals wavelet packet decomposition and reconstruction at different scales were done by combining different noisy seismic signals, in order to achieve noise removal at optimal wavelet decomposition scale. Simulation results and real data experiments showed that the wavelet packet transform method can effectively remove the noise in seismic signals and retain the valid signals, wavelet packet transform denoising is very effective.
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Chakraborty, Avijit, and David Okaya. "Frequency‐time decomposition of seismic data using wavelet‐based methods." GEOPHYSICS 60, no. 6 (1995): 1906–16. http://dx.doi.org/10.1190/1.1443922.
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Spectral analysis is an important signal processing tool for seismic data. The transformation of a seismogram into the frequency domain is the basis for a significant number of processing algorithms and interpretive methods. However, for seismograms whose frequency content vary with time, a simple 1-D (Fourier) frequency transformation is not sufficient. Improved spectral decomposition in frequency‐time (FT) space is provided by the sliding window (short time) Fourier transform, although this method suffers from the time‐ frequency resolution limitation. Recently developed transforms based on the new mathematical field of wavelet analysis bypass this resolution limitation and offer superior spectral decomposition. The continuous wavelet transform with its scale‐translation plane is conceptually best understood when contrasted to a short time Fourier transform. The discrete wavelet transform and matching pursuit algorithm are alternative wavelet transforms that map a seismogram into FT space. Decomposition into FT space of synthetic and calibrated explosive‐source seismic data suggest that the matching pursuit algorithm provides excellent spectral localization, and reflections, direct and surface waves, and artifact energy are clearly identifiable. Wavelet‐based transformations offer new opportunities for improved processing algorithms and spectral interpretation methods.
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Herrera, Roberto H., Jiajun Han, and Mirko van der Baan. "Applications of the synchrosqueezing transform in seismic time-frequency analysis." GEOPHYSICS 79, no. 3 (2014): V55—V64. http://dx.doi.org/10.1190/geo2013-0204.1.
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Time-frequency representation of seismic signals provides a source of information that is usually hidden in the Fourier spectrum. The short-time Fourier transform and the wavelet transform are the principal approaches to simultaneously decompose a signal into time and frequency components. Known limitations, such as trade-offs between time and frequency resolution, may be overcome by alternative techniques that extract instantaneous modal components. Empirical mode decomposition aims to decompose a signal into components that are well separated in the time-frequency plane allowing the reconstruction of these components. On the other hand, a recently proposed method called the “synchrosqueezing transform” (SST) is an extension of the wavelet transform incorporating elements of empirical mode decomposition and frequency reassignment techniques. This new tool produces a well-defined time-frequency representation allowing the identification of instantaneous frequencies in seismic signals to highlight individual components. We introduce the SST with applications for seismic signals and produced promising results on synthetic and field data examples.
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Margrave, Gary F., Michael P. Lamoureux, and David C. Henley. "Gabor deconvolution: Estimating reflectivity by nonstationary deconvolution of seismic data." GEOPHYSICS 76, no. 3 (2011): W15—W30. http://dx.doi.org/10.1190/1.3560167.
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We have extended the method of stationary spiking deconvolution of seismic data to the context of nonstationary signals in which the nonstationarity is due to attenuation processes. As in the stationary case, we have assumed a statistically white reflectivity and a minimum-phase source and attenuation process. This extension is based on a nonstationary convolutional model, which we have developed and related to the stationary convolutional model. To facilitate our method, we have devised a simple numerical approach to calculate the discrete Gabor transform, or complex-valued time-frequency decomposition, of any signal. Although the Fourier transform renders stationary convolution into exact, multiplicative factors, the Gabor transform, or windowed Fourier transform, induces only an approximate factorization of the nonstationary convolutional model. This factorization serves as a guide to develop a smoothing process that, when applied to the Gabor transform of the nonstationary seismic trace, estimates the magnitude of the time-frequency attenuation function and the source wavelet. By assuming that both are minimum-phase processes, their phases can be determined. Gabor deconvolution is accomplished by spectral division in the time-frequency domain. The complex-valued Gabor transform of the seismic trace is divided by the complex-valued estimates of attenuation and source wavelet to estimate the Gabor transform of the reflectivity. An inverse Gabor transform recovers the time-domain reflectivity. The technique has applications to synthetic data and real data.
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Sacchi, Mauricio D., and Tadeusz J. Ulrych. "Estimation of the discrete Fourier transform, a linear inversion approach." GEOPHYSICS 61, no. 4 (1996): 1128–36. http://dx.doi.org/10.1190/1.1444033.
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Spatio‐temporal analysis of seismic records is of particular relevance in many geophysical applications, e.g., vertical seismic profiles, plane‐wave slowness estimation in seismographic array processing and in sonar array processing. The goal is to estimate from a limited number of receivers the 2-D spectral signature of a group of events that are recorded on a linear array of receivers. When the spatial coverage of the array is small, conventional f-k analysis based on Fourier transform leads to f-k panels that are dominated by sidelobes. An algorithm that uses a Bayesian approach to design an artifacts‐reduced Fourier transform has been developed to overcome this shortcoming. A by‐product of the method is a high‐resolution periodogram. This extrapolation gives the periodogram that would have been recorded with a longer array of receivers if the data were a limited superposition of monochromatic planes waves. The technique is useful in array processing for two reasons. First, it provides spatial extrapolation of the array (subject to the above data assumption) and second, missing receivers within and outside the aperture are treated as unknowns rather than as zeros. The performance of the technique is illustrated with synthetic examples for both broad‐band and narrow‐band data. Finally, the applicability of the procedure is assessed analyzing the f-k spectral signature of a vertical seismic profile (VSP).
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Sajid, Muhammad, and Deva Ghosh. "Logarithm of short-time Fourier transform for extending the seismic bandwidth." Geophysical Prospecting 62, no. 5 (2014): 1100–1110. http://dx.doi.org/10.1111/1365-2478.12129.
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Qi, Pengfei, and Yanchun Wang. "Seismic time–frequency spectrum analysis based on local polynomial Fourier transform." Acta Geophysica 68, no. 1 (2019): 1–17. http://dx.doi.org/10.1007/s11600-019-00377-0.
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Wang, Yuqing, Zhenming Peng, Yan Han, and Yanmin He. "Seismic Attribute Analysis with Saliency Detection in Fractional Fourier Transform Domain." Journal of Earth Science 29, no. 6 (2017): 1372–79. http://dx.doi.org/10.1007/s12583-017-0811-z.
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Mohan, N. L., and L. Anand Babu. "A note on 2-D Hartley transform." GEOPHYSICS 59, no. 7 (1994): 1150–55. http://dx.doi.org/10.1190/1.1443670.
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In recent years the application of the Hartley transform, originally introduced by Hartley (1942), has gained importance in seismic signal processing and interpretation (Saatcilar et al., 1990, 1992). The Hartley transform is similar to the Fourier transform but is computationally much faster than even the fast Fourier transform (Bracewell, 1983; Bracewell et al., 1986; Sorensen et al., 1985; Pei and Wu, 1985; Duhamel and Vetterli, 1987; Zhou, 1992). Surprisingly, we have not seen a clear definition of the 2-D Hartley transform in the published literature.
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Song, Xiaolei, and Sergey Fomel. "Fourier finite-difference wave propagation." GEOPHYSICS 76, no. 5 (2011): T123—T129. http://dx.doi.org/10.1190/geo2010-0287.1.
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We introduce a novel technique for seismic wave extrapolation in time. The technique involves cascading a Fourier transform operator and a finite-difference operator to form a chain operator: Fourier finite differences (FFD). We derive the FFD operator from a pseudoanalytical solution of the acoustic wave equation. Two-dimensional synthetic examples demonstrate that the FFD operator can have high accuracy and stability in complex-velocity media. Applying the FFD method to the anisotropic case overcomes some disadvantages of other methods, such as the coupling of qP-waves and qSV-waves. The FFD method can be applied to enhance accuracy and stability of seismic imaging by reverse time migration.
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Ibrahim, Amr, Paolo Terenghi, and Mauricio D. Sacchi. "Simultaneous reconstruction of seismic reflections and diffractions using a global hyperbolic Radon dictionary." GEOPHYSICS 83, no. 6 (2018): V315—V323. http://dx.doi.org/10.1190/geo2017-0655.1.
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We have developed a new transform with basis functions that closely resemble seismic reflections and diffractions. The new transform is an extension of the classic hyperbolic Radon transform and accounts for the apex shifts of the seismic reflection hyperbolas and the asymptote shifts of the seismic diffraction hyperbolas. The adjoint and forward operators of the proposed transform are computed using Stolt operators in the frequency domain to increase the computational efficiency of the transform. This new transform is used, in conjunction with a sparse inversion algorithm, to reconstruct common-shot gathers. Our tests indicate that this new transform is an efficient tool for interpolating coarsely sampled seismic data in cases in which one cannot use small data windows to validate the linear event assumption that is often made by Fourier-based reconstruction methods.
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Li, Wei, Shan You Li, Zhen Zhao, and Zhi Xin Sun. "Research on Applications of Wavelet Transform Threshold Method to Strong Motion Signal Processing." Advanced Materials Research 446-449 (January 2012): 2387–91. http://dx.doi.org/10.4028/www.scientific.net/amr.446-449.2387.
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Fourier transform and short-time Fourier transform are the main methods in signal analysis, which can reflect the spectrum signature of signals in the whole time domain; but they cannot be used in the multi-resolution analysis on the non-stationary signals. However, the wavelet transform overcome the limits of Fourier and short-time Fourier transform, which can be performed in accurate time-frequency analysis of signals. Furthermore, the diversity of wavelet functions makes the wavelet transform more adaptive and flexible. Applying the wavelet transform to seismic signal processing is the complement and improvement of existing processing methods. In this paper, the basic theory of the wavelet threshold denoising method and its application to the strong motion signal processing were mainly introduced. The high-frequency noises were removed, and simultaneously the high-frequency signals were effectively retained.
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Gholami, Ali, and Toktam Zand. "Three-parameter Radon transform based on shifted hyperbolas." GEOPHYSICS 83, no. 1 (2018): V39—V48. http://dx.doi.org/10.1190/geo2017-0309.1.
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The focusing power of the conventional hyperbolic Radon transform decreases for long-offset seismic data due to the nonhyperbolic behavior of moveout curves at far offsets. Furthermore, conventional Radon transforms are ineffective for processing data sets containing events of different shapes. The shifted hyperbola is a flexible three-parameter (zero-offset traveltime, slowness, and focusing-depth) function, which is capable of generating linear and hyperbolic shapes and improves the accuracy of the seismic traveltime approximation at far offsets. Radon transform based on shifted hyperbolas thus improves the focus of seismic events in the transform domain. We have developed a new method for effective decomposition of seismic data by using such three-parameter Radon transform. A very fast algorithm is constructed for high-resolution calculations of the new Radon transform using the recently proposed generalized Fourier slice theorem (GFST). The GFST establishes an analytic expression between the [Formula: see text] coefficients of the data and the [Formula: see text] coefficients of its Radon transform, with which a very fast switching between the model and data spaces is possible by means of interpolation procedures and fast Fourier transforms. High performance of the new algorithm is demonstrated on synthetic and real data sets for trace interpolation and linear (ground roll) noise attenuation.
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Han, Jiajun, and Mirko van der Baan. "Empirical mode decomposition for seismic time-frequency analysis." GEOPHYSICS 78, no. 2 (2013): O9—O19. http://dx.doi.org/10.1190/geo2012-0199.1.
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Time-frequency analysis plays a significant role in seismic data processing and interpretation. Complete ensemble empirical mode decomposition decomposes a seismic signal into a sum of oscillatory components, with guaranteed positive and smoothly varying instantaneous frequencies. Analysis on synthetic and real data demonstrates that this method promises higher spectral-spatial resolution than the short-time Fourier transform or wavelet transform. Application on field data thus offers the potential of highlighting subtle geologic structures that might otherwise escape unnoticed.
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Sinha, Satish, Partha S. Routh, Phil D. Anno, and John P. Castagna. "Spectral decomposition of seismic data with continuous-wavelet transform." GEOPHYSICS 70, no. 6 (2005): P19—P25. http://dx.doi.org/10.1190/1.2127113.
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This paper presents a new methodology for computing a time-frequency map for nonstationary signals using the continuous-wavelet transform (CWT). The conventional method of producing a time-frequency map using the short time Fourier transform (STFT) limits time-frequency resolution by a predefined window length. In contrast, the CWT method does not require preselecting a window length and does not have a fixed time-frequency resolution over the time-frequency space. CWT uses dilation and translation of a wavelet to produce a time-scale map. A single scale encompasses a frequency band and is inversely proportional to the time support of the dilated wavelet. Previous workers have converted a time-scale map into a time-frequency map by taking the center frequencies of each scale. We transform the time-scale map by taking the Fourier transform of the inverse CWT to produce a time-frequency map. Thus, a time-scale map is converted into a time-frequency map in which the amplitudes of individual frequencies rather than frequency bands are represented. We refer to such a map as the time-frequency CWT (TFCWT). We validate our approach with a nonstationary synthetic example and compare the results with the STFT and a typical CWT spectrum. Two field examples illustrate that the TFCWT potentially can be used to detect frequency shadows caused by hydrocarbons and to identify subtle stratigraphic features for reservoir characterization.
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Liu, Yang, and Sergey Fomel. "OC-seislet: Seislet transform construction with differential offset continuation." GEOPHYSICS 75, no. 6 (2010): WB235—WB245. http://dx.doi.org/10.1190/1.3479554.
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Many of the geophysical data-analysis problems such as signal-noise separation and data regularization are conveniently formulated in a transform domain, in which the signal appears sparse. Classic transforms such as the Fourier transform or the digital wavelet transform (DWT) fail occasionally in processing complex seismic wavefields because of the nonstationarity of seismic data in time and space dimensions. We present a sparse multiscale transform domain specifically tailored to seismic reflection data. The new wavelet-like transform — the OC-seislet transform — uses a differential offset-continuation (OC) operator that predicts prestack reflection data in offset, midpoint, and time coordinates. It provides a high compression of reflection events. Its compression properties indicate the potential of OC seislets for applications such as seismic data regularization or noise attenuation. Results of applying the method to synthetic and field data examples demonstrate that the OC-seislet transform can reconstruct missing seismic data and eliminate random noise even in structurally complex areas.
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Stanton, Aaron, and Mauricio D. Sacchi. "Vector reconstruction of multicomponent seismic data." GEOPHYSICS 78, no. 4 (2013): V131—V145. http://dx.doi.org/10.1190/geo2012-0448.1.
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A multicomponent seismic record was transformed to the frequency-wavenumber domain using the quaternion Fourier transform. This transform was integrated into the projection onto convex sets algorithm to allow for the reconstruction of vector signals. The method took advantage of the spectral overlap of components in the frequency-wavenumber domain. The results of the method were compared with standard component-by-component reconstruction. Results were found for synthetic and real data including an example of 5D reconstruction of a converted-wave data set acquired over a heavy oil reservoir. We found an improvement in reconstruction quality producing fully sampled radial and transverse offset-azimuth gathers with a preserved vector relationship.
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Grigoryan, Artyom M. "Representation of the Fourier Transform by Fourier Series." Journal of Mathematical Imaging and Vision 25, no. 1 (2006): 87–105. http://dx.doi.org/10.1007/s10851-006-5150-0.
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Putri, Intan Andriani, and Awali Priyono. "THIN BED IDENTIFICATION IMPROVEMENT USING SHORT – TIME FOURIER TRANSFORM HALF – CEPSTRUM ON “TG” FIELD." PETRO:Jurnal Ilmiah Teknik Perminyakan 8, no. 3 (2019): 103. http://dx.doi.org/10.25105/petro.v8i3.5511.
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<p>Thin Bed Identification is still a difficult task even with the advanced technology of seismic acquisition. Certain high frequency component is necessary and could be obtained through resolution enhancement. Short – Time Fourier Transform Half Cepstrum (STFTHC) is performed to enhance seismic resolution thus a better separation of thin bed could be improved. Basic principal of STFTHC is to replace the frequency spectrum by its logarithm while phase spectrum remains the same. Synthetic seismic was built based on Ricker and Rayleigh criterion. They were used to test the program yielding a better separation of two interfaces under tuning thickness without creating new artifacts. The algorithm was applied to seismic data from TG field. Using post-STFTHC seismic data as input of acoustic impedance inversion, well tie correlation increases by 10% and decreases inversion analysis error by 17,5%. Several thin bed -which once could not- could be identified on acoustic impedance result.</p>
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Jahanjooy, Saber, Ramin Nikrouz, and Nematullah Mohammed. "A faster method to reconstruct seismic data using anti-leakage Fourier transform." Journal of Geophysics and Engineering 13, no. 1 (2016): 86–95. http://dx.doi.org/10.1088/1742-2132/13/1/86.
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Schuster, Gerard T., and Zhaojun Liu. "Seismic array theorem and rapid calculation of acquisition footprint noise." GEOPHYSICS 66, no. 6 (2001): 1843–49. http://dx.doi.org/10.1190/1.1487127.
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The acquisition footprint noise in migrated sections partly consists of migration artifacts associated with a discrete recording geometry. Such noise can corrupt the interpretation of seismic sections for stratigraphic variations, AVO signatures, and enhanced oil recovery operations. We show that the point scatterer response of the far‐field Kirchhoff migration operator, which reveals acquisition footprint noise, is proportional to the stretched Fourier transform of the source and geophone sampling function. Using the array theorem developed by optical and electrical engineers, this Fourier transform for an orthogonal recording geometry can be calculated quickly by a product of 1‐D analytical functions. Thus, the acquisition footprint noise in migrated sections can be calculated efficiently for different recording arrays. By rapid trial and error or by an optimization method, the survey planner can use this theorem to help design a better recording geometry.
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Fu, Li‐Yun. "Wavefield interpolation in the Fourier wavefield extrapolation." GEOPHYSICS 69, no. 1 (2004): 257–64. http://dx.doi.org/10.1190/1.1649393.
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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.
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Çiflikli, Cebrail, and Ali Gezer. "Self similarity analysis via fractional Fourier transform." Simulation Modelling Practice and Theory 19, no. 3 (2011): 986–95. http://dx.doi.org/10.1016/j.simpat.2010.12.009.
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Anderson, Richard G., and George A. McMechan. "Noise‐adaptive filtering of seismic shot records." GEOPHYSICS 53, no. 5 (1988): 638–49. http://dx.doi.org/10.1190/1.1442498.
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Ambient noise can obscure reflections on deep crustal seismic data. We use a spectral subtraction method to attenuate stationary noise. Our procedure, called noise‐adaptive filtering, is to Fourier transform the noise before the first arrivals, subtract the amplitude spectrum of the noise from the amplitude spectrum of the noisy data, and inverse Fourier transform. The phase spectrum is not corrected, but the method attenuates noise if the phase shift between the signal and noise is random. The algorithm can be implemented as a frequency filter, as a frequency‐wavenumber filter, or as two separate frequency and wavenumber filters. Noise‐adaptive filtering is often superior to conventional frequency or frequency‐wavenumber filtering because it adapts to spatial variations in the noise without parameter testing. Noise‐adaptive filters can achieve noise rejection ratios of up to 45 dB; their dynamic range is about 25 dB. These filters work best when the input signal‐to‐noise ratio is on the order of 0 dB and there are significant differences between the frequency‐wavenumber amplitude spectra of the signal and noise. Application of the method to field data can enhance events that are not visible in the input data.
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O'cinneide, Colm Art. "Euler summation for fourier series and laplace transform inversion." Communications in Statistics. Stochastic Models 13, no. 2 (1997): 315–37. http://dx.doi.org/10.1080/15326349708807429.
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Yu, Hao, Jin Song Li, Yan Zhang, and Guang Cheng Xu. "Research and Application of Spectral Decomposition in Carbonate Reservoir and Fault Identification in MaiJie Let Gas Field, Amu Darya Basin." Advanced Materials Research 616-618 (December 2012): 141–44. http://dx.doi.org/10.4028/www.scientific.net/amr.616-618.141.
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Spectral decomposition is to convert seismic signals from the time domain to the frequency domain by mathematical transformation, and analyze amplitude and phase response characteristics of different scale geological bodies. Spectral decomposition could get higher resolution than conventional seismic data. In the identification of the fault system, it is fit for fault interpretation and plane combination of the sections. In the reservoir prediction, it can identify the shape and contour of the reservoir. This document analyzed algorithm and applicability of short time Fourier transform, continuous wavelet transform and S-transform. Using these three methods for carbonate reservoir identification in MaiJie let gas field, Amu Darya Basin, it proves that the frequency division section is more clearly than conventional seismic section in reservoir and fault description. And S-transform gets the best result.
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Li, Duan, John Castagna, and Gennady Goloshubin. "Investigation of generalized S-transform analysis windows for time-frequency analysis of seismic reflection data." GEOPHYSICS 81, no. 3 (2016): V235—V247. http://dx.doi.org/10.1190/geo2015-0551.1.
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The frequency-dependent width of the Gaussian window function used in the S-transform may not be ideal for all applications. In particular, in seismic reflection prospecting, the temporal resolution of the resulting S-transform time-frequency spectrum at low frequencies may not be sufficient for certain seismic interpretation purposes. A simple parameterization of the generalized S-transform overcomes the drawback of poor temporal resolution at low frequencies inherent in the S-transform, at the necessary expense of reduced frequency resolution. This is accomplished by replacing the frequency variable in the Gaussian window with a linear function containing two coefficients that control resolution variation with frequency. The linear coefficients can be directly calculated by selecting desired temporal resolution at two frequencies. The resulting transform conserves energy and is readily invertible by an inverse Fourier transform. This modification of the S-transform, when applied to synthetic and real seismic data, exhibits improved temporal resolution relative to the S-transform and improved resolution control as compared with other generalized S-transform window functions.